A proposal to perform musick in perfect and mathematical proportions containing I. the state of musick in general, II. the principles of present practice ..., III. the tables of proportions, calculated for the viol ... / by Thomas Salmon ... ; with large remarks upon this whole treatise by the reverend and learned John Wallis ...
         Salmon, Thomas, 1648-1706.
      
       
         
           1688
        
      
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             A proposal to perform musick in perfect and mathematical proportions containing I. the state of musick in general, II. the principles of present practice ..., III. the tables of proportions, calculated for the viol ... / by Thomas Salmon ... ; with large remarks upon this whole treatise by the reverend and learned John Wallis ...
             Salmon, Thomas, 1648-1706.
             Wallis, John, 1616-1703.
          
           [13], 42, [2] p., 4 leaves of plates : ill.
           
             Printed for John Lawrence ...,
             London :
             1688.
          
           
             Errata: p. [13].
             Advertisement: p. [12] and p. [1]-[2] at end.
             Reproduction of original in Library of Congress.
          
        
      
    
     
       
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         eng
      
       
         
           Musical temperament.
        
      
    
     
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           A
           PROPOSAL
           TO
           Perform
           Musick
           ,
           IN
           Perfect
           and
           Mathematical
           Proportions
           .
        
         
           CONTAINING
           ,
           
             
               I.
               The
               State
               of
               MUSICK
               in
               General
               .
            
             
               II.
               The
               Principles
               of
               PRESENT
               PRACTICE
               ;
               according
               to
               which
               are
               ,
            
             
               III.
               The
               Tables
               of
               PROPORTIONS
               ,
               calculated
               for
               the
               Viol
               ,
               and
               capable
               of
               being
               Accommodated
               to
               all
               sorts
               of
               Musick
               .
            
          
        
         
           By
           
             Thomas
             Salmon
          
           ,
           Rector
           of
           Mepsal
           in
           the
           County
           of
           BEDFORD
           .
        
         
           
             —
             Exemplaria
             Graeca
          
           
             Nocturnâ
             versate
             manu
             ,
             versate
             diurnâ
             .
          
           
             Hor.
             de
             art
             .
             Poet.
             
          
        
         
           Approved
           by
           both
           the
           Mathematick
           Professors
           of
           the
           Vniversity
           of
           Oxford
           .
        
         
           With
           Large
           REMARKS
           upon
           this
           whole
           Treatise
           ,
           By
           the
           Reverend
           and
           Learned
           
             John
             Wallis
          
           D.
           D.
           
        
         
           IMPRIMATUR
           .
           
             Gilb.
             Ironside
          
           .
           Vicecancel
           .
           Acad.
           Oxon.
           
        
         
           LONDON
           :
           Printed
           for
           
             John
             Lawrence
          
           ,
           at
           the
           Angel
           in
           the
           Poultrey
           .
           1688.
           
        
      
       
         
         
         
           TO
           The
           Valiant
           and
           Learned
           John
           Cutts
           
             Esq
          
           
             Adjutant
             General
          
           in
           the
           Service
           of
           His
           Imperial
           Majesty
           .
        
         
           THIS
           ,
           Sir
           ,
           is
           so
           far
           from
           the
           common
           road
           of
           Dedications
           ,
           that
           at
           first
           sight
           it
           will
           appear
           neither
           fit
           for
           me
           to
           give
           ,
           nor
           you
           to
           receive
           .
        
         
           Shall
           a
           Person
           so
           publickly
           employed
           in
           the
           greatest
           Attempts
           ,
           and
           most
           victorious
           Successes
           ,
           which
           the
           World
           has
           seen
           for
           these
           many
           Ages
           ,
           have
           leisure
           for
           Philosophical
           Speculations
           ,
           or
           divert
           to
           a
           Science
           proper
           for
           ease
           and
           pleasure
           ?
        
         
           Or
           should
           one
           consecrated
           to
           the
           Divine
           Service
           ,
           so
           laboriously
           search
           into
           the
           Intrigues
           of
           Nature
           ,
           and
           assist
           in
           the
           advancement
           of
           an
           
           Art
           ,
           which
           with
           its
           Airy
           Pleasures
           often
           captivates
           the
           Soul
           to
           sensual
           things
           ,
           and
           makes
           it
           more
           devoted
           to
           the
           World
           ,
           that
           is
           to
           be
           conquered
           with
           another
           sort
           of
           Victory
           than
           your
           Arms
           can
           obtain
           ?
        
         
           All
           this
           I
           have
           thought
           on
           ,
           yet
           still
           find
           so
           much
           in
           this
           affair
           ,
           as
           not
           only
           to
           excuse
           me
           ,
           but
           to
           make
           it
           acceptable
           to
           you
           .
        
         
           This
           Mathematical
           Discourse
           is
           indeed
           the
           Anatomy
           of
           MUSICK
           ,
           wherein
           the
           infinite
           Wisdom
           of
           the
           great
           Creator
           appears
           :
           How
           delightfully
           and
           wonderfully
           is
           it
           made
           !
           Marvellous
           are
           thy
           works
           ,
           O
           Lord
           ,
           and
           that
           my
           Soul
           knows
           right
           well
           .
        
         
           All
           the
           best
           Proportions
           ,
           are
           the
           best
           Chords
           of
           Musick
           ,
           and
           strike
           the
           Ear
           with
           a
           pleasure
           agreeable
           to
           the
           dignity
           of
           their
           Numbers
           .
           The
           effects
           of
           this
           the
           Sensualist
           is
           satisfied
           with
           ,
           and
           desires
           to
           seek
           no
           further
           .
        
         
           But
           is
           it
           not
           grateful
           to
           every
           Gentleman
           ,
           who
           is
           ennobled
           with
           such
           a
           
           Soul
           as
           yours
           ,
           to
           know
           the
           divine
           Harmony
           of
           the
           pleasure
           he
           enjoys
           ?
           Is
           it
           not
           the
           duty
           and
           Felicity
           of
           a
           Rational
           Being
           ,
           to
           consider
           how
           the
           whole
           System
           of
           the
           World
           is
           framed
           in
           Consort
           ?
           How
           Musical
           Instruments
           observe
           their
           Arithmetical
           Laws
           ,
           all
           the
           little
           Meanders
           of
           the
           Ear
           faithfully
           conveying
           the
           organiz'd
           sounds
           ,
           and
           the
           Soul
           of
           man
           made
           to
           receive
           the
           delight
           ,
           before
           he
           himself
           knows
           from
           whence
           it
           comes
           ?
        
         
           How
           great
           is
           this
           !
           How
           mean
           am
           I
           ,
           to
           set
           forth
           such
           a
           Divine
           Subject
           !
        
         
           However
           charming
           this
           still
           voice
           may
           be
           ,
           yet
           no
           body
           will
           believe
           it
           can
           be
           heard
           amongst
           Drums
           and
           Trumpets
           :
           Why
           should
           these
           Papers
           hope
           for
           acceptance
           in
           the
           Camp
           ?
        
         
           I
           must
           confess
           I
           should
           think
           them
           unseasonable
           ,
           had
           not
           you
           ,
           my
           Excellent
           Friend
           ,
           told
           me
           that
           your
           
           most
           Renowned
           General
           Lorain
           does
           in
           all
           the
           intervals
           of
           Action
           govern
           his
           Army
           like
           a
           Colledge
           ,
           and
           allow
           time
           for
           the
           repose
           of
           the
           Mind
           ,
           as
           well
           as
           for
           the
           over-running
           subdued
           Countries
           .
        
         
           You
           may
           then
           reflect
           upon
           the
           great
           Creator
           and
           Governour
           of
           the
           World
           ,
           who
           gave
           you
           being
           ,
           and
           now
           preserves
           you
           in
           the
           most
           eminent
           Hazards
           ;
           these
           shall
           help
           you
           to
           contemplate
           the
           infinite
           Wisdom
           ,
           and
           shall
           be
           of
           the
           greatest
           advantage
           to
           you
           ,
           since
           Piety
           is
           the
           best
           support
           of
           Courage
           ,
           and
           gives
           a
           refreshing
           Ease
           amidst
           the
           Rage
           of
           War.
           
        
         
           But
           the
           design
           of
           War
           is
           Peace
           ,
           and
           your
           Friends
           here
           long
           for
           the
           return
           of
           those
           cool
           hours
           ,
           wherein
           you
           may
           not
           only
           have
           leisure
           for
           these
           Theoretical
           Studies
           ,
           but
           advance
           true
           Practical
           Wisdom
           ;
           which
           you
           have
           already
           represented
           to
           us
           in
           the
           most
           advantageous
           dress
           ,
           and
           enflamed
           our
           desires
           to
           have
           her
           interest
           
           promoted
           to
           greater
           heights
           .
        
         
           I
           know
           the
           glory
           of
           the
           Field
           is
           very
           tempting
           ,
           but
           still
           you
           retain
           as
           great
           a
           passion
           for
           Learning
           ;
           this
           hath
           ever
           found
           a
           place
           in
           the
           noblest
           Breasts
           :
           The
           first
           great
           Conqueror
           of
           this
           Island
           is
           as
           glorious
           for
           his
           Pen
           as
           his
           Sword
           ,
           and
           by
           his
           Commentaries
           perpetuated
           his
           Victories
           .
        
         
           We
           have
           incomparably
           more
           of
           Athens
           here
           in
           England
           ,
           than
           your
           Confederates
           the
           Venetians
           this
           year
           got
           possession
           of
           :
           That
           ancient
           Treasury
           was
           long
           ago
           rifled
           ,
           and
           its
           Jewels
           brought
           into
           the
           Western
           part
           of
           Europe
           :
           this
           the
           learned
           Proprietors
           were
           very
           sensible
           of
           .
        
         
           When
           Cicero
           return'd
           home
           from
           Greece
           by
           Rhodes
           ,
           the
           famous
           Orator
           there
           Apollonius
           begg'd
           the
           favour
           of
           a
           Declamation
           :
           All
           the
           Company
           were
           amazed
           ,
           and
           strove
           with
           the
           highest
           Expressions
           to
           acknowledge
           the
           Obligation
           ;
           but
           Apollonius
           sate
           sad
           and
           silent
           :
           which
           when
           he
           perceived
           
           Cicero
           took
           amiss
           ,
           he
           said
           ,
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           .
           
             I
             value
             and
             admire
             you
             ,
             O
             Cicero
             ;
             but
             in
             the
             mean
             time
             I
             must
             pity
             the
             fortune
             of
             Greece
             ,
             since
             those
             excellent
             Goods
             ,
             Learning
             and
             Eloquence
             ,
             (
             which
             were
             all
             that
             was
             left
             )
             are
             now
             by
             you
             brought
             over
             to
             the
             Romans
             .
          
           Plutarch
           ,
           in
           the
           Life
           of
           Cicero
           .
        
         
           The
           Offers
           made
           in
           these
           Papers
           ,
           are
           the
           Musical
           Spoils
           and
           Relicks
           of
           Athens
           ,
           a
           long
           time
           buried
           in
           obscurity
           ;
           and
           though
           some
           years
           ago
           published
           to
           the
           World
           in
           their
           own
           Language
           ,
           yet
           never
           known
           to
           those
           whose
           greatest
           concern
           it
           was
           to
           be
           acquainted
           with
           them
           .
        
         
           And
           indeed
           Musick
           now
           very
           much
           wants
           such
           Patrons
           as
           you
           are
           ,
           whose
           Reverence
           to
           Antiquity
           and
           Learning
           ,
           may
           give
           Preheminence
           to
           the
           Nobler
           part
           ;
           which
           would
           at
           once
           both
           advance
           and
           regulate
           the
           Practick
           Pleasures
           .
        
         
         
           It
           must
           be
           acknowledged
           that
           this
           Divine
           Science
           has
           a
           great
           while
           sunk
           with
           the
           Devotion
           of
           Churches
           ,
           it
           has
           been
           little
           learned
           and
           little
           regarded
           by
           the
           Religious
           ;
           so
           that
           the
           Angelical
           Praises
           ,
           which
           we
           have
           the
           honour
           to
           communicate
           in
           upon
           Earth
           ,
           are
           faintly
           and
           unfrequently
           celebrated
           :
           This
           celestial
           accomplishment
           ,
           which
           God
           ordained
           to
           enliven
           our
           dull
           Affections
           ,
           is
           every-where
           wanting
           ;
           that
           the
           pleasure
           is
           as
           low
           as
           the
           skill
           of
           performing
           this
           most
           grateful
           part
           of
           Worship
           .
        
         
           Hence
           has
           Musick
           of
           late
           sought
           its
           principal
           Glory
           in
           Theatres
           ,
           and
           sensual
           Entertainments
           ,
           too
           mean
           a
           Service
           to
           be
           reckoned
           the
           designe
           of
           such
           an
           excellent
           Art
           :
           The
           infinite
           Wisdom
           created
           it
           for
           better
           purposes
           .
        
         
           And
           when
           God
           shall
           please
           to
           bless
           the
           World
           with
           greater
           degrees
           of
           Love
           ,
           and
           a
           better
           Adoration
           of
           himself
           ,
           He
           will
           raise
           up
           Men
           
           and
           Means
           thus
           to
           promote
           his
           Glory
           .
           Which
           directs
           me
           now
           to
           seek
           your
           Patronage
           to
           the
           Endeavours
           of
           ,
        
         
           
             SIR
             ,
          
           
             Your
             most
             Humble
             and
             most
             Affectionate
             Servant
             ,
             
               THOMAS
               SALMON
            
             .
          
           
             Mepsal
             ,
             
               Nov.
               1.
               1687.
               
            
          
        
      
       
         
         
           Advertisement
           To
           the
           READER
           .
        
         
           I
           Would
           not
           suffer
           this
           Proposal
           to
           be
           published
           ,
           till
           I
           had
           first
           communicated
           it
           to
           the
           most
           eminent
           Professors
           of
           this
           Science
           ,
           because
           having
           omitted
           the
           demonstrations
           I
           rely
           upon
           ,
           the
           Reader
           might
           be
           satisfied
           in
           their
           Testimonies
           .
        
         
           I
           know
           it
           is
           not
           fair
           to
           offer
           any
           thing
           in
           Mathematicks
           ,
           without
           giving
           the
           Demonstration
           with
           it
           as
           we
           go
           along
           ;
           but
           I
           considered
           that
           the
           Principles
           of
           this
           Science
           are
           very
           little
           known
           ,
           and
           would
           not
           have
           been
           much
           regarded
           by
           Gentlemen
           ,
           till
           they
           first
           had
           seen
           the
           use
           and
           necessity
           of
           them
           .
        
         
           As
           I
           thought
           this
           the
           best
           course
           ,
           so
           it
           has
           proved
           much
           better
           than
           I
           expected
           :
           for
           the
           most
           experienced
           Professor
           in
           this
           part
           of
           Learning
           ,
           hath
           sent
           me
           not
           only
           his
           Approbation
           ,
           but
           his
           Demonstration
           of
           my
           Principles
           ;
           from
           whence
           the
           more
           inquisitive
           Reader
           may
           receive
           a
           compleat
           satisfaction
           .
        
      
       
         
         
           ERRATA
           .
        
         
           Page
           6.
           —
           Line
           16.
           for
           intension
           read
           intention
           .
        
         
           7.
           —
           Penult
           .
           for
           Six
           read
           Sixth
           .
        
         
           10.
           —
           24.
           for
           there
           read
           these
           .
        
         
           13.
           —
           Antepenult
           .
           insert
           the
           word
           add
           .
        
         
           23.
           —
           7.
           for
           Numb
           .
           III.
           read
           Numb
           .
           IIII.
           
        
         
           27.
           —
           18.
           insert
           the
           word
           cross
           .
        
         
           
             Ult.
             —
             ult
          
           .
           two
           
             Iota's
             subscripts
          
           are
           wanting
           .
        
      
    
     
       
         
         
           A
           PROPOSAL
           To
           perform
           Musick
           ,
           IN
           Perfect
           and
           Mathematical
           Proportions
           .
        
         
           
             CHAPTER
             I.
             Of
             the
             State
             of
             Musick
             in
             General
             .
          
           
             WHEN
             the
             great
             Empires
             of
             the
             World
             ,
             were
             in
             the
             height
             of
             their
             Glory
             ,
             especially
             the
             Grecian
             and
             Roman
             ,
             (
             whose
             Authors
             have
             left
             us
             lasting
             Monuments
             of
             their
             Excellency
             )
             Then
             did
             all
             sorts
             of
             Learning
             flourish
             in
             the
             greatest
             Perfection
             :
             The
             Arms
             of
             the
             Conquerors
             ever
             carrying
             along
             with
             them
             Arts
             and
             Civility
             .
          
           
             But
             to
             bring
             about
             a
             fatal
             Period
             ,
             did
             the
             North
             swarm
             with
             barbarous
             Multitudes
             ,
             who
             came
             down
             like
             a
             mighty
             Torrent
             ,
             and
             subdued
             the
             best
             Nations
             of
             the
             World
             ;
             which
             were
             forc'd
             to
             become
             Rude
             and
             Illiterate
             ,
             because
             their
             New
             Masters
             and
             Inhabitants
             were
             such
             .
          
           
           
             Amidst
             these
             Calamities
             ,
             no
             wonder
             that
             Musick
             perished
             :
             All
             Learning
             lay
             in
             the
             Dust
             ,
             especially
             that
             which
             was
             proper
             to
             the
             Times
             of
             Peace
             .
          
           
             But
             this
             Darkness
             was
             not
             perpetual
             ;
             The
             Ages
             at
             last
             clear'd
             up
             ;
             and
             from
             the
             Ruines
             of
             Antiquity
             ,
             brought
             forth
             some
             broken
             Pieces
             ,
             which
             were
             by
             degrees
             set
             together
             ;
             and
             by
             this
             time
             of
             day
             are
             Arriv'd
             near
             their
             ancient
             Glory
             .
             Guido
             has
             been
             Refining
             above
             Six
             Hundred
             Years
             .
          
           
             Two
             things
             are
             chiefly
             conducing
             to
             this
             Restoration
             :
             The
             great
             Genius
             of
             the
             Age
             we
             live
             in
             ,
             and
             the
             great
             Diligence
             in
             searching
             after
             Antiquity
             :
             The
             excellent
             Editions
             of
             the
             best
             Authors
             ,
             and
             the
             most
             laborious
             Comments
             upon
             them
             ,
             abundantly
             testifie
             the
             Truth
             of
             this
             .
          
           
             In
             both
             these
             Felicities
             ,
             Musick
             has
             had
             as
             great
             a
             share
             as
             any
             ;
             
               Aristoxenus
               ,
               Euclide
               ,
               Nichomachus
               ,
               Alypius
               ,
               Bacchius
               ,
               Gaudentius
               ,
               Aristides
               ,
               Martian
               ,
            
             have
             been
             with
             a
             great
             deal
             of
             Diligence
             ,
             set
             forth
             by
             
               Marcus
               Meibomius
            
             ,
             at
             Amsterdam
             ,
             in
             the
             Year
             1652.
             
          
           
             And
             above
             all
             ,
             is
             ,
             
               Claudius
               Ptolomaeus
            
             ,
             who
             Corrected
             and
             Reconciled
             the
             Pythagoreans
             ,
             and
             Aristoxeneans
             ,
             the
             speculative
             and
             practical
             Parties
             :
             This
             Author
             was
             Published
             by
             Dr.
             Wallis
             ,
             at
             Oxford
             ,
             in
             the
             Year
             1682
             ,
             who
             added
             an
             Appendix
             ,
             comparing
             the
             Ancient
             and
             Modern
             Harmony
             ;
             which
             is
             as
             the
             Key
             to
             all
             our
             Speculations
             ,
             and
             without
             which
             the
             former
             Authors
             were
             hardly
             Intelligible
             .
          
           
             Nor
             are
             we
             less
             beholding
             to
             the
             Excellent
             Genius
             of
             our
             Modern
             Musicians
             :
             There
             are
             ,
             indeed
             ,
             only
             two
             Fragments
             (
             as
             I
             know
             of
             )
             remaining
             of
             the
             ancient
             Grecian
             Compositions
             ;
             One
             of
             
             Pindar's
             ,
             found
             
             by
             Kircher
             ,
             at
             Messana
             in
             Sicily
             ;
             the
             other
             of
             
             Dionysius's
             ,
             rescued
             by
             Dr.
             Bernard
             ,
             from
             lying
             hid
             amongst
             some
             Papers
             of
             Arch-Bishop
             
             Vsher's
             ;
             both
             Published
             with
             Chilmeads
             Notes
             in
             the
             end
             of
             Aratus
             ,
             at
             the
             
               Oxford
               Theatre
            
             :
             These
             are
             very
             short
             ,
             and
             very
             imperfect
             ,
             and
             therefore
             we
             cannot
             make
             any
             Judgment
             of
             their
             Songs
             or
             Lessons
             .
          
           
             But
             by
             all
             that
             we
             can
             discern
             from
             their
             Harmonical
             Treatises
             ,
             There
             never
             was
             such
             regularity
             in
             the
             designing
             of
             Keys
             ,
             such
             a
             pleasing
             sweetness
             of
             Air
             ,
             such
             a
             various
             contexture
             of
             Chords
             ,
             as
             the
             Practical
             Musicians
             are
             at
             this
             day
             Masters
             of
             .
          
           
             It
             may
             seem
             now
             ,
             that
             there
             remains
             nothing
             to
             be
             added
             ,
             or
             to
             be
             learn'd
             out
             of
             those
             Eminent
             Authors
             I
             have
             here
             Recited
             ;
             And
             the
             mighty
             Power
             of
             Musick
             ,
             Recorded
             by
             the
             most
             Grave
             and
             Authentick
             Historians
             ,
             may
             be
             lookt
             upon
             as
             Romance
             ,
             since
             all
             the
             Excellencies
             now
             perform'd
             ,
             cannot
             conquer
             the
             Soul
             ,
             and
             subdue
             the
             Passions
             as
             has
             been
             done
             of
             Old.
             
          
           
             But
             before
             we
             quit
             the
             Testimonies
             of
             what
             Musick
             has
             done
             ,
             and
             despair
             of
             any
             further
             Advancement
             ;
             Let
             us
             enquire
             whether
             there
             be
             not
             something
             very
             Considerable
             still
             wanting
             ,
             something
             Fundamental
             very
             much
             amiss
             ,
             even
             that
             which
             the
             forementioned
             Philosophers
             were
             likely
             to
             be
             most
             Excellent
             at
             ,
             when
             the
             Learned
             and
             Practical
             Part
             were
             met
             in
             the
             same
             Persons
             :
             Whether
             this
             be
             not
             the
             Accurate
             Observation
             of
             Proportions
             ,
             which
             the
             Soul
             is
             from
             Heaven
             inform'd
             to
             Judge
             of
             ,
             and
             the
             Body
             in
             Union
             with
             it
             ,
             must
             Submit
             to
             .
          
           
             Surely
             ,
             I
             need
             not
             prove
             ,
             that
             all
             Musick
             consists
             in
             Proportion
             ;
             that
             the
             more
             exact
             the
             Proportions
             ,
             
             the
             more
             Excellent
             the
             Musick
             :
             This
             is
             that
             ,
             all
             the
             World
             is
             agreed
             in
             .
             For
             this
             ,
             I
             have
             every
             Man
             of
             my
             side
             ,
             that
             except
             the
             Voice
             ,
             the
             Instrument
             be
             well
             in
             Tune
             ,
             the
             best
             Composition
             that
             was
             ever
             made
             ,
             will
             never
             please
             ;
             And
             what
             is
             it
             to
             be
             in
             Tune
             ,
             but
             for
             every
             Note
             to
             bear
             a
             due
             Proportion
             to
             one
             another
             ?
          
           
             Indeed
             ,
             the
             Proportions
             of
             Musick
             are
             twofold
             ;
             First
             ,
             In
             respect
             of
             Tune
             ,
             and
             Second
             ,
             In
             respect
             of
             Time
             :
             The
             latter
             of
             these
             ,
             which
             Dr.
             Vossius
             contends
             so
             much
             about
             ,
             is
             certainly
             very
             considerable
             ;
             that
             the
             Musick
             should
             agree
             with
             the
             Poetical
             Prosodia
             ;
             that
             all
             the
             Variety
             of
             Rythmical
             Feet
             should
             have
             their
             proper
             Movements
             .
          
           
             Then
             would
             the
             Sense
             be
             favoured
             by
             such
             Measures
             ,
             as
             were
             most
             fit
             to
             Excite
             or
             Allay
             the
             Passions
             aim'd
             at
             ;
             and
             the
             Words
             of
             a
             Song
             would
             be
             capable
             of
             a
             more
             easie
             and
             intelligible
             Pronunciation
             .
          
           
             Since
             Musicians
             have
             not
             undert●ken
             to
             be
             Poets
             ,
             and
             Poets
             have
             left
             off
             being
             Musicians
             ;
             this
             now
             disjoynted
             Work
             ,
             of
             making
             Words
             ,
             and
             setting
             Tunes
             to
             them
             ,
             has
             not
             been
             so
             exactly
             done
             as
             formerly
             ,
             when
             the
             same
             Authour
             perform'd
             both
             .
          
           
             But
             were
             it
             never
             so
             well
             done
             for
             Time
             ,
             and
             the
             Proportions
             of
             Tune
             neglected
             ;
             it
             could
             signifie
             nothing
             :
             None
             will
             pretend
             to
             make
             Musick
             by
             playing
             good
             Time
             ,
             except
             the
             Instrument
             and
             Voice
             be
             in
             Tune
             .
          
           
             However
             ,
             till
             both
             these
             Fundamental
             Points
             be
             observed
             with
             such
             Exactness
             and
             Excellency
             ,
             as
             the
             Ancients
             took
             care
             of
             ;
             we
             must
             not
             say
             we
             do
             all
             they
             did
             ,
             or
             that
             they
             could
             not
             prevail
             more
             than
             we
             can
             ;
             all
             the
             Modern
             Excellencies
             may
             be
             rendred
             Ineffectual
             ,
             
             by
             tolerating
             so
             many
             unproportionate
             imperfections
             ,
             as
             are
             every
             where
             found
             amongst
             us
             .
          
           
             I
             shall
             not
             here
             give
             an
             account
             of
             all
             those
             accurate
             Proportions
             ,
             which
             the
             Ancients
             contended
             for
             ,
             nor
             their
             little
             enharmonical
             Distances
             ,
             whereof
             their
             more
             curious
             Musick
             did
             consist
             ;
             but
             only
             of
             what
             is
             now
             practised
             amongst
             us
             ,
             that
             the
             certain
             Knowledge
             of
             our
             Fundamental
             Principles
             may
             produce
             Performances
             ,
             much
             more
             exact
             and
             powerful
             .
          
        
         
           
             CHAP.
             II.
             The
             present
             Practice
             of
             Musick
             .
          
           
             THE
             Hours
             of
             Study
             are
             tedious
             to
             some
             and
             precious
             to
             others
             :
             I
             cannot
             therefore
             suppose
             any
             man
             will
             search
             into
             the
             demonstrative
             Reasons
             ,
             or
             acquaint
             himself
             with
             the
             Mathematical
             Operations
             belonging
             to
             this
             Proposal
             ,
             till
             he
             be
             first
             assured
             of
             the
             truth
             and
             usefulness
             of
             it
             .
          
           
             So
             that
             what
             is
             purely
             Speculative
             shall
             be
             reserv'd
             at
             present
             :
             This
             offers
             nothing
             but
             the
             Principles
             of
             continual
             Practice
             ,
             whereby
             the
             Reader
             may
             be
             lead
             into
             the
             knowledge
             of
             what
             he
             is
             always
             to
             design
             ;
             and
             taking
             the
             String
             of
             any
             Instrument
             ,
             may
             give
             his
             Eye
             ,
             and
             his
             Ear
             ,
             and
             his
             Reason
             ,
             an
             immediate
             satisfaction
             ,
             in
             all
             that
             is
             here
             dictated
             to
             him
             .
          
           
             Before
             we
             compose
             or
             perform
             any
             Musick
             ,
             two
             things
             must
             be
             provided
             for
             .
          
           
             I.
             That
             we
             have
             some
             little
             gradual
             Notes
             ,
             which
             
             may
             (
             whilst
             the
             Voice
             rises
             or
             falls
             )
             succeed
             one
             another
             in
             the
             best
             Proportions
             possible
             ;
             whereof
             (
             as
             of
             so
             many
             Alphabetical
             Elements
             )
             the
             whole
             Musick
             must
             consist
             .
          
           
             II.
             These
             gradual
             Notes
             must
             be
             placed
             in
             such
             order
             ,
             that
             the
             greater
             Intervals
             (
             compounded
             of
             them
             )
             may
             in
             the
             best
             Proportions
             possible
             arise
             out
             of
             them
             ,
             and
             be
             come
             at
             with
             the
             greatest
             conveniency
             :
             That
             in
             all
             the
             Points
             ,
             where
             the
             single
             Notes
             determine
             ,
             there
             the
             larger
             Chords
             may
             be
             exactly
             coincident
             ;
             if
             it
             was
             not
             for
             this
             ,
             there
             could
             be
             no
             Consort-Musick
             .
          
           
             To
             set
             forth
             this
             ,
             we
             may
             as
             well
             use
             the
             first
             seven
             Letters
             of
             the
             Alphabet
             ,
             as
             all
             the
             hard
             Names
             of
             
             Guido's
             Gamut
             ;
             because
             they
             were
             framed
             long
             before
             Musick
             was
             brought
             into
             that
             good
             order
             wherein
             it
             now
             stands
             ,
             and
             the
             first
             intension
             of
             them
             is
             not
             agreeable
             to
             the
             present
             practise
             .
          
           
             Only
             this
             will
             be
             worth
             our
             Observation
             ,
             that
             whereas
             in
             the
             Scale
             of
             Musick
             ,
             there
             are
             three
             Octaves
             ,
             (
             besides
             the
             double
             Notes
             and
             Notes
             in
             Alt
             )
             viz.
             the
             Base
             ,
             Mean
             ,
             and
             Treble
             ,
             we
             may
             use
             Three
             Sizes
             of
             Letters
             in
             a
             greater
             ,
             middle
             and
             lesser
             Character
             :
             as
             will
             be
             found
             in
             the
             Tables
             of
             Proportions
             .
          
           
             For
             understanding
             the
             two
             things
             pre-required
             ,
             we
             suppose
             the
             proportion
             of
             one
             gradual
             Note
             to
             be
             contained
             between
             A
             and
             B
             ,
             then
             between
             B
             and
             C
             the
             proportion
             of
             another
             gradual
             Note
             ,
             though
             much
             lesser
             ;
             these
             two
             single
             proportions
             ,
             viz.
             that
             of
             A
             B
             ,
             and
             that
             of
             B
             C
             being
             added
             together
             ,
             must
             exactly
             constitute
             a
             lesser
             Third
             ;
             the
             proportions
             of
             the
             two
             gradual
             Notes
             must
             determine
             in
             that
             point
             ,
             where
             the
             compounded
             interval
             may
             be
             coincident
             with
             them
             .
          
           
           
             To
             proceed
             ,
             if
             we
             add
             another
             gradual
             Proportion
             from
             C
             to
             D
             ,
             then
             must
             arise
             the
             exact
             proportion
             of
             a
             Fourth
             ,
             from
             the
             first
             given
             A
             to
             the
             Note
             D
             :
             if
             one
             more
             be
             added
             from
             D
             to
             E
             ,
             there
             must
             be
             found
             the
             exact
             proportion
             of
             a
             Fifth
             ,
             from
             A
             to
             E
             ,
             and
             of
             a
             greater
             Third
             from
             C
             to
             E.
             
          
           
             Thus
             must
             the
             Gradual
             Notes
             be
             contrived
             to
             be
             exactly
             subservient
             to
             the
             greater
             intervals
             thorough
             all
             the
             Octaves
             :
             and
             if
             at
             any
             time
             this
             cannot
             be
             (
             as
             may
             happen
             in
             two
             or
             three
             instances
             )
             such
             particular
             Chords
             must
             be
             esteem'd
             inconcinnous
             and
             inconvenient
             ,
             but
             they
             are
             very
             few
             ,
             and
             lye
             much
             out
             of
             the
             way
             .
          
           
             If
             we
             settle
             one
             Octave
             ,
             the
             whole
             work
             is
             as
             good
             as
             done
             ;
             all
             the
             rest
             is
             only
             repetition
             of
             the
             same
             Notes
             in
             a
             larger
             or
             more
             minute
             figure
             :
             for
             the
             eight
             Notes
             which
             are
             used
             in
             constant
             practise
             ,
             proceeding
             gradually
             ,
             take
             up
             just
             half
             the
             string
             ,
             from
             the
             sound
             open
             to
             the
             middle
             of
             it
             :
             And
             if
             we
             have
             occasion
             to
             go
             further
             ,
             't
             is
             but
             just
             the
             same
             over
             again
             .
          
           
             The
             great
             concern
             is
             in
             what
             order
             our
             gradual
             Notes
             (
             which
             are
             of
             different
             sizes
             )
             must
             stand
             ,
             from
             the
             Key
             or
             Sound
             given
             ,
             till
             we
             arrive
             at
             the
             Octave
             ;
             for
             there
             will
             be
             a
             great
             variety
             ,
             according
             as
             the
             lesser
             gradual
             Notes
             are
             placed
             sooner
             or
             later
             :
             This
             must
             be
             lookt
             upon
             as
             the
             internal
             constitution
             of
             an
             Octave
             ,
             which
             practical
             Musicians
             commonly
             understand
             by
             their
             Flat
             or
             Sharp
             ,
             that
             is
             ,
             their
             greater
             or
             lesser
             Third
             .
          
           
             But
             as
             much
             as
             I
             can
             observe
             from
             the
             Compositions
             of
             the
             most
             Eminent
             Masters
             for
             these
             last
             Twenty
             years
             ,
             this
             internal
             constitution
             of
             an
             Octave
             is
             but
             twofold
             :
             either
             with
             a
             greater
             Third
             ,
             Sixth
             and
             Seventh
             ;
             or
             a
             Lesser
             Third
             ,
             Six
             and
             Seventh
             :
             In
             the
             same
             composition
             all
             are
             lesser
             ,
             or
             all
             greater
             .
          
           
           
             There
             needs
             then
             only
             this
             twofold
             constitution
             of
             the
             Octave
             to
             be
             considered
             by
             us
             ,
             the
             two
             Keys
             A
             and
             C
             :
             all
             the
             rest
             serve
             only
             to
             render
             the
             same
             series
             of
             Notes
             in
             different
             pitches
             ;
             which
             is
             demonstrable
             by
             transposing
             Tunes
             from
             one
             Key
             to
             another
             :
             The
             Tune
             remains
             the
             same
             ,
             only
             the
             compass
             of
             the
             Voice
             or
             Instrument
             is
             better
             accommodated
             .
          
           
             These
             two
             Keys
             A
             and
             C
             are
             called
             Natural
             ,
             because
             the
             Proportions
             ,
             originally
             assigned
             to
             each
             Letter
             ,
             keep
             those
             proper
             places
             ,
             which
             either
             Guido
             the
             first
             restorer
             or
             immemorial
             Custom
             hath
             allotted
             to
             them
             ;
             Whereas
             by
             taking
             other
             Keys
             ,
             as
             suppose
             G
             for
             A
             ,
             the
             proportions
             or
             different
             sizes
             of
             the
             gradual
             Notes
             are
             forced
             to
             shift
             their
             quarters
             ,
             and
             by
             flats
             or
             sharps
             to
             straiten
             or
             widen
             their
             usual
             distances
             .
          
           
             'T
             is
             sufficient
             demonstration
             for
             all
             this
             ,
             that
             when
             any
             Tune
             is
             transposed
             into
             A
             or
             C
             ,
             it
             wants
             nor
             flats
             nor
             sharps
             ,
             whatever
             it
             did
             before
             .
          
           
             I
             shall
             in
             the
             first
             place
             give
             you
             the
             natural
             order
             of
             the
             gradual
             Notes
             as
             they
             stand
             in
             the
             Key
             A
             ,
             where
             we
             have
             a
             lesser
             Third
             Sixth
             and
             Seventh
             ,
             exactly
             coincident
             with
             the
             Third
             Sixth
             and
             Seventh
             gradual
             Note
             .
          
           
             You
             have
             between
             every
             Letter
             ,
             that
             part
             or
             proportion
             of
             the
             string
             assigned
             which
             each
             gradual
             Note
             requires
             :
             underneath
             you
             have
             the
             proportion
             of
             each
             compounded
             interval
             ,
             what
             part
             of
             the
             string
             it
             's
             stop
             must
             be
             when
             compared
             with
             the
             whole
             string
             open
             from
             the
             Nut
             to
             the
             Bridge
             .
          
           
           
             
               
                 The
                 Constitution
                 of
                 the
                 Key
                 A.
                 
              
               
                 
                    
                
                 
                   A.
                   
                
                 
                   9.
                   
                   B.
                   
                
                 
                   16.
                   
                   C.
                   
                
                 
                   10.
                   
                   D.
                   
                
                 
                   9.
                   
                   E.
                   
                
                 
                   16.
                   
                   F.
                   
                
                 
                   9.
                   
                   G.
                   
                
                 
                   10
                   .
                   a
                   .
                
              
               
                 
                   A
                   Lesser
                   Third
                
                 
                    
                
                 
                    
                
                 
                   6
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Fourth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   4
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Fifth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   3
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Lesser
                   Sixth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   3
                   /
                   8
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Lesser
                   Seventh
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   4
                   /
                   9
                
                 
                    
                
              
               
                 
                   An
                   Eighth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   2
                
              
            
          
           
             The
             Experiment
             must
             be
             thus
             :
             You
             are
             to
             take
             any
             one
             String
             ,
             and
             suppose
             it
             to
             be
             the
             Key
             A
             ,
             when
             it
             is
             open
             :
             then
             measure
             the
             9th
             part
             of
             it
             ,
             you
             will
             have
             B
             or
             one
             gradual
             Note
             .
             Not
             that
             the
             first
             Fret
             must
             stand
             there
             ,
             but
             the
             second
             ;
             for
             we
             are
             not
             reckning
             according
             to
             Tableture
             ,
             but
             Notes
             specified
             by
             the
             first
             Letters
             of
             their
             hard
             names
             :
             the
             half
             Notes
             shall
             be
             considered
             afterwards
             .
          
           
             From
             the
             place
             of
             B
             measure
             the
             16th
             part
             of
             the
             remaining
             String
             ,
             there
             will
             be
             C
             ,
             the
             least
             gradual
             Note
             :
             And
             there
             you
             will
             arrive
             at
             the
             6th
             part
             of
             the
             whole
             String
             ,
             which
             is
             the
             proportion
             of
             the
             Lesser
             Third
             ;
             and
             the
             Ear
             will
             acknowledge
             it
             to
             be
             so
             .
          
           
             From
             the
             place
             of
             C
             take
             the
             10th
             part
             of
             the
             remaining
             String
             ,
             there
             will
             be
             D
             ,
             another
             gradual
             Note
             ,
             much
             wider
             than
             the
             last
             ,
             between
             B
             and
             C
             ,
             but
             something
             less
             than
             the
             first
             between
             A
             and
             B.
             At
             the
             place
             of
             D
             ,
             you
             will
             arrive
             at
             the
             4th
             part
             of
             the
             whole
             String
             ,
             which
             is
             the
             proportion
             of
             a
             practical
             Fourth
             .
          
           
           
             Here
             ,
             to
             prevent
             all
             perplexity
             and
             mis-understanding
             ,
             the
             Reader
             must
             carefully
             distinguish
             in
             the
             Terms
             of
             Art
             :
             The
             Practical
             Musician
             reckons
             how
             many
             gradual
             Notes
             he
             has
             gone
             over
             from
             his
             Key
             or
             Sound
             given
             ,
             and
             accordingly
             calls
             his
             Intervals
             a
             Third
             ,
             Fourth
             ,
             and
             Fifth
             ,
             as
             having
             so
             many
             gradual
             Notes
             contained
             in
             them
             ;
             but
             the
             Mathematician
             regards
             only
             the
             parts
             of
             the
             String
             ,
             what
             proportion
             the
             part
             stopped
             bears
             to
             the
             String
             open
             .
          
           
             Here
             indeed
             the
             Practical
             and
             Mathematical
             Terms
             are
             the
             same
             ,
             a
             fourth
             part
             of
             the
             String
             mathematically
             measured
             ,
             is
             a
             practical
             fourth
             ;
             but
             in
             all
             other
             Chords
             they
             differ
             ,
             as
             we
             have
             seen
             a
             Lesser
             Third
             to
             be
             the
             sixth
             part
             of
             a
             String
             .
          
           
             From
             the
             place
             of
             D
             take
             the
             9th
             part
             of
             the
             remaining
             String
             ,
             (
             which
             is
             a
             gradual
             Note
             of
             the
             same
             proportion
             with
             the
             first
             ,
             between
             A
             and
             B
             ;
             )
             here
             will
             be
             E
             :
             and
             here
             you
             will
             find
             you
             are
             arrived
             at
             the
             third
             part
             of
             a
             String
             ,
             which
             is
             the
             grateful
             proportion
             of
             a
             Practical
             Fifth
             .
          
           
             The
             proportions
             of
             the
             Lesser
             Sixth
             and
             Seventh
             ,
             viz.
             ⅜
             and
             4
             /
             9
             ,
             are
             of
             a
             different
             sort
             from
             the
             rest
             ;
             the
             former
             Chords
             arise
             from
             the
             natural
             division
             of
             an
             Octave
             or
             Duple
             proportion
             ,
             there
             are
             formed
             by
             an
             artificial
             addition
             of
             a
             second
             or
             a
             third
             to
             the
             Fifth
             :
             The
             former
             proportions
             are
             called
             by
             Arithmeticians
             Super-particular
             ,
             these
             are
             super-partient
             .
          
           
             I
             believe
             the
             Reader
             will
             not
             desire
             to
             be
             troubled
             with
             the
             nature
             of
             them
             here
             ,
             but
             only
             to
             be
             informed
             how
             to
             measure
             them
             for
             his
             present
             satisfaction
             :
             He
             is
             to
             know
             then
             ,
             that
             he
             must
             not
             take
             the
             upper
             number
             3
             of
             3
             /
             8
             for
             the
             third
             part
             of
             the
             whole
             String
             ,
             for
             then
             a
             
             Lesser
             Sixth
             would
             be
             the
             same
             as
             a
             Fifth
             ;
             but
             he
             is
             to
             devide
             the
             whole
             String
             into
             eight
             parts
             ,
             as
             the
             lower
             number
             specifies
             ,
             and
             then
             where
             three
             of
             those
             parts
             determine
             from
             the
             Nut
             ,
             there
             will
             be
             a
             Lesser
             Sixth
             .
          
           
             This
             is
             the
             addition
             of
             the
             16th
             part
             of
             the
             remaining
             String
             from
             E
             to
             F
             :
             for
             A
             open
             to
             E
             was
             the
             third
             part
             of
             the
             String
             ,
             that
             is
             a
             Practical
             Fifth
             ;
             from
             A
             open
             to
             F
             will
             be
             three
             parts
             of
             the
             whole
             String
             divided
             into
             eight
             parts
             ,
             which
             is
             a
             practical
             Lesser
             Sixth
             .
          
           
             A
             Lesser
             Seventh
             is
             produced
             by
             taking
             a
             9th
             part
             of
             the
             remaining
             String
             from
             F
             to
             G
             ,
             which
             is
             a
             Lesser
             Third
             above
             E
             ,
             this
             will
             be
             found
             to
             determine
             at
             4
             when
             the
             whole
             String
             is
             divided
             into
             9
             parts
             ,
             and
             therefore
             is
             the
             proportion
             4
             /
             9.
             
          
           
             From
             G
             take
             a
             10th
             part
             of
             the
             remaining
             String
             ,
             you
             will
             arrive
             at
             a
             ,
             the
             precise
             middle
             of
             the
             whole
             String
             ,
             so
             that
             an
             Octave
             is
             a
             duple
             proportion
             ;
             the
             fullest
             and
             most
             perfect
             satisfaction
             that
             can
             be
             given
             to
             the
             Ear.
             And
             by
             this
             is
             the
             whole
             proceeding
             demonstrated
             to
             be
             right
             ,
             because
             not
             only
             by
             the
             way
             ,
             every
             interval
             was
             exactly
             found
             in
             its
             proper
             place
             ,
             but
             at
             last
             this
             Chord
             ,
             the
             sum
             total
             of
             all
             Musick
             ,
             does
             just
             contain
             all
             its
             Particulars
             .
          
           
             After
             the
             same
             manner
             may
             the
             internal-constitution
             of
             an
             Octave
             in
             the
             Key
             C
             be
             demonstrated
             :
             I
             shall
             set
             it
             down
             without
             any
             explication
             ,
             because
             the
             Experiment
             and
             Reason
             of
             both
             are
             alike
             .
          
           
           
             
               
                 The
                 Constitution
                 of
                 the
                 Key
                 C.
                 
              
               
                 
                    
                
                 
                   C.
                   
                
                 
                   10.
                   
                   D.
                   
                
                 
                   9.
                   
                   E.
                   
                
                 
                   16.
                   
                   F.
                   
                
                 
                   9
                   .
                   g
                   .
                
                 
                   10
                   .
                   a
                   .
                
                 
                   9
                   .
                   b
                   .
                
                 
                   16
                   .
                   c
                   .
                
              
               
                 
                   A
                   Greater
                   Third
                
                 
                    
                
                 
                    
                
                 
                   5
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Fourth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   4
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Fifth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   3
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Greater
                   Sixth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   2
                   /
                   5
                
                 
                    
                
                 
                    
                
              
               
                 
                   A
                   Greater
                   Seventh
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   7
                   /
                   15
                
                 
                    
                
              
               
                 
                   An
                   Eighth
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                   2
                
              
            
          
           
             Though
             we
             have
             all
             along
             supposed
             a
             Monochord
             or
             single
             String
             ,
             to
             make
             this
             demonstration
             more
             evident
             ,
             and
             to
             shew
             that
             all
             the
             gradual
             Notes
             of
             an
             Octave
             put
             together
             ,
             arrive
             just
             at
             the
             middle
             of
             the
             String
             ;
             yet
             the
             progress
             of
             the
             Proportions
             is
             the
             same
             ,
             when
             we
             take
             some
             of
             them
             upon
             one
             String
             ,
             some
             upon
             another
             .
             For
             each
             String
             is
             tuned
             Unison
             to
             some
             part
             of
             that
             which
             went
             before
             ;
             so
             that
             't
             is
             all
             one
             whither
             the
             Proportions
             go
             along
             upon
             the
             same
             String
             ,
             or
             go
             on
             to
             the
             next
             ,
             when
             we
             come
             at
             the
             place
             of
             tuning
             Unisons
             .
          
           
             As
             suppose
             upon
             the
             Viol
             the
             Fourth
             String
             be
             C
             open
             ,
             when
             you
             come
             to
             E
             ,
             or
             the
             Fourth
             Fret
             ,
             you
             have
             a
             greater
             Third
             ;
             then
             't
             is
             all
             one
             whether
             you
             take
             the
             16th
             part
             of
             the
             same
             String
             to
             make
             F
             at
             the
             fifth
             Fret
             ,
             or
             the
             16th
             part
             of
             the
             next
             whole
             String
             to
             make
             F
             upon
             the
             first
             Fret
             :
             'T
             is
             all
             one
             ,
             because
             the
             
             third
             String
             open
             is
             tuned
             Unison
             to
             E
             ,
             or
             the
             fourth
             Fret
             upon
             the
             fourth
             String
             .
          
           
             As
             I
             would
             avoid
             troubling
             my
             Reader
             with
             needless
             difficulties
             ,
             so
             I
             would
             not
             omit
             any
             thing
             of
             necessary
             information
             ;
             this
             last
             consideration
             makes
             me
             here
             add
             a
             discourse
             of
             Seconds
             ,
             which
             is
             the
             name
             whereby
             the
             gradual
             Notes
             are
             commonly
             called
             :
             for
             reckning
             inclusively
             in
             Musick
             ,
             one
             Interval
             ,
             which
             must
             needs
             be
             contain'd
             between
             two
             Sounds
             ,
             is
             term'd
             a
             Second
             .
          
           
             It
             is
             best
             to
             treat
             of
             them
             in
             that
             method
             ,
             which
             our
             Authors
             used
             in
             the
             Classical
             times
             ,
             because
             't
             is
             their
             Perfection
             we
             are
             now
             aiming
             at
             :
             They
             divided
             their
             Musick
             into
             three
             sorts
             ,
             
               Diatonick
               ,
               Chromatick
               ,
               Enharmonick
            
             ,
             which
             was
             so
             diversified
             by
             those
             several
             sorts
             of
             Seconds
             or
             gradual
             Proportions
             they
             used
             therein
             .
          
           
             1.
             
             In
             Diatonick
             Musick
             ,
             the
             foregoing
             constitution
             of
             an
             Octave
             discovers
             three
             several
             sizes
             of
             Seconds
             ,
             viz.
             the
             9th
             part
             of
             a
             String
             ,
             the
             10th
             part
             ,
             and
             the
             16th
             part
             .
             I
             would
             satisfie
             the
             Reader
             in
             this
             variety
             ,
             because
             he
             will
             think
             much
             to
             enter
             upon
             an
             Observation
             ,
             not
             yet
             received
             ,
             except
             he
             knows
             some
             necessity
             for
             it
             .
          
           
             We
             must
             have
             the
             Proportion
             of
             the
             16th
             part
             of
             the
             String
             between
             B
             and
             C
             ,
             as
             also
             between
             E
             and
             F
             ,
             or
             we
             cannot
             bring
             our
             gradual
             Notes
             into
             the
             form
             of
             an
             Octave
             ,
             into
             the
             compass
             of
             a
             duple
             proportion
             ;
             this
             is
             already
             acknowledged
             both
             by
             speculation
             and
             practice
             :
             No
             one
             ever
             yet
             pretended
             to
             rise
             or
             fall
             eight
             Notes
             one
             after
             another
             ,
             all
             of
             the
             same
             size
             .
          
           
             To
             this
             16th
             part
             we
             must
             a
             9th
             part
             ,
             or
             we
             can
             never
             have
             an
             exact
             Lesser
             Third
             ,
             which
             is
             the
             6th
             part
             of
             the
             whole
             String
             ;
             but
             if
             we
             add
             another
             Note
             of
             the
             
             same
             size
             ,
             viz.
             a
             9th
             part
             to
             make
             up
             that
             Lesser
             Third
             a
             Fourth
             ,
             we
             shall
             find
             that
             we
             have
             a
             great
             way
             over-shot
             the
             fourth
             part
             of
             the
             String
             ,
             and
             without
             taking
             the
             10th
             part
             ,
             we
             can
             never
             hit
             it
             ;
             as
             will
             appear
             by
             the
             former
             demonstrations
             upon
             the
             Monochord
             ,
             in
             many
             instances
             .
          
           
             I
             must
             confess
             ,
             this
             is
             so
             contrary
             to
             the
             common
             Opinion
             of
             Practical
             Musicians
             ,
             that
             I
             would
             not
             insist
             upon
             it
             ,
             did
             not
             necessity
             compel
             me
             ,
             did
             not
             the
             greatest
             Reason
             and
             Authority
             assure
             me
             ,
             that
             it
             will
             not
             be
             hereafter
             denied
             :
             Of
             these
             three
             sizes
             of
             Seconds
             does
             the
             whole
             progress
             ,
             from
             the
             Key
             to
             the
             Octave
             ,
             consist
             in
             the
             forementioned
             order
             ,
             being
             all
             along
             exactly
             coincident
             with
             the
             larger
             Intervals
             .
          
           
             My
             Authorities
             are
             
             Cartes's
             Musick
             ,
             
             Gassendus's
             Introduction
             ,
             
             Wallis's
             Appendix
             ,
             and
             all
             other
             Learned
             men
             ,
             who
             have
             in
             this
             last
             Age
             reviewed
             the
             Harmonical
             concerns
             .
             'T
             is
             time
             certainly
             to
             receive
             into
             practice
             those
             Improvements
             ,
             which
             the
             greatest
             Modern
             Philosophers
             in
             the
             World
             have
             afforded
             Musick
             .
          
           
             And
             indeed
             't
             is
             in
             vain
             to
             stand
             out
             ,
             Nature
             always
             acknowledged
             and
             received
             them
             ;
             a
             good
             Voice
             performing
             by
             it self
             ,
             a
             faithful
             Hand
             guided
             by
             a
             good
             Ear
             upon
             an
             unfretted
             unconfined
             Instrument
             ,
             exactly
             observes
             them
             :
             All
             that
             I
             contend
             for
             is
             ,
             that
             the
             Practiser
             may
             know
             what
             he
             does
             ,
             and
             may
             always
             make
             that
             his
             design
             ,
             which
             is
             his
             excellency
             .
          
           
             When
             we
             have
             thus
             much
             granted
             ,
             then
             may
             the
             last
             Chapter
             of
             this
             Proposal
             be
             very
             acceptable
             ;
             which
             puts
             into
             his
             hands
             the
             Tables
             of
             Proportions
             calculated
             for
             every
             Key
             ,
             that
             he
             may
             perform
             them
             upon
             those
             Instruments
             ,
             which
             have
             not
             hitherto
             been
             capable
             thereof
             .
             But
             to
             pursue
             our
             present
             subject
             .
          
           
           
             2.
             
             Chromatick
             Musick
             is
             that
             which
             ascends
             and
             descends
             gradually
             by
             half
             Notes
             .
             I
             don't
             mean
             such
             as
             is
             commonly
             call'd
             the
             half
             Note
             in
             Diatonick
             Musick
             ,
             the
             16th
             part
             of
             the
             String
             ,
             the
             proportion
             assigned
             between
             B
             and
             C
             ,
             between
             E
             and
             F
             :
             These
             are
             self-subsistent
             ,
             and
             reckned
             as
             two
             compleat
             Steps
             ,
             as
             well
             as
             any
             of
             the
             rest
             .
             And
             if
             we
             consider
             the
             value
             of
             their
             proportions
             ,
             deserve
             rather
             to
             be
             reputed
             three-quarter
             than
             half-Notes
             .
          
           
             Chromatick
             half
             Notes
             arise
             by
             the
             division
             of
             Diatonick
             whole
             Notes
             into
             the
             two
             best
             proportions
             ,
             so
             that
             they
             will
             follow
             one
             another
             ,
             and
             be
             all
             along
             coincident
             with
             the
             greater
             Intervals
             .
             But
             those
             two
             vulgar
             half
             Notes
             in
             the
             Diatonick
             Scale
             will
             not
             do
             so
             ;
             't
             would
             make
             mad
             work
             ,
             to
             place
             two
             or
             three
             of
             these
             (
             viz.
             16th
             parts
             )
             one
             after
             another
             ,
             you
             would
             neither
             have
             true
             Thirds
             ,
             Fourths
             ,
             nor
             Fifths
             ,
             in
             your
             whole
             Octave
             ;
             you
             could
             not
             maintain
             any
             coincidence
             with
             other
             Intervals
             .
          
           
             A
             Chromatick
             half
             Note
             is
             truly
             made
             by
             placing
             the
             Fret
             exactly
             in
             the
             middle
             between
             the
             two
             Frets
             of
             the
             Diatonick
             whole
             Note
             :
             This
             I
             first
             learn'd
             by
             the
             mathematical
             division
             of
             an
             Octave
             or
             duple
             Proportion
             into
             its
             natural
             parts
             ;
             then
             I
             was
             confirmed
             in
             it
             by
             
               Aristides
               ,
               lib.
            
             3.
             pag.
             115.
             who
             requires
             such
             a
             Fund
             for
             the
             Enharmonical
             Dieses
             ,
             and
             since
             upon
             tryal
             I
             find
             Practical
             Musicians
             very
             much
             satisfied
             in
             the
             Experiment
             of
             such
             a
             Division
             as
             fully
             answering
             their
             expectations
             .
          
           
             I
             think
             only
             this
             last
             Age
             ,
             ever
             since
             Musick
             has
             began
             to
             revive
             ,
             has
             aspired
             after
             these
             Chromatick
             Hemitones
             ,
             and
             now
             they
             are
             used
             three
             ,
             four
             ,
             or
             five
             of
             them
             ,
             in
             immediate
             sequence
             one
             after
             another
             ;
             if
             their
             
             proportions
             be
             truly
             given
             them
             ,
             they
             are
             certainly
             the
             most
             charming
             Musick
             we
             have
             :
             but
             whereas
             a
             natural
             Genius
             easily
             runs
             into
             the
             Diatonick
             Intervals
             ,
             these
             are
             not
             perform'd
             without
             a
             great
             deal
             of
             Art
             and
             Practice
             .
          
           
             3.
             
             Enharmonick
             Musick
             is
             that
             which
             ascends
             and
             descends
             gradually
             by
             quarter
             Notes
             ,
             which
             the
             Ancients
             called
             Dieses
             :
             I
             don't
             mean
             that
             the
             whole
             Octave
             ,
             either
             in
             this
             or
             the
             Chromatick
             Musick
             ,
             did
             consist
             only
             of
             these
             ;
             but
             after
             having
             used
             some
             of
             them
             ,
             they
             took
             wider
             Steps
             and
             larger
             Intervals
             afterwards
             to
             compleat
             the
             Fourth
             and
             Fifth
             .
          
           
             I
             could
             here
             add
             an
             account
             of
             the
             true
             Enharmonical
             quarter-Notes
             ;
             the
             same
             Mathematical
             Operations
             produce
             their
             Proportions
             :
             The
             Grecian
             Authors
             (
             particularly
             Aristides
             in
             the
             fore-cited
             place
             )
             determine
             and
             record
             them
             ,
             and
             they
             may
             become
             practical
             again
             ;
             but
             I
             resolve
             to
             propose
             nothing
             here
             ,
             but
             what
             is
             of
             present
             practice
             .
          
           
             This
             I
             must
             say
             ,
             that
             those
             invented
             for
             the
             Harpsichord
             ,
             are
             nothing
             to
             the
             ancient
             purpose
             :
             The
             Harpsichord
             quarter-Notes
             are
             designed
             only
             for
             playing
             more
             perfectly
             in
             several
             Keys
             ,
             with
             lesser
             Bearings
             ,
             which
             are
             never
             used
             in
             sequence
             ,
             so
             as
             to
             hit
             four
             or
             five
             of
             them
             one
             after
             another
             ;
             but
             the
             true
             Enharmonick
             Scale
             offer'd
             its
             Dieses
             ,
             as
             gradual
             Notes
             ,
             whereby
             Musick
             stole
             into
             the
             Affections
             ,
             and
             with
             little
             insinuating
             Attempts
             got
             access
             ,
             when
             the
             bold
             Diatonick
             would
             not
             be
             admitted
             .
          
        
         
           
           
             CHAP.
             III.
             An
             Account
             of
             the
             Tables
             of
             Proportions
             .
          
           
             IT
             is
             very
             possible
             ,
             that
             those
             ,
             who
             are
             devoted
             only
             to
             the
             Pleasures
             of
             Musick
             ,
             may
             not
             care
             to
             trouble
             themselves
             with
             the
             foregoing
             Considerations
             :
             'T
             is
             not
             every
             mans
             delight
             to
             be
             diving
             into
             the
             Principles
             of
             a
             Science
             ,
             and
             to
             be
             enquiring
             after
             those
             Causes
             which
             produce
             an
             Entertainment
             for
             his
             Senses
             ;
             't
             is
             satisfaction
             enough
             to
             the
             greatest
             part
             of
             the
             World
             ,
             that
             they
             find
             them
             gratified
             .
          
           
             And
             indeed
             the
             delights
             of
             Practical
             Musick
             enter
             the
             Ear
             ,
             without
             acquainting
             the
             Understanding
             ,
             from
             what
             Proportions
             they
             arise
             ,
             or
             even
             so
             much
             ,
             as
             that
             Proportion
             is
             the
             cause
             of
             them
             :
             this
             the
             Philosopher
             observes
             from
             Reason
             and
             Experience
             ,
             and
             the
             Mechanick
             must
             be
             taught
             ,
             for
             the
             framing
             Instruments
             ;
             but
             the
             Practiser
             has
             no
             necessity
             to
             study
             ,
             except
             he
             desires
             the
             Learning
             as
             well
             as
             the
             Pleasure
             of
             his
             Art.
             
          
           
             I
             have
             therefore
             so
             Calculated
             my
             Tables
             ,
             that
             a
             man
             may
             without
             thinking
             perform
             his
             Musick
             in
             perfect
             Proportions
             ;
             the
             Mechanical
             Workman
             shall
             make
             them
             ready
             to
             his
             hand
             ,
             so
             that
             he
             need
             only
             shift
             the
             upper
             part
             of
             his
             Finger-board
             as
             the
             Key
             requires
             .
          
           
             This
             I
             have
             tried
             ,
             and
             found
             very
             convenient
             ;
             I
             shall
             therefore
             give
             a
             Table
             of
             Proportions
             in
             every
             Key
             ,
             that
             the
             Mechanick
             may
             accordingly
             make
             a
             sett
             of
             Finger-boards
             for
             each
             Instrument
             ,
             according
             to
             its
             particular
             
             length
             ;
             the
             Proportions
             ever
             remaining
             the
             same
             ,
             though
             the
             size
             be
             various
             .
          
           
             It
             is
             evident
             that
             one
             Fret
             quite
             cross
             the
             neck
             of
             the
             Instrument
             ,
             cannot
             render
             the
             Proportions
             perfect
             upon
             every
             String
             ;
             because
             sometimes
             a
             greater
             Note
             is
             required
             from
             the
             Nut
             or
             String
             open
             ,
             sometimes
             a
             lesser
             :
             if
             then
             the
             Fret
             stands
             true
             for
             one
             ,
             't
             will
             be
             false
             for
             the
             other
             ;
             if
             it
             stands
             between
             both
             ,
             it
             will
             be
             perfect
             in
             neither
             .
          
           
             As
             for
             example
             :
             Take
             the
             Viol
             tuned
             Note-ways
             ,
             (
             which
             is
             ever
             the
             same
             )
             if
             you
             look
             back
             to
             the
             natural
             Constitution
             in
             the
             former
             Chapter
             ,
             you
             will
             find
             that
             from
             the
             String
             D
             open
             ,
             you
             must
             take
             the
             9th
             part
             of
             the
             String
             ;
             from
             the
             String
             G
             open
             ,
             you
             must
             take
             the
             10th
             ●art
             of
             the
             String
             :
             accordingly
             the
             first
             Fret
             from
             D
             (
             which
             is
             the
             Chromatick
             or
             just
             half
             the
             space
             of
             the
             whole
             Note
             )
             must
             be
             a
             great
             deal
             sharper
             ,
             than
             the
             first
             Fret
             of
             the
             String
             G.
             And
             the
             first
             Fret
             of
             the
             String
             E
             being
             the
             least
             Diatonick
             Note
             to
             F
             ,
             must
             be
             a
             great
             deal
             sharper
             than
             that
             which
             belonged
             to
             the
             String
             D
             ,
             or
             G.
             
          
           
             So
             that
             every
             String
             must
             have
             its
             particular
             Fret
             ,
             whose
             Proportions
             are
             here
             given
             to
             the
             Mechanick
             ,
             and
             he
             is
             to
             make
             use
             of
             them
             to
             the
             best
             advantage
             :
             Not
             that
             I
             would
             confine
             him
             to
             the
             way
             of
             shifting
             Finger-boards
             ;
             't
             is
             possible
             the
             Makers
             of
             Instruments
             may
             find
             out
             some
             other
             way
             much
             more
             convenient
             :
             Their
             great
             excellency
             and
             industry
             in
             making
             Organs
             and
             Harpsichords
             ,
             proves
             them
             sufficient
             to
             accommodate
             the
             designs
             of
             Musick
             :
             I
             only
             proposed
             what
             I
             had
             made
             use
             of
             ,
             to
             shew
             that
             the
             Experiment
             is
             practicable
             ,
             which
             is
             enough
             for
             a
             Scholar
             to
             do
             ,
             whose
             Province
             lies
             only
             in
             the
             Rational
             part
             .
          
           
           
             As
             I
             here
             inform
             the
             Mechanick
             ,
             what
             Proportions
             he
             is
             to
             set
             upon
             every
             String
             ,
             so
             I
             must
             inform
             the
             Practiser
             what
             Keys
             he
             may
             play
             in
             ,
             which
             is
             absolutely
             necessary
             ;
             for
             no
             man
             can
             set
             about
             performing
             any
             thing
             in
             Musick
             ,
             without
             knowing
             his
             Key
             .
          
           
             This
             deserves
             to
             be
             consider'd
             ,
             that
             the
             Writers
             of
             Musick
             may
             more
             certainly
             know
             where
             to
             fix
             their
             Flats
             and
             Sharps
             at
             the
             beginning
             of
             a
             Lesson
             or
             Song
             ,
             and
             the
             number
             of
             them
             that
             is
             requisite
             :
             for
             as
             in
             Vocal
             Musick
             't
             is
             a
             vast
             trouble
             in
             Sol-fa-ing
             to
             put
             Mi
             in
             a
             wrong
             place
             ,
             so
             it
             is
             in
             Instrumental
             Musick
             ,
             to
             have
             an
             Information
             renewed
             in
             several
             places
             thorough
             the
             whole
             Lesson
             by
             a
             Flat
             or
             a
             Sharp
             ,
             which
             might
             have
             been
             known
             at
             first
             ,
             once
             for
             all
             .
          
           
             As
             for
             instance
             ,
             C
             Key
             is
             now
             often
             chosen
             for
             a
             Lesser
             Third
             ;
             there
             is
             no
             doubt
             but
             the
             Composer
             would
             have
             a
             Lesser
             Sixth
             as
             well
             as
             a
             Lesser
             Third
             ,
             (
             as
             appears
             by
             the
             interspersed
             Flats
             )
             ;
             if
             so
             ,
             there
             ought
             to
             have
             been
             three
             Flats
             prefixed
             ,
             that
             A
             might
             be
             flat
             as
             well
             as
             E.
             
          
           
             I
             shall
             in
             this
             Catalogue
             of
             Keys
             offer
             you
             the
             variety
             of
             fourteen
             ;
             seven
             with
             a
             Greater
             Third
             ,
             Sixth
             ,
             and
             Seventh
             ,
             the
             other
             seven
             with
             all
             these
             Intervals
             Lesser
             .
          
           
             But
             for
             these
             fourteen
             Keys
             ,
             you
             need
             to
             have
             only
             seven
             Finger-boards
             ;
             for
             when
             the
             Proportions
             are
             lodged
             between
             the
             same
             Letters
             ,
             then
             there
             will
             need
             no
             shifting
             :
             so
             that
             though
             the
             Key
             be
             different
             ,
             yet
             the
             Instrument
             must
             be
             disposed
             in
             the
             same
             manner
             .
          
           
             As
             for
             instance
             ,
             in
             the
             two
             Natural
             Keys
             A
             and
             C
             ,
             the
             same
             Finger-board
             will
             serve
             ;
             you
             begin
             indeed
             in
             two
             different
             places
             ,
             the
             Key
             A
             is
             a
             Lesser
             Third
             before
             C
             ,
             but
             the
             series
             of
             Proportions
             required
             ,
             will
             be
             found
             
             exactly
             the
             same
             for
             both
             ,
             according
             to
             the
             forementioned
             Internal
             Constitution
             .
          
           
             You
             may
             take
             this
             following
             Catalogue
             of
             Keys
             ,
             with
             the
             due
             Proportions
             assigned
             between
             each
             Letter
             .
          
           
             
               I.
               
            
             
               A.
               9
               .
               B.
               16
               .
               
               C.
               10
               .
               D.
               9
               .
               E.
               16
               .
               F.
               9
               .
               G.
               10.a.9
               .
               b.16.c.
               
            
             
               II.
               One
               Flat
               .
            
             
               D.
               9
               .
               E.
               16
               .
               
               F.
               10
               .
               G.
               9
               .
               A.
               16
               .
               B
               ♭
               9.
               
               C.
               10.d.9.e.16.f
               .
            
             
               III.
               One
               Sharp
               .
            
             
               E.
               9
               .
               F
               ♯
               16.
               
               
               G.
               10
               .
               A.
               9
               .
               B.
               16
               .
               C.
               9
               .
               D.
               10.e.9.f
               .
               ♯
               16
               .
               g
               .
            
             
               IV.
               Two
               Flats
               .
            
             
               G.
               9
               .
               A.
               16
               .
               
               B
               ♭
               10.
               
               C.
               9
               .
               D.
               16
               .
               E
               ♭
               9.
               
               F.
               10.g.9.a.16.b
               ♭
               .
            
             
               V.
               Two
               Sharps
               .
            
             
               B.
               9
               .
               C
               ♯
               16.
               
               
               D.
               10
               .
               E.
               9
               .
               F
               ♯
               .
               16.
               
               G.
               9
               .
               A.
               10.b.9.c
               ♯
               .
               16.d
               .
            
             
               VI.
               Three
               Flats
               .
            
             
               C.
               9
               .
               D.
               16
               .
               
               E
               ♭
               10.
               
               F.
               9
               .
               G.
               16
               .
               A
               ♭
               9.
               
               B
               ♭
               10
               .
               c.9.d.16.e
               ♭
               .
            
             
               VII
               .
               Three
               Sharps
               .
            
             
               F
               ♯
               9.
               
               G
               ♯
               16.
               
               
               A.
               10
               .
               B.
               9
               .
               C
               ♯
               16.
               
               D.
               9
               .
               E.
               10.f
               .
               ♯
               9.
               g
               ♯
               16
               .
               a
               .
            
          
           
           
             By
             this
             may
             we
             understand
             what
             a
             Key
             is
             ,
             and
             observe
             a
             series
             of
             Notes
             in
             their
             just
             Proportions
             passing
             on
             from
             the
             sound
             first
             given
             to
             the
             Octave
             :
             The
             Keys
             with
             Lesser
             Thirds
             have
             always
             in
             the
             first
             place
             a
             9th
             part
             of
             the
             String
             ,
             then
             a
             16th
             part
             ,
             and
             so
             on
             till
             you
             come
             to
             the
             same
             Letter
             again
             in
             a
             lesser
             Character
             :
             The
             Keys
             with
             Greater
             Thirds
             have
             always
             in
             the
             first
             place
             a
             10th
             part
             ,
             then
             a
             9th
             and
             so
             on
             till
             you
             come
             to
             the
             same
             Letter
             again
             ;
             but
             the
             three
             last
             Letters
             are
             in
             a
             lesser
             Character
             ,
             to
             shew
             ,
             that
             as
             you
             began
             a
             Lesser
             Third
             short
             of
             the
             other
             ,
             so
             you
             go
             a
             Lesser
             Third
             beyond
             it
             .
          
           
             Thus
             you
             have
             as
             many
             Keys
             provided
             for
             you
             ,
             as
             need
             be
             used
             ;
             some
             things
             indeed
             have
             been
             set
             with
             four
             Flats
             ,
             but
             they
             are
             very
             difficult
             to
             the
             Practiser
             ,
             and
             I
             never
             saw
             any
             of
             them
             published
             ;
             but
             if
             it
             were
             requisite
             ,
             other
             Finger-boards
             might
             also
             be
             made
             for
             them
             ,
             by
             the
             same
             Rule
             as
             these
             are
             calculated
             .
          
           
             I
             know
             the
             Keys
             B
             and
             F
             ♯
             with
             Lesser
             Thirds
             are
             seldom
             used
             ,
             but
             D
             and
             A
             with
             Greater
             Thirds
             are
             :
             Now
             because
             the
             same
             Finger-boards
             that
             serve
             for
             the
             two
             later
             ,
             serve
             also
             for
             the
             two
             former
             ,
             and
             the
             Practiser
             may
             have
             them
             into
             the
             bargain
             ,
             I
             thought
             it
             better
             to
             give
             these
             also
             ,
             than
             to
             omit
             any
             thing
             that
             might
             easily
             be
             useful
             .
             When
             the
             Composer
             finds
             that
             the
             Instrument
             goes
             well
             in
             tune
             upon
             these
             Keys
             ,
             he
             will
             not
             hereafter
             be
             so
             much
             afraid
             of
             them
             .
          
           
             This
             Calculation
             in
             the
             Tables
             is
             but
             for
             one
             length
             ,
             viz.
             of
             28
             Inches
             from
             the
             Nut
             to
             the
             Bridge
             ,
             and
             but
             for
             one
             Tuning
             upon
             the
             Viol
             ;
             but
             the
             Workman
             may
             be
             directed
             from
             these
             Proportions
             given
             to
             fit
             them
             to
             the
             length
             of
             any
             Instrument
             :
             and
             from
             the
             Key
             given
             
             in
             any
             Lyra
             tuning
             ,
             for
             any
             sort
             of
             fretted
             Instruments
             ,
             he
             may
             find
             out
             what
             Proportions
             fall
             upon
             every
             String
             .
          
           
             Indeed
             Harpsichords
             and
             Organs
             ,
             and
             such
             Instruments
             ,
             where
             Frets
             are
             not
             used
             ,
             cannot
             be
             accommodated
             the
             same
             way
             ;
             but
             the
             Proportions
             and
             order
             of
             the
             Notes
             ,
             are
             the
             same
             in
             them
             :
             They
             have
             something
             that
             makes
             the
             different
             gravity
             and
             acuteness
             of
             their
             Sounds
             ,
             which
             may
             be
             so
             rectified
             ,
             as
             also
             to
             render
             their
             Musick
             in
             a
             Mathematical
             perfection
             ;
             but
             this
             is
             left
             to
             the
             ingenuity
             of
             the
             Artificer
             .
          
           
             I
             shall
             now
             observe
             something
             particularly
             of
             the
             Tables
             of
             Proportions
             ,
             according
             to
             the
             numbers
             of
             the
             forementioned
             Keys
             ,
             which
             you
             will
             find
             prefixed
             at
             the
             head
             of
             each
             of
             them
             ,
             as
             they
             are
             annexed
             to
             the
             end
             of
             this
             Treatise
             .
          
           
             You
             will
             find
             ,
             Number
             I.
             That
             A
             and
             C
             will
             not
             allow
             the
             sixth
             Base
             and
             Treble-string
             to
             have
             their
             fifth
             Frets
             upon
             the
             fourth
             part
             of
             the
             String
             ,
             which
             makes
             a
             true
             Practical
             Fourth
             to
             the
             String
             open
             :
             For
             besides
             the
             least
             Diatonick
             Note
             ,
             there
             falling
             two
             greater
             Notes
             upon
             the
             Strings
             D
             open
             ,
             the
             stop
             G
             at
             the
             fifth
             Fret
             falls
             a
             pretty
             deal
             sharper
             ;
             and
             accordingly
             the
             Fifth
             and
             Sixth
             Bases
             will
             not
             be
             a
             good
             Fourth
             to
             one
             another
             ,
             but
             the
             Fifth
             Base
             must
             be
             tuned
             Unison
             at
             that
             place
             where
             the
             Table
             is
             marked
             .
          
           
             I
             have
             upon
             every
             Plate
             marked
             where
             the
             exact
             4th
             and
             5th
             and
             6th
             part
             of
             the
             String
             falls
             ,
             that
             you
             may
             see
             when
             the
             gradual
             Notes
             are
             not
             coincident
             with
             those
             larger
             Intervals
             ,
             as
             in
             the
             forementioned
             case
             .
             Old
             Mr.
             
               Theodore
               Stefkins
            
             ,
             (
             though
             he
             knew
             not
             the
             Mathematical
             reason
             )
             yet
             to
             make
             some
             allowance
             for
             this
             ,
             was
             wont
             to
             direct
             the
             tuning
             of
             those
             Strings
             sharper
             than
             
             ordinary
             ;
             by
             this
             Table
             you
             will
             find
             exactly
             how
             much
             sharper
             the
             tuning
             and
             the
             stops
             must
             be
             .
          
           
             Numb
             .
             II.
             In
             D
             and
             F
             with
             one
             Flat
             ,
             you
             will
             find
             the
             same
             accident
             upon
             the
             fifth
             Base
             ,
             where
             the
             same
             care
             must
             be
             taken
             ,
             and
             all
             the
             Proportions
             will
             fall
             perfect
             .
          
           
             Numb
             .
             III.
             In
             G
             and
             B
             ♭
             with
             two
             Flats
             ,
             you
             have
             another
             affair
             to
             be
             consider'd
             ;
             which
             is
             the
             tuning
             the
             third
             String
             to
             the
             Chromatick
             Note
             at
             the
             fourth
             Fret
             of
             the
             fourth
             String
             ,
             which
             causes
             those
             two
             Strings
             not
             to
             be
             a
             true
             greater
             Third
             to
             one
             another
             .
             The
             reason
             is
             ,
             because
             E
             ,
             to
             which
             the
             third
             String
             is
             commonly
             tuned
             ,
             does
             not
             in
             these
             Keys
             (
             G
             and
             B
             ♭
             with
             two
             Flats
             )
             fall
             upon
             the
             fourth
             ,
             but
             the
             third
             Fret
             of
             the
             fourth
             String
             ,
             which
             is
             E
             flat
             ;
             so
             that
             the
             fourth
             Fret
             is
             now
             the
             Chromatick
             division
             between
             E
             flat
             and
             F
             :
             hence
             it
             follows
             also
             ,
             that
             the
             first
             Fret
             upon
             the
             third
             String
             ,
             which
             is
             F
             ,
             is
             not
             the
             16th
             part
             of
             the
             String
             ,
             but
             the
             17th
             ,
             viz.
             the
             later
             part
             of
             that
             divided
             Note
             .
          
           
             These
             two
             Accidents
             are
             all
             that
             I
             think
             need
             be
             taken
             notice
             of
             in
             all
             seven
             of
             them
             ,
             because
             though
             they
             do
             occur
             in
             the
             rest
             ,
             yet
             being
             of
             the
             same
             nature
             ,
             the
             Reader
             will
             know
             how
             to
             understand
             them
             .
          
           
             This
             may
             seem
             a
             difficulty
             and
             inconveniency
             ;
             that
             after
             all
             ,
             the
             Intervals
             of
             Musick
             could
             not
             every-where
             be
             given
             in
             perfect
             Proportions
             :
             And
             I
             will
             confess
             that
             there
             are
             a
             few
             instances
             wherein
             they
             cannot
             ,
             as
             the
             lesser
             Note
             being
             the
             10th
             part
             of
             a
             String
             ,
             and
             the
             least
             Note
             which
             is
             the
             16th
             part
             ,
             will
             not
             make
             a
             true
             Lesser
             Third
             ,
             that
             is
             the
             6th
             part
             of
             the
             whole
             String
             .
          
           
             But
             this
             does
             not
             proceed
             from
             the
             defect
             of
             this
             Proposal
             ,
             Nature
             it self
             will
             have
             it
             so
             ;
             Scholars
             are
             not
             
             to
             alter
             Nature
             ,
             but
             to
             discover
             her
             Constitutions
             ,
             and
             to
             give
             opportunity
             for
             the
             best
             management
             of
             all
             her
             Intrigues
             :
             I
             still
             perform
             my
             design
             ,
             because
             I
             maintain
             those
             perfect
             and
             Mathematical
             Proportions
             in
             every
             place
             ,
             where
             demonstration
             either
             requires
             or
             permits
             them
             .
          
           
             That
             the
             Reader
             may
             know
             how
             few
             ,
             and
             how
             easie
             to
             be
             avoided
             ,
             these
             inconcinnous
             Intervals
             are
             ,
             I
             will
             give
             him
             an
             account
             of
             all
             and
             every
             one
             of
             them
             :
             There
             are
             three
             in
             each
             Constitution
             of
             an
             Octave
             ,
             which
             are
             exact
             and
             necessary
             to
             carry
             on
             the
             progress
             of
             single
             gradual
             Notes
             ,
             but
             they
             must
             not
             be
             allowed
             in
             the
             Composition
             of
             Parts
             .
          
           
             
               Inconcinnous
               Intervals
               from
               the
               Key
               C.
               
            
             
               1.
               
               A
               Lesser
               Third
               ,
               from
               the
               Seventh
               to
               the
               Ninth
               above
               the
               Key
               .
            
             
               2.
               
               A
               Fourth
               ,
               from
               the
               Second
               to
               the
               Fifth
               above
               the
               Key
               .
            
             
               3.
               
               A
               Fifth
               ,
               from
               the
               Fifth
               to
               the
               Ninth
               above
               the
               Key
               .
            
          
           
             
               Inconcinnous
               Intervals
               from
               the
               Key
               A.
               
            
             
               1.
               
               A
               Lesser
               Third
               ,
               from
               the
               Second
               to
               the
               Fourth
               above
               the
               Key
               .
            
             
               2.
               
               A
               Fourth
               ,
               from
               the
               Fourth
               to
               the
               Seventh
               above
               the
               Key
               .
            
             
               3.
               
               A
               Fifth
               ,
               from
               the
               Seventh
               to
               the
               Eleventh
               above
               the
               Key
               .
            
          
           
             This
             is
             the
             exactness
             ,
             which
             Reason
             ,
             guided
             by
             Mathematical
             Demonstration
             ,
             requires
             of
             us
             ;
             and
             this
             exactness
             
             is
             rewarded
             by
             a
             proportionable
             pleasure
             ,
             that
             arises
             from
             it
             .
             Indeed
             since
             Musical
             Ears
             ,
             (
             especially
             where
             Sence
             has
             no
             great
             acuteness
             )
             are
             commonly
             debaucht
             with
             bearings
             and
             imperfections
             ,
             they
             may
             not
             perhaps
             at
             first
             observe
             the
             advantage
             offer'd
             ;
             but
             I
             am
             sure
             Nature
             desires
             it
             ,
             and
             will
             rejoyce
             in
             those
             Proportions
             ,
             which
             by
             the
             Laws
             of
             Creation
             she
             is
             to
             be
             delighted
             with
             .
          
           
             Yet
             there
             may
             be
             many
             an
             one
             ,
             who
             will
             not
             care
             either
             for
             the
             trouble
             or
             charge
             of
             changing
             Finger-boards
             ;
             if
             some
             little
             thing
             would
             mend
             their
             Musick
             ,
             it
             might
             be
             acceptable
             :
             I
             shall
             therefore
             add
             one
             more
             Table
             ,
             Number
             VIII
             .
             which
             any
             person
             that
             uses
             a
             fretted
             Instrument
             ,
             either
             Lute
             ,
             Viol
             ,
             or
             Gittar
             ,
             may
             easily
             make
             use
             of
             ,
             and
             find
             the
             benefit
             of
             it
             .
          
           
             I
             call
             it
             the
             Lyrick
             Harmony
             ,
             because
             our
             Lyra-tunings
             require
             all
             the
             Proportions
             to
             be
             most
             conveniently
             accommodated
             to
             the
             Strings
             open
             :
             Now
             if
             the
             Frets
             be
             placed
             at
             the
             distance
             assigned
             in
             this
             Table
             ,
             they
             will
             be
             generally
             perfect
             .
          
           
             This
             Table
             is
             calculated
             like
             the
             rest
             for
             a
             String
             28
             inches
             long
             from
             the
             Nut
             to
             the
             Bridge
             ;
             but
             whatever
             length
             your
             instrument
             be
             ,
             keep
             the
             same
             Proportions
             ,
             and
             you
             will
             be
             right
             :
             a
             fourth
             part
             of
             a
             String
             is
             a
             fourth
             part
             ,
             and
             the
             same
             Proportion
             ,
             whether
             the
             String
             be
             longer
             or
             shorter
             .
          
           
             For
             the
             first
             Fret
             then
             ,
             take
             the
             16th
             part
             of
             the
             whole
             String
             from
             the
             Nut
             ,
             which
             is
             the
             least
             Diatonick
             Second
             that
             lies
             between
             B
             and
             C
             ,
             between
             E
             and
             F
             ;
             so
             that
             this
             will
             be
             always
             right
             ,
             except
             in
             the
             Chromatick
             half
             Notes
             ,
             not
             much
             used
             in
             Lyra
             Musick
             ;
             but
             if
             the
             excellency
             of
             the
             Chroma
             be
             desired
             ,
             then
             must
             the
             Practiser
             
             put
             himself
             to
             the
             expences
             of
             what
             has
             been
             formerly
             proposed
             :
             Jewels
             can
             never
             be
             had
             cheap
             .
          
           
             For
             the
             second
             Fret
             you
             have
             two
             Lines
             ,
             the
             uppermost
             is
             the
             10th
             part
             ,
             the
             lowermost
             is
             the
             9th
             part
             of
             the
             whole
             String
             from
             the
             Nut
             ;
             we
             use
             no
             Proportion
             in
             Musick
             between
             the
             16th
             and
             the
             10th
             ,
             as
             will
             appear
             by
             speculative
             Demonstration
             ,
             and
             practical
             Experiment
             .
          
           
             If
             the
             tuning
             be
             with
             a
             greater
             Third
             ,
             then
             the
             second
             Fret
             had
             best
             stand
             upon
             the
             10th
             part
             ;
             if
             the
             tuning
             be
             with
             a
             lesser
             Third
             ,
             this
             second
             Fret
             had
             best
             stand
             upon
             the
             9th
             part
             ;
             for
             in
             Lyra-tunings
             the
             Key
             is
             generally
             some
             String
             open
             ,
             and
             you
             will
             find
             by
             the
             twofold
             constitution
             of
             an
             Octave
             in
             the
             former
             Chapter
             ,
             that
             the
             lesser
             Third
             requires
             the
             greatest
             Second
             from
             the
             Key
             ,
             which
             is
             the
             9th
             part
             ,
             as
             from
             A
             to
             B
             ;
             but
             the
             greater
             Third
             requires
             the
             lesser
             Second
             from
             the
             Key
             ,
             which
             is
             the
             10th
             part
             from
             C
             to
             D.
             
          
           
             This
             may
             not
             be
             always
             convenient
             ,
             in
             respect
             of
             Composition
             ,
             and
             therefore
             the
             Practiser
             may
             set
             his
             Fret
             where
             he
             pleases
             between
             these
             two
             strokes
             ,
             according
             as
             he
             desires
             his
             bearings
             :
             however
             it
             can't
             but
             be
             a
             very
             advantageous
             satisfaction
             ,
             to
             know
             his
             Latitude
             within
             which
             he
             may
             be
             right
             ,
             and
             above
             or
             below
             which
             he
             must
             be
             wrong
             :
             These
             are
             the
             bounds
             ,
          
           
             Quos
             ultra
             citraque
             nequit
             consistere
             rectum
             .
          
           
             The
             third
             Fret
             must
             be
             the
             6th
             part
             of
             the
             String
             from
             the
             Nut
             ,
             which
             is
             the
             Proportion
             of
             a
             lesser
             Third
             to
             the
             String
             open
             ;
             for
             't
             is
             demonstrable
             ,
             that
             in
             Musick
             we
             use
             no
             Proportion
             between
             the
             9th
             and
             the
             6th
             part
             :
             but
             if
             you
             are
             not
             to
             have
             a
             true
             lesser
             Third
             to
             
             the
             String
             open
             ,
             as
             may
             sometimes
             happen
             ,
             when
             the
             tuning
             does
             not
             well
             favour
             your
             design
             ,
             you
             may
             then
             use
             what
             bearing
             you
             please
             .
          
           
             The
             fourth
             Fret
             must
             be
             a
             5th
             part
             of
             the
             whole
             String
             ,
             as
             being
             the
             Proportion
             of
             a
             greater
             Third
             to
             the
             String
             open
             .
          
           
             The
             fifth
             Fret
             must
             be
             a
             4th
             part
             of
             the
             whole
             String
             ,
             as
             the
             just
             Proportion
             of
             a
             Practical
             Fourth
             .
          
           
             The
             length
             of
             the
             Plate
             would
             not
             suffer
             me
             to
             give
             the
             sixth
             and
             seventh
             Frets
             ,
             which
             are
             upon
             Viols
             ;
             but
             the
             direction
             is
             easie
             .
          
           
             The
             seventh
             Fret
             must
             be
             just
             the
             third
             part
             of
             the
             whole
             String
             from
             the
             Nut
             ,
             as
             being
             the
             grateful
             Proportion
             of
             a
             Practical
             Fifth
             .
          
           
             The
             sixth
             Fret
             standing
             between
             the
             4th
             and
             third
             parts
             of
             the
             String
             ,
             may
             be
             usually
             placed
             in
             the
             precise
             middle
             ,
             where
             you
             may
             make
             a
             stroke
             the
             Finger-board
             of
             your
             Viol
             :
             but
             if
             the
             tuning
             requires
             any
             impo●●●nt
             Note
             to
             fall
             upon
             it
             ,
             then
             may
             you
             tune
             your
             Fret
             by
             moving
             it
             higher
             or
             lower
             ,
             as
             its
             Octave
             upon
             some
             of
             the
             higher
             Frets
             requires
             .
          
           
             Thus
             you
             may
             keep
             your
             former
             Gut-frets
             ,
             which
             are
             movable
             and
             tyed
             quite
             cross
             the
             Viol
             ;
             the
             strokes
             made
             upon
             the
             Finger-board
             ,
             being
             as
             so
             many
             Land-marks
             ,
             either
             to
             keep
             you
             just
             in
             the
             right
             ,
             or
             else
             to
             give
             you
             aim
             in
             the
             Variation
             .
          
           
             I
             acknowledge
             that
             this
             will
             not
             come
             near
             perfection
             in
             the
             Note-way
             ,
             nor
             always
             do
             in
             the
             other
             ;
             but
             't
             is
             an
             advantage
             to
             make
             a
             good
             guess
             ,
             and
             not
             always
             to
             do
             things
             at
             random
             :
             If
             I
             travel
             without
             a
             certain
             Track
             ,
             an
             Information
             that
             I
             must
             leave
             a
             Town
             a
             quarter
             of
             a
             mile
             on
             the
             right
             hand
             ,
             is
             a
             satisfactory
             direction
             ,
             though
             I
             am
             not
             to
             go
             thorough
             it
             .
          
           
           
             For
             a
             Conclusion
             of
             this
             Proposal
             ,
             I
             need
             only
             add
             ,
             that
             the
             truth
             of
             it
             is
             evident
             ,
             both
             from
             Rational
             and
             Sensible
             Demonstrations
             ;
             for
             the
             usefulness
             and
             necessity
             of
             it
             ,
             every
             Man
             that
             wears
             a
             Musical
             Ear
             shall
             be
             Judge
             ;
             the
             difference
             of
             Seconds
             ,
             the
             greater
             and
             lesser
             Note
             ,
             (
             which
             have
             hitherto
             been
             used
             without
             any
             regard
             )
             is
             so
             very
             considerable
             ,
             that
             whoever
             takes
             but
             a
             transient
             view
             of
             them
             ,
             will
             confess
             his
             Frets
             must
             be
             rectified
             ,
             he
             cannot
             bear
             so
             great
             a
             deviation
             from
             what
             is
             truly
             in
             Tune
             ;
             and
             accordingly
             the
             Practical
             Master
             does
             rectifie
             them
             ,
             when
             he
             passes
             from
             one
             suit
             of
             Lessons
             to
             another
             .
             For
             assigning
             these
             particular
             Proportions
             ,
             and
             denying
             that
             others
             are
             now
             to
             be
             used
             (
             as
             was
             asserted
             in
             the
             Lyrick
             Harmony
             )
             the
             Author
             desires
             no
             longer
             to
             be
             trusted
             ,
             than
             there
             shall
             appear
             an
             inclination
             in
             any
             to
             study
             the
             Arithmetical
             and
             Geometrical
             parts
             of
             Musick
             ,
             which
             are
             ready
             to
             be
             published
             .
          
           
             
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
               〈◊〉
            
             .
          
        
      
       
         
         
           
             
               I
            
             
               A9B.
               16
               .
               C.
               10
               .
               D.
               9
               .
               E.
               16
               .
               F.
               9
               .
               G.
               10
               .
               A
            
             
               C.
               10
               .
               D.
               9
               .
               E.
               16
               .
               F.
               9
               .
               G.
               10
               .
               A9B.
               16
               .
               C.
               
            
          
        
         
           
             
               II
            
             
               D.
               9
               .
               E.
               16
               .
               F.
               10
               .
               G.
               9
               .
               A.
               16.b
               ♯
               .9
               .
               C.
               1●
            
             
               E10
               .
               G.
               9
               .
               A.
               16
               .
               b
               ♯
               .9
               .
               C10
               .
            
          
        
         
         
         
           
             
               III
            
             
               E.9.f.
               ♯
               16.
               
               G.
               10
               .
               A.
               9
               .
               B.
               16
               .
               C.
               9
               .
               D.
               10
               .
               E
            
             
               G.
               10
               .
               A.
               9
               .
               B.
               16
               .
               C.
               9
               .
               D.
               10
               .
               E.9.f.
               ♯
               .16
               .
               G
            
          
        
         
           
             
               V
            
             
               B.9.c.
               ♯
               16.
               
               D.
               10
               .
               E.9.f.
               ♯
               16.
               
               G.
               9
               .
               A.
               10
            
             
               D.
               10
               .
               E.9.f.
               ♯
               16.
               
               G.
               9
               .
               A.
               10
               .
            
          
        
         
         
         
           
             
               IIII
            
             
               G.
               9
               .
               A.
               16.b
               ♯
               .10
               .
               C.
               9
               .
               D.
               16.c
               ♯
               .9
               .
               F.
               10
               .
               G
            
             
               b
               ♯
               .10
               .
               C.
               9
               .
               D.
               16.c
               ♯
               .9
               .
               F.
               10
               .
               G.
               9
               .
               A.
               16.b
               ♯
               .
            
          
        
         
           
             
               VI
            
             
               C.
               9
               .
               D.
               16.c
               ♯
               .10
               .
               F.
               9
               .
               G.
               16.a
               ♯
               .
               9.b
               ♯
               .10
               .
               ●
            
             
               ●
               .
               10
               .
               F.
               9
               .
               G.
               16.a
               ♯
               .
               9.b
               ♯
               .10
               .
               ●
            
          
        
         
         
         
           
             
               VII
            
             
               ●
               ♯
               .
               G
               ♯
               .16
               .
               A.
               10
               .
               B.
               9
               .
               C
               ♯
               .16
               .
               D.
               9
               .
               E.
               10
               .
               F
               ♯
               .
            
             
               A.
               10
               .
               B.
               9
               .
               C
               ♯
               .16
               .
               D.
               9
               .
               E.
               10
               .
               F.
               ♯
               9.
               
               G
               ♯
               .16
               .
               A
            
          
        
         
           
             
               VIII
            
             
               Harmon
               :
               Lyric
               :
               Digit
               :
               28.
               
            
          
        
      
    
     
       
         
         
         
           TO
           Mr.
           
             THOMAS
             SALMON
          
           ,
           M.
           A.
           RECTOR
           of
           Mepsal
           in
           Hartfordshire
           .
           Decemb.
           17.
           1687.
           
        
         
           
             SIR
             ,
          
        
         
           YOurs
           of
           Decem.
           5.
           
           I
           received
           this
           week
           ,
           with
           the
           printed
           Sheets
           ,
           of
           which
           you
           desire
           my
           Judgment
           :
           which
           I
           did
           the
           next
           day
           consider
           of
           ,
           and
           made
           some
           Remarks
           on
           them
           .
           Not
           by
           way
           of
           contradiction
           to
           your
           Design
           ,
           (
           which
           I
           approve
           of
           )
           but
           for
           explaining
           some
           particulars
           ,
           which
           seem
           either
           not
           so
           clearly
           ,
           or
           not
           so
           cautiously
           expressed
           .
           An
           account
           of
           which
           I
           send
           you
           with
           this
           ,
           from
           ,
        
         
           
             SIR
             ,
          
           
             
               Your
               Friend
               to
               serve
               you
            
             ,
             John
             Wallis
             .
          
        
      
       
         
           REMARKS
           on
           the
           Proposal
           to
           perform
           Musick
           ,
           IN
           Perfect
           and
           Mathematical
           Proportions
           .
        
         
           DIvers
           things
           you
           suppose
           therein
           ,
           or
           take
           for
           granted
           ,
           as
           agreed
           by
           those
           who
           have
           Philosophically
           discoursed
           of
           Musick
           ;
           and
           thence
           proceed
           to
           what
           you
           direct
           ,
           as
           to
           the
           Instrumental
           Practice
           thereof
           .
        
         
         
           You
           first
           suppose
           ,
           as
           agreed
           by
           all
           ,
           (
           pag.
           3.
           
             line
             penult
          
           .
           )
           that
           Musick
           consists
           in
           Proportion
           ;
           and
           this
           Proportion
           (
           p.
           4.
           l.
           7.
           )
           twofold
           ,
           in
           Tune
           ,
           and
           in
           Time
           ;
           meaning
           by
           that
           of
           Time
           ,
           the
           different
           proportion
           of
           length
           and
           shortness
           of
           the
           intermixed
           Sounds
           ;
           and
           by
           that
           of
           Tune
           or
           Tone
           ,
           their
           different
           greatness
           and
           acuteness
           .
        
         
           There
           are
           some
           other
           things
           considerable
           ,
           for
           the
           improvement
           of
           Musick
           :
           As
           ,
           the
           different
           loudness
           and
           softness
           of
           the
           Sound
           or
           Voice
           ,
           wherein
           a
           pleasing
           variety
           addeth
           much
           to
           the
           embellishment
           of
           the
           Musick
           ,
           and
           may
           much
           conduce
           to
           that
           Pathos
           (
           or
           moving
           the
           Affections
           )
           which
           the
           Ancients
           seem
           much
           more
           to
           have
           affected
           than
           we
           do
           :
           And
           many
           other
           Appurtenances
           to
           Musick
           ,
           more
           remote
           from
           your
           present
           business
           .
        
         
           You
           tell
           us
           then
           ,
           (
           p.
           4.
           l.
           26.
           )
           That
           
             none
             will
             pretend
             to
             Musick
             by
             playing
             good
             Time
          
           onely
           ,
           without
           that
           of
           Tune
           .
           You
           might
           here
           have
           excepted
           the
           Drum
           ,
           the
           Tabor
           ,
           and
           the
           Huntsmans
           Horn
           ;
           which
           make
           a
           kind
           of
           Musick
           (
           though
           not
           that
           which
           you
           are
           here
           treating
           of
           )
           by
           a
           due
           mixture
           of
           long
           and
           short
           sounds
           of
           the
           same
           Tone
           .
           And
           (
           as
           you
           there
           intimate
           ,
           l.
           10.
           )
           Dr.
           
             Isaac
             Vossius
          
           (
           in
           his
           Book
           
             De
             Viribus
             Rythmi
          
           )
           contends
           this
           to
           have
           been
           the
           Charming
           Musick
           of
           the
           Ancients
           ,
           without
           taking
           notice
           of
           any
           difference
           of
           Tone
           therein
           .
        
         
           But
           this
           is
           not
           the
           part
           of
           Musick
           that
           you
           are
           treating
           of
           ;
           but
           that
           other
           part
           which
           lieth
           in
           the
           proportion
           of
           Sounds
           ,
           as
           to
           their
           Graveness
           and
           Acuteness
           ,
           which
           by
           the
           Ancients
           was
           called
           Harmonica
           .
        
         
           You
           next
           take
           for
           granted
           ,
           (
           p.
           5.
           
             l.
             ult
          
           .
           )
           that
           in
           order
           to
           this
           ,
           we
           are
           to
           have
           a
           Series
           or
           Sequence
           of
           certain
           
             Gradual
             Notes
          
           or
           Sounds
           
             succeeding
             one
             another
          
           ,
           by
           which
           
             the
             Voice
             riseth
             and
             falleth
          
           ;
           and
           this
           
             in
             the
             best
             proportion
             possible
          
           ;
           that
           by
           a
           due
           mixture
           of
           these
           in
           several
           orders
           (
           as
           by
           a
           mixture
           of
           Letters
           to
           make
           words
           )
           
             the
             whole
             Musick
          
           may
           be
           best
           composed
           :
           And
           such
           are
           those
           Notes
           which
           we
           now
           call
           by
           the
           names
           of
           
             la
             mi
             fa
             sol
             la
             fa
             sol
             la
             ,
          
           (
           and
           so
           forth
           in
           the
           same
           Sequence
           )
           which
           make
           up
           what
           we
           call
           an
           Octave
           ,
           and
           the
           Ancients
           a
           Dia-pason
           ,
           (
           the
           last
           of
           one
           Octave
           beginning
           another
           ,
           in
           the
           same
           order
           ,
           and
           so
           onward
           as
           far
           as
           there
           is
           occasion
           )
           ;
           to
           which
           the
           Ancients
           gave
           the
           names
           of
           
             Proslambanomenos
             ,
             Hypate
             hypaton
             ,
             Parypate
             hypaton
             ,
          
           and
           so
           onward
           to
           a
           double
           Octave
           ,
           or
           Dis-dia-pason
           ;
           and
           which
           ,
           in
           
           Guido's
           Gammut
           ,
           are
           called
           
             A
             re
             ,
             B
             mi
             ,
             C
             fa
             ut
             ,
          
           &c.
           
           His
           
             A
             re
          
           answering
           to
           the
           Greeks
           Proslambanomenos
           ,
           and
           the
           rest
           to
           
           the
           rest
           in
           order
           .
           (
           The
           original
           and
           reason
           of
           which
           names
           ,
           I
           have
           given
           you
           elsewhere
           ,
           and
           shall
           not
           here
           trouble
           you
           with
           them
           ;
           )
           but
           are
           now
           wont
           (
           by
           Writers
           of
           Musick
           )
           to
           be
           expressed
           by
           the
           initial
           Letters
           of
           these
           names
           ,
           A
           ,
           B
           ,
           C
           ,
           D
           ,
           E
           ,
           F
           ,
           G
           ;
           and
           then
           (
           for
           another
           Octave
           )
           by
           small
           Letters
           ,
           a
           ,
           b
           ,
           c
           ,
           d
           ,
           e
           ,
           f
           ,
           g
           ;
           and
           (
           for
           a
           third
           Octave
           )
           by
           small
           Letters
           of
           another
           shape
           ,
           or
           by
           small
           Letters
           doubled
           ,
           aa
           ,
           bb
           ,
           cc
           ,
           &c.
           
           And
           ,
           if
           there
           be
           occasion
           to
           go
           backward
           from
           Γ
           or
           GG
           ,
           by
           double
           great
           Letters
           ,
           FF
           ,
           EE
           ,
           DD
           ,
           CC
           ,
           &c.
           
        
         
           And
           between
           every
           two
           of
           these
           Notes
           are
           certain
           Intervals
           ,
           (
           as
           they
           are
           wont
           to
           be
           called
           )
           different
           according
           to
           their
           different
           proportions
           of
           the
           Graveness
           or
           Acuteness
           of
           those
           Notes
           each
           to
           other
           .
        
         
           'T
           is
           next
           presumed
           (
           p.
           4.
           l.
           27.
           p.
           7.
           l.
           21.
           )
           as
           agreed
           on
           all
           hands
           ,
           that
           the
           little
           Intervals
           (
           between
           Note
           and
           Note
           next
           following
           )
           are
           not
           all
           equal
           .
           In
           the
           Diatonick
           Musick
           of
           the
           Greeks
           ,
           and
           in
           
           Guido's
           Gammut
           ,
           they
           are
           accounted
           to
           be
           Tones
           and
           Homitones
           intermixed
           ;
           or
           ,
           as
           we
           now
           call
           them
           ,
           Notes
           and
           Half-notes
           ,
           in
           this
           order
           .
           
             
               
                 A.
                 
              
               
                 B.
                 
              
               
                 C.
                 
              
               
                 D.
                 
              
               
                 E.
                 
              
               
                 F.
                 
              
               
                 G.
                 
              
               
                 a.
                 
              
               
                 b.
                 
              
               
                 c.
                 
              
               
                 d.
                 
              
               
                 e.
                 
              
               
                 f.
                 
              
               
                 g.
                 
              
               
                 aa
                 .
              
               
                 &c.
                 
              
            
             
               
                 la.
                 
              
               
                 mi.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 mi.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 &c.
                 
              
            
             
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                  
              
               
                  
              
            
          
           Not
           that
           they
           are
           exactly
           such
           ,
           but
           near
           the
           matter
           .
           And
           what
           is
           the
           exact
           proportion
           of
           each
           ,
           we
           are
           after
           to
           consider
           .
           What
           it
           is
           in
           the
           Chromatick
           or
           Enarmonick
           of
           the
           Greeks
           ,
           comes
           not
           into
           our
           present
           Enquiry
           .
        
         
           And
           here
           we
           are
           to
           take
           notice
           of
           the
           ambiguous
           or
           double
           sence
           of
           the
           word
           Note
           .
           Before
           ,
           it
           signified
           one
           particular
           Sound
           or
           Note
           ,
           as
           
             la
             ,
             mi
             ,
             fa
          
           ,
           &c.
           but
           here
           ,
           when
           we
           speak
           of
           Notes
           and
           Half-notes
           ,
           it
           signifies
           an
           Interval
           between
           Note
           and
           Note
           .
           The
           Greeks
           had
           two
           words
           for
           it
           ,
           Phthongus
           ,
           and
           Tonus
           ;
           but
           we
           use
           Note
           promiscuously
           for
           both
           .
        
         
           In
           an
           Octave
           thus
           constituted
           ,
           from
           mi
           to
           mi
           (
           or
           from
           B
           to
           b
           ,
           which
           is
           the
           natural
           place
           of
           mi
           )
           there
           are
           five
           Tones
           ,
           and
           two
           Hemitones
           ;
           (
           fa
           rising
           every-where
           
             half
             a
             Note
          
           ,
           and
           the
           rest
           a
           
           
             whole
             Note
          
           :
           )
           and
           ,
           therein
           two
           Tetrachords
           or
           Dia-tessaron's
           ,
           or
           (
           as
           we
           call
           them
           )
           Fourths
           ;
           where
           ,
           from
           mi
           or
           la
           ,
           we
           rise
           by
           
             fa
             sol
             la
          
           ,
           (
           that
           is
           ,
           
             mi
             fa
             sol
             la
          
           ,
           or
           
             la
             fa
             sol
             la
          
           ,
           according
           as
           mi
           or
           la
           come
           next
           before
           
             fa
             ;
          
           )
           and
           moreover
           ,
           one
           Tone
           ,
           or
           whole
           Note
           ,
           
             la
             mi
          
           :
           which
           added
           to
           the
           Tetrachord
           ,
           either
           next
           before
           it
           ,
           or
           next
           after
           it
           ,
           makes
           a
           Pentachord
           or
           Dia-pente
           ,
           which
           we
           call
           a
           Fifth
           ;
           whereby
           the
           Octave
           is
           near
           equally
           divided
           into
           a
           Fourth
           and
           a
           Fifth
           ,
           as
           BE
           and
           Eb.
           
        
         
           And
           the
           same
           we
           have
           ,
           for
           quantity
           ,
           in
           any
           other
           Octave
           ;
           as
           from
           A
           to
           a
           ,
           from
           C
           to
           c
           ,
           &c.
           though
           not
           in
           the
           same
           order
           :
           for
           the
           same
           Notes
           still
           return
           after
           one
           Octave
           ,
           in
           the
           same
           sequence
           as
           before
           ,
           the
           last
           Note
           of
           one
           Octave
           being
           the
           first
           of
           the
           next
           ;
           so
           that
           what
           is
           cut
           off
           at
           the
           one
           end
           ,
           is
           supplied
           at
           the
           other
           ,
           at
           what
           place
           soever
           we
           begin
           :
           but
           the
           order
           is
           different
           .
           If
           we
           begin
           at
           A
           ,
           then
           is
           
             la
             mi
          
           (
           the
           odd
           Note
           )
           in
           the
           first
           place
           ,
           (
           before
           the
           two
           Tetrachords
           :
           )
           If
           at
           B
           ,
           't
           is
           in
           the
           last
           place
           (
           after
           both
           :
           )
           If
           at
           E
           ,
           't
           is
           just
           in
           the
           middle
           between
           both
           ,
           (
           and
           will
           make
           a
           Pentachord
           with
           either
           .
           )
           If
           at
           any
           other
           place
           ,
           one
           of
           the
           Tetrachords
           will
           be
           divided
           ;
           and
           what
           of
           it
           is
           wanting
           at
           the
           one
           end
           ,
           will
           be
           supplied
           at
           the
           other
           end
           of
           the
           Octave
           .
        
         
           Sutably
           hereunto
           ,
           the
           Greeks
           observed
           seven
           Species
           or
           sorts
           of
           Octaves
           ,
           (
           all
           equal
           in
           quantity
           ,
           but
           different
           in
           order
           )
           according
           to
           their
           different
           beginnings
           .
           The
           first
           Aa
           ,
           and
           second
           Bb
           ,
           the
           third
           Cc
           ,
           &c.
           of
           which
           each
           had
           its
           peculiar
           name
           .
        
         
           The
           first
           of
           them
           begins
           at
           A
           ,
           which
           was
           their
           Proslambanomenos
           ,
           (
           a
           sound
           given
           ,
           or
           assumed
           ,
           )
           answering
           to
           what
           our
           Musicians
           now
           call
           the
           Key
           :
           A
           Note
           at
           which
           (
           in
           the
           Base-part
           )
           the
           Song
           begins
           ,
           and
           to
           which
           great
           respect
           is
           had
           through
           the
           whole
           .
           The
           other
           Species
           had
           their
           several
           beginnings
           ,
           or
           respective
           Keys
           .
        
         
           Which
           ,
           and
           how
           many
           of
           these
           ,
           they
           did
           make
           use
           of
           in
           their
           Musick
           ,
           is
           hard
           to
           say
           .
           Our
           Ancestors
           ,
           about
           
           Guido's
           time
           ,
           seem
           chiefly
           to
           have
           affected
           that
           of
           G
           ;
           as
           appears
           by
           their
           Scale
           beginning
           at
           Gamm-ut
           ;
           and
           their
           Series
           of
           Notes
           ,
           
             Vt
             re
             mi
             fa
             sol
             la
             :
          
           where
           we
           have
           two
           Notes
           lower
           than
           mi.
           And
           our
           Musick-Masters
           ,
           to
           this
           day
           ,
           when
           they
           teach
           to
           sing
           ,
           begin
           commonly
           with
           
             sol
             la
             mi
             fa
          
           ,
           &c.
           as
           if
           ut
           ,
           or
           the
           nearest
           sol
           below
           mi
           ,
           were
           the
           Key
           or
           Note
           most
           regarded
           .
           Beside
           which
           ,
           they
           seem
           (
           by
           the
           Gamm-ut
           )
           to
           have
           had
           two
           more
           (
           upon
           a
           remove
           )
           but
           of
           the
           
           same
           constitution
           ,
           at
           C
           ,
           and
           F
           ,
           where
           (
           in
           the
           Gamm-ut
           )
           the
           Note
           ut
           returns
           again
           .
        
         
           
             For
             twenty
             years
             last
             past
          
           ,
           you
           tell
           us
           
             (
             p.
          
           7.
           l.
           29.
           p.
           8.
           l.
           2.
           )
           
             in
             the
             compositions
             of
             the
             most
             eminent
             Masters
             ,
             they
             scarce
             make
             use
             of
             any
             other
             Key
             (
             as
             to
             the
             internal
             constitution
             of
             the
             Octave
             )
             than
          
           A
           and
           C.
           That
           is
           ,
           their
           Key
           or
           leading
           Note
           ,
           at
           which
           their
           Base-part
           begins
           ,
           is
           either
           la
           next
           before
           mi
           ,
           or
           fa
           next
           above
           mi.
           And
           ,
           because
           you
           are
           much
           better
           acquainted
           with
           the
           eminent
           Masters
           of
           Musick
           ,
           and
           their
           Compositions
           ,
           than
           I
           am
           ,
           I
           take
           it
           ,
           as
           to
           matter
           of
           fact
           ,
           so
           to
           be
           .
           But
           at
           whatever
           Note
           they
           please
           to
           begin
           ,
           and
           make
           it
           their
           Key
           ,
           it
           is
           ,
           as
           to
           this
           point
           ,
           much
           one
           .
        
         
           If
           these
           Keys
           be
           transferred
           to
           any
           other
           place
           (
           to
           accommodate
           the
           Voice
           or
           Instrument
           )
           mi
           is
           transferred
           also
           ,
           (
           by
           help
           of
           Flats
           and
           Sharps
           )
           and
           the
           other
           Sequence
           of
           Notes
           with
           it
           ;
           which
           alters
           the
           Pitch
           but
           not
           the
           
             Tune
             (
             p.
          
           8.
           l.
           3
           ,
           12.
           )
           that
           is
           ,
           the
           same
           Tune
           is
           sung
           by
           a
           higher
           or
           lower
           Voice
           .
        
         
           As
           for
           instance
           :
           If
           I
           would
           remove
           these
           Keys
           from
           A
           and
           C
           ,
           to
           B
           and
           D
           ,
           (
           that
           is
           ,
           if
           I
           would
           set
           the
           whole
           Tune
           a
           Note
           higher
           )
           then
           must
           la
           (
           which
           is
           now
           at
           A
           )
           remove
           to
           B
           ,
           and
           mi
           (
           from
           B
           )
           to
           C
           ,
           and
           fa
           to
           D
           ,
           and
           so
           forth
           .
           But
           then
           I
           find
           ,
           that
           BC
           (
           an
           Half-note
           )
           is
           not
           wide
           enough
           to
           receive
           
             la
             mi
          
           ,
           (
           a
           whole
           Note
           :
           )
           and
           therefore
           C
           (
           by
           a
           Sharp
           )
           must
           be
           raised
           half
           a
           Note
           higher
           ,
           that
           B
           C
           ♯
           may
           be
           fit
           to
           receive
           
             la
             mi
          
           ;
           and
           thence
           to
           D
           ,
           will
           remain
           half
           a
           Note
           for
           
             mi
             fa
          
           ;
           thence
           to
           E
           ,
           an
           whole
           Note
           for
           
             fa
             sol
          
           ;
           but
           thence
           to
           F
           ,
           is
           but
           half
           a
           Note
           ,
           not
           capable
           of
           
             sol
             la
          
           ,
           an
           whole
           Note
           :
           and
           therefore
           F
           (
           by
           another
           Sharp
           )
           must
           be
           advanced
           half
           a
           Note
           higher
           ,
           that
           E
           F
           ♯
           may
           receive
           
             sol
             la
          
           ;
           and
           F
           ♯
           G
           receive
           
             la
             fa
          
           ,
           (
           an
           Half-note
           ;
           )
           and
           G
           a
           receive
           
             fa
             sol
          
           ;
           a
           b
           receive
           
             sol
             la
          
           ;
           and
           then
           b
           c
           ♯
           receive
           
             la
             mi
          
           ,
           and
           so
           forth
           .
           So
           that
           C
           and
           F
           must
           every-where
           have
           a
           Sharp
           ,
           but
           the
           other
           Notes
           remain
           as
           before
           ,
           in
           this
           form
           .
        
         
           
             
               
                 B.
                 
              
               
                 C
                 ♯
                 .
              
               
                 D.
                 
              
               
                 E.
                 
              
               
                 F
                 ♯
                 .
              
               
                 G.
                 
              
               
                 a.
                 
              
               
                 b.
                 
              
               
                 c
                 ♯
                 .
              
               
                 d.
                 
              
               
                 &c.
                 
              
            
             
               
                 la.
                 
              
               
                 mi.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 mi.
                 
              
               
                 fa.
                 
              
               
                 &c.
                 
              
            
             
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 h.
                 
              
               
                  
              
               
                  
              
            
          
        
         
         
           In
           like
           manner
           ,
           if
           I
           would
           bring
           back
           A
           and
           C
           ,
           to
           Γ
           and
           B
           ,
           that
           is
           ,
           if
           I
           would
           set
           the
           Tune
           a
           Note
           lower
           ;
           then
           must
           la
           come
           back
           to
           Γ
           ,
           and
           mi
           to
           A
           ,
           and
           fa
           to
           B
           ,
           and
           so
           forth
           .
           But
           because
           A
           B
           (
           an
           whole
           Note
           )
           is
           too
           wide
           for
           
             mi
             fa
          
           (
           an
           Half-note
           )
           there
           B
           (
           by
           a
           Flat
           )
           must
           be
           taken
           down
           half
           a
           Note
           ;
           that
           so
           A
           B
           ♭
           may
           just
           receive
           
             mi
             fa
          
           ,
           and
           (
           instead
           of
           B
           C
           an
           Half-note
           )
           B
           ♭
           C
           (
           an
           whole
           Note
           )
           receive
           
             fa
             sol
          
           ;
           and
           (
           for
           a
           like
           reason
           )
           another
           Flat
           at
           E.
           And
           so
           every-where
           at
           B
           and
           E.
           The
           rest
           remaining
           as
           before
           ,
           in
           this
           form
           .
           
             
               
                 Γ.
                 
              
               
                 A.
                 
              
               
                 B
                 ♭
                 .
              
               
                 C.
                 
              
               
                 D.
                 
              
               
                 E
                 ♭
                 .
              
               
                 F.
                 
              
               
                 G.
                 
              
               
                 a.
                 
              
               
                 b
                 ♭
                 .
              
               
                 &c.
                 
              
            
             
               
                 la.
                 
              
               
                 mi.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 fa.
                 
              
               
                 sol
                 .
              
               
                 la.
                 
              
               
                 mi.
                 
              
               
                 fa.
                 
              
               
                 &c.
                 
              
            
             
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 h.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 t.
                 
              
               
                 h.
                 
              
               
                  
              
               
                  
              
            
          
           And
           the
           like
           in
           other
           cases
           ;
           of
           which
           you
           give
           us
           the
           particulars
           ,
           at
           pag.
           20.
           
        
         
           And
           it
           is
           ,
           in
           effect
           ,
           no
           other
           than
           the
           common
           Rules
           by
           which
           Learners
           are
           taught
           to
           find
           mi
           :
           Only
           there
           ,
           by
           the
           Flats
           and
           Sharps
           we
           find
           the
           place
           of
           mi
           ;
           and
           here
           ,
           by
           the
           assigned
           place
           of
           mi
           ,
           you
           find
           the
           Flats
           and
           Sharps
           ,
           if
           any
           be
           .
        
         
           Of
           those
           two
           Keys
           ,
           at
           A
           and
           C
           ,
           in
           their
           natural
           position
           ,
           you
           observe
           aright
           (
           p.
           7.
           l.
           31.
           p.
           8.
           l.
           21.
           p.
           21.
           l.
           3.
           )
           that
           if
           you
           begin
           at
           A
           ,
           then
           are
           C
           ,
           F
           ,
           G
           ,
           the
           lesser
           Third
           ,
           Sixth
           ,
           and
           Seventh
           ,
           from
           the
           Key
           :
           If
           at
           C
           ,
           then
           are
           E
           ,
           a
           ,
           b
           ,
           the
           greater
           third
           ,
           Sixth
           ,
           and
           Seventh
           ,
           from
           the
           Key
           .
           And
           therefore
           the
           Composer
           ,
           if
           he
           design
           those
           ,
           should
           chuse
           the
           Key
           A
           ;
           if
           these
           ,
           the
           Key
           C
           :
           or
           the
           equivalents
           of
           these
           transferred
           (
           as
           is
           shewed
           )
           to
           other
           places
           .
        
         
           And
           thus
           far
           we
           have
           pursued
           the
           Language
           of
           Aristoxenes
           and
           his
           Followers
           the
           Aristoxeneans
           ,
           and
           the
           Practical
           Musicians
           ,
           who
           content
           themselves
           with
           the
           names
           of
           Notes
           and
           Half-notes
           ,
           without
           enquiring
           into
           the
           Proportions
           of
           each
           Interval
           .
        
         
           But
           Pythagoras
           and
           his
           Followers
           ,
           whom
           you
           call
           the
           Speculative
           Musicians
           ,
           enquired
           further
           into
           the
           Proportions
           (
           of
           Graveness
           and
           Acuteness
           )
           at
           several
           Intervals
           .
        
         
         
           And
           here
           it
           was
           first
           discovered
           by
           Pythagoras
           ,
           and
           since
           admit●ed
           by
           all
           ,
           (
           which
           therefore
           you
           suppose
           as
           granted
           )
           that
           the
           Proportion
           for
           an
           Octave
           (
           as
           A
           a
           ,
           or
           B
           b
           ,
           )
           is
           Double
           ,
           or
           as
           2
           to
           1
           :
           That
           for
           a
           Fifth
           ,
           (
           as
           A
           E
           ,
           or
           E
           b
           ,
           )
           Sesquialter
           ,
           or
           as
           3
           to
           2
           :
           That
           for
           a
           Fourth
           ,
           (
           B
           E
           ,
           or
           E
           a
           ,
           )
           Sesquitertian
           ,
           or
           as
           4
           to
           3
           :
           And
           consequently
           ,
           that
           for
           the
           Tone
           (
           A
           B
           ,
           or
           a
           b
           ,
           )
           Sesquioctave
           ,
           or
           as
           9
           to
           8.
           
        
         
           Hence
           Euclide
           (
           in
           his
           
             Musical
             Introduction
          
           ,
           and
           
             Section
             of
             the
             Canon
          
           )
           taking
           all
           Tones
           to
           be
           equal
           ,
           computes
           ,
           that
           for
           the
           
             Ditoue
             ,
             fa
             sol
             la
          
           ,
           to
           be
           as
           81
           to
           64
           ,
           (
           which
           is
           the
           duplicate
           of
           9
           to
           8
           ;
           )
           and
           therefore
           the
           Hemitone
           
             mi
             fa
          
           ,
           or
           
             la
             fa
          
           ,
           to
           be
           so
           much
           as
           that
           wants
           of
           a
           Fourth
           ;
           that
           is
           ,
           256
           to
           243
           :
           which
           is
           somewhat
           less
           than
           half
           a
           Tone
           ,
           or
           the
           sub-duplicate
           of
           9
           to
           8.
           
           And
           so
           Boethius
           (
           after
           him
           )
           and
           others
           downward
           ,
           till
           that
           (
           about
           an
           hundred
           years
           ago
           )
           Zarline
           revived
           the
           Doctrine
           of
           Ptolomy
           in
           this
           point
           .
        
         
           That
           of
           Ptolomy
           (
           in
           his
           Harmonicks
           )
           is
           this
           :
           The
           Proportions
           for
           the
           Octave
           ,
           the
           Fifth
           ,
           and
           Fourth
           ,
           he
           retains
           as
           before
           ▪
           and
           that
           of
           the
           Dia-zeuctick
           Tone
           
             la
             mi
          
           ;
           which
           ,
           together
           with
           the
           two
           Fourths
           ,
           compleats
           the
           Octave
           ;
           and
           doth
           ,
           with
           either
           of
           them
           ,
           compleat
           the
           Fifth
           ▪
           (
           And
           so
           far
           Pythagoras
           proceeded
           ,
           and
           no
           further
           .
           )
           But
           the
           Ditone
           
             fa
             sol
             la
          
           ,
           (
           which
           is
           the
           greatest
           Third
           )
           he
           reckons
           to
           consist
           of
           two
           Tones
           ,
           but
           unequal
           .
           The
           one
           as
           9
           to
           8
           ,
           (
           which
           is
           called
           the
           major
           Tone
           ,
           and
           is
           equal
           to
           that
           of
           
             la
             mi
          
           ;
           )
           the
           other
           as
           10
           to
           9
           ,
           (
           called
           the
           minor
           Tone
           :
           )
           and
           therefore
           the
           Compound
           of
           both
           ,
           as
           5
           to
           4
           ;
           (
           which
           is
           a
           much
           more
           Musical
           Interval
           ,
           than
           81
           to
           64
           :
           )
           And
           consequently
           the
           Hemitone
           or
           Half-note
           
             mi
             fa
          
           ,
           or
           
             la
             fa
          
           ,
           (
           so
           much
           as
           5
           /
           4
           wants
           of
           4
           /
           3
           )
           as
           16
           to
           15
           ,
           (
           which
           is
           somewhat
           more
           than
           half
           a
           Note
           :
           )
           and
           this
           together
           with
           a
           major
           Tone
           ,
           (
           which
           make
           the
           lesser
           Third
           )
           as
           6
           to
           5
           ;
           as
           
             la
             mi
             fa
          
           (
           from
           A
           to
           C
           ,
           )
           or
           
             la
             fa
             sol
          
           (
           from
           E
           to
           G
           ,
           )
           or
           
             mi
             fa
             sol
          
           (
           from
           B
           to
           D
           )
           taking
           
             fa
             sol
          
           for
           a
           major
           Tone
           ,
           and
           therefore
           
             sol
             la
          
           for
           a
           minor
           Tone
           .
        
         
           These
           Proportions
           setled
           by
           Ptolomy
           ,
           (
           for
           the
           Diatonick
           kind
           )
           were
           afterward
           disused
           ,
           till
           revived
           by
           Zarline
           about
           a
           hund●●d
           years
           since
           ;
           who
           is
           followed
           by
           
             Kepler
             ,
             Mersennus
             ,
             Cartes
             ,
             Gassend●●
             ,
          
           and
           others
           ,
           
             (
             p.
          
           14.
           l.
           15.
           )
           and
           generally
           admitted
           by
           Specula●●●●
           Musicians
           since
           
           Zarline's
           time
           .
           And
           these
           you
           pre-suppose
           (
           up●n
           their
           Authorities
           )
           referring
           to
           their
           Reasons
           for
           it
           .
        
         
         
           You
           pre-suppose
           further
           ,
           (
           which
           Ptolomy
           had
           shewed
           at
           large
           ,
           and
           is
           since
           admitted
           )
           that
           (
           in
           the
           same
           or
           like
           String
           )
           the
           degree
           of
           Acuteness
           is
           in
           counter
           -
           proportion
           to
           the
           length
           :
           That
           is
           ,
           if
           the
           Acuteness
           of
           the
           sound
           be
           as
           2
           to
           1
           ,
           (
           as
           in
           the
           upper
           Note
           of
           an
           Octave
           )
           the
           length
           of
           the
           String
           is
           as
           1
           to
           2
           ,
           (
           the
           Sound
           twice
           as
           sharp
           ,
           the
           String
           half
           so
           long
           .
           )
           And
           where
           the
           Acuteness
           of
           the
           Sound
           is
           as
           3
           to
           2
           ,
           (
           as
           in
           the
           upper
           Note
           of
           a
           Fifth
           )
           the
           length
           is
           as
           2
           to
           3.
           
           Where
           that
           is
           as
           4
           to
           3
           ,
           (
           as
           in
           a
           Fourth
           )
           this
           is
           as
           3
           to
           4.
           
           Where
           that
           is
           as
           5
           to
           4
           ,
           (
           as
           in
           the
           greater
           Third
           )
           this
           is
           as
           4
           to
           5.
           
           Where
           that
           ,
           as
           6
           to
           5
           ,
           (
           as
           in
           the
           lesser
           Third
           )
           this
           ,
           as
           5
           to
           6.
           
           Where
           that
           ,
           〈◊〉
           9
           to
           8
           ,
           (
           as
           in
           the
           greater
           Tone
           )
           this
           ,
           as
           8
           to
           9.
           
           Where
           that
           ,
           as
           10
           to
           9
           ,
           (
           as
           in
           the
           lesser
           Tone
           )
           this
           ,
           as
           9
           to
           10.
           
           And
           thus
           every-where
           ,
           the
           length
           of
           the
           String
           is
           in
           the
           counter-proportion
           to
           the
           height
           or
           sharpness
           of
           the
           Sound
           .
        
         
           These
           things
           being
           premised
           ,
           or
           pre-supposed
           ,
           you
           proceed
           to
           the
           Constitution
           of
           the
           Keys
           A
           and
           C
           ,
           
             (
             p.
          
           9.
           and
           p.
           12.
           )
           where
           ,
           by
           9
           ,
           16
           ,
           10
           ,
           you
           mean
           1
           /
           ●
           ,
           1
           /
           16
           ,
           1
           /
           10
           ,
           (
           which
           ,
           I
           suppose
           ,
           you
           so
           expressed
           ,
           lest
           the
           small
           Fraction-figures
           should
           not
           be
           so
           easily
           seen
           and
           read
           .
           )
           And
           so
           ,
           by
           6
           ,
           4
           ,
           3
           ,
           2
           ,
           (
           the
           portion
           of
           the
           String
           from
           the
           Nut
           to
           the
           Fret
           ,
           for
           the
           Key
           A
           )
           you
           mean
           1
           /
           6
           ,
           1
           /
           4
           ,
           1
           /
           ●
           ,
           1
           /
           2
           ,
           (
           sutable
           to
           3
           /
           ●
           and
           4
           /
           9
           :
           )
           And
           (
           for
           the
           Key
           C
           )
           by
           5
           ,
           4
           ,
           3
           ,
           2
           ,
           you
           mean
           1
           /
           5
           ,
           1
           /
           4
           ,
           1
           /
           3
           ,
           1
           /
           ●
           ,
           (
           in
           the
           same
           sence
           with
           2
           /
           ●
           and
           7
           /
           15
           ;
           )
           according
           the
           Exposition
           your self
           gives
           ,
           pag.
           9
           ,
           10
           ,
           11.
           
        
         
           I
           should
           rather
           have
           chosen
           to
           express
           the
           place
           of
           the
           Fret
           ,
           by
           the
           length
           of
           the
           other
           part
           of
           the
           String
           ,
           from
           the
           Bridge
           to
           the
           Fret
           ,
           than
           from
           the
           Nut
           to
           the
           Fret
           :
           (
           For
           it
           is
           that
           ,
           not
           this
           ,
           that
           gives
           the
           Sound
           .
           )
           In
           this
           manner
           .
           
             
               
                  
              
               
                  
              
               
                 8
              
               
                  
              
               
                 5
              
               
                  
              
               
                 3
              
               
                  
              
               
                 2
              
               
                  
              
               
                 5
              
               
                  
              
               
                 5
              
               
                  
              
               
                 1
              
            
             
               
                 A
              
               
                 8
                 /
                 9
              
               
                 B
              
               
                 15
                 /
                 16
              
               
                 C
              
               
                 9
                 /
                 10
              
               
                 D
              
               
                 8
                 /
                 9
              
               
                 E
              
               
                 15
                 /
                 16
              
               
                 F
              
               
                 8
                 /
                 9
              
               
                 G
              
               
                 9
                 /
                 10
              
               
                 a.
                 
              
            
             
               
                  
              
               
                  
              
               
                 9
              
               
                  
              
               
                 6
              
               
                  
              
               
                 4
              
               
                  
              
               
                 3
              
               
                  
              
               
                 8
              
               
                  
              
               
                 9
              
               
                  
              
               
                 2
              
            
          
           
             
               
                  
              
               
                  
              
               
                 9
              
               
                  
              
               
                 4
              
               
                  
              
               
                 3
              
               
                  
              
               
                 2
              
               
                  
              
               
                 3
              
               
                  
              
               
                 8
              
               
                  
              
               
                 1
              
            
             
               
                 C
              
               
                 9
                 /
                 10
              
               
                 D
              
               
                 8
                 /
                 ●
              
               
                 E
              
               
                 1●
                 /
                 16
              
               
                 F
              
               
                 8
                 /
                 9
              
               
                 G
              
               
                 9
                 /
                 10
              
               
                 a
              
               
                 8
                 /
                 9
              
               
                 b
              
               
                 15
                 /
                 10
              
               
                 c.
                 
              
            
             
               
                  
              
               
                  
              
               
                 10
              
               
                  
              
               
                 5
              
               
                  
              
               
                 4
              
               
                  
              
               
                 3
              
               
                  
              
               
                 5
              
               
                  
              
               
                 15
              
               
                  
              
               
                 2
              
            
          
           
           That
           is
           ,
           from
           the
           Bridge
           to
           B
           ,
           is
           8
           /
           9
           of
           what
           it
           is
           to
           A
           ;
           and
           from
           the
           Bridge
           to
           C
           ,
           is
           15
           /
           16
           of
           what
           it
           is
           to
           B
           :
           and
           so
           elsewhere
           .
           Again
           ,
           from
           the
           Bridge
           to
           B
           ,
           is
           8
           /
           9
           of
           that
           whole
           String
           ;
           from
           the
           Bridge
           to
           C
           ,
           is
           1
           /
           6
           of
           the
           whole
           .
           And
           so
           of
           the
           rest
           .
           But
           the
           sense
           is
           the
           same
           with
           that
           of
           yours
           .
        
         
           But
           herein
           I
           dissent
           from
           you
           ;
           namely
           ,
           whereas
           you
           say
           ,
           C
           10
           D
           9
           E
           ;
           I
           would
           rather
           say
           ,
           C
           9
           D
           10
           E.
           That
           is
           ,
           of
           
             fa
             sol
             la
          
           ,
           I
           would
           make
           
             fa
             sol
          
           the
           bigger
           Tone
           ,
           and
           
             sol
             la
          
           the
           lesser
           ;
           at
           C
           D
           E
           ,
           as
           well
           as
           at
           F
           G
           a.
           And
           then
           my
           Proportions
           would
           stand
           thus
           :
           
             
               
                  
              
               
                  
              
               
                 8
              
               
                  
              
               
                 5
              
               
                  
              
               
                 20
              
               
                  
              
               
                 2
              
               
                  
              
               
                 5
              
               
                  
              
               
                 5
              
               
                  
              
               
                 1
              
            
             
               
                 A
              
               
                 8
                 /
                 9
              
               
                 B
              
               
                 15
                 /
                 16
              
               
                 C
              
               
                 8
                 /
                 9
              
               
                 D
              
               
                 9
                 /
                 10
              
               
                 E
              
               
                 15
                 /
                 10
              
               
                 F
              
               
                 8
                 /
                 9
              
               
                 G
              
               
                 9
                 /
                 10
              
               
                 a.
                 
              
            
             
               
                  
              
               
                  
              
               
                 9
              
               
                  
              
               
                 6
              
               
                  
              
               
                 ●7
              
               
                  
              
               
                 3
              
               
                  
              
               
                 8
              
               
                  
              
               
                 9
              
               
                  
              
               
                 2
              
            
          
           
             
               
                  
              
               
                  
              
               
                 8
              
               
                  
              
               
                 4
              
               
                  
              
               
                 3
              
               
                  
              
               
                 2
              
               
                  
              
               
                 3
              
               
                  
              
               
                 8
              
               
                  
              
               
                 1
              
            
             
               
                 C
              
               
                 8
                 /
                 9
              
               
                 D
              
               
                 9
                 /
                 10
              
               
                 E
              
               
                 15
                 /
                 16
              
               
                 F
              
               
                 8
                 /
                 9
              
               
                 G
              
               
                 9
                 /
                 10
              
               
                 a
              
               
                 8
                 /
                 9
              
               
                 b
              
               
                 15
                 /
                 16
              
               
                 c.
                 
              
            
             
               
                  
              
               
                  
              
               
                 9
              
               
                  
              
               
                 5
              
               
                  
              
               
                 4
              
               
                  
              
               
                 3
              
               
                  
              
               
                 5
              
               
                  
              
               
                 15
              
               
                  
              
               
                 2
              
            
          
        
         
           The
           reason
           why
           you
           did
           otherwise
           at
           C
           D
           E
           ,
           that
           at
           F
           G
           a
           ,
           is
           (
           I
           presume
           )
           that
           A
           D
           might
           be
           a
           true
           Fourth
           to
           the
           Key
           A.
           Whereas
           otherwise
           20
           /
           27
           is
           a
           false
           Fourth
           .
           Though
           very
           near
           a
           true
           one
           :
           for
           21
           /
           21
           would
           be
           a
           true
           Fourth
           ,
           being
           the
           same
           with
           3
           /
           4.
           
        
         
           My
           Reasons
           to
           the
           contrary
           ,
           are
           these
           .
           First
           ,
           a
           Fourth
           (
           now
           a
           days
           )
           is
           scarce
           allowed
           as
           a
           good
           Concord
           ;
           and
           therefore
           the
           less
           to
           be
           regarded
           :
           especially
           at
           this
           place
           .
           For
           ,
           I
           suppose
           ,
           it
           is
           not
           usual
           ,
           from
           A
           the
           Key
           ,
           to
           rise
           a
           Fourth
           at
           one
           step
           ;
           but
           rather
           a
           Fifth
           ,
           or
           a
           Third
           .
           And
           when
           from
           the
           Key
           you
           rise
           an
           Octave
           at
           two
           steps
           ,
           the
           Fifth
           always
           begins
           ,
           and
           the
           Fourth
           follows
           ,
           (
           not
           first
           a
           Fourth
           ,
           and
           then
           a
           Fifth
           :
           )
           For
           a
           Fifth
           being
           much
           the
           sweeter
           ,
           this
           is
           first
           chosen
           ;
           and
           the
           Fourth
           ,
           which
           follows
           ,
           (
           though
           not
           of
           it self
           so
           sweet
           )
           is
           helped
           by
           being
           an
           Octave
           to
           the
           Key
           ;
           which
           is
           in
           fresh
           memory
           ,
           as
           being
           the
           Sound
           last
           heard
           but
           one
           .
           (
           And
           of
           an
           Interval
           never
           used
           ,
           we
           need
           have
           the
           less
           regard
           .
           )
           And
           if
           from
           any
           other
           place
           we
           
           move
           to
           D
           ,
           the
           Note
           from
           whence
           we
           move
           (
           which
           was
           last
           heard
           )
           is
           more
           to
           be
           regarded
           ,
           than
           the
           Key
           ,
           which
           had
           not
           been
           heard
           for
           some
           while
           .
        
         
           And
           then
           ,
           by
           preserving
           this
           Fourth
           at
           A
           D
           ,
           you
           spoil
           a
           good
           Third
           (
           which
           is
           more
           considerable
           )
           at
           D
           F
           ,
           and
           again
           at
           B
           D
           ,
           and
           a
           fourth
           at
           D
           G
           ,
           and
           a
           Fifth
           at
           D
           a
           ,
           and
           again
           at
           Γ
           D
           ;
           and
           so
           leave
           no
           place
           from
           whence
           to
           move
           to
           this
           D.
           And
           if
           perhaps
           you
           will
           say
           from
           d
           ;
           at
           that
           it
           is
           as
           hard
           to
           come
           as
           at
           this
           D.
           So
           that
           your
           D
           will
           be
           of
           no
           use
           at
           all
           .
        
         
           And
           for
           such
           reasons
           ,
           it
           was
           a
           Rule
           with
           the
           Greeks
           ,
           that
           the
           two
           conjoyned
           Tetrachords
           ,
           
             mi
             fa
             sol
             la
          
           ,
           at
           B
           C
           D
           E
           ;
           and
           
             la
             fa
             sol
             la
          
           ,
           at
           E
           F
           G
           a
           ;
           should
           ever
           be
           divided
           in
           the
           same
           manner
           .
           For
           though
           they
           had
           their
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           ,
           (
           a
           Transite
           from
           one
           kind
           of
           dividing
           the
           Tetrachord
           to
           another
           ;
           )
           yet
           they
           would
           have
           this
           Transite
           to
           be
           always
           at
           the
           Disjunctive
           Tone
           
             la
             mi.
          
           Allowing
           to
           the
           Tetrachord
           above
           it
           ,
           a
           different
           division
           from
           that
           below
           it
           ;
           but
           not
           a
           different
           division
           to
           the
           two
           conjunct
           Tetrachords
           above
           it
           .
        
         
           Again
           ,
           if
           you
           make
           your
           Key
           at
           C
           ,
           and
           thence
           sing
           
             fa
             sol
             la
          
           ;
           it
           is
           most
           natural
           for
           the
           greater
           Tone
           to
           begin
           ,
           and
           the
           lesser
           to
           follow
           .
           (
           Upon
           the
           same
           account
           as
           when
           we
           rise
           an
           Octave
           at
           twice
           ,
           the
           Fifth
           leads
           ,
           and
           the
           Fourth
           follows
           ;
           and
           when
           at
           two
           steps
           we
           would
           rise
           a
           Fifth
           ,
           it
           is
           most
           proper
           for
           the
           greater
           Third
           to
           begin
           ,
           and
           the
           lesser
           to
           follow
           .
           )
           And
           therefore
           here
           ,
           not
           to
           make
           
             fa
             sol
          
           the
           lesser
           Tone
           ,
           and
           
             sol
             la
          
           the
           greater
           ;
           but
           ,
           that
           the
           greater
           ,
           and
           this
           the
           lesser
           ;
           as
           your self
           do
           at
           F
           G
           a.
           
        
         
           And
           if
           ,
           for
           these
           reasons
           ,
           you
           give
           the
           same
           division
           to
           C
           D
           E
           ,
           that
           you
           do
           to
           F
           G
           a
           ;
           it
           will
           relieve
           all
           those
           
             inconcinnous
             Intervals
          
           mentioned
           pag.
           24.
           
        
         
           I
           have
           now
           done
           with
           the
           Constitution
           of
           your
           two
           Keys
           at
           A
           and
           C
           ,
           (
           and
           indeed
           of
           any
           other
           Key
           )
           in
           the
           natural
           Constitution
           .
           And
           consequently
           ,
           with
           all
           the
           Essential
           Flats
           and
           Sharps
           ,
           which
           serve
           onely
           to
           remove
           it
           to
           a
           higher
           or
           lower
           pitch
           ,
           without
           changing
           the
           sequence
           of
           the
           Notes
           :
           And
           which
           are
           wont
           to
           be
           noted
           at
           the
           beginning
           of
           the
           Tune
           ,
           so
           as
           to
           influence
           the
           whole
           ,
           without
           repeating
           them
           at
           the
           several
           Notes
           .
        
         
         
           But
           ,
           besides
           these
           ,
           there
           are
           some
           Accidental
           Flats
           and
           Sharps
           ,
           which
           occur
           in
           the
           middle
           of
           the
           Musick
           ,
           affecting
           some
           one
           Note
           .
           These
           you
           call
           
             Chromatick
             Half-notes
             ,
             (
             p.
          
           15.
           l
           10
           ,
           20.
           p.
           18.
           l.
           16.
           )
           which
           ,
           you
           say
           ,
           are
           
             truly
             made
             by
             placing
             the
             Fret
             exactly
             in
             the
             middle
             between
             the
             two
             Frets
             of
             the
             Diatonick
             whole
             Note
          
           wherein
           it
           falls
           .
           Which
           is
           not
           so
           cautiously
           expressed
           as
           not
           to
           be
           liable
           to
           a
           mistake
           .
           The
           Reader
           ,
           by
           
             exactly
             in
             the
             middle
          
           ,
           will
           be
           apt
           to
           understand
           an
           Arithmetical
           middle
           :
           as
           for
           instance
           ,
           if
           F
           G
           be
           an
           Inch
           ,
           F
           F
           ♯
           and
           F
           ♯
           G
           should
           be
           each
           of
           them
           half
           an
           Inch.
           Or
           ,
           a
           Geometrical
           middle
           ,
           (
           which
           we
           call
           a
           
             mean
             Proportional
          
           ;
           )
           as
           ,
           if
           the
           proportion
           for
           F
           G
           be
           as
           8
           to
           9
           ,
           that
           for
           F
           F
           ♯
           and
           F
           ♯
           G
           should
           be
           as
           28
           to
           29
           (
           the
           square
           Root
           of
           8
           to
           the
           square
           Root
           of
           9.
           )
           Neither
           of
           which
           are
           your
           meaning
           .
           But
           you
           mean
           (
           I
           presume
           )
           a
           kind
           of
           Musical
           middle
           ,
           which
           is
           thus
           to
           be
           taken
           :
           Suppos●●g
           the
           Proportion
           for
           F
           G
           to
           be
           as
           8
           to
           9
           ,
           that
           is
           ,
           (
           doubling
           both
           numbers
           )
           as
           16
           to
           18
           ;
           this
           is
           to
           be
           divided
           by
           help
           of
           the
           middle
           number
           17.
           
           So
           that
           the
           Proportion
           for
           F
           F
           ♯
           shall
           be
           16
           to
           17
           ;
           and
           for
           F
           ♯
           G
           ,
           17
           to
           18.
           
           Which
           ,
           together
           ,
           make
           F
           G
           as
           16
           to
           18
           ,
           or
           8
           to
           9.
           
           In
           like
           manner
           ,
           if
           G
           a
           be
           as
           9
           to
           10
           ,
           that
           is
           as
           18
           to
           20
           ;
           then
           is
           G
           G
           ♯
           as
           18
           to
           19
           ,
           and
           G
           ♯
           a
           as
           19
           to
           20
           :
           which
           ,
           together
           ,
           make
           18
           to
           20
           ,
           or
           9
           to
           10.
           
           And
           this
           ,
           I
           presume
           ,
           (
           though
           the
           Book
           be
           not
           at
           hand
           )
           is
           the
           meaning
           of
           Aristides
           at
           the
           place
           cited
           ;
           and
           your
           meaning
           here
           .
           And
           this
           ,
           I
           suppose
           ,
           may
           do
           pretty
           well
           in
           most
           places
           .
        
         
           But
           if
           we
           would
           be
           exact
           ,
           we
           must
           ,
           in
           each
           place
           ,
           consider
           ,
           what
           is
           the
           particular
           design
           we
           aim
           at
           in
           such
           a
           Flat
           or
           Sharp
           ;
           and
           make
           the
           division
           accordingly
           .
           As
           for
           instance
           ,
           if
           to
           the
           Key
           A
           ,
           instead
           of
           a
           lesser
           Sixth
           at
           F
           ,
           I
           would
           have
           a
           greater
           Sixth
           at
           F
           ♯
           ;
           I
           must
           not
           so
           much
           aim
           at
           such
           equal
           division
           (
           or
           near-equal
           )
           of
           F
           G
           ;
           as
           ,
           to
           take
           out
           of
           it
           so
           much
           as
           will
           make
           E
           F
           ♯
           a
           minor
           Tone
           ,
           (
           whatever
           chance
           to
           be
           the
           remaining
           part
           to
           G.
           )
           Which
           will
           make
           for
           F
           F
           ♯
           ,
           (
           not
           16
           to
           17
           ,
           but
           )
           24
           to
           25.
           
           (
           For
           this
           ,
           with
           15
           to
           16
           ,
           for
           E
           F
           ,
           will
           make
           that
           for
           E
           F
           ♯
           as
           9
           to
           10
           ;
           and
           for
           A
           F
           ♯
           as
           3
           to
           5
           ,
           a
           greater
           Sixth
           ;
           instead
           of
           A
           F
           as
           5
           to
           8
           ,
           a
           lesser
           
           Sixth
           .
           )
           In
           like
           manner
           ,
           if
           I
           would
           ,
           from
           C
           ,
           rise
           a
           Fourth
           to
           F
           ,
           at
           two
           near-equal
           steps
           ,
           (
           as
           when
           we
           rise
           an
           Eighth
           by
           a
           Fifth
           and
           Fourth
           ;
           or
           a
           Fifth
           ,
           by
           greater
           and
           lesser
           Third
           ;
           )
           that
           is
           ,
           if
           I
           would
           divide
           the
           Proportion
           of
           3
           to
           4
           ,
           or
           6
           to
           8
           ,
           into
           two
           near-equals
           ;
           those
           are
           to
           be
           6
           to
           7
           ,
           and
           7
           to
           8
           :
           And
           therefore
           C
           D
           ♯
           as
           6
           to
           7
           ,
           and
           D
           ♯
           F
           as
           7
           to
           8
           ,
           (
           whatever
           chance
           thereby
           to
           be
           the
           division
           of
           DE.
           )
           And
           the
           like
           for
           other
           cases
           .
           So
           that
           for
           instance
           ,
           the
           same
           D
           ♯
           or
           F
           ♯
           ,
           as
           to
           different
           purposes
           ,
           shall
           signifie
           differently
           .
           And
           such
           Arts
           we
           must
           make
           use
           of
           ,
           if
           we
           would
           revive
           the
           
             Greeks
             Chromatick
          
           and
           Enarmonick
           Musick
           .
           But
           the
           Speculation
           is
           too
           nice
           for
           most
           of
           our
           present
           Practisers
           .
        
         
           To
           return
           therefore
           to
           our
           Diatonick
           Notes
           ,
           in
           their
           Natural
           or
           Primitive
           Constitution
           ,
           together
           with
           their
           Essential
           Sharps
           and
           Flats
           incident
           upon
           removing
           the
           Key
           to
           an
           higher
           or
           lower
           pitch
           :
           having
           once
           fixed
           their
           Proportions
           as
           to
           the
           Monochord
           ,
           (
           as
           supposing
           them
           all
           set
           off
           upon
           one
           String
           ;
           )
           it
           is
           easie
           to
           transfer
           them
           to
           as
           many
           Strings
           as
           you
           please
           ,
           (
           by
           the
           substitution
           of
           a
           smaller
           String
           ,
           instead
           of
           a
           shorter
           distance
           )
           as
           you
           direct
           ,
           p.
           12.
           l.
           9.
           
           And
           from
           thence
           to
           the
           
             Finger-board
             ,
             p.
          
           17.
           l.
           23.
           p.
           19.
           l.
           25.
           p.
           21.
           l.
           17.
           
        
         
           Nor
           do
           I
           take
           exception
           to
           any
           of
           your
           Numbers
           herein
           ;
           save
           that
           I
           think
           D
           with
           its
           Octaves
           (
           or
           the
           Equivalent
           upon
           removing
           the
           Key
           to
           an
           higher
           or
           lower
           pitch
           )
           should
           be
           tuned
           somewhat
           sharper
           than
           you
           direct
           .
           And
           therefore
           you
           may
           ,
           if
           you
           please
           ,
           cause
           a
           Mark
           to
           be
           made
           for
           it
           ,
           in
           each
           of
           your
           Finger-boards
           ,
           (
           as
           is
           already
           done
           in
           that
           at
           Numb
           .
           VIII
           .
           )
           a
           little
           below
           yours
           ,
           in
           such
           proportion
           as
           I
           have
           before
           directed
           :
           That
           so
           the
           Practiser
           may
           ,
           at
           his
           discretion
           ,
           make
           use
           of
           yours
           or
           mine
           ,
           as
           his
           Ear
           shall
           direct
           him
           .
        
         
           Your
           Reader
           also
           may
           be
           advertised
           ,
           That
           though
           your
           Measures
           be
           fitted
           ,
           on
           the
           Plates
           ,
           (
           p.
           21.
           l.
           29.
           )
           to
           a
           String
           of
           28
           Inches
           ;
           yet
           ,
           on
           the
           Paper
           ,
           they
           may
           chance
           to
           be
           somewhat
           less
           than
           so
           :
           For
           (
           being
           printed
           upon
           wet
           Paper
           )
           the
           Measures
           will
           shrink
           ,
           as
           the
           Paper
           dries
           .
        
         
         
           And
           ,
           because
           (
           p.
           18.
           l.
           29.
           )
           you
           do
           not
           confine
           your
           Artificer
           to
           this
           Method
           (
           of
           distinct
           Finger-board
           )
           onely
           ;
           you
           will
           give
           me
           leave
           to
           propose
           another
           .
           Which
           is
           (
           without
           changing
           the
           Finger-board
           )
           to
           have
           the
           Frets
           ,
           for
           each
           String
           ,
           to
           slide
           up
           and
           down
           in
           a
           Groove
           ,
           (
           whereby
           they
           may
           be
           removed
           from
           place
           to
           place
           as
           you
           please
           )
           with
           Lines
           or
           Marks
           on
           the
           Finger-board
           (
           of
           several
           Colours
           ,
           as
           Black
           ,
           White
           ,
           Red
           ,
           &c.
           )
           at
           the
           place
           where
           every
           Fret
           is
           to
           stand
           for
           each
           Tuning
           .
           By
           help
           of
           which
           ,
           the
           Practiser
           may
           at
           pleasure
           set
           his
           Frets
           to
           any
           of
           these
           Tunings
           ,
           (
           or
           to
           any
           other
           upon
           emergent
           occasion
           )
           with
           what
           exactness
           he
           please
           ;
           as
           well
           as
           in
           any
           unfretted
           Instrument
           .
        
         
           Nor
           do
           we
           herein
           depart
           from
           the
           Ancients
           :
           For
           there
           are
           manifest
           places
           in
           Ptolomy
           ,
           that
           their
           Frets
           (
           
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
             〈◊〉
          
           )
           were
           movable
           ,
           not
           in
           Tuning
           onely
           ,
           but
           even
           in
           Playing
           .
        
         
           FINIS
           .
        
      
       
         
         
           To
           the
           Reverend
           Mr.
           SALMON
           .
        
         
           
             Reverend
             ,
          
        
         
           I
           Thank
           you
           for
           the
           great
           kindness
           of
           your
           Letter
           ,
           of
           which
           I
           am
           altogether
           undeserving
           :
           And
           as
           for
           your
           good
           contrivance
           in
           the
           Art
           of
           Musick
           ,
           I
           wish
           that
           the
           use
           of
           the
           Masters
           of
           that
           Art
           may
           not
           hinder
           the
           best
           Method
           to
           obtain
           .
           However
           ,
           you
           have
           left
           a
           Specimen
           to
           Posterity
           ,
           of
           what
           might
           have
           been
           done
           ;
           and
           also
           by
           this
           Present
           ,
           much
           obliged
           ,
        
         
           
             Reverend
             ,
          
           
             Your
             Humble
             and
             Affectionate
             Servant
             ,
             
               E.
               BERNARD
            
             .
          
           
             Oxon.
             
               Dec.
               15.
               1687.
               
            
          
        
      
       
         
         
           A
           Catalogue
           of
           Books
           ,
           Printed
           for
           ,
           and
           Sold
           by
           
             John
             Lawrence
          
           ,
           at
           the
           Angel
           in
           the
           Poultrey
           ,
           Bookseller
           .
        
         
           
           GELL's
           Remains
           :
           being
           sundry
           Pious
           and
           Learned
           Notes
           and
           Observations
           on
           the
           
             New
             Testament
          
           ,
           Opening
           and
           Explaining
           it
           :
           Wherein
           JESUS
           CHRIST
           ,
           as
           yesterday
           ,
           to
           day
           ,
           and
           the
           same
           for
           ever
           ,
           is
           Illustrated
           .
           By
           that
           Learned
           and
           Judicious
           man
           Dr.
           
             R.
             Gell
          
           ,
           late
           Rector
           of
           St.
           
             Mary
             Aldermary
             ,
             London
          
           .
           In
           Folio
           .
        
         
           The
           History
           of
           the
           Affairs
           of
           Europe
           in
           this
           Present
           Age
           ;
           but
           more
           particularly
           of
           the
           Republick
           of
           Venice
           .
           Written
           in
           Italian
           by
           
             Battisti
             Nani
          
           ,
           Cavalier
           and
           Procurator
           of
           St.
           Mark.
           Englished
           by
           Sir
           
             Robert
             Honeywood
          
           ,
           Knight
           .
           Folio
           .
        
         
           The
           Merchant
           Royal.
           A
           Sermon
           preached
           before
           the
           King
           at
           the
           Nuptials
           of
           an
           Honourable
           Lord
           and
           his
           Lady
           .
           Quarto
           .
        
         
           Several
           Excellent
           Discourses
           :
           viz.
           1.
           
           Of
           Purity
           and
           Verity
           .
           2.
           
           Of
           Repentance
           .
           3.
           
           Of
           seeking
           first
           the
           Kingdom
           of
           God.
           By
           that
           Eminent
           Divine
           
           
             Hezekiah
             Burton
          
           ,
           D.
           D.
           late
           Rector
           of
           Barns
           near
           London
           ,
           and
           Prebendary
           of
           Norwich
           .
           Published
           by
           the
           Reverend
           Doctor
           Tillotson
           ,
           Dean
           of
           
             Canterbury
             .
             Octavo
          
           .
        
         
           The
           Antiquity
           of
           China
           :
           or
           ,
           an
           Historical
           Essay
           ,
           endeavouring
           a
           Probability
           that
           the
           Language
           of
           the
           Empire
           of
           China
           is
           the
           Primitive
           Language
           spoken
           through
           the
           whole
           World
           ,
           before
           the
           Confusion
           of
           Babel
           .
           Wherein
           the
           Customs
           and
           Manners
           of
           the
           Chineans
           are
           presented
           ,
           and
           Ancient
           and
           Modern
           Authors
           consulted
           with
           .
           By
           
             John
             Webb
          
           of
           Burleigh
           in
           the
           County
           of
           Somerset
           ,
           
             Esq
          
           Octavo
           .
        
         
           A
           Relation
           of
           the
           Invasion
           and
           Conquest
           of
           Florida
           by
           the
           Spaniards
           ,
           under
           the
           Command
           of
           
             Fernando
             de
             Soto
          
           .
           Written
           in
           Portuguese
           by
           a
           Gentleman
           of
           the
           Town
           of
           Elvas
           .
           Now
           Englished
           .
           To
           which
           is
           subjoyned
           two
           Journeys
           of
           the
           present
           Emperour
           of
           China
           into
           Tartary
           ,
           in
           the
           Years
           1682
           ,
           and
           1683.
           
           With
           some
           Discoveries
           made
           by
           the
           Spaniards
           in
           the
           Island
           of
           California
           in
           the
           Year
           1683.
           
           
             In
             Octavo
          
           .
        
         
           The
           Life
           and
           Actions
           of
           the
           late
           Renowned
           Prelate
           and
           Souldier
           
             Chr.
             Bernard
             van
             Gale
          
           ,
           Bishop
           of
           Munster
           ,
           Prince
           of
           the
           holy
           Empire
           ,
           &c.
           
           In
           which
           is
           an
           Account
           of
           the
           most
           considerable
           Actions
           of
           Europe
           in
           his
           time
           .
           
             In
             Octavo
          
           .
        
         
           The
           Triumphs
           of
           Grace
           :
           or
           ,
           the
           last
           Words
           and
           Edifying
           Death
           of
           the
           Lady
           
             Margaret
             de
             la
             Musse
          
           ,
           a
           Noble
           French
           Lady
           ,
           who
           died
           in
           May
           1681
           ,
           aged
           but
           Sixteen
           Years
           .
           
             In
             Twelves
          
           .