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THE LIBRARY
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THE UNIVERSITY
OF CALIFORNIA
RIVERSIDE
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A
TREATISE ON HARMONY
\^ BY THE
Eev. Sir F^ A. GOEE OUSELEY, Bart., MA., Mus. Doc.
PROFESSOR OF MUSIC IX THE UNIVERSITY OF OXFORD
AT THE CLARENDON PRESS
M.DCCC.LX^^[II
MTyo
€Iar.entr0iT ^ress Btxm
PRmCIPLES OF HARMONY
OUSELEY
Hoution
MACMILLAN AND CO.
FUBLISHEBS TO THE UNIVEBSITT OF
xioxti
PREFACE.
The object aimed at in this volume is the combination of true philosophical prin-
ciples with simplicity of explanation. It also endeavours to include all necessary-
details in the smallest possible compass.
So many treatises on Harmony have appeared since the beginning of this
century, that some apology is perhaps due for adding yet one more to the number.
The author must plead as his apology, the conviction that although the existing
treatises on the subject contain much very valuable matter, yet all seem to him
to be either founded on erroneous principles, or faulty in arrangement. Some mix
up together the elements of Harmony, Counterpoint, and Pianoforte-practice; others
start from principles not based upon nature, but too often contradicted by the
now better ascertained phenomena of acoustics; others repudiate physical science
altogether, and treat of Music as though it were only an emotional art. The
present volume tries to avoid these and other similar errors.
Wherever existing works explain or illustrate any point with clearness, they
have been quoted freely and acknowledged with gratitude. But in other and
more frequent cases it has been necessary to take up entirely fresh ground, and
to employ new explanations and illustrations. It is confidently hoped that the
system thus evolved may prove useful to the student, by giving him natural and
rational explanations of the facts of Music, and of the rules deduced from those
facts. The author has aimed throughout at a consistent theory, founded in nature,
progressively expanded, and involving no purely arbitrary rules. He now lays the
results of his endeavour's before the public.
viii PREFACE.
His warmest thanks are due to many who have aided him in the production
of this work, especially to Professor Pole, Mus. Doc, for his admirable illustration
of the comparative magnitude of intervals, and the lucid explanations accom-
panying it, which invest the work with a scientific value it would not otherwise
have possessed.
In conclusion, a suggestion is offered as to the best way of using this little
work. The student is recommended to study Harmony and Counterpoint con-
currently, working at them a little at a time, alternately. He will find that
neither can be perfectly mastered without the other. With this view, the author
proposes, before long, to bring out a Treatise on Counterpoint, based on those of
Fux, Mai'pm'g, Reicha, and Cherubini, which will be so constructed as to run
parallel to the present volume, as far as possible ; and he desu*es that the two
works may be regarded as pai-ts of one whole, for the instruction of such as desire
to grasp the subject in its completeness.
St. MiCH-iEL-'s College, Tenbuey,
January, 1868.
TO
THE REV. G. W. KITCHIN, M.A.
THIS TREATISE
IS AFFECTIONATELY DEDICATED
BY HIS OLD FRIEND
THE AUTHOR
EKRATA.
Page 32. First stave of music from top of page, under the last two chords,
6 6 B6 be
for 4 4 read 4 4
2 2 b2 b2
Page 34. Second stave of music from top, bar 2, for
Page 47. Second stave of music from bottom, bar 5, for
^
:cz
— read
-^
221
4^±-
Page 53. Second stave from top, bar 3
, for g
-.'S>-
^^ read y?W=?^_
-^-
read
E
-<S>-
Page 85. Bar 6 of example, bass stave, for V^ ^^j, — read V^ y^^y
Page 88. Line 4 from bottom, for t]4 read :5 ; and in line 2 for B5 read D4
3 3 b b
Page 89. Second stave from top, bar 5, for - V ^ l rj I'ead 4^
'^Q rJ
Page 97. Second stave from top, for
read
..88
Page 108. Last bar but one of Exercise on Dissonances, for - read 5
Page 114. Second stave of music, bar 1, for
£
't
d^ read / 4 s &-
F
eJ ^"
Page 129. Last stave of music, in bars 4 and 7, for
Page 136. First stave, first bar, for
ySi rea
•q^^ read @ ^-f^
ad ^7~^ ^
Page 139. Second stave from top, in bar 1, for '^ read *'^; in bar 3 for jfs
b6 T . , (js be
read 4^; and in bar 5 for Jfo read 44-
2 b b
Paere 149. Lowest stave, bar S, for \ ^^ read $ bs
D
9 D
Page 163. In second bar of example, for
-if —
hn
^gtd
V 7 u,*^^
J a^
/m\' I
le;.
v^
1 ^
1
read
5
c
Page 169.
Page 174.
Page 178.
Page 192.
Last stave, bar S, for ^^ read '^^
In fourth chord of example, for B read A
I
2^=
Second stave, bar 6, for - /W-
-&
read
fe
e/
The Jt in the signature of the example should be omitted
Page 20O. Upper stave, bar 2, for ^^^ fj
„ Lowest stave, bar 4, for y^-g!— ^
,ead ^r: p^1=p:
I— J— read vfiv- Sl— S *! ^ -
r f^^
Page 205.
Page 210.
Lines 1 and 2, for " subdominant " read ^^ supertonic
First bar of exercise, for
-tS>-
r^-H <S' — read
)■ — ^2:^ — @ —
-&»-
^
Page 212. Second stave, bar 1, for
f
St
read
fcdE^-3
i
Page 217. Last bar but one, for C in the bass read CJ
Page 219. Under fourth bar from top, for J read ^
Page 220. Exercise 13, bar 6, for ^ read | ; and in bar 10, for | read |
Page 221. Exercise 14, bar 3, under last note insert 6
Page 225. In exercise 18, bar 2, for ^ read 6
-(S>
:?:2
- read
rJ rJ
Page 227. Second stave from bottom, bar 3, for
„ Last stave, bar 2, for ^ read ^
Page 248. Bar 4 should be the same as bar 3
Page 262. Line 10, for "or as the fourth below A'^ read "or as a fifth below A.'
TABLE OF INTERVALS.
THE SAME COUNTING DOWNWASDS.
fttJ I f J j ^^rr 1''' ^
I I
jzin
a
^^^^^
^i^- ^kHi
s^^s
j ff?rrbj:^
^^ Trjfr^
^
^ti=^
*3 p
This nomenclature will be strictly adhered to throughout this treatise.
A SHORT TREATISE
ON THE
PEINCIPLES OF HARMONY,
CHAPTER I.
P^'eliminary Remarks.
1. It is presumed that the student who uses this book is abeady
acquainted with what are commonly considered the first elements of music.
A very brief summary of these will, therefore, be all that is required
here, before we begin to treat of harmony strictly so called. And it
will be sufficient for our pui'pose to indicate most of these elementary
matters in a tabular form, as our chief object in so doing is merely to
set forth the system of nomenclature which will be adhered to in these
pages.
2. In the first place, it is important that the student should have
clear views as to musical notation. For this purpose he is referred to
Hullah's admirable little Treatise on the Stave, than which notliing can
be more definite and intelhgible.
B
THE PRINCIPLES OF HARMONY.
He will there learn, that our ordinary stave of five lines is derived
from what is called the great stave of eleven —
The three lines to which the letters G, C, and F are here appended,
represent fixed sounds of definite pitch : from these all others are counted.
Formerly, the whole eleven lines were occasionally used, and sometimes
six, seven, four, or three. But five lines have been found sufficient for
all purposes, and that number is now universally employed.
It is obvious that in selecting the five adjacent lines which we shall
use for our stave, we must be guided by a consideration of the notes we
have to write, and their place in the scale or range of musical sounds.
If we are writing for violins, flutes, or other high-toned instruments,
or for treble voices, or for the upper portion of keyed instruments, it
will be necessary to select the topmost fines of the great stave. These
five lines do not include either the fixed note C or the fixed note F, but
they do include the fixed note G, which is represented by the figure j[^
called the G clef — or treble clef — or violin clef, and which will be ^
placed so that the principal curl of it shall fall on the second line
counting upwards; thus q
If we are writing for bass instruments or voices, it will be necessary
to select the five lines at the bottom of the great stave. These will not
include either the fixed notes G or C, but only the fixed note F, which is
represented by the figure ^: or ^ , called the bass clef, and which will
THE PRINCIPLES OF HARMONY. 3
be placed so that the two dots shall fall above and below the fourth line
counting upwards ; thus ^ ~@'
Instruments of large range, such as the organ, pianoforte, or harp,
require the combination of these two staves ; thus
and on comparing this with the great stave, it will be seen that it
contains the whole of it except the middle line, which represents the
fixed note C.
If we are writing for instruments of medium range, such as the viola,
or the alto and tenor trombones, or for contralto or tenor voices, neither
of the above staves will be found convenient ; the one being too high
and the other too low in the scale : therefore we must select some other
set of five liaes, to suit the requirements of the case. Whichever set we
select will include always two and sometimes the three fixed notes ui
the great stave ; and as we have already appropriated the clefs repre-
senting G and F to the highest and lowest staves respectively, it will
conduce to clearness and distinction if we use the symbol of the fixed
note C, which is thus formed, u , or H , to distinguish our medium
staves.
Of such staves only three are now in use.
The most acute is called the soprano, and is used in instrumental
scores of oratorios, and in old Church music, for the soprano or treble
voices. It consists of the 6th, 7th, 8th, 9th, and 10th lines of the great
stave, countiag upwards, and is expressed thus „ , — ^-G.
B 2
THE PRINCIPLES OF HARMONY.
The next is the truly medium stave, and consists of the 4th, 5th, 6th,
7th, and 8th lines of the great stave, counting upwards. It is used
for the viola, or alto violin, for the alto trombone, and for the alto
voice parts in scores. It is expressed thus C-
F
The last is called the tenor clef (or, more properly, the tenor stave),
and is used for the tenor voices in scores, as well as for the tenor
trombone, and sometimes also for the higher notes of the violoncello
and bassoon. It consists of the 3rd, 4th, 5th, 6th, and 7th lines of the
great stave, counting upwards, and is expressed thus
For further details the student is referred to Hullah's book above-
mentioned, or to Marx's " Theory and Practice of Musical Composition,"
(translated by Saroni. New York, 1852).
3. It will be sufficient to give a list of the different kinds of notes
and rests, an ordinary acquaintance with them being presupposed.
A Breve |Io|| is equal to
two Semibreves
o
G>
or to
four Minims
1
1
p
1
f P
1 •
or to
eight Crotchets p
1
p
1
p f
or to
0":^. r^ cc It ti cr G' tm- -'»
sixty-four Demisemiquavers, and so forth.
THE PRINCIPLES OF HARMONY. 5
The corresponding 7'ests (to represent intervals of silence) are the
breve
, the semibreve ^^^^
, the minim ---- , the crotchet [*, the
quaver *|, the semiquaver sj, and the demisemiquaver ^.
A dot after a note or rest makes it half as long again : thus o*, or
a dotted semibreve, is equivalent to three minims, or six crotchets, or
twelve quavers : [O*, or a dotted minim, is equivalent to tlnree crotchets,
or six quavers, or twelve semiquavers.
If a second dot is added, it will lengthen the note or rest half as much
again as the first dot: thus i***, or a doubly dotted crotchet, is equi-
valent to seven semiquavers, or foiu-teen demisemiquavers ; for it consists
of the value of a crotchet, a quaver, and a semiquaver, i. e. two quavers,
one quaver, and a semiquaver, i. e. four semiquavers, two semiquavers,
and one semiquaver, i. e. 4+2 + 1 = 7 semiquavers.
When a note or a rest is to be lengthened indefinitely, this figure ^^,
called a ijcmse, is placed \uider or over it.
For further details and exercises on this branch of musical knowledge
we must refer to more elementary treatises.
4. Supposing the student to be famihar with the use of bars and
double bars, with the theory of accent, and the art of beating time, it
wiU be sufficient for our purpose to give a time-table, to shew the system
and arrangement adopted in this book.
THE PRINCIPLES OF HARMONY.
a. Common or Duple Time — divided into
i. Simple, and ii. Compound.
in a bar.
(g or Double Common, 4 rzt
(1^ or Alia Breve, 2 |C5 in a bar.
(3 01* - or Common, 4 |# in a bar.
4 I
or 2 # in a bar.
r
— or 4 |0» in a bar.
4 I
]2
"8"
6
4
_ or 2 (#• in a bar.
8 I
or 4 !#• in a bar.
I
or 2 |0* in a bar.
jS. Triple Time — divided into
i. Simple,
- or 3 o or 1 o* in a bar.
2 I
3
- or 3 |# or 1 fO* in a bar.
4 I I
3
- or 3 |# or 1 !#• in a bar.
8 I I
16
or 3 |# or 1 !#• in a bar
:
and
ii.
Compound.
9
4
or
3p
1
• in a bar.
9
8
or
3 p.
1
in a bar.
9
16
or
1/
in a bar.
5. It may be taken for granted tliat the student knows tlie meanings
of the figures J|;, b, Q, x, bb, and the use of them, both as accidentals
and in the signature. If not, we must again refer Mm back, to the
elementary treatises, contenting ourselves with a tabular list of the keys,
major and minor, with their several signatures.
THE PRINCIPLES OF HARMONY.
Major keys with shai-ps.
7 sharps CJf
6 sharps FJf
5 sharps B|^
4 sharps Ep
3 sharps Ap
2 sharps Dp
] sharp Gp
Minor keys with sharps.
7 sharps Aj(
6 sharps
• n
5 sharps G|j!
4 sharps C Jf
3 sharps Fjf
2 sharps Bp
1 sharp Ep
Major natural key . . Cp
Minor natural key
AP
-d
i\Iajor keys with flats.
1 flat Fp
2 flats Bb
3 flats Eb
4 flats Ab
5 flats Db
6 flats Gb
7 flats
Cb
fm^
Minor keys with flats.
1 flat Dp
2 flats Gp
3 flats cp
4 flats Fp
5 flats Bb
6 flats Eb
7 flats Ab
THE PRINCIPLES OF HARMONY.
Each of these major keys is supposed to be nearly related to that
minor key which has the same signature, and stands on a Hne with it
in the above table : hence the terms relative major and relative minor.
6. It will be necessary to give a table of intervals, although it is
supposed that the student has already mastered them before using this
book, because some of them have been variously named by different
writers, and it is essential, as a preliminary, to fix our own nomenclature,
which will be strictly adhered to in the following pages.
Intervals are divided into consonant and dissonant intervals, (or, as
they are sometimes called, concords and discords).
Consonant intervals are of two kinds, perfect and imperfect.
Imperfect consonant intervals are subdivided again into major and
minor.
Perfect consonances cannot be so subdivided.
Dissonant intervals also, like the imperfect consonances, are either
major or minor.
All intervals are susceptible likewise of certain alterations, called
augmentation and diminution; excepting only that major intervals can-
not be diminished, and that minor intervals cannot be augmented.
Perfect consonances alone can both be diminished and augmented.
Intervals
I
Consonant
Dissonant
Perfect
Imperfect
major mmor
h
major minor
§• B
c
p
g
&-
T
c
p
p
p
'ft-'
p.
P^
B
p
p
B
c-t-
eri-
g
fl>
rt>
B
»
2
p
1-S
i-j
a
n>
rt>
tjD
O'
CI-
&-
ft-
ft-
Cr"
n>
fD
Q^
ft.
ft-
THE PRINCIPLES OF HARMONY. 9
The smallest unaltered interval is the minor second, or semitone : and
it is convenient to compare and measure all larger intervals by the
number of semitones they contain, or to which they are equivalent.
The perfect consonant intervals, according to the usual computation,
are the octave and the fifth ; to which may be added the fourth also,
though only under certain restrictions, to be considered hereafter.
The imperfect consonant intervals are the major and minor third, and
the major and minor sixth.
The dissonant intervals are the second and the seventh, each of which
may also be major or minor.
If the interval of a semitone be subtracted from any perfect or minor
interval, by means of a sharp or flat, to alter the pitch of the lower or
upper notes respectively, such interval is said to be diminished.
If by the same means a perfect or major interval be enlarged to the
same extent, such interval is said to be augmented
All augmented or diminished intervals so produced are called chro-
matic dissonances, except the two which occur in the diatonic major
scale, i. e. the augmented fourth or tritone between the 4th and 7th
degree, and the diminished fifth between the 7th degree and the 4th
in the next octave. Of all which more will be said hereafter.
If an octave be added to any interval, its original character (as re-
gards divisibility into major, minor, augmented, or diminished) remains
the same ; only that in the case of the 2nd, 3rd, 4th, 5th, and 6th, they
are in that case sometimes designated as the 9th, 10th, 11th, 12th, 13th,
as will be more fully explained in a future section.
We here annex a list of all the intervals, illustrated in the key of
C natural.
Note. Some writers call augmented intervals " extreme." Others call augmented perfect
intervals " pluperfect." Others call diminished fifths " false fifths." Others call augmented
fourths " tritones." Others call minor sevenths and seconds " flat." Others call major
10
THE PRINCIPLES OF HARMONY.
sevenths and seconds " sharp." Others call diminished fourths " flattened." Others call
augmented fifths "sharp."
But every one of these terms is unsystematic and unphilosophical, and some of them
are absolutely incorrect.
7. There are yet remaining a few technical terms which ought to be
explained in this preliminary chapter.
(a) A cliord is the simultaneous sounding of several different notes,
selected according to certain fixed principles and rules.
(j8) A triad is a chord of three notes consisting of a bass with its
third and fifth, e.g.
Triads are of three kinds : —
i. Major; ii. Minor; iii. Imjperfect, or diminished.
A major triad consists of a major third and perfect fifth.
A minor triad consists of a minor third and perfect fifth.
An imperfect triad consists of a minor third and diminished fifth.
(7) If the octave of the bass is added above a major or minor triad,
it constitutes what is called a common cliord.
^OTE. The imperfect triad being (as its name imports) not formed after the same perfect
model as the major and minor triads, cannot be converted into a common chord by the
addition of the octave of its lowest note.
{Examples) fL ^
* A major triad.
§ Imperfect triad.
-^s-
m
-^
'^^
t
X
§
1[
t A minor triad.
If Common chord major.
X Imperfect triad.
II Common chord minor.
THE PRINCIPLES OF HARMONY.
11
Besides these triads we occasionally meet with what is called an
augmented triad, which consists of a major third and augmented fifth,
or of two major thirds superposed ; thus
M
^
8. It may be as well here to explam one or two terms which belong
rather to counterpoint than to harmony, but to wliich reference must
necessarily be made in treating of the resolution of discords, and in
some other cases also.
i. Motion is of three kinds : — similar, obhque, and contrary.
a. Similar motion is said to exist between any two or more parts,
or voices, which ascend or descend simultaneously, but not in
unison ; e. g.
^^g:
&C.
j8. Oblique motion is when one part remains without moving while
another ascends or descends ; e. g.
ici
xz
7. Contrary motion is when two parts, or voices, move in opposite
directions ; e. g.
^^
^^
:jrj=^
Avr r r
ii. Consecutive fifths or octaves are produced when two parts move so
as to produce the same interval (of a fifth or an octave, as the
c 2
12
THE PRINCIPLES OF HARMONY.
case may be) between tbem in successive chords. There are several
kinds of such consecutive fifths and octaves.
a. By similar motion ; e. g.
^
-(S^-
:,^=^
a
or
/8. By contrary motion ; e. g.
^^
-(S>— '
-tS>-
:^d
or
5^^
r
-p^ I
"?:^
7. Hidden fifths or octaves, which occur when an unaccented note
intervenes without any change of fundamental harmony ; e. g.
--i^i
By the laws of strict counterpoint, every consecutive fifth or octave,
of any of the above kinds, is altogether forbidden.
Consecutive major thirds have been hkewise forbidden by some ancient
authorities, though modem composers use them very freely. Still it must
be admitted that several unaccompanied major thirds in succession do
produce a very disagreeable effect. Any one playing the following notes
on a keyed instrument will perceive their badness : —
^ ^ ^ C^ —
Consecutive fourths are also strictly forbidden, unless accompanied
THE PRINCIPLES OF HARMONY. 13
by thirds below them, when they become perfectly correct and pleasant
to the ear ; thus
^^^
:^
-(S»-
is bad, while
Wff
is good.
It is unnecessary to go any further now into this matter, as the
harmonic principle involved will be duly explained in its proper place.
CHAPTER 11.
Fii'st Principles of Harmony.
1. The origin of harmony must be sought in natural phsenomena. This involves an
acquaintance with the science of acoustics, and is therefore more or less foreign to the
peculiar subject of this treatise. Still, it will be necessary to obsei-ve here that the primary
chord given us by nature is the following : — root, octave, twelfth, fifteenth, seventeenth, and
nineteenth.
As an example of such a natural and primary chord, we wiU assume C as our root, or
generator, and give the notes which result from it as natural harmonics : —
^
3
q:
root
octave 12th
15th
-&■
17th 19th
The interval of an octave is so perfect a concord, that it may be regarded as almost
equivalent to an unison. It is therefore possible for us to omit for the present the root
and its octave ■^^^-
2!2
"3r
, as they are sufficiently represented by its double octave
(or fifteenth),
^
Similarly, we may for the present omit the twelfth of the root,
sufficiently represented by its own octave
_ , as it is
, which is the nineteenth, or octave-
twelfth, or double-octave-fifth, of the original generator.
THE PRINCIPLES OF HARMONY.
15
We get then, as our residuum, the major triad
harmony of nature.
which is the primary
2. If we were to continue the natural series of harmonic sounds as they occur in nature,
we should arrive at some which would lead us out of the key in which we began, and which
in fact do not belong to that key at all, but to others related to it in a peculiar way, which
will be explained when we come to speak of modulation.
For instance, starting from the generator C, as before, we should find the following
sequence of notes succeeding those given above : —
r^^ ^ P -
^^"p- ^ .. ^^
^2:
W^-
^-
^^=^2=^
^
-j:2.
i?
±zz±
O X o
o
Now of these there are four, marked x , which not only are foreign to the key of C,
but are out of tune in any key.
Four others, marked © , are merely repetitions of the intervals of the major triad in an
upper octave. They may therefore be considered as identical with the primary chord we
have already obtained.
Two only remain, D and B, which we can make use of for our present purpose.
On carefully examining these two notes, and combining them with the note G which
we already possess, we find that they constitute together a major triad.
To shew this more clearly, let us take the highest G in the above figure.
the B next above it,
■^
^2_
written beyond the double bar
^i
: and let us take the higher octave of the D, which is
and this gives us the major triad
«y
16 THE PRINCIPLES OF HARMONY.
of which G may be regarded as the generator, just as C is the generator of the triad
3. Seeing that octaves may be neglected in these considerations, it will be convenient to
write this new major triad an octave lower, thus
; and setting out the whole
harmonic series of G, of which it forms the principal part, we produce this scheme —
cz
i
:&
n 'P "-
3^#^
No. i.
VI. Vll. VIU. IX.
XU. Xlll. XIV. XV. XVI.
Here it will be observed, that every note belongs to the key of C till we come to the
double bar : and although the note F, marked x , is not perfectly in tune, yet we can sub-
stitute a really true F without at all materially disturbing our new series of sounds.
Omitting, then, numbers i, ii, iii, viii, x, (and of course all beyond the double bar,) as we
did in the case of the harmonic series of the generator C, we get as our residuum the chord
, which is called the " dominant chord of nature," being based on the fifth of the
key; which fifth is called the, dominant, because it exercises the most powerful influence on
the harmony.
Reducing our two chords to the same octave, and writing them in juxtaposition,
, our ear at once detects a close relationship between them, and on
hearing the former of the two, it immediately feels a desire to hear
the latter also, and feels relieved and satisfied when it has been
sounded.
4. From the preceding section it will have been seen that while the dominant harmony
suggests the idea of chcmge or motion, and tends to the primary major triad, that primary
major triad suggests no such idea, but rather induces rest, and in fact determines the key
in which the music is written ; gives, as it were, the chai'acteristic tone to the music : hence
THE PRINCIPLES OF HARMONY.
17
the original generator is called the Tonic, and its triad or common chord is called Tonic
harmony.
If we were to be confined to tonic harmony alone, we should be like a i)risoner within
the four walls of a gaol — we could never get out of one groove. Our fatigue would become
unbearable.
d:
^
S^
s=
i-y=j-^
^h=^
^
'n^^t^rr
^
^
rrf^==rrf
=^
5S=
If this passage be played over several times, it will give a fair idea of the irksomeness of
purely tonic harmony.
On the other hand, if we were confined to dominant harmony, not only should we have
a continually unsatisfied craving after a Tonic chord, but we should be even more wearied
than in the above case. An example will sujfficiently prove this : —
^3
-(S>-
^ij=d:
-<s>-
-<s>-^
\r^f-^^
2=k
-m
JPL
-^
rr
'^'
d^r^d-^^^
d:
S
-fS*-
-Gh
-&
_»__
1^2
S^
'^^•
.«•
Play over this exclusively dominant piece, and it will remind you of a traveller wandering
homeless from place to place, seeking a welcome, and finding none.
18
THE PRINCIPLES OF HARMONY.
But an alternation of tonic and dominant chords will always excite and satisfy the ear,'
just as alternate activity and rest are salutary and pleasant to the body.
5, It may, then, be taken as proved, that the key-note or tonic, with
its third and fifth, satisfies the ear, and leads it to desire no further
change. For this reason, the close or " perfect cadence " of a piece of
music must always terminate in a tonic chord.
It may also be taken as proved, that the fifth of the tonic, which
is called the dominant, with its third, fifth, minor seventh, and major
ninth, does not satisfy the ear, but leads it to desire a change to the
tonic harmony. For this reason the dominant harmony never can
end a piece, but should precede the tonic.
And it is from this characteristic feature of the dominant harmony
that the whole system of the resolution of fundamental discords is
derived, of which we shall soon have to speak.
This is a most important first principle, and should be thoroughly
understood and mastered by the pupil before going any further. For
which object it is desirable that he should strike the following dominant
chords, pausing after each, and realizing the unsatisfactory impression
they leave on the ear : —
22:
~JZZ.
M^
b^
-<s
-ci-^
\:^
HS>-
\i^
"<>>
\y
-Gh
-^
S>-
v^
\U
\:^
THE PRINCIPLES OF HARMONY.
19
To shew how the ear may be satisfied by a tonic chord succeeding
each of these dominant chords, let the student play the following : —
jQ_
s
^.
^
i±
Z21
-Gh
Si^
!2 a
T2--
^
:22:
^ --^
C2:
T^
-"Gh
X^
^
-<st-
x^
is>-
s>-
"^?"
M=n=3B
=g:
iq:
^^
g
;^^
-!?(S^
-^^^
"^7"
-iZf:^
12^:2 -_ !2^
i2^
J^L
e
172^
-•S)
-ts^
-?»^-
t54Si'-
^Is*-
-9|S>-
-c^-
^s^
and he will perceive that in each case the tonic chord, which here suc-
ceeds the dominant, at once satisfies the ear, and produces the sensation
of rest.
6. It must next be observed, that it is not every note in the domi-
nant harmony which necessarily possesses the peculiarity of leading thus
to the tonic. For if we simply take the major triad of the dominant
root, there is notliing in it to shew that it is not a tonic triad : for
instance.
g=
^
may be simply the tonic harmony of the key of G.
But the moment we add the next note in order which belongs to
the dominant harmony, which in this case would be F, thus ^
^
we preclude all impressions of the key of G, or of G as a tonic root, for
D 2
20 THE PEINCIPLES OF HARMONY.
tlie F here is natural, and therefore out of the scale of G, which of course
requires the F to be sharpened.
Moreover, the intei'val of a minor seventh, from G to F, and of a
diminished fifth, from B to F, at once destroy all the rest and jpei^ma-
nence of the chord.
This minor seventh, then, is clearly the characteristic note which
invests the chord with its distinctively dominant character. Hence this
chord is generally called the " chord of the dominant seventh."
Note. This chord is often called the " added seventh," because it is composed of a seventh
added to a major triad. It is also sometimes called the " fundamental seventh," to distin-
guish it from other chords of the seventh. Likewise some writers call it the " minor seventh"
chord. None of these designations are incorrect, but in this work it will be invariably styled
the " chord of the dominant seventh."
7. Tliis chord, as we have just seen, contains discords, and these are
made to satisfy the ear by means of the chord of the tonic, which must
follow immediately. When the discords have thus been rendered agree-
able to the ear, they are said to be resolved ; and this resolution of dis-
cords forms the most important element of the science of harmony.
The rule for resolving the chord of the dominant seventh is a very-
simple one : " Each of the discordant notes leads to and is resolved into
that note in the succeeding tonic chord which is nearest to it in pitch,
whether that note be above or below it on the scale."
Thus, in the chord i ^~"8 — , which is resolved into 7/K ^ > the F
goes to E, and the B to C, the bass note G of course goes to the bass
note C (root to root), while the D, being equidistant from C and E, may
go to either.
It appears, then, that the only notes (beside the root) which have
a compulsory resolution, are the third, B, and the seventh, F. Of these,
THE PRINCIPLES OF HARMONY.
21
the " third of the dominant" always goes to the octave of the tonic root,
and is therefore called the " leading note/' while the seventh always falls
to the third of the tonic.
This is a most important rule, and leads to many essential conse-
quences. It should therefore be thoroughly learnt and appreciated at
this early stage of the student's progress. We accordingly give a few
examples in different keys, by way of illustration : —
8. It would seem that the tendency to a tonic resolution, which, as
we have seen, is the characteristic feature of the chord of the dominant
seventh, is attributable mainly to the discordant interval which exists
between the third and seventh, and which is either a diminished fifth, (if
the third be below the seventh,) or an augmented fourth, (if the seventh
be below the third,) as may be seen by a careful examination of the
preceding examples, where the intervals are purposely placed in various
positions.
Note. Any interval within an octave is susceptible of what is called " inversion." Which
may be thus explained : —
22 THE PRINCIPLES OF HARMONY.
If the lower of the tvvo notes forming any interval be changed into its upper octave, e. g.
I^ changed to /L ~r^ - J , the interval is said to be inverted, or, in other words.
the new interval thus formed is an inversion of the former : thus if the interval
^
be given, then its inversion will be
m
:^
The same result will be obtained if the upper note be taken down an octave : thus —
P
It will be seen by the subjoined table that the inversions of 'perfect intervals are perfect;
of major, ai*e minor; of minor, are major; of augmented, are diminished; of diminished,
are augmented.
The student is recommended to copy out this list, and name all the upper and lower
intervals according to the table given in Chap. I. sect. 6.
Indeed, if a diminished fifth or an augmented fourth be played, alone
and unaccompanied, the same craving after resolution will ensue. Thus, if
2^^y be struck, the ear craves the regular resolution f^ ^
-&
J
and if the inversion
the regular resolution
THE PRINCIPLES OF HARMONY. 23
be played, a similar craving is felt for
This effect, however, is enhanced by the discord which exists between
the minor seventh and the root, and more strongly still between the
minor seventh and the octave, where the interval of a major second is
produced. Thus i fr)—^^ — g— or ^f
IQ
^-
-is»-
9. The dominant harmony contains yet another note besides those
we have been considering, as will be seen by a reference to Section 3
of this Chapter. It is called the major ninth, and it is occasionally
added to the chord of the dominant seventh, to strengthen, vary, and
enhance its effect, although it is not an essential element of that chord.
When thus enhanced, the complete chord is appropriately called the
" chord of the added ninth," or the " chord of the fundamental ninth," to
distinguish it from certain other chords of the ninth which will be ex-
plained hereafter.
The addition of this new dominant interval to the chord of the
dominant seventh does not alter the resolution of the two essential
notes of that chord (i. e. the third and the seventh). The third rises to
the octave of the tonic root, and the seventh falls to the third of the
tonic, just as they would were the ninth absent.
The ninth itself, being as it were a coadjutor and strengthener of
the seventh, pursues a similar course to that which is peculiar to that
24
THE PRINCIPLES OF HARMONY.
interval. While the seventh falls to the third of the tonic, the ninth
falls to its fifth. Thus
-&
:^3:
Here we see by an example the necessary resolution of the dominant
root, the third, the seventh, the ninth ; but how about the fifth, D %
If the ninth were not there, this note might either rise to E, or fall to 0,
being (nearly) equidistant from both. But the addition of the ninth. A,
renders it impossible for the D to proceed downwards to C, without
violating that rule of counterpoint which forbids consecutive fifths (see
Chap. I. sect. 8. no. ii.), for their joint progress would be
^
:P2:
Therefore the fifth of the dominant root (here D) is forced in this
case to adopt the alternative of rising to the third of the tonic (here E).
To shew this more clearly, we will arrange the notes of the chord
in the following position : —
m
:ezz2:
122115"
t:?~i
-1— (S>-
m
Note. Logier does not appear to recognize any dominant interval of harmony beyond
the dominant minor seventh. He regards the major and minor dominant ninths as no
more than substituted notes.
THE PRINCIPLES OF HARMONY.
In this Fetis agrees with him in his valuable Treatise on Harmony.
But this seems to be a very unphilosophical view of the case. For nature supplies us
with both these intervals, (the major and minor ninth,) one perfectly, and the other almost
perfectly, in tune, as will be shewn hereafter (Chaps. IV and VI), whereas the natural
dominant minor seventh is by no means so perfect.
Then again, it is not philosophical to account for an interval by the hypothesis of a
substituted note, and yet to allow of the coexistence and simultaneous use of that note and
of the one for which it is supposed to be substituted : e. g.
8 > Q
^
-3— Sit
IBZZZS
where the A according to these theorists would be substituted for G, and yet the G is
allowed to be sounded in another octave.
Surely it is more consistent with analogy, and more agreeable to the phjenomena of nature,
to regard the ninth here as an added harmonic, derived from the chord aflforded us by nature,
XT-
as will be more fully explained hei-eafter (vide Chaps. Ill, IV, VI).
E
CHAPTER 111.
1. From what has been abeady said, it will have appeared that the
intervals of a chord may be inverted amongst themselves, and their
order of acuteness interchanged, without thereby altermg either their
relations to the root or to each other.
This may be illustrated by taking the triad or common chord of C,
and the chords of the dominant seventh, and of the added ninth, and
arranging them variously.
^
^
.CL
e
■^
^
"7^r~ ^-1~r^ H— ^
^
:^
?2:
-^
iS»-
-^
:g=U
:^--gr
^^^
^^y
--&
&c., &c.
@:
?2:
P^zif^
T^
rJ rj
y^-rj rJ
^ \^j ^~r5 g2
^
rj f j zp r^ r j^
:B
:^i
^
S
^-
-sf
-^ — f^ — s
&
-cP=g^
-^hT-
S=^
:&
-^2.
-f=^
-^^
-|Q-
&C.J &c.
P=r
S» lis — & — ts-
3^
-&-
THE PRINCIPLES OF HARMONY. 27
But we may now go a step further, and omit the root altogether,
substituting for it occasionally its octave in an upper part.
The effect of this will obviously be that one of the other notes of
the chord will be at the bottom of the harmony.
But although the root be in such a case unheard, yet it must be
always imagined.
All the other notes of the chord are equally derived from it, and
dependent on it. And the chords thus modified are called inverted
chords — or inversions — and their constituent intervals will require the
same treatment, for the most part, as though the root were sounded.
The effect on the ear, however, will not be the same, especially in the
inversions of the tonic harmony, for the sensation of rest and fixity will
be absent : and therefore " every piece of music must end with an un-
inverted tonic chord."
2. The tonic triad consisting of three notes, and either of the upper
ones being capable of being taken as the lowest note in the harmony,
by inversion, as above described, it is clear that only two such inversions
are possible — first, when the third is in the bass, and secondly, when the
fifth is in the bass.
When the third is in the bass, it is called the " first inversion of the
common chord ;" and as the octave of the root then forms the interval
of a sixth with the third which is placed in the bass, this chord is also
called the " chord of the sixth." For instance, deriving it from C, / ■i s Cj
E 2
28
THE PRINCIPLES OF HARMONY.
it will appear iii the following forms, according to the collocation of the
upper notes of which it is composed :
-f^
-Q-
-f^
C^
-^
t^ r^
:^
^2:
q:
e
:?2:
-*s-
-<s
=^
-Gh
s»-
kS^-
J-
&C.J &c.
p rj r^ p r^ r^-z— ^
^^
TZL
^2:
^2:
where it may be observed that either the octave of the root or its fifth,
(i. e. C, or G,) may be doubled, appearing in two octaves. Likewise,
the G may be left out, but C must not, as it is the characteristic note
of the chord.
When the fifth is in the bass, it is called " the second inversion of
the common chord," and, as the octave of the root then forms the interval
of a fourth with the fifth which is placed in the bass, and as, moreover,
the third of the root forms, with tliis same bass note, the interval of a
sixth, — this chord is also called the " chord of the fourth and sixth,"
or, more concisely, the " chord of the six-four." For example, taking C
as oiu- root as before,
foUows :
IQ
--gr
, this chord will appear variously as
i
XT
-O-
P^ ffi
2^
e
2^
122:
*i
23:
S
±=a
p::2_
'rj rJ^
^2:
^
^
s^
-Gh-
--g:
■4S>-
-»
-o-
-iS
r ^ r^ r^
^
2~r> q:
'f^ r^ r:t r:^
r^ r n — r^ ^^ ^'^
THE PRINCIPLES OF HARMONY.
29
This second inversion is less satisfactory than the first, and should be
more sparingly employed.
3. And perhaps this will be the most fitting place to introduce the
subject of what is known as " thorough-bass-figming."
This is a kind of musical short-hand, of no great value, but occa-
sionally convenient, by which figures, placed under or over the bass notes
of a piece of music, are used to mdicate, vaguely but concisely, the kind
of harmony to be played with the given bass.
A bass note with no figures attached to it, indicates a simple imaltered
triad or common chord. If, however, one of the notes of the harmony
has to be modified by an accidental sharp or flat, such modification is
indicated by an accidental instead of a figure, (in the case of the third,)
or by an accidental added to a figure, (in the case of the fifth). Thus
the chords
^
-^ ^^'-
"-^^
:^
^&r
-<^-
:p2:
^±
^-
^-^
may be thus expressed
z:±
-G^
3
P
«5
-S* — "
The chord of the dominant seventh is always expressed by the
figure 7, either alone or with any accidental natural or flat which it
may require. If the fifth or third in the chord require it, accidentals
30
THE PRINCIPLES OF HARMONY.
may be used for them as in the case of the common chord. Thus the
chords
-^
e=»3
fS>-
&-
-^
^^m
zd:
-&
j::z.
would be figured as follows
m
-s^
T^
I
C7
3
-<Si
~JZZ.
b7
i
'i
The first inversion of the common chord is figured with a 6. The
second inversion of the common chord with a ^. Thus
i
3
IS p—^p r^^
Pf^
2±
±mr^
-t^-
9^ ^&<Sl-
m
T^
-^ ft ^
:c2:
S
e
:^i=:Jt^
would be indicated as follows
6 J "-« "■«
^f
°p^
bt
^ p ft^
- ^ gp -
1^
:P2:
Ifp^
:^
where it should be observed that it is usual to indicate a sharp sixth,
fourth, or second by a Hne drawn through the figure : thus 6, 4), %,
THE PRINCIPLES OF HARMONY. 31
These s)niibols are of course perfectly arbitrary, and are merely used
for convenience and brevity, and to save an extra stave in certain
cases.
4. The chord of the dominant seventh consists, as we have seen, of
four notes. It is consequently susceptible of three inversions.
The first inversion has the third in the bass. The octave of the
root forms with this bass note the interval of a sixth, the dominant
seventh forms the interval of a diminished fifth, and the fifth of the
root forms the interval of a third. This chord is called the " chord of
the fifth and sixth," or, more concisely, the " chord of the six-five." It
is most correct, however, to designate it as the " first inversion of the
dominant seventh," to avoid confusion with other chords of similar
appearance but different character. It is figured ^, with any accidentals
which may be necessary.
The second inversion has the fifth in the bass. The octave of the
root forms the interval of a fourth, — the seventh, that of a third, — and
the third, that of a sixth, — with the bass note. It is most correctly
designated as the " second inversion of the dominant seventh ;" but it is
also frequently called the " chord of the six-four-three," from the intervals
of which it consists. It is figured 4, with whatever accidentals may be
requisite.
The third inversion has the seventh in the bass. The octave of the
root forms with this bass the interval of a second ; the fifth forms that
of a sixth ; and the third that of an augmented fourth. It is called
properly the " third inversion of the dominant seventh ;" but often also
the " chord of the six-four-two," or, the " chord of the second and fourth."
It is figured 4, with the requisite accidentals.
2
32
THE PRINCIPLES OF HARMONY.
We will now give examples of all these inversions, each with its
proper thorough-bass figuring : —
P ^b 3 2 2 2
Very often it wiU be a sufficient indication to figure the second inver-
sion ^, omitting the sixth; and to figure the third inversion |, or even
2 only. This may be done whenever no ambiguity can thence arise.
5. The inversion of a dominant chord does not in anywise change
the natural characteristics of the notes of which it is composed. The
same rules which held good, therefore, in the resolution of the funda-
mental chord, in its original condition, will equally hold good in the
case of its inversions.
Accordingly, in the first inversion, which has the leading note in
the bass, the bass note has a compulsory resolution upwards to the tonic
root, while the diminished fifth, which is the minor seventh of the domi-
nant root, is compelled to fall to the third of the tonic. For example —
^
^fP
-<s-
s>-
^-^ = 3— r
S
rJ jfr^ ^— ^ g^-
1^
The second inversion has the fifth in the bass, and consequently this
THE PRINCIPLES OF HARMONY.
33
note may either ascend to the thii'd of the tonic or descend to the
tonic itself, according to circumstances, the fifth of the dominant
having no comptdsory resohition. The third in this chord, being the
seventh of the root, must of course descend to the third of the tonic ;
while the sixth, which is the leading note, is compelled to rise to the
octave of the tonic root.
The tliird inversion has the dominant seventh itself in the bass,
which is therefore compelled to fall to the third of the tonic. But the
third of the tonic in the bass constitutes the first inversion of the common
chord : therefore the third inversion of the chord of the dominant seventh
must always be followed by the first inversion of the tonic common chord.
The augmented fourth in this chord is the leading note, and goes of
course to the octave of the tonic root.
In all the inversions, the octave of the dominant root remains without
motion, being converted into the fifth of the tonic. The next examples
will shew all this more clearly.
m
'-^^
-&-
2i
Z2:
f-
-<st
^w^-
t^
Z2:
^
-(S>-
iq:
XF'rj-^ g:
Z2:
"Cr
^
^
'IZ2L
•^~
r
T^
^G)-
2i:
ici:
-G^
^in^i
b6
F
34
THE PRINCIPLES OF HARMONY.
We will now give a longer example, including all the inversions of
the common chord and the chord of the dominant seventh, with their
various resolutions and coiTect thorough-bass figuring.
H
_c^
:d
JOL.
:g=^
fe
-^-
jCk-
-G>-
iS"-
Z2:
HS"-
-s>-
g
-«s>-
:e
=P^
1!S>-
q:
-Q_
-<s>-
£i
^z
z::i
lS>-
gE^
S^
-^
:^
22:
-«s^
Z2:
-.C2-
-(S*-
:s
=P2:
^=^^^=0
-«s^
|S>-
?::^
1^-
:z:2
-s>-
1^2:
-(S»-
z±
"C5"
-«S>-
i
^
-^
-J^2L
«=^
-e>-
-G^
S^3
-s^
-<S^
-^
=3=
-<s>-
_Q „
^
ZZ
-(S-
if^
3^
-!S»-
-^s — <s-
:?:?
-<s>-
-&-
The student is recommended to copy this out, and to analyse each
chord carefully, putting the letter D over those chords that are dominant,
and T over those that are tonic, and drawing Hues from all the discordant
notes to those which resolve them, according to the foregoing rules.
THE PRINCIPLES OF HARMONY. 35
6. In the chord of the dominant seventh, in its original position, neither
the root nor the seventh can be omitted. The leading note may be, on
an emergency, but it is always better to avoid this. The fifth may be
omitted whenever it is convenient to do so, as it is not an essential note
in the chord.
In the first inversion of the dominant seventh the leading note is the
bass, and of course cannot be omitted. The seventh of the root, which
here becomes a diminished fifth, is the characteristic note, and therefore
cannot be spared. The octave of the root, which here is a sixth, may
occasionally be omitted ; but if this is done, the chord is converted into
a dimmished triad, and becomes weak and ambiguous, as we shall see
hereafter ; still, when necessary, it may be omitted. The fifth of the root,
which here is a third, may be omitted at pleasure, when convenient.
In the second inversion, the fifth of the root becomes the bass, and
cannot be omitted ; the fundamental seventh, which is here the third,
cannot be omitted, for the same reasons as in the former cases. The
leading note, which is the sixth in this inversion, ought not to be left
out, except when absolutely necessary. But the octave of the root not
only may be omitted, but it is generally better that it should be, because
of the harsh effect of the interval of a perfect fourth which it makes
with the bass.
In the third m version, the fundamental seventh being in the bass is
essential, and must remain. The octave of the root should hardly ever
be omitted. The leading note, which forms the augmented fourth, cannot
well be dispensed with ; but the fifth of the root, which is here a sixth,
may be omitted, whenever it is convenient.
These rules about omissions will be found specially useful when the
student begins to write music in less than four-part harmony.
F 2
36 THE PRINCIPLES OF HARMONY.
Examples of imperfect chords, with the various omissions allowed-
-<s>-
zz
-s>-
-<s>-
-^-
lor.
22:
-s>-
-<s>-
-<s»-
s
-s>-
321
-<S>-
-<s>-
S:
-<s^
i^z:
hS>-
i^z:
-(S>-
22:
-s>-
1221
05
-7 1
rzi
^--i
n
Xl .-^ <"-^
^-^ r^
>5
C^ 1
1^^ '^-^
r^ 1
^"i ^ r^
1
c^
II
y>\) ^-^ ^-^
'^-^
1
r^
rj
^-. w
J ^ ^
/'i^• • — -
i'-:) '
rj
r^
ift)-
rj
rj
r^ 1
V-^
1
r^
r^
1
A
7. It is allowable, and indeed often necessary, to donhle a part, i. e.
to let the same note be heard at once in two different octaves.
Now it is obvious that if we thus double a note whose progression
is compulsoiy, we must either break the compelling rule or fall into the
great contrapuntal error of consecutive octaves. For instance, if we double
the leading note "^
^<S)-
and resolve the chord regularly, both
the notes E will go to F ; thus
E
producmg the consecutive octaves y cL-b^
221
^
Q r^
therefore the first general rule is, that " the leading note and the domi-
nant seventh must not be doubled,"
THE PRINCIPLES OF HARMONY.
37
The root and the fifth, however, may be doubled whenever it is con-
venient. For the dominant root may either go up a fourth or down a fifth
to the tonic root. And its octave has, besides this, the power of remaining
without motion. Moreover, either the root and its octave may also,
under certain circumstances, go to the third of the tonic, though this
is necessarily of very rare occiu-rence, for contrapuntal reasons. And
the fifth of the dominant root may either ascend or descend one degree
in its resolution ; when the dominant chord is inverted, moreover, it may
go up or down by a skip to the fifth of the tonic root.
In the second inversion of the chord of the dominant seventh, a licence
is allowed ; namely, that of doubling the fundamental seventh itself, when
the octave of the root is omitted ; and in that case it is considered suf-
ficient for one part, which has this interval, to resolve it regularly by
descending to the third of the tonic : the other, which is the double of
it, may then rise to the fifth of the tonic ; thus —
hSH
I^
--^
-Gt 1
^
=^
22:
It is hetter in this case to resolve the upper note regularly, and let the
loiver one rise instead of falling ; though sometimes even this caution
is not strictly observed.
The fifth of the dominant root may either resolve upwards to the
third of the tonic, or downwards to the tonic itself or its octave. It
may also go by a skip to the fifth of the tonic, whenever such a pro-
gression Mill not involve consecutive fifths.
The best opportunity for this will be in the third inversion, when
38
THE PRINCIPLES OF HARMONY.
the progression of the bass is not to the tonic root, but to its first in-
version. The annexed examples will shew the various ways m which
the root and the fifth may be doubled —
-(S>-
_Q^2.
S5^
iS
^^=g:
.jQ_
-Gf-
rj fj
zz:
1^2:
rjrj
'^ZiL
S:
^2:
-<Sf-
jL^
^rj,rj rj-_
-Gh-
1^2:
"C^
-s-
-s>-
-o-
-^
^
^
&e., &c.
1^21
-(S>-
:s2:
3:
~rjr- r\
IZH
-<s><s>-
8. There is yet one case which remains to be noticed, which is when
the chord of the dominant seventh is resolved by the second inversion
of the tonic triad, instead of the original common chord of the tonic.
This occurs only in the course of a piece of music, never at its close ;
for the second inversion is no position of rest, but quite the contrary.
Example-
i
The student's ear will at once tell him that something must follow this
ere rest can be gained.
9. We must now speak of the chord of the added ninth and its
inversions.
This chord has five notes in it, and is therefore susceptible of four
inversions. It is fovmd, however, in practice, that the fourth of these is
seldom available. Still it will be as well to give the whole here, and
then to state the cautions and limitations which are required.
THE PRINCIPLES OF HARMONY.
39
The first inversion has the third in the bass ; thus
I
F
The second inversion has the fifth in the bass ; thus
m
fS»-
The third inversion has the seventh in the bass ; thus
(S^
And the foiu"th inversion has the ninth in the bass ; thus
7=v (S> 1
© — F= 3
Now it will be perceived, on examining these chords, that all the
intervals except the ninth (A) are precisely the same as in the inversions
40
THE PRINCIPLES OF HARMONY.
of the dominant seventh. Consequently the resohition of these notes is
the same as it is in the former case ; except where the new interval of
the added ninth might cause consecutive fifths to occur. We have already
shewn that such is the case in the original position of the chord of the
added ninth : for if the fifth be resolved downwards to the tonic or its
octave, it makes consecutive fifths with the ninth ; thus
-&
:r±
Therefore in this case the fifth is of necessity resolved by ascending to
the third of the tonic. And this holds good also in the inversions.
The first inversion is thus resolved —
^=rjr:^''
-^
or else thus
-^±
-is> — ei>&
in
(where the D skips downwards to the G). The latter plan is, on the
whole, preferable.
In the second inversion, the fifth, being in the bass, cannot go to
the tonic root, but is forced to go to the third of the tonic ; thus
:z2:
THE PRINCIPLES OF HARMONY.
41
In the third inversion of course the same resolution must take place,
(i. e. to the first inversion of the tonic common chord,) because the domi-
nant seventh is in the bass ; thus
l(S>-
P^
This is perhaps the pleasantest of these inversions, and the easiest to
manipulate.
The fourth inversion is crude and harsh, and should be avoided. In
it the ninth is in the bass : consequently its natural resolution is into
the second inversion of the tonic triad ; thus
^
T2:
In this case, the ninth being below the fifth, no consecutive fifths
are produced, and therefore the fifths may either ascend or descend.
10. The ninth may go into its resolution before the rest of the
chord ; thus
-Gh
^±
r±
e=n
=s
-<s>-
«=^
Ty
:^±
-o-
i^z:
-(S>-
-^
-o-
Z2:
-«s^
T^
-^
G
42
THE PRINCIPLES OF HARMONY
or thus —
-f- S r3 n
y
^
r^
rv n
((\ >^ r^ II
11 ; c^ c^ W
tJ
— 1^—
iP^
-^—
zp:_
_^^ — H
V^ ' 1 II
II
This renders the fourth inversion rather less unwieldy, but still it is
very harsh and awkward.
11. Probably this harshness results simply from the fact that the
interval of the ninth is not susceptible of regular inversion, as it is
beyond the limits of the octave. In the chord of the added ninth, the
added note is essentially a ninth and not a second, seeing that it is
originally added on above the seventh. Therefore it must always be
kept at the distance of a ninth from the root, or the octave of the root,
in order to preserve its essential character. But in the fourth inversion
of the chord of the added ninth this feature is destroyed by the position
of the ninth in the bass. Hence the harshness and awkwardness of this
ugly inversion.
12. The thorough-bass figuring of the added ninth is §. The first
766 7
inversion is figured 6; the second 5; the third |; and the fourth §;
^32 2
together with any accidental flats or sharps which may be required.
13. The chord of the added nmth is so full of notes, and consequently
of dissonances, that it is greatly improved by omissions and curtailments.
These must now be considered in order.
When this chord is uninverted, the best note to omit is the fifth.
THE PRINCIPLES OF HARMONY. 43
both because it is not essential to the character of the chord, and also
because by this omission all danger of consecutive fifths with the ninth
is avoided ; and in the resolution the third of the tonic triad need not
be doubled, as it otherwise must be. The leading note may also be
omitted, as the seventh and ninth are sujBB.cient without it to give a
distinctively dominant character to the chord. But neither the root, the
seventh, nor the ninth can be omitted.
In the first inversion it is always desirable to omit the octave of the
root, as this note forms very harsh discords both with the seventh and
ninth. The fifth may also be omitted freely, for the reasons given
before. The leading note, being in the bass, is essential, and of coiu"se
cannot be omitted.
Neither the seventh nor the ninth of the root can be ever left out.
Note. This rule may perhaps be occasionally relaxed in tlie case of tlie seventh, when
it could not be introduced without contravening the rules of counterpoint. But such licence
is not recommended.
In the second inversion, the fifth, being in the bass, cannot be dis-
pensed with. The octave of the root, however, is better away. The
leading note, seventh, and ninth of the root cannot be omitted.
Note. Here again some relaxation of the nile is sometimes necessitated in the case of
the seventh. But the chord when thus weakened loses much of its dominant character.
In the tliird inversion, which has the seventh in the bass, (and
therefore to be retained as essential,) the octave of the root may be
omitted, and so may the fifth, although the latter omission renders the
chord somewhat bare. The leading note and the ninth can on no
account be dispensed with.
In the fourth inversion, which has the ninth in the bass, it is almost
always necessary to leave out the octave of the root ; indeed the chord
G 2
44
THE PRINCIPLES OF HARMONY.
is hardly ever seen in its complete form, on account of its extreme
harshness. The Jifth may also be omitted, and even the seventh and
leading note, though these two last omissions almost divest the chord
of its dominant character : whichever of these two is omitted, therefore,
the other must always be retained.
The following are the usual forms of this chord, with the omission
of intervals, and with resolution and figuring : —
The chord of the added ninth with the fifth omitted.
All the resolutions in this case are compulsory.
m
ts>-
T^
-^
B=z^
S-
2±
The first inversion, with the octave of the root
omitted. The fifth resolved by a skip, to avoid
consecutive fifths.
Note. The leading note is in French called "la note sensible;" and the first inversion of
the added ninth with the octave of the root omitted is therefore called " la septieme de sen-
sible." But inasmuch as this designation ignores the true derivation of the chord, it has
not been adopted in this work.
The first inversion, with the fifth of the root
omitted.
r:?
-&-
:a
-^±.
THE PRINCIPLES OF HARMONY.
45
The first inversion, with the octave and fifth of the
root both omitted.
-Gi-
Bl
-s^
zz
The first inversion, with the dominant seventh omit-
ted. This should generally be avoided.
fefe
HS>-
-<S>L
1^2:
The second inversion, with the octave of the root
omitted.
I
m
-s>-
-fS>-
~X2.
The second inversion, with the octave and seventh
of the root both omitted. This, however, is not
recommended.
16
THE PRINCIPLES OF HARMONY.
:^==^
The third inversion, with the octave to the root
omitted.
^— f^^
|S>-
T^-
^
'^-
The third inversion, with the fifth of the root
omitted.
-Q-
:b ^~-
-P rj
The third inversion, with the octave and fifth of the
root both omitted.
da
?^ gd -^^ The fourth inversion, with the octave of the root
omitted. Even thus it is too harsh to be used,
^ ^^ ' except in very rare cases.
'-^'-
THE PRINCIPLES OF HARMONY.
47
-Gh
ZZ
S
Z2:
The fourtli inversion, with the octave and fifth of
the root both omitted. This is just tolerable, and
may be occasionally used, with caution.
We can also in every case allow the ninth to fall to its resolution
before the other notes of the chord. This very much facilitates the use
of the fourth inversion. Examples —
S J J J ,1 J j
?2
^Bgp
^
J^
u
^
A
^
^-
T2L
T^
-4^
-<2.
is>-
(S>-
-^
^-
9 8
7 -
7 6
5 -
I '-
6-6
5 4
6 -
5 4
fc^iLd:
J=^
_C^_
(S»-
^^
f^
s
is>-
&c., &c.
PSEE^
'¥=^
^
:q
-^-
-1^-
^.
6 7
6 7 3
14. Some of the above chords have a strong resemblance, on j^9aj9er,
though not when sounded, to the dominant seventh and its inversions.
The thorough-bass figuring is also very often similar or identical. It
48
THE PRINCIPLES OF HARMONY.
will be as well, therefore, to shew how to distingLiish the chords belonging
to one series from those which belong to the other.
I
which is the first inversion of the chord of the added
#
_C2_
ninth, with the octave of its root, G, omitted, might at first sight be
mistaken for a chord of the dominant seventh on the root B. But the
chord of the dominant seventh on B would require D and F to be
sharp, not natural, as here, and would be written thus
-^2-
f
For in the chord of the dominant seventh the tliird, or leading note, is
always major, and the fifth perfect : whereas in the first inversion of
the chord of the added ninth, the third is minor and the fifth
diminished.
Again,
which is the second inversion of the chord of
the added ninth with the octave of its root, G, omitted, might at first
THE PEINCIPLES OF HARMONY.
49
sight be mistaken for the first inversion of a dominant seventh on the
dominant root B flat. But such a chord would require both the B and
the A to be flattened, and not natural, as here ; thus
*
^--
^^S
For in the first inversion of the chord of the dominant seventh, the sixth
is minor and the fifth diminished : whereas in the second inversion of
the chord of the added ninth, the sixth is major and the fifth perfect.
Again,
which is the third inversion of the chord of
(S» —
the added ninth, with its root, G, omitted, might at first sight be mis-
taken for the second inversion of a chord of the dominant seventh on
the root B flat. But such a chord would require both the B and the
A to be flattened, and not natural, as here ; thus
m
-o-
^l
50
THE PRINCIPLES OF HARMONY.
For in the second inversion of the chord of the dominant seventh, the
fourth is perfect and the third minor : whereas in the third inversion of
the chord of the added ninth, the fourth is augmented and the third
major.
Again,
which is the fourth inversion of the chord of
m
the added ninth, with its root, G, omitted, might at first sight be mis-
taken for the third inversion of a chord of the dominant seventh on the
root B natural. But such a chord would require both the D and the
F to be sharp, and not natural, as here ; thus
m
ife
--^
For in the third inversion of the chord of the dominant seventh, the
sixth is major and the fourth augmented : whereas in the fourth inver-
sion of the chord of the added ninth, the sixth is minor and the fourth
perfect.
Before going any further, we would strongly recommend the student
to transpose all the examples m this Chapter into several other keys.
In no other way can he impress all the intervals, chords, and resolutions
THE PRINCIPLES OF HARMONY.
51
so firmly on his memory. We also subjoin a longer example, wliich he
is advised first to analyse and then to transpose —
zt
£Eg
:2^
S
:^=£
-<si
22
-(S«-
=s
i^:^
22:
^
-^2 — c^^
-G>
-j:^
^ ^
f^^9-
r^^
^
=2=25
l6
b5
i-^t
23
:i^3
22
t22:
rJ rJ
J^-
-(S^-
r±:^E=^
3
S
^21
fci
T2L
21:
^±=^
-<s>
'i
-o
^^m
-oi
^
zi:
22
22-
3:
47 — r2-
22:
:^
22t
-fS'-
f
22:
:^
to:
22:
22_
-(S>-
jdzz M=^
^ 5^
:^
3^
S
I o -
rJ r J
i
-O-
-^-
H 2
-fS"-
Z2]
S=°
-<S^
22"
7
52
THE PRINCIPLES OF HARMONY.
Note. It is needless to give any rules about the doubling of the notes of the chord
of the added ninth, as that chord is so fiiU in itself that it seldom admits of such doubling.
It may be as well, however, to state that the same rules which were given as to the doubling
of notes in the choi'd of the dominant seventh and its inversions, will equally apply in this
case also ; with this one addition, that the ninth itself must never on any account be
doubled.
"We may also remark that it is allowable occasionally to interchange dissonant notes in
a fundamental discord, provided they afterwards are resolved according to rule, and that
the root does not alter while the interchange is being made ; for example —
It is always desirable to let the interchanging parts proceed by contraiy motion, as here.
Often, too, a licence is granted ; a seventh or ninth is allowed to skip to the leading note
on the same bass, without any interchange of parts ; thus
J-
^-
i
d:
1^21
22:
122:
22ir:
"8 ~'^
-<s>-
^
-^
is:
-<s>-
122:
22:
:zz:
Z2:
zz:
1^21
:^z:
Z2:
Nay, more, the seventh may skip sometimes to the fifth on the same bass, instead of
being resolved regularly; thus —
THE PRINCIPLES OF HARMONY.
53
fl
J^^
/ d
^^ ^
r^o*
(f3 ^
t/
rm\' <"^
l^-
v_^
c^
though of course this is regarded as a licence.
Another relaxation of the rule is permitted sometimes when another note on the same
bass intervenes between a dissonant interval arid its resolution; thus
-t^-
J-
S
^
-C2-
z:^:
j^^
s
f^-
<s>-
-o-
r-
iq:
-<s>-
— Gh-
isz
1^21
-eS*-
-(^
irz:
Z2:
:z2:
-K^
15. In thorough-bass figuring it is usual, when a note is resolved on
the same bass, to figure the intei^al to which it passes, even if it be an
octave, a fifth, or a third ; thus
fctd:
:q:
or
#£
-Gh-
^=d
or
m^
-Gt-
or
@5
icz:
-f^
r^ -D
/
p*
((^
V y r^ ' 1
r»xiL '
1^- ft
1
>~^ 1
1
And when the other parts do not move, the fact is usually indicated
by horizontal lines drawn from them ; thus —
54
THE PRINCIPLES OF HARMONY.
ii^
S
-o-
■^
m
r
-<s>-
i^z:
and the same lines are drawn when the bass moves, while the other
parts stand still ; thus
i
w-
-&-
::?:5:
'C>
m^.
0| 7 ^ ^
-<S>-
which last example might also have been figured thus
m^
m
3
-Gh-
We will now give a figured bass and treble, and the student is to
fill in two inner parts, according to the figiu-es, carefully avoiding conse-
cutive fifths and octaves, and scrupulously resolving every dissonance
according to the preceding rules :
^
^=ii
r J ^ '
'ZJl.
'&-
-&-
@^
la:
22:
±
-o^
-^ — ^-
22:
-&-
-o-
Z2:
■is>-
zz:
i
THE PRINCIPLES OF HARMONY.
55
*i
-s>-
:izL
^2:
-^±
Z2
-<S'
221
-^-
32:
S^
-eS-
la
:^ — p
I
-«s^
¥
:z2:
:s2
i^z:
f
N. B. The student is advised now to go back to the beginning of Chap. II, and study
the sections printed in small type, before he advances any further.
c^^k::Hy€-^o
CHAPTEE IV.
i
Batio of vibrations in a given time —
12 3 4 5 6
10 ''
,„ , 13 I 14 ^15
12 !.-.b,Q.fl^
16
-^
■^p-fe^Qo. ip:
"n^^^
19 20
:9^
7zr
-iS>-
^&-
:^-
9J
I — f lT-
2,n^ 1
l>
-«
r-H
e3
fl
f3
o
<M
O
rt
^
r^
•rH
-i-i
r3
■-rt
ITi
(Ti
^
-^=
CO
CJ
h
»?^
o
a
id
CO
o
CO
<u
-Ti
>
n
^
(M
o
o
o
Oi
10
o
o
CO
CO
11
CO
^
6X)
12
>
CO
13
CO
14
CO '^
tH CO
o
15
CO
a a
16
17
CI
05
18
>
>
cS
n
-t?
-;j
o
C)
o
o
,£3
,d
-4-3
"^
•^
d
d
05
19
>
d
CO
o
d
20
-(J
rd
-4-3
-^
d
This paradigm of harmonics has been placed at the head of this
Chapter, because frequent reference will have to be made to it. The
THE PRINCIPLES OF HARMONY.
57
student is advised to study and copy it, as upon it all our superstructure
will be built.
2. '" In the first place, it will be remarked that in this series of har-
monics the distances or intervals between the adjacent notes become
progressively less as we proceed upwards.
Thus : the interval between the generator and the first harmonic is
an octave — the next interval is a perfect fifth above that — the next is
a j)erfect fourth — the next is a major third— the next a minor thkd —
and so on.
Now, rejectmg those sounds which are out of the key, B flat and F
sharp, let us take the three notes which come between them, C, D, and E,
and we find that they form a real diatonic progression (i. e. the fii'st three
notes of the diatonic scale of C major).
Let us then take these three notes as a melody to be harmonized.
For this purpose, the fii'st thing will be to discover the fundamental
basses.
Now, seemg that C and E are harmonics of the root C (as above
given), and that D is not only so, but is also the second harmonic sound
produced by the root G, as we have seen in Chapter II ; let us take
these sounds, C, G, C, as oiu- basses, and we shall at once effect oiu*
object.
8
5
3
]/
^ — 1
rv
A
r^
rrn
"W
J
^'
Ov^
1
^'^-^
V^ r^ 1
n
.
* This section is entirely derived from Logier's admirable " System of the Science of Music ;" London,
1827, pp. 48, et seqq.
58 THE PRINCIPLES OF HARMONY.
The fundamental basses being thus discovered, we can now proceed
to add harmonies, according to the rules laid down in this book.
Our result is as follows : —
r-f ^ ^ -1
>
rj
s^^
rv
ff\
^*^
^ J "r^ "^ ^^ '
t/
/S^- ^^ 1
(€>■
v_^
rj
r^
Here, then, we get a harmonized scale of three notes. How are we
to get beyond these ?
Let us turn to the general paradigm at the head of this Chapter,
and we shall find that the harmonic corresponding to no. 7 (or ^ of that
string or tube whose whole length would give the generator C) is B flat.
This note is not a part of the scale of C, but it is its fundamental
minor seventh ; and when it is added to the triad of C, the immediate
result is that C ceases to be a tonic root and becomes a dominant.
Then the ear is immediately seized with an irresistible craving after
a tonic resolution.
To satisfy this, the seventh must descend and the third must rise,
as we have already explained in Chapter II, and thus we find om'selves
landed in the key of F.
This process is called a moclidatio7i, and will be further explained
in a future Chapter.
We have by this means added one more note to our original scale
of three ; and that note is F natural.
Taking F, then, for our new tonic generator, and treating it exactly
as we treated C, it mil supply three notes, F, G, and A, corresponding
THE PRINCIPLES OF HARMONY.
to the C, T>, and E which we took for our original scale of three
notes.
Combining these two series, and connecting them harmonically by
means of the dominant seventh, B flat, we shall produce a scale of six
notes, or a hexachord, as follows : —
:c2:
:S2:
&S=;
-O-
S:
S:
:g--
Z2_
hi
Z2:
:z2:
1^21
122:
"O-
-^5'
Our scale of melody, taken alone, belongs to the key of C, but our
harmonies belong partly to the key of C and partly to that of F.
If we wish to add three more notes according to this system, and
by a similar process, we shall modulate into the key of B flat, of which
F is the dominant. And in the same way, by pursuing a similar course,
we should next modulate into the key of E flat, of which B flat is the
dominant. To this process there is no limit.
z^ — <o- — ■ " -^<s>-
^
2Z
'ZJ-
S:
-(S*-
&c.
SES
^Q-
-^-
-Gf-
^Q
L£^ ^ :
^&-
221
-Gh
^&-
-<S>-
"C5-
b7
hi
bl
1)7
&c.
itM
h ^ CDW
2
Generator 1
I 2
60
THE PRINCIPLES OF HARMONY.
From these premises Logier * draws the conclusion that " no scale
can naturally consist of more than three sounds, for which there are only
two fundamental basses required, viz. the tonic and dominant."
He also shews that the " subdominant" (or the fifth below the tonic,
or the fourth of the scale ascending) is in reality no true root of any
part of a natural scale, but only the generator, or tonic, of a new one.
From all wliich he sums up the following general rule : —
" Whenever we use a [fundamental] seventh, and thus proceed to a
new key, we modulate into that key."
3. From the discoveries in the preceding section, it has been seen
how to form a compound scale which shall modulate continually and
progressively into new keys : but it has not yet been shewn how to
derive from nature a true diatonic scale which shall begin and end in
the same key.
As far as the sixth note of the scale, we have to a certain extent
succeeded in our endeavour to form and harmonize the scale ; but there
we broke away altogether, and hopelessly.
Having wandered into the key of the subdominant, our only way of
returning must be by a modulation into the original tonic again.
But, by the above rule of Logier's, this is to be effected by the inter-
position of the chord of the dominant seventh.
* "System of the Science of Music," pp. 50, 51.
THE PRINCIPLES OF HARMONY. 61
It will be observed here, that at the sixth note of this scale the
E flat (or dominant seventh leading to the key of B flat) has not been
introduced. The sixth chord may then be considered as a tonic chord,
and at rest.
But our object now is not to rest there, but to complete the scale of
C in our melody, and at the same time to modulate back from the key
of F into that of C by means of the dominant harmony on G. The
minor seventh of G, being F, here supplies a combining note for the
harmony, (indicated in the above example by a bind,) otherwise these
two chords would be totally disjoined.
It will also be remarked that the introduction of the leading note,
B natural, in the seventh place of the scale, causes the semitones to faU
between the 3rd and 4th, and 7th and 8th, according to the rule of all
diatonic major scales.
By this forcible introduction of the dominant harmony of G the
ascending major scale of C has been harmonized.
But still there remains an awkward harmonic disconnection between the 6th and 7th
of the scale, which can never be entirely got over.
The fact seems to be, that the leading note has so strong a tendency upwards to the
octave of the tonic that it cannot bear reference to any lower interval. Its appropriate
place would rather appear to be helow the key-note from which the scale is started.
^
-^
-^ — < " ^
as then every chord by which the scale is harmonized would be naturally and strongly
connected with the next.
If such be a true view, then the leading note would be simply a preparatory note,
introducing a scale of the compass of a hexachord — ascending. This view of the natural
formation of a harmonized ascending scale will be seen to be of some value when we come
to consider the minor scale. But that must be i-eserved for the present.
62
THE PRINCIPLES OF HARMONY.
4. By the use of the inversions of the common chord, and of the
chord of the dominant seventh, great variety can be introduced into
the harmonic accompaniment of the ascending major scale.
It will be a very useful exercise for the student to take the scale
of and harmonize it according to the following varied basses and
figures. And after that he can still further improve himself by trans-
posing his work into other keys.
No. 1.
3
:c2:
-B
-.r±
i^:
22:
iq:
Z2:
-&
-^^
IQ
-^-
'r^ r ^
e
:^
C2:
:^
1^-
:^2:
:^
4.
/mr \ ^^ f-^ ^^ II
l^-
h-'
fZi
rj
h^
v»^
_rJ _
_ .
L^_
\
1
^
rj II
' L_ L_ '. \ L
l6
b5
5.
(S>-
^-
^2:
^
t:2i
f^-
T2--
-iSh-
'.ZJl
5. The descending diatonic scale now comes for consideration.
Here, of course, the leading note will not help us, as it rises but
THE PRINCIPLES OF HARMONY.
63
does not fall. By treating the seventh of the scale as simply part of
the common chord of the tonic G, we can get over this difficulty ; thus
^
IQ
-Gi
-^-
22:
:z2
The next stage in our proceedings will be to consider ourselves in the
key of G, and to regard the three notes B, A, and G as a natural scale
in that key, just as we did with respect to the scale of three notes
ascending. Our result will be as follows : —
-jr^-
-<S(-
-^^-
-Gh
-^
W
:^
-Gh
3E3
I
All we require now is a connecting link between these last chords
and the three lowest notes of the scale of C, viz, E, D, and C, which
notes we shall of coiurse harmonize according to the same method.
i
22
"?:3
-C
s^=R
^
64
THE PRINCIPLES OF HARMONY.
Now the fourth note of the scale being F, and F being also the very
dominant seventh which we require, let us adopt it, and thus we shall
get the descending diatonic scale of C complete, with correct basses and
harmonies.
tonic tonic dominant tonic dominant tonic dominant tonic
1^21
-(S*-
-O-
^
1^21
Generators 1 C
^
~ZJ'
■i^
-G>^^
:5^
-^-
-«s>-
-s>-
-<s-
G
"C2_
Z2:
-Gf-
Z2:
C7
-^
=^
6. Another way of harmonizing this descending scale is by regarding
the note A as a part of the chord of the added ninth, in which case
the result will be —
T.
T.
D.
T.
D.
T.
D.
T.
221
:SE
:c2=
~G^
-E^-
"g^
22:
"c?-
ICZ.
c>
-&
zc?
22:
-s^
B:
S
-s>-
G
:^2:
-^
-(S*-
-@-
Generators I C
(The lines drawn from note to note indicate the resolutions of discords.)
(The dotted lines indicate the crossing of the parts to avoid consecutive octaves.)
THE PRINCIPLES OF HARMONY. 65
This last method answers best when inversions are used ; thus —
:q;
:?2:
'7:j-
:q:
-ZJlL
Sf
Z2:
'^U"
-Ci:
1^:2:
-Gt-
-e^
:r2:
-G>-
-<s>-
V2:
s
^-
There are many other ways of harmonizing both the ascending and
descending diatonic scale, but they involve certain rules of harmony
which have not yet been reached in this Treatise, and are therefore for
the present postponed.
K
CHAPTEE Y.
1. We must now consider the origin of what is called the "minor mode." {Scarcely
any subject in the whole science of music has so much perplexed theorists.
The explanation here offered is therefore put forward with some diffidence, although the
author is himself convinced of its coiTcctness.
The necessity for such an explanation arises from the conviction that " nothing which
is agreeable to the ear can be contrary to nature : " but " the minor mode is agreeable to
the ear ;" therefore " the minor mode is not contrary to nature" — L e. it is derived from
natural phaenomena, and, consequently, can be explained by deductions from those phsenomena.
In order to render such explanation intelligible, it is necessary here to make a digi'ession,
for which, it is hoped, no further apology is needed.
2. Referring to the paradigm of harmonics at the head of Chap. IV, and having regard
to those which cori'espond to prime numbers, i.e. 1, 2, 3, 5, 7, 11, 13, 17, 19, it will appear
that their degree of ferfection gradually decreases as they ascend in pitch, at least so far
as No. 13.
To explain this by examples. The octave, or the ratio 1:2, is absolutely perfect.
The fifth above this, which is at the distance of a twelfth from the root, or the ratio
1:3, is also generally called perfect. And so it is in a certain sense.
But it has one imperfection, viz. that if we continually ascend by intervals of a fifth,
we shall never exactly reach an upper octave of the root. For the cycle of twelve fifths,
C, G, D, A, E, B, FJ, C#, G#, DJ, A#, EJJ, and B#, is not the same in its results as seven
octaves of C : in other words, after i-educing the two results to the compass of one octave,
it is found that B# is sharper than the octave of the root, giving (instead of the ratio 1 : 2)
an interval which is represented by 2^® : 3^^
THE PRINCIPLES OF HARMONY. 67
Now, the true octave is iu the ratio 1 . 2, while this interval is 2^* : 3^^, or 2G2144 :
531441. The difference, then, between them will be found by comparing these ratios : which
is done by multiplying the 1 : 2 by 262144 ; i. e. 1 : 2 = 262144 : 524288 : whence it appears
that the interval arrived at by ascending in fifths exceeds the true octave by the fraction
5^4088 ' ^ residuary interval, which is called the " Pythagorean comma."
For this reason all the fifths, in tuning keyed instruments, are tuned a little flatter than
perfection, by one twelfth of the Pythagorean comma, an interval so minute that the ear
cannot detect it.
The next interval corresponding to a prime number is the major third, which occurs in
the second octave, and thus bears the ratio 1 : 5 to the root.
This interval is much more imperfect than the fifth, and, in tuning a keyed instniment,
will require much gi-eater alteration, or tempering (as it is called). For the cycle of three
major thirds, C, E, Gtf, and BJJ, falls short of the true octave ; giving the ratio 4^ : 5^,
instead of the ratio of the octave, 1 : 2, which exceeds it by the comma yff , or " enhar-
monic diesis."
This imperfection is obviously much more appreciable than that of the fifth, especially
when it is remembered that in " tempering" it has to be equally distributed among no more
than three major thirds.
The next prime number in the series (no. 7) gives the interval of the fundamental minor
seventh. But, as has been already observed, it gives it very much too flat. So much so,
that most theorists have demurred to accepting it as the origin of dominant discords and
of modulation.
Rameau *, who, I believe, first noticed it among the acute harmonics of a vibrating
sti'ing, called it " le son perdu," the lost sound, and passed it over as insignificant.
But, even in the last century, there were some writers who to a certain extent recognized
its place in the harmonic system f.
It is admitted by Chladui | to be midway between consonances and dissonances, although
he deprecates its use.
He, however, uses these remarkable words : " On pent cependant presumer que la cause,
pour laquelle I'accord le la septiSme, (ut, mi, sol, si ^), et celui de la sixte supei-flue, (ut mi>
sol, laj), ne sont pas aussi desagreables ^ I'oreille, que Ton pourrait le croire, d'api-es leurs
* " G^n^ration Hannonique;" Paris, 1737, (vide Chap, i.)
+ Vide Pizzati, " La Scienza de' Suoni e dell' Armonia;" Venice, 1782, (Part. IV, Chap, ix.) Also
Tartini, "Trattato di Musica j" Padua, 1754, (p. 126.)
X "Traits d' Acoustique ;" Paris, 1809, (p. 28.)
K 2
68 THE PRINCIPLES OF HARMONY.
nombres compliqu6s, tient k ce que I'oreille substitue h. ces nombres les rapports 4:5:6:7,
dans lesquels Fintervalle ^ differe de la septieme -- du comma g|, et de la sixte supei-flue
du comma encore plus petit jff •" But surely, as we admit of the temperament of fifths
aud thirds, without denying the genuineness of those found in nature, so we may regard
the ordinaiy minor seventh as a tempered modification of the fimdamental seventh found
among the harmonic sounds of nature.
Similarly, we may regard the harmonics corresponding to the prime numbers 11 and 13
in our paradigm as repi'esenting the augmented fourth and the minor sixth in the third
octave : — although the former is much too flat, being nearly midway between the perfect
and augmented fourth ; and the latter is much too sharp, being also intermediate between
the major aud minor sixth.
The next prime number, 1 7, however, gives us the interval of a minor ninth, very nearly
in tune, being only a very little too flat, and almost identical with that intei-val on an equally
tempered instrument*. This number, then, we may at once admit among our fundamental
discords, and use as freely as the major ninth or the minor seventh.
The next prime number is one of the utmost importance (No. 19), as it gives us the
minor third of nature, and may be regarded as the source of the whole system of the
minor mode.
It is almost in tune. Indeed it is more satisfactory to the ear than the minor third as
usually represented by the ratio 5 .; 6.
Reducing it to the first octave, it is represented by the ratio 16 : 19, and only falls short
of the usual ratio 5 : 6 by the very small comma gV.
Regarding the major triad as 16 : 20 : 24, and the minor triad, as here derived, as
16 : 19 : 24, nothing can be more simple and natural than their relations one to the other.
Even Chladni admits this t, where he says " Peutetre quand on se sert quelquefois de I'accord
parfait mineur au lieu du majeur, ou du majeur an lieu du mineur, I'oreille est raoins hlessee{'.)
parce qu'elle substitue 'k la tierce mineur | I'intervalle yf, en entendant une variety de
rapports, comme 16 : 19 : 24, et 16 : 20 : 24."
Let us, then, assume the fundamental minor third of nature to be |-|, or that produced
by taking |g- of the length of a string or tube.
It will only remain to shew why the usual figures, 5 : 6, cannot be taken for this pur-
* This derivation of the minor ninth is duly recognized by Catel, in his well-known " Traits d' Har-
monie;" Paris, 1802, (p. 6.)
t Chladni, " Traits d' Acoustique ;" (p. 29.)
THE PRINCIPLES OF HARMONY. 69
pose — in other words, why the minor triad is not correctly represented by the ratios
10 : 12 : 15.
3. Referring once more to our general paradigm at the head of Chap. IV, and seeing
which are the notes represented therein by the ratios 10, 12, and 15, we find them to be
E, G, and B, or the minor triad of E. But the root of all the notes in this series is C,
not E. And C cannot be the root of the minor triad of E. Therefore the numbers 10,
12, and 15 do not correctly produce a genuine minor triad. Therefore the ratio 10 : 12 (or,
which is the same, 5 : 6), does not represent a real fundamental minor third.
But it does really give the interval between the thii-d and fifth of the major triad, as
the paradigm will shew.
There are, then, in nature two different minor thirds, only one of which is fundamental ;
or, as it may be expressed, one of them, ^, is the root with its minor third, the other, |, is
the third and fifth with the root omitted.
4. It may, then, from these arguments, be fau-ly assumed that the
mmor triad has its origin in nature, analogously to the major, and that
it is part of the tonic series.
The dominant must always have its third or leading note major ;
otherwise it would be too far from the note to which it leads, and
from which it is necessarily a semitone distant.
5. On referring to Chapter I, it will be seen that every major key
has a minor key connected with it, called its " relative minor." This
connection can hardly be said to be of natural origin, inasmuch as the
harmonics of the root of the major key do not give us the common chord
of its relative minor. But as several of the intervals of their scales
are common to both, and as their " signature" (or the flats or shai'ps in
the stave) is the same, and as, moreover, it is very easy and very usual
to go either by modulation or by harmonic progression, (which term
shall be explained hereafter,) from one to the other ; it will be most
70
THE PRINCIPLES OF HARMONY
convenient on the whole to ilhistrate the derivation and harmonization
of the minor mode by starting from the key-note A, and taking as our
model the process adopted for the scale of C major in the preceding
Chapter.
6. For this purpose we will shew the scale of three notes in the
major and minor modes, side by side —
Major. Minor.
a
122:
-<s>-
-<s»-
-©-
i^:
-G-
1^21
1^2:
-g--
-(S-
-<s»-
m
Z2:
Z2:
Z2:
Now although these two scales look very similar at first sight, yet
are they essentially unlike ; and the minor sounds eminently unsatis-
factory.
The reason of this is, that the harmony of the second note of the
major scale is essentially dominant, seeing that it will admit of the
addition of a seventh; thus —
I
i^z:
-s>-
-o-
-o-
22
f^-
s
"^z:
221
Z2:
whereas the harmony of the second note of the minor scale, as written
here, is not, and cannot be, dominant, inasmuch as it contains no leading
note, the third being minor. It is simply the minor triad of E, and has
no connection with the chords with which it is here associated.
THE PRINCIPLES OF HARMONY.
71
To correct this, it is merely necessary to put a sharp before the G,
when it immediately becomes a leading note : —
iq:
^-
^^
hS»-
xz:
1^2:
icz:
to which we may add the seventh if we please —
.<s>-
:g^
^&
^
1^21
'JZZ.
:_ tizi^z
s
In forming the harmonized major scale, the fii'st three notes were
followed by three others, similar, and similarly harmonized, in the key
of the subdominant. Pursuing the same course, as far as possible, with
the minor scale, the result will be as follows —
22:
--g--
s
s
"^IT"
-/S>-
?^s
m
-jr:2i
:z2:
221
hS-
-^-
221
i i
This will do very well. But it differs from the scale in the major,
inasmuch as there is no modulation into the key of D betw^een the third
and fourth note. This is inevitable, because the tliird note of the scale
72
THE PRINCIPLES OF HARMONY.
is necessarily a minor third to the root A, and therefore cannot be used
as part of a dommant chord. Still, there is sufficient connection to satisfy
the ear, and a real modulation does take place at the next note, where
the leading note, G% is introduced.
As we have thus got out of the key of A into that of D minor, we
are obhged to modulate back again to our original key : and tliis obliges
us to introduce the leading note, Gj$, into the melody as the seventh note
of the scale. The whole will then stand as follows —
#
-«s>-
-o-
22:
-s>-
-Gh-
-<S>-
1^2:
Z2:
}
It will not escape notice, that between the sixth and seventh notes
of the minor scale, as here shewn, there is the interval of an augmented
second. This constitutes the chief characteristic of a regular minor
scale.
7. In Chapter IV, towards the end of section 3, it is shewn how awkwardly the sixth
and seventh notes of the diatonic scale hang together. This is even more apparent in the
minor scale, in consequence of the introduction of the very dissonant interval of the aug-
mented second between those notes. If the leading note be, as it were, prefixed to the
minor hexachord (or scale of six notes), all this awkwardness vanishes at once ; thus —
:i^=^:^=^:
P:
:£
^2:
:e
Every chord connects itself with those next it, without difficulty, and without harshness.
There is therefore no small i-eason to surmise that this is the most natural form of the scale.
THE PKINCIPLES OF HARMONY. 73
8. Before proceeding to harmonize the descending minor scale, it will
be necessary to introduce a new dominant interval, or fundamental dis-
cord — the minor ninth. This is one of the most important elements of
modulation, and affords a greater variety of resources than any other
known combination of notes to the experienced harmonist. It will be
well, therefore, to devote a Chapter to it alone.
CHAPTER VI.
1. It will be seen on reference to the paradigm prefixed to Chapter
lY, that the 17th harmonic gives us the minor ninth in the fourth
octave from the root. It is therefore a natural harmonic.
But it is clearly no part of the tonic harmony, for it is out of the
key. Nor is it in the hey of the dominant ; for the minor ninth, for
example, of the dominant G is A flat, which neither belongs to the
scale of G nor to that of C major.
But it does belong to the scale of C minor, although it is a harmonic
of the dominant G.
It therefore is a very important addition to the dominant harmony,
because it only belongs to it as such, and by no means belongs to the
same root considered as a tonic, being necessarily and essentially domi-
nant, and leading downwards to the fifth of the tonic quite as irresistibly
as the leading note leads upwards to the octave of the tonic, or the
dominant seventh downwards to its third.
When the tonic is of the minor mode, the force of the dominant
seventh is weakened, because, instead of falhng a semitone to its reso-
lution, it has to fall a whole tone, since the tonic third is minor. Compare
the two —
THE PRINCIPLES OF HARMONY.
-9 1^—
75
sa
'Gh-
SEE
semitone
r
whole tone
:^2
But the minor ninth at once makes up for this defect, by its own
strong dominant tendency to the fifth of the tonic.
i
-f^-
-is-
^i
^
isi:
The two together form as strong a dominant chord as can be required.
And the additional indication of the minor mode which this mterval
gives, renders it of especial value when we wish to introduce and fix
that mode.
The major ninth is not suitable for the minor mode, on account of
its incongruity with the sixth note of the minor scale, which is mmor, as
we have seen, and with that same note considered as the third of the
subdominant triad, which, as we have also seen, is minor also. Therefore
the minor ninth must be substituted for the major in the minor mode.
Its resolutions, inversions, and figuring are analogous to those of the
chord of the added major ninth, only that the accidental flats and sharps
involved will of course be different ; and in tliis difference there is also
this advantage, i. e. that there is no danger of mistaking the inversions
L 2
76
THE PRINCIPLES OF HARMONY.
or figuring of the chord of the minor nmth for those of the dominant
seventh ; for in the former the accidentals are always combined of a con-
trary Tcind'^f which is never the case in the latter.
A few examples wiU shew all these points at one view.
m
^
a
-&-
22:
221
The chord of the minor ninth uninverted, properly-
resolved and figured.
J The same, with the ninth resolved to the octave
on the same bass, before the resolution of the
'^ ' other parts.
The next two examples give the ninth in an inner part, and below
the leading note. This is always allowable with the minor ninth, and
gives it a great advantage over the major, — where such a position is
not recommended.
-(^3-
:^
s
r-
-<s> '
-<s» — 1
1221
5^
22-
S
-(S>-
-SH
'jr^L.
I
* i. e. shai'ps against flats ; thus
^-
is easily to be distinguished fi-om
THE PRINCIPLES OF HARMONY.
77
S=
^
i
-<s»-
:«^
I
P5
-«S»-
-^-
:i^:
D5
5^
-<^
P5
s
s;
-("S*-
-«^-
s=
■B:
-^s-
i2:
-(S*-
i
-^ 1
<v
The first inversion of the chord of the minor
ninth, — where observe the forced movement of
the fifth B up to C, to avoid consecutive fifths.
As, however, one of the fifths is diminisliedy
this precaution is sometimes neglected by mo-
dern composers.
^ The same, with the ninth resolving before the
rest. Here the progression of the fifth is per-
fectly free and safe.
The second inversion. Here the fifth, being in
the bass, must not descend to the tonic, as the
consecutives would be between the extreme
parts, and therefore offensive.
:^
±t
-^
or.
-Gh
'JZL
i-J^
fe
-<S»-
hS>-
1221
When the ninth is resolved in this way, the hidden consecutives between
the extreme parts may be tolerated (as in bar 2), but it is very much
better as in bar 1.
78
THE PRINCIPLES OF HARMONY.
^
s
'^ Q '^ o I This method of resokition takes away all difficulty
about the descent of the bass. It may equally
apply to the first inversion if required.
-<s^-
zz:
I
-«s>-
32:
&
-o-
The third inversion, regularly resolved.
-s>-
CJ
1^
s
S:
-^
-O-
i^z:
The same, with the ninth resolved before the
rest.
6-6
f" 2
iJ
1^2^21
-«S-
-<S^
Z2:
The same, resolved so as to enable the fifth to
descend.
6-6
2
THE PRINCIPLES OF HARMONY.
79
^
S
-^^-
iq:
The fourth inversion. This should be avoided,
because of its harshness and unsatisfactory reso-
hition on the ?•
4
^
I
or.
-rj r j z:
-<s>-
q:
^
I
Two ways are here shewn of improving the resolution of this ugly chord.
The former is not very satisfactory, but the latter is good. In tliis
inversion, the ninth being below the fifth, no consecutive fifths can
occur.
2. Hitherto we have been speaking of the chord of the minor ninth
and its inversions, without omitting any interval. Like the chord of
the added ninth, however, (see Chap. III. 13,) this chord is all the better
for such curtailment ; and the general rules for the omission of its various
intervals in each inversion are similar to those given in the case of the
chord of the added ninth.
But when the octave of the root is omitted from the chord of the
minor ninth, a very remarkable and important series of chords is dis-
covered, which demand special consideration.
80
THE PRINCIPLES OF HARMONY.
3. The first inversion of the chord of the minor ninth, omitting the
octave of the root, is known as the " chord of the diminished seventh,"
and is so named after its characteristic interval —
It is susceptible of three inversions, all of them being of the greatest
use m harmonizing —
=^
■o-
m.
T
3
-^
-^
^l
6
f
Now on examining the chord of the diminished seventh, it will be
seen that it consists entirely of minor thirds superposed — GltfB, BD, and
DF. And on examming the inversions, they will be found to consist
in each case of two minor thirds and one augmented second ; thus — BD,
DF, and FGJI; DF, FGjf, and GflB; and FGJf, GJfB, and BD.
On all keyed instruments, the same key stands for Gil and Ab, And
although they are not the same note really^ yet they are so nearly the
same that one may be always substituted for another with impunity.
Indeed, the construction of our instruments, the method of tuning now
universally prevalent, and the requirements of free modulation, render
such interchanges imperatively necessary.
Note. The interval which actually exists between two such notes is named the " Enhar-
monic Diesis," and is represented by the fraction i|4.
Sec short Examples at the end of this work, Nos. 6, 7, and 12.
THE PRINCIPLES OF HARMONY.
Any note, then, may be changed into that which is nearest it in
pitch, and which is represented by the same key on all keyed instru-
ments, e.g. Gjl into Ab, Aj$ into Bb, Bfl into CQ, CJ|: into Db, DJ into
Eb, M into FQ, F^ into Gb ; and conversely, Gb into FJf, Fb into EQ,
Eb into m, Bb into C^, Cb into Bt3, Bb into Aj?, Ab into Gjf.
This sort of interchange is called an "Enharmonic Change!' The ear
is, as it were, deceived by it, just as it is by the tempering applied to
the fifths, thirds, and sevenths, as described in Chapter V. Whence it
follows that the chord of the diminished seventh and all its inversions
may be alike regarded as in fact composed of three minor thirds ;
thus —
1st minor third <
Gft
F
or Ab
F
B or
Ab
Cb or B
Ab or GJf
D or Ebb
B or Cb
2nd minor third \
^ B
F
D
F
D
Ab
F
Ab or GJj:
F or E#
B or cb
Gj$ or Ab
3rd minor third 4
B
D
B
F
D
F or EJf
D or Cx
GJf or Ab
Eif or F
The minor thirds may also be reckoned the other way
(i) F to Gft or Ab;
(2) G$f or Ab to B or cb ;
(3) B or cb to D or Ebb; &c.
For every one of the intervals composing this chord may be enharmoni-
cally changed, as is here shewn.
M
82
THE PRINCIPLES OF HARMONY.
Hence it may be seen how very useful this chord is in modulating
from key to key. For example —
d=
B F3-
^.
-^^
t
m
-jtX
-<s
&-
23
^-
s»-
HS>-
it
-^
~C7
-"^^
4p:
(S»-
:^^
=^==g^
-^
-x±
Roots A
E
A
A
D
A
Keys
D
t X.
fa^
^2:
:<:2:
-is>-
4S?-
-^
-s*-
51
-^
-^'
KS'-
Bb
Eb
#=
J2 p ^ ^^nzqp^Jgz:::
^21
ipzizzp^
:|^
^
Roots Eb BQ EQ B
KeysEb Ep
E
E
A
B
E
E
A
;
E
In this example —
Bar i. contains the minor triad of A, and the first inversion of the
dominant triad of E, with the seventh omitted.
Bar ii. contains the minor triad of A ; and the third inversion of the
chord of the minor ninth of A without the octave of the root :
being therefore the second inversion of the chord of the dimi-
nished, seventh, leading to the next chord.
THE PRINCIPLES OF HARMONY. 83
Bar iii. contains the first inversion of the minor triad of D ; and the
first inversion of the chord of the minor ninth of A, without
the octave of the root — being the chord of the diminished
seventh.
Bar iv. contains an enharmonic alteration of the last chord, whereby
it is converted into the fom'th inversion of the minor ninth of
C, without the octave of the root — being therefore the third
inversion of the chord of the diminished seventh. In the last
half of the bar, the bass resolves the minor ninth by descenduig
to the root, and the rest of the chord remains as a chord of the
dominant seventh on C.
The student ^vill now be able to carry on this process of analysis, by
the aid of the rules already given. He is begged, however, to take
especial notice of the enharmonic change in bar vi., which he vd]\ have
to examine very carefully. He will do well, also, to observe the roots
and keys, as indicated in capital letters below the bass stave.
4. In bar x. of the preceding example, the common chord of E major
is marked thus f . This has been done to draw attention to the fact
that here the chord of the minor ninth has been followed by a major
triad. This licence is always allowable; although the character of tliis
chord is more essentially minor than major, as has been sufficiently shewn.
By means of this licence, an even more extended usefidness is gained,
and the composer is enabled by it to modulate stiU more freely.
5. It is then possible from one chord of the diminished seventh
to modulate directly into a great variety of keys : for instance, taking
M 2
84
THE PRINCIPLES OF HARMONY.
our start from A minor, the following resolutions of the diminished
F
seventh ^ will at once appear : —
^^
3
~i:jl
s
22
^=1
<S-
:^=to
-<s>-
-^
«E
:tt?2:
&=J|^z±f^=^:
©
A major
C minor
C major
-A == 1 I J V \ \ -
m r~'^r=^
o-
«p=
#
-^<<s>-
Fjl minor
FJf major
D jf major
i^i=fe
w5^^r$^S^g=#=^
fl^
J-^S*-
f
<s^
4
ite:
^=ffR
|(S>-
fezzz^
<s>-
itcz
^
"g^-
5!2:
D Jj! minor
Eb mi
mmor
Eb
major
^
^^^=|i^WE^i^
3
12^21
:^g:
^^
z:?"
izs:
B^: — b ^
[221
Gb
major
Gb
minor
THE PRINCIPLES OF HARMONY.
85
At f , bars viii. and ix., the equivalent notes have been omitted, and
the enharmonic change taken for granted, This is the usual way of
writing such modulations, for the sake of simplicity.
We may here quote a very good example from Catel (Traite
d'Harmonie), which clearly exhibits the enharmonical resources of the
chord of the diminished seventh —
>Q_
11.
^^
111.
4f^
4tQ_
4fQ.
ife:
n:2L
^m-
\&-
-G>-
-Gh
-G»-
m^
Z2:
'-^-
=S^
g; ivv -
-^-
-«s>-
-<s>-
-f^^- :
3^2:
b5
Roots
>Q_
b5
6
t
b7
5
«5
>E=i
22:
:(=z
:^^;
*=:
\'
1
1
C9
85
In this example the diminished seventh >h and all its inversions
are successively presented in bars ii. — vi., and all the various roots given
from which they are severally derived. Yet, if played on a keyed
instrument, the chords in these five bars remain unchanged, as will be
at once perceived on playing them.
6. Before proceeding any further, it will be well to give a rule by
which to discover the root of any fundamental harmony.
86 THE PRINCIPLES OF HARMONY.
For this purpose it will be necessary to refer to the general table
of the keys with their signatures, given in Chapter I, sect. 5.
Now, it will be found a convenient plan, to class the key-notes
according to their signatiu-es, calhng that key the sharpest key which
has the greatest number of sharps ; and, generally, in comparing two or
more keys together, calling that the sharpest which has the most sharps
or the fewest flats in its signature.
And we may apply the same mode of speaking to the ludiviclual
notes themselves : for instance, E Q may be called a sharper note than
Dp, because the signature of that key of which it is the tonic has two
more sharps than belong to the key of D.
Similarly, F may be called a sharper note than B b ; and, generally,
we may call the notes in the subjoined Hst sharper or less sharp accord-
ing to the order in which they stand; those to the left hand being
sharper than those to the right. Thus, in the list of notes —
Ctt, FjJ, B, E, A, D, G, C, F, Bb, Eb, Ab, Db, Gb, Cb ;
CJt is the sharpest of all notes; then comes F^, and so on, decreasing
in sharpness till we reach Cb, which is the flattest of all.
This is simply a convenient mode of naming the various notes by
way of comparison, the special advantage of wliich we shall now proceed
to shew.
7. " In every fundamental chord, the leading note is the sharpest note
to be found."
To this rule there is no exception, and by means of it the root of
every fundamental chord can easily be found.
THE PRINCIPLES OF HARMONY.
87
For example, let us try to discover the root of the following
chord —
M
:q
On reference to the Hst or table in the last section, it will be seen that
of the notes composing this chord the sharpest is A ; therefore A is the
leading note ; therefore F is the root ; and it is evidently the first inver-
sion of the chord of the minor ninth without the octave of the root,
otherwise called the chord of the diminished seventh. And from this
it follows, in the regular course, that the chord belongs to the key of
Bt^ minor.
This rule is so compendious, so simple, and of such general apphca-
tion, that it ought to be thoroughly mastered, and continually applied, by
every student of harmony.
To it may also be added the following subsidiary rule —
" If there is a minor ninth in any chord, that ninth will be the
flattest note."
Thus, in the above example the flattest note is Gb, which is thereby
known to be the miaor ninth of F, the root.
If, however, there be no minor ninth in a dominant chord, then the
flattest note will be the dominant seventh.
88
THE PRINCIPLES OF HARMONY.
Thus, in the annexed chord, the flattest note is D, which is the
dominant seventh of E, and the whole chord is a chord of the added
ninth on that root :
=i^
.^2_
Z2:
99
i
8. Let us now apply these rules to the enharmonic variations of the
chord of the diminished seventh and its inversions —
^^-
'^^
&-
^
,«s»-
-m-
"oio^-
"gg:
In No. i. the sharpest note is CJf, therefore the root is AQ, and the
L
chord is a diminished seventh ^J.
In No. ii. the sharpest note is EQ, therefore the root is CQ, and the
.6
chord is in its last inversion 04 .
%
In No. iii. the sharpest note is Ajf, therefore the root is Y%, and the
6
chord is the first inversion of a diminished seventh Q4 .
3
In No. iv. the sharpest note is Gp, therefore the root is Eb, and the
chord is the second inversion of a diminished seventh Q5 .
In the first case the chord leads to D minor (or major) ; in the second
THE PRINCIPLES OF HARMONY.
89
to F minor (or major) ; in the third to B minor (or major) ; and in the
fourth to Ab minor (or major).
The student will now analyse the following exercise, detecting the
roots, and describing every chord —
S^
2i
i
--^^=^
®
:§fl=4
-S»-
Gh-
.J.
-&-
g^^
£i
1
-^
~jzi:
^
±^
i^
:Jf^
:^=3P
-Gh
"1
6
f
t —^^J^
■^Gh
i
ist
:^
«s<s>-
-«si — &i-
^fe^
l^it^^^^g:
i^ &^^~b3
E
■b a<jQ.g, -g ^B
--g--
122:
-^^
:?2:
(S^
-r^ — r^
-^
-G>
1^21
:rj:
-^s*-
i^
b5
-«s
b7
b5
t> b7 b
d:
/^Ts
i
22 aS
[^ — g zzLz^
-^
^^=
"c^
^^ -^- b^
-(S>-
S
is>-
^^=B^=
2:±
:^
:^z
3i
:zii:
^
Itc^'
-iS^
"^>"
^O'
9 8 I b7
Note. If the root and leading note be both omitted from the chord of the minor
ninth, it is reduced to an imperfect or diminished triad —
i
«y
N
00
THE PRINCIPLES OF HARMONY
In this form it is impossible to detect its root, excei)t by a consideration of the
succeeding cliord.
If that chord be on the tonic A, the preceding example will of course be an imperfect
chord of the minor ninth on the dominant root E.
-S-
E
A
But if the succeeding chord be on the tonic C, then the doubtful chord must be
regarded as an imperfect chord of the dominant seventh on G.
f^
W=^^
G
C
It is therefore the most equivocal and unsatisfactory of all the simple chords in
music.
CHAPTER VII.
1. Having introduced and explained the chord of the minor ninth
and its derivatives in the last Chapter, we may now proceed to apply it
to the harmonization of the descending minor scale.
In Chapter V, section 6, the ascending minor scale was worked out :
and it was shewn in the following section, that between the sixth and
seventh degrees of this scale the chromatic interval of an augmented
second occurs.
In forming and harmonizing the descending minor scale, care will be
required in the treatment of this awkward interval.
^^^^EEP^^^^EEg
:^±
^AEEEEAEA
On setting out the melody, it will be observed that all the notes of
it belong to the dominant chord of E, except the first, sixth, and eighth.
We need not, then, unless we choose, have any dealings with the
subdomlnant D, or with any harmony but that of the tonic and
dominant.
N 2
92
THE PRINCIPLES OF HARMONY.
:a:
g— ^-eg:
22:
"C5-
1^21
-s>-
1221
-(S*-
:e2:
12:2:
22:
:z2:
Roots
A
A
E
E
E
6
E
E
A
Keys A E A
But here there appears a somewhat irregular treatment of the domi-
nant seventh between the third and fourth chords ; thus —
fS>-
:2:2:
5E^=
-f^
:g =p= F
And although this progression is tolerated by licence, on account of the
contrary motion between the extreme parts, yet it is by no means elegant.
It will therefore be better to look on the note E in this scale as part
of the tonic harmony, and to harmonize thus —
E
-IS»-
q=ip2:
g=^
-<Sh
^P^
3S
-Gh-
Roots
Keys
A
A
%
E
E
-&-
E
1?:^=:^:
O"
?=2:
5
6
A
E
:^i=t
E
:?^
A
THE PRINCIPLES OF HARMONY.
93
Or, if we prefer to introduce the subdominant triad D, we can do
so with good effect, as follows —
J^^^^^l^
S
m
C2:
--^
i^
:q
€J u
Roots
Keys
A
E
E
6
f
E
A
D
D
6
4
A
i
E
A
A
And this can be varied by using different inversions.
On the sixth chord we shall have some observations to make in a
subsequent Chapter, as, in such cases as this, it is not, strictly speaking,
so much an inversion of the tonic triad, as a double dissonance by sus-
pension ; on which see Chapter VIIT, section 7.
2. It is often convenient to alter the minor scale, so as to avoid
the augmented second between its sixth and seventh degree. This is
usually done by sharpening the sixth m ascending, and by flattening the
seventh in descending.
The scale then departs from nature, and becomes entirely artificial.
It will then stand thus —
i^:
4t^
-<S'-
-^-a^
-^-
1^2:
-(S>-
:q:
:q:
-<s>-
:??^E^
1^2:
94
THE PRINCIPLES OF HARMONY.
Such an alteration necessitates a total change in the harmonization.
The following appears to be the best method : —
I
iS
-<s>-
:q:
-^^-IQ-
S
s
i^SzIg:
JUL
-O-
- rj rj 7^ —7-J-
^
Z^^Z
"r^
s
:z^ ry
--g:
e> — <^-
-<&-
-o-
_c*t-
-(S^
-e>-
-s*-
#
122:
22:
-<s-
1^
-s>-
'IZL.
4J^2Zz^zttQ:
Z2:
Z2:
221
6
Q5
I
3. It will be useful now to give examples of the harmonizing of
scales minor and major when the scale is given to the bass part. As
an exercise to the student, only the bass and figures will be given, and
it will be for him to supply the three upper parts, and to assign the
proper roots of each chord :
#a
-^-
-f^^-
-s>-
4Q-ip
jQ_
J^lJ2^
1^=22
■tS)-
:^2z
6
B5
6 6
' f
m
-&-
I^Z
-o-
Z2:
^^=-^--
:^=^:
=^=p=
-t^-
- f^—^—^ _
D6
■^—&-
221
6 6
-^'
-Gi-
rj rjr_
:i^:±=S
1^21
6
4
6
^=3=
r^
CHAPTER VIII.
1. All the discords hitherto described have belonged to one class,
viz. " fundamental discords." That is, they have all been parts of the
dominant harmony, as derived from nature. But besides the funda-
mental discords, there are others of various kinds, some of which it is
time to explain.
2. It may be well, in this place, before going fiurther, to define a
few terms more accurately than has yet been done in this Treatise.
i. A Concord is a combination of root, third, and fifth, such as the
major and minor triads — (and perhaps their inversions also, though
they are imperfect concords).
ii. The sounds of which a concord consists are called Consonances.
iii. When any one of the sounds composing a concord is removed,
and some other sound substituted in its place, the perfection and
satisfactoriness of the concord is destroyed, and a different and
contrasting effect produced. The intruded new sound which pro-
duces this result is called a Dissonance.
iv. The chord in which the dissonance is heard is called a Discord.
96
THE PRINCIPLES OF HARMONY.
3. The discords and dissonances of fundamental dominant harmony
liavc been discussed. It is necessary now to speak of Dissonances by
Suspension ■*'.
Let the following melody be played, accompanied only by its funda-
mental bass. It is satisfactory to the ear, though tame and bald :
I
_Q_
'G>-
Z2:
-G>-
S
i^z:
-s»-
1221
:z2:
But let the sound G in the melody be continued through part of
the succeeding bar, without altering the bass (as in the following
example), and the ear will immediately begin to long for the delayed
note F. And if each note in this melody be similarly treated, we shall
have
22:
«s>-
221
2IZ
izz:
Wlien an interval of a melody (or of an inner part) is thus kept back
in descending, it is called a susjoension.
Thus, in the preceding example,
G suspends F,
F suspends E,
E suspends D.
* This is principally derived from Logier's "System," ut sup. pp. 62-G5.
THE PRINCIPLES OF HARMONY.
Or, if viewed in relation to their bass notes,
The ninth suspends the eighth,
The fourth suspends the third,
The sixth suspends the fifth.
These include all the dissonances by suspension.
The following are examples of each : —
97
-<S-
"C5"
:c2i
-«s*-
'C?"
s
-iS^
2^=^=^:
n-R
-ts>-
''Z±k
.Q_
^?^:^^B
m
i^z:
:q:
22:
icz:
221
:q:
X2_
i^z
t z
\ 3
Here the dissonance of the fourth is produced by suspending the third.
-s*-
-<s-
~rT
-e>-
;i
Q_
^±
J^
(S>-
Z2:
Si
,Si
-«s>-
S— n
-<S' iS>-
-<s>-
-<s>-
-o-
-Gf-
22:
izz:
:22:
Z2:
-<s>-
-<s^
-<s^
-«s>-
9 8
9 8
Here the dissonance of tlie ninth is produced by suspending the eighth.
'TJ — C
22:
:g:
<o> r rJ
-(S>-
^zi
^^f^
-C^
-<S'-
g=E
-S^
H^
-(S>
^^^
-!S>-
T^
22:
22:
22:
22:
221
22:
22:
Z2:
6 5 6 5 6 5
Here the dissonance of the sixth is produced by suspending the fifth.
98
THE PRINCIPLES OF HARMONY.
4. The suspension, then, being produced by a lagging note — a note
behind the rest in the progression from one chord to another, — it follows
that these dissonant notes must exist as consonances in the previous
chord, before they became dissonances by being, as it were, left behind.
The rule, then, may be thus stated : —
"All dissonances must be introduced by consonances," or, in other
words, " The sound which constitutes the dissonance must first be heard
in the preceding chord as a consonance."
And tliis is called preparing a dissonance.
And it is also evident from this, that in whatever part (treble, bass, or
inner) the dissonance occurs, in that same part it must also be prepared.
It will also be noted, that the dissonance always descends one degree
upon the following consonance.
And this is called resolving a dissonance.
A few examples will make this clearer : —
221
-@-
-O-
-<S>-
m-
1221
-O-
Z2:
If we wanted to introduce the dissonance of the fourth into this har-
mony, we might do so, perhaps, as follows : —
:c2E
,<s»-
=?2
-fS>-
Z2:
:^2:
-<^
THE PRINCIPLES OF HARMONY.
99
But on inspecting the progression of these parts, it will soon be seen
that the fourth is not i^repared.
To remedy this defect, it must be introduced into the same part in
the preceding bar ; thus —
F
Z2T2
(S>-
E
-G>-
rU
-S^
m
22:
-<s>-
Z2:
And the same process would enable us to introduce the dissonance of
the ninth into the last bar ; thus —
:22ZC2
^
^2
-<s>-
o-
ICZ
■^:2:
m
Z2:
-s>-
:q:
And the dissonance of the sixth should be prepared and resolved in
the same manner.
The following rules will be useful : —
I. The dissonance of the fourth may be introduced whenever the
fundamental bass (or root) descends a fourth or ascends a fifth.
II. The dissonance of the ninth may be introduced whenever the fun-
damental bass ascends a fourth or descends a fifth.
o 2
100 THE PRINCIPLES OF HARMONY.
The preceding example will serve to illustrate both these rules.
5. The preceding remarks refer only to dissonances by suspension
occurring in uniiiverted chords. By employing inverted basses, a great
variety of effects by suspension may be obtained.
In every case care must be taken " never to let the suspending note
be heard together with the note it suspends.^' Thus
^
:^
:^
:^:
-cr
:^2=
would be wrong ; as the suspended B is heard in the tenor part. In
a free style, however, such a combination is occasionally met with.
Still, it is necessary for the student to keep to the rule ; excepting
only in the case of the ninth by suspension, wbich may be used with
the eighth, 2')'^^ovided the parts proceed by coiitrary motio7i, and only
then. Indeed, even in this allowed case, the licence should be sparingly
used. Thus, for instance,
would bo allowed by licence, while the following —
THE PRINCIPLES OF HARMONY.
101
is*-
^-
i
;^E^^
-G>~
^
ITj-
-^
would be altoo'ether inadmissible. For it is evident that hidden coiise-
cutive octaves would be produced —
-<s^
iH
fc?^
6. In the first inversion of the common chord, the fundamental dis-
sonance of the ninth becomes a seventh to the inverted bass, as follows : —
Roots
*y
-&
-G>-
I^
y«^-
1221
Here the leading note, B, is allowed by licence to fall to G, to avoid
the transgression of the above nile for the resolution of dissonances.
In this inversion the fLmdamental dissonance of the sixth becomes
a fourth to the inverted bass ; as thus —
102
THE PRINCIPLES OF HARMONY.
Roots
-y- — 1
1 —
1
1
1 1 n
^^ — ^
— ^H
J — ^
feh-fe
—
_cJ eL- \—^—
_c-^ (S'— ■
tJ
. ... f
-J
f^^—
f^—
r^
j-^
rj
v^
4 3
I
4 3
/m^' -^
vfj-
— -^
V-^
^-^
rj
1
— & — 1
6 5
6 6
In the second case here given, the dominant seventh, F, has been allowed
to fall to C instead of E, in order to avoid breaking the same rule.
In this inversion the fundamental dissonance of the fourth necessi-
tates the suspension of the inverted bass itself ; thus —
Roots
^
-C2_
-O-
g f^-D
-^
^
22:
:q:
-Gf-
i^-
cn
Z2:
for the inverted bass is always to be looked upon as an upper melody
transposed into the lowest place, and therefore to be treated in all
respects as such.
THE PRINCIPLES OF HARMONY.
103
In the second inversion of the common chord, the fundamental
dissonance of the fourth becomes a seventh to the inverted bass,
thus —
Roots
g
JCZ.
:^:
r
iq:
&-
122:
In this inversion the fundamental dissonance of the ninth becomes
a fifth to the inverted bass, thus —
Roots
#^^^^ A-
-r^
-&
^
-<s>-
In this inversion the fundamental dissonance of the sixth in-
104
THE PKiNCIPLES OF HARMONY.
volves the suspension of the inverted bass itself, producing the fol-
lowing : —
Roots
-i
-r±
Z2:
-o-
- ^— r^
-.■s*-
1^21
In the inversions of the chord of the dominant seventh the various
dissonances will appear as follows. The student will understand them
without further explanation.
^^
-P
-Gh
^
~rjr:z^
-Gf-
:gr"-sep»-^
Z5:
122
^2=jz^=:t
^
d=d:
I^
gd] -ri- b^=Hzg2
1^21
--^-g-
ici
22:
1^2
22:
:q;
-«s>-
JZ^
Z2
Z2
Z2:
"-—(S*-
7 -
t 8
:^S> '
6 -
5 -
4 3
^
z:2
221 :zq:
:22:
ziz:
:z2
22i:r^z
::<S>-
:^S>-
-<S>-
-<S>-
Roots
4 3
7
t 3
THE PRINCIPLES OF HARMONY.
105
m^
-^
:q:
-o-
-«s»-
iq:
Z2:
:?^=;^
1^21
Z2:
7 6
6 - 6
5 4
7 6
i =
:r±:
Z2
z:2:
^i?3:
zz:
Z2
i^z:
Z2:
:^-
B/oots
:<S>-
-<S>-
:«S>-
i I
3 3 - 4 3
In none of these cases has the dissonance of the ninth been admitted,
because it becomes identical, practically, with the chord of the added
ninth, already described ; or, if minor, with the chord of the minor ninth.
The student will now be able to add dissonances by suspension to the
chords of the added and minor nmths and their inversions, by carefully
adliering to the rules given above. We will therefore give a bass with
figures, and with the roots indicated, for him to harmonize.
To render this easier, however, the following rules will be useful. The
first and second have already been given in other words.
I. When the fundamental bass (or root) descends a foui'th, or ascends
a fifth, the dissonance of the fourth may be introduced, prepared
by the octave.
i
=F
12:2:
'Z^
^m
2^
-G>'
?<S>-
lOG
THE PRINCIPLES OF HARMONY.
II. When the root ascends a fourth, or descends a fifth, the dis-
sonance of the ninth may be introduced, prepared by the fifth.
:C2:
-G>-
=F
W^
S>z
1^21
-«S>-
III. When the root descends a fourth, or ascends a fifth, the dis-
sonance of the sixth may be introduced, prepared by the third.
ly. Wlien the root ascends a second, the dissonance of the ninth,
prepared by the third, and that of the fourth, prepared by the
fifth, may be introduced.
i
-^
s
^:
-(S>-
-«s>-
-1^-
~r^
5 —
'JIJZ
-G^
1^21
-<S>-
V. Wlien the root ascends a third, the dissonance of the sixth, pre-
pared by the third, may be introduced.
THE PRINCIPLES OF HARMONY
107
m
'j:^
2:^:
VI. When the root ascends a sixth, or descends a third, there can
be no dissonance.
VII. When the root descends a second, or ascends a seventh, the
dissonance of the sixth, prepared by the fifth, or that of the
fourth, prepared by the third, may be introduced.
- "^ . .^1 J
iq:
-iS>-
-G^-
-^■
A-
-JZll
-fE^-
-^
"C?"
-^2_-
*^2=H
-iSi-
jf5
[When inverted basses are used, and dominant harmonies
added, certain additional rules may be added.]
VIII. When the root of preparation bears a dominant seventh, and
ascends a fourth, or descends a fifth, the dissonance of the fourth,
prepared by the seventh, may be mtroduced.
-St
r^
T^
-G>-
A
r
-IS'T
22^21
P 2
108
THE PRINCIPLES OF HARMONY.
IX. When the root of preparation bears an added ninth or a minor
ninth, and ascends a fourth, or descends a fifth, the dissonance
of tlie sixth, prepared by the ninth, may be introduced.
fc^
=^
:^2:
hS^
-^-
9 ^— 6 I
^
0=
^iSh
221
:P=
1^21
-(^^
1221
Exercise on Dissonances.
)h-t
-G^
'-9-
-Gt^
1^2:
-<s>-
G
-&
~«^
-Gt-
9 8
G
-s>-
G
D
D
7 6
G
D
F=P=
1^-
:r=t
<r>
-<s?-
-s^
:^
i^:
-s^
-^s*
i^
G
G
7 6
c
t¥
5
52
D
G
D
rrs
^
1^2:
-«s>-
hS»-
-(S*-
-<s>-
z:i
-<s>-
3
6
-v —
G
^v_
D
6 7 6
)
c
8 7
% -
D
6 - 5
5 4 4
V
G
D
-H'&H-
t 3
G
7. When two dissonances can be regularly introduced on the same
root, according to the foregoing rules, they may be introduced simul-
THE PRINCIPLES OF HARMONY.
109
taneously. They are then called double dissonances, and the chord into
wliich they are introduced is called a double discord.
Thus we may have the double dissonance of the fourth and ninth.
For example —
i
XT
3
-^Eif-
^
-^S-
-f^-
iq:
We may likewise have the double dissonance of the sixth and fourth.
For example —
J.
-^±
1^2:
It is important to observe carefully the identity of the notes
composing this last double discord with those forming the second in-
version of the common chord. In nine cases out of ten, a 9 must be
4
treated as a dissonance, and not as an inverted triad; i.e. the dissonant
notes must be prepared and resolved according to the rules given in
this Chapter.
Whenever a J is followed by a § on the same bass, we may be sure
it is a dissonance, and must act accordingly. The root will then, of
110
THE PRINCIPLES OF HARMONY.
course, be different from what it would be, were the chord the inversion
of a triad. Thus the root of the first note in this bar,
"^7
^
is not G, but D ; and bears the double dissonance of the sixth and
fourth. Therefore both the B and the G must be sounded, in the
same parts, in the previous chord ; thus —
-«SI
-!&'-
"^
z-tS
^.
-^
T^~
Ib7
5
">%
i. The dissonance of the ninth may be resolved on the sixth, if
the bass ascends a third —
1221
(S»-
.^
'^^-
~jrjr.
ii. The dissonance of the sixth may be resolved on the third, if the
bass ascends a third —
i
--g--
id:
IC^I
THE PRINCIPLES OF HARMONY.
] 1 1
iii. The dissonance of the sixth, may be resolved on the sixth, if the
bass descends one degree —
m
C21
-B
-TTJ-
iv. The dissonance of the fourth may be resolved on the sixth, if the
bass descends a fourth, or ascends a fifth —
-«s-
g
icz:
Z2:
-c;^
-<©-
-&-
T^
:^
4-21
i^
:z2:
1
9. It Tvdll also be useful to remember that while fundamental dis-
cords are resolved on another bass, discords by suspension are resolved
on the sa^ne hass. Also it is well to bear m mind that while funda-
mental discords require no |9repa?Tt^zo?i, but only resolution, discords
by suspension require both ^^j^e^^ara^io?^ and resolution.
CHAPTER IX.
1. In" Chapter VIII, section 3, it was said that " When an interval
of a melody (or of an inner part) is kept back in descending, it is called
a suspension.
It remains now to treat of the other case : " When an interval
of a melody (or of an inner part) is kept back in ascending, it is called
a retardation!'
The principle of the two is the same, only that the name of
suspension is not, strictly speaking, suitable to the case of a note kept
from asce7iding.
2. When the leading note is retarded, the dissonance of the major
seventh hy retardation is produced.
i
XT
221
-fSh-
-<^-
-G^
1^2:
m^
^
22:
A
1^21
This is prepared by the tliird, and can be introduced whenever the
root falls a fifth, or rises a fourth.
THE PRINCIPLES OF HARMONY.
113
When the fifth is retarded, the dissonance of the second hy retardation
is produced.
I
:g:
=F
z:±
is:
-C2.
-G>-
1^2:
■^
id:
:^2"
-Gh-
i^z:
This is prepared by the fifth, and can be introduced whenever the
root falls a fifth, or rises a fourth ; if the fifth is so placed as to be
able to rise a degree.
These are the only fundamental retardations.
The rules for their preparation and resolution are much the same
as those which refer to suspensions.
3. When the two retardations are used together, a douhle discord
hy retardation is produced ; thus —
-A
-Gh-
T^
1^2:
The student will be able to work out the application of these dis-
sonances to chords with inverted basses, by reference to the method
employed in the case of suspensions.
Q
114
THE PRINCIPLES OF HARMONY.
4. Eetardations and suspensions may likewise be variously combined
of which we will now give some examples.*
i^tett^
^Eg
gj , — &^
^m
'JOH
f^-
:^:
:JJ^
-et-
'JUl-
-^-
TIT"
9 §
^
P5
-^
-G>-
t^-=^ 3
-&r
;*i
-Gh-
.■q:
-<s^-
:^
i^.
'-P- -B- ^
-(^-
P2:
iS
n^z
-(S*-
s
-/^
-&-
:JtP
e^
-<s>-
-<^-
-JZT-
%
S!
-^
-fS'
"O"
-<S>-
^^ C^
1^
m
^-=^
2d=^
:c2:
-TD
rcz:
53
-<Si-
■•s*-
-<^-
12:2:
1^21
9 8 9 8
* See short Examples at the end of this work, No. 14.
THE PRINCIPLES OF HARMONY.
115
1^2:
q:
(S^
:p^
-«s^
-<s-
z:2:
J,
-(S*-
:z2:
«=^
-^-
<s>-
-<s>-
^— ^
rD— \—r:3
P=^
(S>-
:^
:?2:
T^~
iS>-
-(S*-
7 6 2 6
* This chord will be explained in a future Chapter.
-<^y(S^S(S/t)^y^>-^
Q 2
CHAPTER X.
1. In Chapter IV, section 2, Logier's rule was quoted, that " Wlien-
ever we use a [fundamental] seventh, and thus proceed to a new key,
we modulate into that key." And in the same place a regularly pro-
gressive modulation of this kind was given.
We will now give it in a more concise form.
g
^^^
-Gh
7^
a
a^g
^aiS^:
22__L^
fe^=^i
-^^
s^
K>g3 l^'Sj^-
lu^ r:^
a
122^
^51
e:
ip2=S
m
(S>-
(S>-
:z2
^
isi
6
b5
b7
be
b5
be b7
b
.be
b5
b
^
-t9-
:^2:
't^.
3§^
-<5<-
Z2"
'=is:
-m^m^
:^2z:bs^:
s-^
^
«:
t^
«S— "
i^
3^
^3^
^e^«i
^!S>-
xz
1^
t^
122:
tz=±
:^±
:^
e
4
i
D7
57
THE PRINCIPLES OF HARMONY.
117
This may be still further curtailed by the following method, whereby
the octave changes to a dominant seventh in each successive chord.
^^
r r
:^*£=^W
^
fer
'^^
v^-
VSh-
m£
'IZL.
--^
tec
^
i*:
b5
be
b.5
be
/rs
1^^-
^m-
S*— jt?
tl*-
l3:
It-i
g
_a
t:;^
-(S«-
-^^
^
I^X
z^:
I ^^ — '1 I I
-^
be
-^— r-^i-
iS:
1f^
4j-
2
41-
2
(i
C7
\:^
But the progression can even further be shortened by omitting all
the tonic triads and inversions, and regarding each successive chord of
the dommant seventh, or its inversion, as embracing the proper resolution
of the preceding one.
In this form the result will be as follows : —
:^^=^;
^E
fc
^&- -&-
&g=5g
%gHg='S=fi^
Jld
-d-
A
^S
I^S
^21
2?:^
*!^-
^«S-
b5
D6
b5
where we may observe, 1st, that the leading notes all appear as though
118
THE PRINCIPLES OF HARMONY.
they were resolved downwards, iii defiance of nile. But the ear supplies
the omitted intervening tonic chord in each case, and thus is satisfied.
Moreover, the progression of the roots, each becoming a dominant to the
next, quite overi'ides the effect of the UTegular resolutions of the leading
notes, and carries the ear with it headlong in its downward course. There-
fore this mode of resolving leading notes is allowed, as a licence. We
observe, 2ndly, that the bass and treble both proceed chromatically by
similar motion, and at the interval of a diminished fifth. And we
observe, 3rdly, that the first and third inversions are used alternately.
Another form of this series is the fundamental position, thus —
'J-l^^-
(S»-
E^i^S
^^ ^m ^^^
1C5:
:il^
m
Z2L
G^ —
t'^
'ZH.
^^
i^
i^:
-cr
D7
D
b7
b7
b7
b7
D7
E7
&c., &c., &c.
Such a series is called a " sequence of dominant sevenths."
2. Chords of the added ninth, as well as of the minor ninth, are
susceptible of similar treatment. The latter particularly so, in the
form of diminished sevenths, thus —
sd^
i
BADE
where, by the enharmonic substitution of Gtt for A^ in the fourth chord
THE PRINCIPLES OF HARMONY.
119
we change the bass from G (which it would have been) to E, and
thus confine our progression of roots to the three, E, A, and D. By
similar enharmonic alterations of the intervals of any of the chords, we
can of course introduce any roots we please, as was shewn in Chapter IV,
section 5.
We may call this series a " sequence of minor ninths," or, if we
please, a " sequence of diminished sevenths."
3. If we wish to modulate in the contrary direction, i.e. from
tonic to dominant continually, our process of curtailment will not be
quite so complete as that just described.
It will be well to begin our work thus —
^-
^:
-H — e*
— 1± — 1^1^.
*^p:
-& — «i^.
:2:
e>-
Z2^
sa
os^
i gU-4^— 3
i
T^
B-
_^_
IQ-
cz:
(S>~
'^-
\?±.
1^
-^-
^£S=ii=^a=^s
^^S?i
M-=jf
q:
S3i
#^
i^:
@^p^=^
jzi
-:^=^$^-
<r ^ 1
I?^
W-
i^
■ t
jf5
«5 -
t -
b7
&c., &c., &c.
120
THE PRINCIPLES OF HARMONY.
This progression may be curtailed as follows-
^S=i0
-G»
1^
:p
:«^
-^--
i^zzd
#5
%
This may be called a " reversed sequence of dominant sevenths."
Also, such a phrase as —
±^
^_
^m
:fe
%^
53^^^
'm
&c., &c.
may be called a " reversed sequence of minor ninths," or a " reversed
sequence of diminished sevenths."
All these sequences necessarily involve perpetual modulation from
key to key. Hereafter it will be necessary to explain another kind of
sequence which never modulates out of the key at all.
It will be a good exercise, meanwhile, for the student to introduce
dissonances by suspension and retardation into the various sequences
we have now described, and to vary the inversions employed in as many
ways as he possibly can.
THE PRINCIPLES OF HARMONY. 121
4. It is now time that we should give a few general rules for the
harmonizing of melodies, which wiU be found useful to the student.
They have, for the most part, been selected from Looier
i. When the fifth of the dominant chord is in the melody, it is
a good plan to take the third of the root in the bass, and
harmonize it as a §.
5
ii. When the thii'd of the dominant chord is in the melody, it is
best to take the seventh of the root for the bass, and har-
. ., 6
monize it as a 4.
iii. When the third is m the melody (whether of dominant or
tonic), and the seventh cannot be taken as the bass, adopt
the fundamental bass.
N.B. These three rules may be reversed, i.e. the intervals of the
bass and melody may change places.
iv. When the seventh is in the melody, the fifth may be taken
as the bass, and harmonized 4.
3
V. When the fifth is in the melod}^, the seventh may be taken
as the bass, and harmonized 4.
N.B. Whenever an accidental sharp occurs in a melody, we may
regard it as indicating a modulation, and it will usually be
the leading note ascending to the new tonic. Similarly an
accidental flat may be regarded in most cases as a minor
seventh descending to the third of the new tonic. Hence the
use of the following rules.
122 THE PRINCIPLES OF HARMONY.
vi. A note of modulation which ascends a semitone, modulates either
to the key which lies a semitone above it, whether major
or minor, or to the relative minor of the above-named major
key.
vii. A note of modulation which descends a semitone, modulates
either to the key of which it will be the major third on so
descending, or to the relative minor of that key,
viii. A note of modulation which descends a whole tone, modulates
either to the key which lies a whole tone below it, or to the
relative minor of that key, or to the key of the fifth below
the note to which it descends.
ix. A note of modulation which ascends a whole tone, can only
modulate to one key, to which it will be a major third after
thus ascending.
X. When any note is repeated, it may modulate to the key of
which it is the fifth.
ix. A note of modulation which ascends a fourth or descends a fifth,
modulates to a key to which it will be the octave, or to a
key to which it will be a fifth. In either case the modulation
may lead to a major or minor key.
CHAPTER XL
1. Hitherto in deriving the various intervals of tonic and dominant harmony from
the natural harmonics of a root, we have confined ourselves to prime numbers, except in
the solitary instance of the major ninth.
This intei-val occurs in the third octave,* and is produced by i of the length of
a pipe or string, the ratio of the velocity of its vibrations being to those of the root
as 9:1.
Compared with the harmonic No. 3 (which is the twelfth of the root or the fifth in
the first octave), it is as 9 : 3, or as 3:1. Accordingly its pitch is the twelfth (or octave
fifth) above that intei'val.
It is on account of that intermediate link between the major ninth and the root
that it is a very pleasant intei'val to the ear, more so indeed than the untempered minor
seventh of nature, which is a prime number, 7.
2. The fifth of the root, or dominant, has been made the basis of the whole super-
structure of harmony; as has been sufficiently shewn in the preceding pages, especially
in Chapter II, sections 2 and 3,
Supposing now that it were found necessary to have a second derivative root, in order
to explain the formation or origin of certain fundamental discords, what harmonic of the
original root would ofier the best hope of a satisfactory solution?
Clearly that one which afibrds new secondary harmonics the least remote from the
tonic root.
* See Paradigm at head of Chapter IV.
B 2
124 THE PRINCIPLES OF HARMONY.
The first which would suggest itself would probably be the major third, which occurs
in the second octave, and is represented by the ratio 5:1. But on examining the
harmonics of it, Ave find that its perfect fifth is the same note (No. 15) as we already
possess as our dominant leading note, and that the next foreign interval it gives us
(i.e. its major third) is the augmented fifth of the root, or the augmented octave of the
dominant ; which would be represented by the No. 25, if the paradigm in Chapter IV
were carried far enough.
It is indeed possible that this interval so derived may occasionally be of use to account
for the introduction of the augmented triad, for instance —
and for certain progressions and modulations to be mentioned in their place. But on the
whole we may conclude that the major third of the root will not answer our purpose
as a supplementary or secondary root, since its available harmonics are both few and
remote.
Still less will the fundamental minor seventh answer ; for not only is it still more
remote, but it also produces intervals, every one of which would require the same great
amount of alteration and tempering as is required in its own case.
We are therefore driven to the major ninth, which will be found to answer well.
It is not so remote in reality as it at first sight appears to be, because it is produced
by a square number 3x3 = 9, and is therefore, as an interval, the fifth of the fifth of
the root ; or rather, as we should say, the dominant of the dominant.
It is therefore as intimately connected with the dominant, as the dominant is with
the tonic.
Moreover, as will be shewn when we come to speak of "the tonic pedal," all the
harmonics of this secondary root, even up to its minor and major ninths, will sound well
when combined with the tonic and dominant roots, separately or in conjunction.
"WTienever then discords of a fundamental character occur, which cannot be accounted
for by referring them to a dominant root, they must be classed as derived fi-om the
major ninth of the root, which is the dominant of the dominant, and which, from its
place in the diatonic scale as the second degi-ee, is named "the supertonic."
THE PRINCIPLES OF HARMONY.
125
3. But it may be asked, ' Do cases often occur which require tliis expedient to solve
them ? ' To which the reply is, That they are of constant occurrence, and that it is mar-
vellous that theorists should have gone on confusing themselves with inadequate explanations
for generations past, without having recourse to so simple a method of accounting for
difficulties which were continually arising.
The earliest treatise in which recourse has been had to this method of explaining
certain choi'ds (as far as the author is aware), is Day's " Treatise on Harmony," London,
1845. He only aj^plies the principle to the "chord of the augmented sixth," but it is
really susceptible of a much wider application.
However, as this is perhaps the most obvious case, let us consider it first in order.
4. If we take the second inversion of a dominant seventh, such as
^
Z2:
and lower the bass-note a chromatic semitone by an accidental flat, thus-
:lfe
zu-
[^2-
we shall produce a discord which is called "" a " French sixth " by many
writers, and is very common in modern music.
* Vide Crotch, "Elements of Musical Composition," pp. 71, 72 (4ti)., London, 1812).
126 THE PRINCIPLES OF HARMONY.
Again, if we take the second inversion of a dominant seventh-
=#^
z:^:
omitting the octave of the root, and lower the bass note as before —
m
dm:
we shall produce a discord which has been called an " Italian sixth"
This* has been in use in Italy and elsewhere for more than 150 years.
Again, if we take the second inversion of the chord of the minor
ninth, omitting the root, thus —
?=^tfQ,
<s> '
22:
and then lower the bass note, as before-
* Logier in his " System of the Science of Music " calls this chord simply the " chord of the sharp
sixth." But as the sixth is not sharp only, but augmented, it is better to call it, as Day does, the
" chord of the augmented sixth."
THE PRINCIPLES OF HARMONY.
127
i
t^
^
e/
15^21
we produce a discord,"' which has been called by the same authorities
the " German sixth."
These names are very unmeaning and unsystematic; but as they are
frequently employed, it is well for tlie student to make himself familiar
with them, and with the chords they are used to denote.
5. Now it is clear that in each of these chords every note, except
the loioest tvhich has been altered, is derived from the root D, and
belongs to that key of which D is the dominant, viz. G.
But if the chord be played several times in succession, alternately
with the chord of G, thus —
S^
-&■
T T T T T T r f " r
T^
if
r
f
r
IS
r
r
-G>-
the ear will be anything but satisfied, and will desii'e to hear the har-
mony of C ; as thus —
* Logier Ciills this tlie •' compuuiid sharp sixtli."
128
THE PRINCIPLES OF HARMONY.
-et-
■x^
r
-^Ijr
:^:
1^2:
ii^
-&
:^:
12^
-s»-
This is enough to excite a suspicion that G is here rather a dominant
than a tonic root.
And this suspicion is strengthened by the fact that, after the chord
of the augmented sixth, the ear will not tolerate the minor common
chord of G ; as thus —
:Jf^
\y^-
-o-
-&-
In this case, the ear imperatively calls for the majo?'; thus proving that
the leading note is required, which is the only third in the dominant
harmony.
We may therefore fairly conclude that the chord of the augmented
sixth must be followed by a dominant chord. Therefore its root (so far
as it has been discovered hitherto by us) must be the dominant of a
dominant, i.e. the supertonic. Therefore it belongs to the key of which
it is the supertonic, and of which the chord into which it resolves
is the dominant.
THE PRINCIPLES OF HARMONY.
120
6. But now the question arises : What is the root of tliat altered
bass note which could not be derived from the supertonic root 1
Clearly it is the minor ninth of the dominant, and being itself
consequently an essentially dominant interval, it is the cause of that
tendency to the real tonic wliich was pointed out in the last section.
7. It will be seen that the interval of the augmented sixth is
formed between the minor ninth of the dominant root, and the leading
note of the supertonic —
and that the only interval derived from the dominant root is its minor
ninth. All the other intervals of the chord, whether it be in the forms
commonly called the French, the Italian, or the German, are obviously
derived from the supertonic root. Examples —
3
^
^:^
-G>-
-G^
s
-<s^
-<5h-
^^-
1^21
22:
Z2;
7
m
^
icz:
-T2L
:z2:
-<s>-
fS^
%
i^z:
-<s»-
f&
m
121
-<5f-
Roots. '^9
Ib9 « 'I
S
ib9
d7
5
t
130
THE PRINCIPLES OF HARMONY.
Where there are two roots, the figures of the secondary root are
written above, and those of the primary root below.
8. The following rules may be found useful in resolving this
discord : — *
i. If in the resolution both notes forming the augmented sixth
move, the lower one must fall, and the upper one rise a
minor second, to a note which is either the octave or the
fifth to the root of the next chord. Example —
:^
-^
-^.^^-— B-!
#2:
-f5> —
-<s»-
^
'-^^
i
#
ii. If in the resolution one note only moves, while the other
remains still, the moving note may approach the other by a
clwomatic semitone. Example —
i
^
*=t
""C3"
><e-
r
1^
I
iii. The intervals derived from the supertonic root are treated
(except as restricted above) just as if it were a simple
dominant, and as though the minor ninth of the other root
* Vide "Treatise on Harmony," by Alfred Day, M.D. (8vo., London, 1845), p. 124.
THE PKINCIPLES OF HARMONY.
131
were away : provided only that they make no false pro-
gressions with that ninth.
We will now give some of the many different ways of resolving the
chord of the augmented sixth, whether accompanied by a third, a fourth,
or a fifth ; in other words, whether Italian, French, or Geiman.
No. 1.
No. 2.
No. 3.
^=4^:
:^
:i^
-«^
f
1^2:
-o-
»p=^
Z2:
--gr
r^
:^
:ff^ — r~^
Z2:
-G>-
-^^-
m-
-fer-
'-^-
^-^ A
e
1221
iq:
:cz
8 D7
8 a?
8-pD7
E/OOts. 7
^:
-<sf
-s^
122:
221
b9
-^-
^-
b9 8 C7
07
bo 8 C^
No. 4.
No. 5.
No. 6.
^
g
i
2:2:
3?:
IS
"^21
:it^
-<^
*i
-iS'-
-(S>-
^^sJ-^
1^21
^S*-
-(S>-
:^
:q:
]2to:
zz:
r=r
jis-
Roots. ''9
3
b5
D6
4
:^
b9
-&
1^21
-cz:
-^
b9 \
S 2
b9
-O-
22:
132
THE PRINCIPLES OF HARMONY.
No. 7.
No. 8.
No. 9.
e
(S>-
T^
1^21
Roots. 7
-<s
3
l.- ^ l I
:^^
«^=^
rr
:g:
:flg=r3
rf
fs>-
^21
'TTJ'
^ t
HSt
s
Z2:
^
g — ^
B 6 07
B4 4 6
-^
3
^21
^g--
122:
zz:
:^2:
-e?
b9 o -
4 3
b9 6
4
-^■
b9 6 d7
4 5
No. 10.
No. 11.
No. 12.
i
Z3=^
3
i^
2:3:
3
1^21
:22Z
1^2:
«r^
:js
-(Si-
"C7"
:S^
rr
^^
:?=:
•TTT
r=fe
^2:
:^:
^
^21
-©-
-s^
Z2:
t 3
Roots. 7
1)
b9
22:
:^
22:
b9
b9
22:
THE PRINCIPLES OF HARMONY.
133
No. 13.
No. 14.
No. 15.
^
— <^ —
s
22:
-^x
-^^
-rj
^
k^-
r
■^-
=»s^
-(S*-
T:^
Z2Z
^
rr-
: y-R
^
?2:
^0-^
I — I I —
^
irz.
'i::iL.
- 8 07
Roots. 7
^-
n4 -
t 3
? -
G4 -
S
iq:
:^
:^z:
icz:
•59 8 C:
59
4 3
(59
6 5
4 3
No. 16.
No. 17.
No. 18.
^a
*=^=R=
^r-q»-
*»-5
^
zz
-iS'-
e
221
Z2:
P
-TTT
8 D7
Roots.''? =
« -.
^
Z2:
8 07
b5 -
^? =
59
1221
5 —
4 3
bo
b9 -
-« #-
99
fl6
4
'% i
Z2:
134
THE PRINCIPLES OF HARMONY.
No. 19.
No. 20.
No. 21.
No. 22.
No. 23.
No. 24.
^
~r2L
1ZL
^='^:
:z2:
-Gh-
icz:
^__&jJ_J_^_g_tii:if^— 5^
~-^y-^-^-=
:b^i=^
Z2:
i^z:
i^
221
E2
6 B7
Roots. 7
^
22:
G7
Q7
7
h9 57
^2:
1^21
6 B7
-<S-
-(S' &
b9 £7
122:
THE PRINCIPLES OF HARMONY.
13;
No. 25.
No. 26.
No. 27.
^
-<s»-
i^3— -^Qizt:^
Z2:
^rX
:^^
1^2:
32:
4;^
a
2iz:
"c:?
221
^<^-
j&-
^
--^
^^
^^
J^2.
:z2:
ifi
icT
:^z:
Z2:
hS^
-lS»-
|S7
b9
B7
B7
'9o
Roots. 7
b?
8
r»\'
1
l^' r-
,
v^ cj
rj
r^
'-A
r^
^
<si—
^
<si— J
— s' —
(»H
b9
B7
b9
No. 28.
No. 29.
No. 30.
1^2:
I^X
:^
221
-»Sf-
-«s
22:
-'^^^ St
^-g--
°=3S=^
^-g:-
^^-
icz:
^s>-
1^21
^S-
221
B6
4)-
Roots. 7
%
b9
C7
be
4
221
D6
7
%
b9
'I
221
07
7
^-
b9
"b7~
be
4
2z:
136
THE PRINCIPLES OF HARMONY.
No. 31.
No. 32.
No. 33.
-JTJlZ
^-g--
-^x
-«s-
:^i$^^2zte:
_cz;
^(S>
221
^<^-
iq:
^<s-
i^z:
B6
B6
4
Roots. 7
b5
b9
7
be
4
b5
D6
4
-<S1
221
-SI
:q:
-(S
:^
22:
b9
-<Si^
B7
b9
-&■
D7
b9
D7
&c., &c., &c.
With reference to these examples, it will be well to observe that,
in every one of them, the supertonic discord is first resolved, and,
after that, the dominant discord.
In Nos. 3, 4, 26, and 27, the Eb descends to C before going to
its proper resolution D, in order to avoid consecutive fifths with
the bass.
In Nos. 5, 8, 9, 12, 15, 18, 20, and 21, the dissonance of the sixth
by suspension is not duly prepared. But it may be remarked, that
the rule about preparation is often relaxed in the case of the dis-
sonance of the sixth ; especially if it have the fourth with it, and
if this latter dissonance be duly prepared, which is the case here,
all through.
In Nos. 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, and 33, the inter-
vening octave to the dominant root is omitted, for the sake of curtail-
THE PRINCIPLES OF HARMONY.
137
ment, and the minor seventh introduced by chromatic descent of the
leading note, as explained in Chapter X, section 1, shewing how a
chord of the augmented sixth may form part of a regular sequence.
In Nos. 28, 29, 30, 31, 32, and 33, by the use of the fourth
inversion of the chord of the minor ninth, a resolution is effected on
the second inversion of the tonic triad. But this is hy no means
recommended, although it is strictly speaking allowable.
9. By the use of an enharmonic change, the augmented sixth and
the minor seventh become interchangeable ; by wliich means great
variety of modulation may be obtamed : for instance —
Roots
b7 b9
where these intervals are enharmonically substituted for each other in
two places with good effect.
A particidarly good instance of this way of using the enharmonic
change, in treating the chord of the augmented sixth, may be found
in Dr. Crotch's Oratorio of "Palestine," in the chorus "Let Sinai tell."*
* See Examples at the end of the work, Nos. 4 and 8.
T
138
THE PRINCIPLES OF HARMONY.
By combining this plan with the alternate use of the chords of
the diminished seventh and augmented sixth, the following curious
chromatic sequence may be constructed —
d.
i^gP
-Q.
&^Jgl
l2^
:^#:^
:t3^:^=fe
m
--^
fS>-
^t^^it;^
-&
b7 ,6
5 B5
6 ^6 ^6 6
4 B^ B5 4
I
,b6
b4
b5 b£
4 5 B5 4
&c., &c., &c.
This formula may be found useful when a sudden modulation to a
remote key is required.
10. The chord of the augmented sixth can be inverted-
FiTst Inversion.
^
< g^ %
Roots.
1%
D
G
-<s>-
U-
^
-&-
j^.
-TTT
bfi
4
G
-t
b
C
i
-^^
=i^
b5
D
G
:i
^3=^
tted:
:z2:
G
-^
iq:
B7
THE PRINCIPLES OF HARMONY.
139
U
^
s;
:z2:
■^-
'^
-^-
^gzz=t:g:
^-
^ '^
^
:z2:
€r-^p -
ie
^
b7
Roots.
1
DT
G
i%
D
G
G
Second Inversion.
i
_^
r^z
^E^
^^
-^
-C^
^ P P-
:?2:
-^^ — ^
:f^
-C^
i?^
-<s>-
^
v^
^^
i^i:
^
Z2:
-fe-
Roots.
D
G
G
ii n
G
G
«'6
D
G
6
G
i
^§=t
r^
<^
(S I 'T^ L^'
Roots.
D
G
G
Third Inversion.
-^
Z2:
or,
^^=^-
Roots.
-s»-
D
G
C7
G
T 2
140 THE PRINCIPLES OF HARMONY.
Note. — Dr. Day denies that the interval of the augmented sixth can be inverted, on
the ground that the harmonics derived from the secondary root must not be placed below
those derived from the primary.
But in the first place we may imagine the roots to be distant as many octaves below
as we please, so that we can always suppose the two roots to be at their proper interval
apart, viz. a twelfth.
And, in the next place, the chord has been used in an inverted position by many of
the most esteemed composers with excellent effect. The earliest instance being in Weldon's
Anthem " Hear my crying," near the end of the concluding movement, (Vide Boyce's
Collection of Cathedral Music, vol. ii. p. 218, of the editions of 1768 and 1788.)
Weldon was a pupil of Henry Purcell, and died in 1736.*
11. As this seems the most fitting place for explaining the meaning
of what is called " false relation," or " cross relation f it may be stated
that by these terms are signified certain harmonic incongruities between
two different parts or voices, which are exceedingly offensive, and
generally forbidden.
The general rule is, that " when a note of the same name occurring
in two successive chords is altered by an accidental, it must be sounded
by the same part or voice, otherwise forbidden false relations will
ensue." Thus —
-23=1 IS wrong, but fm — 3 . ^~" i^ right.
-^-
-^r
The reason of which is, that the false relation gives the impression
of two different keys simultaneously used; for, in the former and incorrect
example here given, the treble —
C2:
• See Short Examples at the end of this work, Nos. 1 and 2.
THE PRINCIPLES OF HARMONY.
is part of a melody iii C minor, while the other parts —
141
2!>
-&
clearly belong to C major.
If, however, the note to be altered is doubled, it need only progress
accordmg to the above rule in one of the parts, otherwise consecutive
octaves would ensue —
—9 f^-
S
E
-iS*-
P<S>-
i
By licence false relations are allowed in purely dominant modu-
lations, when occuiTing between different symmetrical phrases or groups;
thus —
•tS»-
^
r^rr
-is>^
5?
i
-tr
r r T r
where the two phrases are enclosed in brackets, and the false relation
of the G and Gitt is not harsh or unpleasant.
Where the roots proceed regularly -from dominant to tonic in a
descending cycle of fifths, or where such regular cycle is symmetrically
reversed, false relations may sometimes be tolerated: for example —
:^:
=z±
^
or
z:^
iSh-
-^1
G
D
' — 3j^
D
G
142 THE PRINCIPLES OF HARMONY.
because "the reo-iilar resolution of the dissonant notes in each case con-
soles the ear, and mitigates the harshness of the false relation.
Great composers have occasionally used considerable licence in the
matter of false relations ; but we do not recommend the young com-
poser to follow their example. We have given some curious and
exceptional instances at the end of this work.*
It may also be observed that in many cases the interposition of an
intervening chord will not save the false relation : for instance —
^
or even /m;— ^rL:*r^
In each of these cases the harsh effect of the false relation remains,
because the intervening dominant chords belong equally to the major
and minor mode, and therefore do not interfere with the impression
produced by the two incongruous tonic chords between which they are
placed.
In strict counterpoint the rules against false relations are even more
exactly enforced. The above observations, however, will suffice for the
student's ordinary guidance.
* See Short Examples, Nos. 16, 17.
CHAPTER XII.
1. In dealing Avitli the chord of the augmented sixth, it was only-
found necessary to make use of one interv^al belonging to the dominant
root, i.e. the minor ninth; all the rest bemg derived from the secondary
or supertonic root.
But whenever we find a chord with a minor third and a minor
seventh, we may be sure it comes from two roots.
For the minor seventh demands a majoi' third, or leading note, to
enable it to resolve regularly.
If we substitute a minor third, our root is necessarily altered.
Thus —
Js
,G>^-=^
«
:g:
%-
-&-
7
is a regular resolution of the chord of the dominant seventh of D to
the tonic common chord of G.
Here, of course, the leading note FJf rises to the octave of the
tonic, G.
144
THE PRINCIPLES OF HARMONY.
But let us substitute F natural, and a totally different effect will
be produced on the ear —
-<s>-
I^X
ziz:
For here, evidently, we must regard the F natural as the minor
seventh of the dominant root G, while the A and the C are as evidently
the fifth and minor seventh of the supertonic or secondary root D.
This chord then has clearly a double root —
-Gf-
S:
-^-
■<s»-
-iS>-
'CJ'
s
-&-
-<s>-
"Z2;
Z2:
7
D
G
7
G
and the supertonic discord is resolved first, and then the dominant dis-
cord, just as was found to be the case in the chord of the augmented
sixth.
2. This is a most important chord in several ways ; for, in the first
place, its first inversion has been known to almost all theorists and
didactic writers on harmony as the " chord of the added sixth," or
" great sixth " —
THE PRINCIPLES OF HARMONY.
145
^q:
:^2
-<Sf-
and they treat of it as though it were the triad of the subdorainant
with the arbitrary addition of this "added sixth," which is an interval
entirely foreign to that root.
It is submitted that such an explanation is wholly unwarranted and
unphilosophical.
For the harmonic sound No. 13 ( see paradigm, Chapter lY) is
clearly connected with donfiinant harmony, if with any, and is most
remote from that part of the harmonic series which gives the tonic
triad. Therefore, if tliis chord belong to the root F, it can only do
so on the supposition that F is a dominayit root, and that it belongs
to the key of B flat.
But as the chord in question does not belong to B flat, and does
belong to the key of C, it follows that F cannot be the root of the
chord.
Nor can D be the root, as we have abeady shewn, because the
third and seventh from such a root camiot botli be minor, as they
would be in this case.
C, again, cannot be the root, as the F would be a fourth, and the
A a sixth — intervals which are not, strictly speaking, natural harmonics.
And even supposing we regarded the F and A as represented by those
very imperfect and doubtful sounds, Nos. 11 and 13 in the series of
harmonics, we should still not find them to be connected with C as
u
140
THE PRINCIPLES OF HARMONY.
tlieir root, for tliey would then be only applicable to C if it were
regarded as a dominant root ; but this is not the case, for the chord
does not belong to the key of F, but to that of C.
Again, Dr. Day argues at length to prove that the root is the
dominant G, bearmg its fifth, seventh, ninth, and eleventh.
But to this explanation there are several grave objections.
The first is that the F in this chord, especially when in the bass,
has a habitual tendency to rise, and is quite devoid of that peculiar
doiivncard character which marks the dominant seventh : nor is it a
sufiicient answer to this objection to reply that this downward tendency
is destroyed by the absence of a leading note; for an ordinary dominant
seventh with the thii'd omitted, still retains the tendency to descend
very unmistakeably, as the following examjole will sufficiently shew —
S
-s>-
-»s-
-Gh-
' 7^-
Of course it may be said that this argument also applies to the
explanation given in this section. But it should be borne m mind
that we hioio the downward tendency of the fundamental seventh,
and connect it in our minds with the great flatness of that interval
as it a^Dpears in nature ; whereas we know that a real perfect fourth
or eleventh does not exist anywhere among the harmonics, and that
the interval which occupies the eleventh degree of the series, and is
supposed to represent the fundamental eleventh, is so very sharp as to
be much more like an augmented fourth or eleventh than a perfect one.
THE PRINCIPLES OF HARMONY.
147
Therefore it is against the indications of nature to treat (jf the
fundamental eleventh as thougli its natural tendencies could form the
basis of any argument.
The next objection is the unnecessary intrusion of the eleventh as
a fundamental harmony.
For in all cases in which this interval is supposed to exist, it may
equally well be considered to be either the minor seventh of a secondary
root, or a dissonance by suspension. Wliy then intrude it here ?
3. Without going further into this matter, let it then be assumed
that the " chord of the added sixth," as it is generally and erroneously
called, is nothing more than the first inversion of the chord of the
minor seventh with a minor third ; and that it is derived from two
roots, the dominant and the supertonic.
This last chord has three inversions —
^
-<s
■-3
^
22:
g-
/I
1 r-'
' 1
<rv
1
D
G
i
-B-
s^
"C7
7^C2.
1^21
^ G
G
When used in the minor mode, the A, which is the ninth of the
root G, becomes a minor ninth, A flat.
Or, taking the key of A minor, as before, for our examples, Ave shall
have the chord And its inversions as follows —
u 2
148
THE PRINCIPLES OF HARMONY.
i
•7
m
-s>-
7
B
iP
^^_
_c:z:
7
E
-o-
C2z:
-^
Z35:
_Q
T^
B
IE/
1*^
t
?:^
_^
:?2:
A
=^^
B
VE
T^-
7
E
l^^
-(S>-
i^:
^
A
6
4
2
B
6
5
E
-<s>-
4. Sometimes, especially in modem music, the regular resolution of
the first inversion of this chord is curtailed by the omission of the
dominant chord, which usually intervenes between the double -root chord
and its final tonic resolution. This curtailment may take place either in
its major or minor variety. Thus —
S'
B
^
IS"
221
or
-s>-
1^21
-&:
^
icz:
1^21
-^
m-
-Gh-
221
5
F
D
5
D
6
5
B
E
Sometimes the minor discord is followed by a major tonic triad,
after the manner of what is called a " Tierce de Picardie " (see Chapter
XIII, 3), when the follomng result is produced — *
* See Short Examples at the end of the work, No. 18.
THE PRINCIPLES OF HARMONY.
149
'Gh
^
S
-^
^
122:
And sometimes this occurs when the whole passage is in the major
key, and when the preceding chord is also itself major, thus —
^^
-G^
221
In all these cases the chord has been preceded by that of the sub-
dominant; but this is far from being always the case. Moreover, the
uninverted chord of the minor seventh and third, as well as its second
and third inversions, may occasionally be treated after a similar manner;
as will be seen by the following examples —
n tf
1 1
1 I
^ J
y ft u. ^ r^r-i
^^
<-^ 1 '^-J,^-)
^^
>^ f^^i^^
s<
■ -f-j
/T it H f^^
c^
r^ hr^^^
•^^
^^
if\ ^ ^ <!--^
rj>
+f^ h^r^
<^ ■■
fA r^
rj
~ C^
\5 ; r J
ft<^ w <^
rD
CJ o
c^
(^
c;i'
/^' ik ] r^
(*3. 5_y. ■ -^
^^.^^
rj
r^
r:,*
V^ tt
&^
■-0
— ■ — m '
■ c>-
B
E
D
7 b5
« D
E
A
1
E
D
Bad.
150
r
(S>-
m
<^
-G>-
THE PRINCIPLES OF HARMONY
-<s»-
:^
IS
:g^
3g=g=t=5g
as:
&=;^
>s>-
6
44
'6
-(S^
:^
-«©-
Bad.
'^-<r<&y(S'<$!kSyf>^^^~^^^
CHAPTEH XIII.
Of Cadences or Closes.
1. We are now sufficiently advanced to speak of Cadences, or, as
they are sometimes called, Closes.
It is most important in a piece of music that every period, or
separate clause, and especially the conclusion of the whole, should tell
the ear exactly what it is intended to convey. It may be compared
to ordinary speaking in this; for it is evidently essential to good
speaking that the inflexions of the voice, as well as the words
employed, should indicate whether any sentence be intended to be
a final conclusion, or only a temporary but incomplete rest in the
sense, or an exclamation, or an interrogation. So in writing, it is
necessary carefully to observe correct punctuation for the same reason :
the sense will often depend in a considerable degree on the right use
of the various stops, the comma, the semicolon, the colon, the full stop,
and the symbols of exclamation and interrogation. In music the same
sort of result depends on the right use of Cadences; and they are
accordingly various in kind, and in their application.
2. The perfect cadence, or full close, is composed of a dominant
triad, or a chord of the dominant seventh succeeded by a regular
152
THE PRINCIPLES OF HARMONY.
resolution into the tonic common chord, which last should always be
placed on the strong place or " down-beat " of the measure. Example —
-^^
w
--^
-Gh
z:^:
:q."
22:
It is still considered a perfect cadence, even when dissonances and
retardations are introduced into the chord ; although they somewhat
weaken its effect, in most cases. Example —
The perfect cadence corresponds exactly to a full stop in writing,
and shoidd be employed whenever any portion of a piece of music is
brought to its completion, and perfect rest is therefore called for.
3. If the piece is serious and solemn, and it is desirable to intensify
the effect of the final perfect cadence, it is usual, especially in sacred
pieces, to add to it a plagal cadence.
This cadence is composed of the major or minor triad of the sub-
dominant, followed by the major triad of the tonic. Examj)le —
THE PRINCIPLES OF HARMONY.
153
"T j-
122:
Z2;
-&*-
%■
Both triads, properly speaking, should be introduced on the down-beat
of the measure.
It is often allowable and desirable to modulate regularly into the
key of the subdominant, so as to introduce the plagal cadence ; and here
also dissonances are admissible on the final bass-note. Example —
i
2^
tJ^ g^-
:^
e
:z2:
zz
-Gh
'^ ~^zj-
-g - ^-
-/S
r
-i::^
T^
:t^
fS*-
:i^
-<s>-
-<s>-
^
2i:
f
where the first four notes contain a perfect cadence in A minor ; after
which follows a regular modulation into the key of the subdominant
D minor, including the dissonance of the ninth by suspension, and thus
forming a regular plagal cadence to the final chord of A major.
When the major third is thus mtroduced into the final chord of
a piece in a minor key, it is called the " Tierce de Picardie," from the
district in which it is said to have been first used in this manner.
The plagal cadence may be used also without a j)revious perfect
cadence, provided the impression of the key in which the piece is to
conclude is sufficiently strong on the ear.
X
154
THE PRINCIPLES OF HARMONY.
This precaution is requisite, to avoid a craving after the key of
the subdominant after the piece is concluded, which is certain to ensue
if the final chord partake in the least of a dominant character.
It is allowable to substitute inversions of the subdominant triad for
its fundamental position, if convenient. Example —
32:
s=
-e
:z2:
r^r
or
ms^
&-
-«s>-
-(S>-
@^
32
321
32:
But the effect of this ending is not quite satisfactory to the ear.
It should, therefore, be used with discretion.
4. When it is wished to make a kind of rest or division in a piece
of music, so as to lead the ear to desire a resumption of the movement,
after, as it were, taking breath, it is usual to employ what is called
the Imperfect Cadence, or half-close.
This is the reverse of the perfect cadence, and consists of the major
common chord of the dominant, preceded by the major or minor common
chord of the tonic, either plain, thus —
:q:
s
-&
321
or varied by dissonances by suspension-
THE PRINCIPLES OF HARMONY.
155
a=^
a
Z2:
r^
i
SE
p¥
a
=s^
122:
-Gf-
-TTT
5 -
4 3
or by Inversions of the tonic chord, thus —
i
-?^
icz:
or thus-
@^
-(S»-
-^'
zz:
^
221
r F
^
tS^-
r±
:q:
t 3
When a melody is regularly divided into two parts by a double
bar, it is very usual to let the former part end with an imperfect
cadence, and the latter part with a perfect one. As an example of
this we will quote a well-known single chant —
1 CL
"^ rzr
23 rj'
^
221
-©-
-Gf-
-&-
'C7-
22"
22"
"C2"
^-
221
221
221
3!
ZIZ22:
Z2:
221
221
6 6-
Imperfect
Cadence.
5 - V.
4 3
V
Perfect
Cadence.
These three are the only regular kinds of cadence or close, — the
perfect, the imperfect, and the plagal.
X 2
156
THE PRINCIPLES OF HARMONY.
5. The following passage from Logier'^' will clearly explain how to
introduce a perfect cadence.
"When the chord of the fundamental seventh, or dominant harmony,
proceeds direct to the tonic, it is called a ' perfect cadence,' as at {a)
in the following example —
za=^
^
:r:^
s:
-o-
n-
-<s>-
1221
:s^
®
-(Sl-
f^
-<s>-
-<Si-
^g
:g=t
-^
"O'
-(S^
c>
=g-~
i^:
:^
:^
e
izz:
22
-/^
22zz:
1^2:
-<s>-
22
-<S^
be
" It must have been observed, that by a continued modulation from
key to key, we are kept in a state of constant excitement, approaching
even to a painful sensation, so that the ear becomes desirous of rest.
Therefore, when we have modulated for some time, it becomes necessary
either to return to the key from which we set out, and there conclude ;
or, if we wish to proceed still further, first to make a close in the key
at which we have arrived, and, after modulating for some time longer,
to come at last to a final close.
" We must not, however, stop upon any tonic at which we may
have arrived [as at (6) in the above example] ; for, as the great object
of a cadence is to lead the ear to a quiet state, an abrupt termination
must destroy the efiect intended. It might be supposed that the per-
fect cadence, described above, would be sufficient for this purpose, yet
we find that this is really not the case. It is true that proceedmg
* " System of the Science of Music," p. 82.
THE PRINCIPLES OF HARMONY.
157
direct by the chord of the fundamental seventh to its tonic forms a per-
fect cadence, with which the ear would be sufficiently satisfied, where
modulations have not recently occurred [as at (c) in the above example];
yet the frequency of its occiurrence in a course of modulation, although
it may not destroy, materially weakens the decisive and concluding
effect it would naturally produce under other circumstances. Therefore,
when we arrive at the tonic of any key to which we have modulated,
and desire to come to a decided and satisfactory close, the ear must
be gradually soothed into a quiescent state by the introduction of a few
chords, so constructed that they shall not only have a tendency to con-
duct the ear to a state of rest, but shall also be calculated to produce
a strong impression of the key m which it is intended the close shall
take place.
" The chords best calculated for this purpose are those of the sub-
dominant and dominant; for the intervals of these chords (including
also those of the tonic) embrace the whole of the diatonic scale : so that,
in fact, by hearing those three chords at the close of a modidation,
we receive an impression of every interval of the key in which we
thus conclude. See (d) in the following example —
(d) {e) (/)
■n — I '-h ^ I I , — -n ! !V - I
1^21
z:2:
-(S»-
g-±=g=Uzgz=a
J^=^=^
-<s»-
Z2:
B
T2~rj
-iS>-
iq:
2:±
Z2:
■jr±.
1^21
I^Z
" However, a frequent recurrence of the simple chords of which this
cadence is constructed, would produce a heavy and rather monotonous
158
THE I'lilNClPLES OF HARMONY.
effect, whicli is much relieved by the introduction of the dissonance of
the fourth upon the dominant, as at (e).
" On account of the frequent occurrence of the final cadence, com-
posers have been induced to seek for every possible variety, and great
liberties have been taken for this purpose. The fourth, as it appeared
in the last example, at (<?), was properly prepared; but a sixth is also
sometimes introduced, which, it will be perceived, cannot be prepared,
and must be considered as a licence, as at (/)."
But that is not all.
Still further to secure variety, the double-root chord of the minor
seventh and minor third, especially its first inversion (which Logier
calls the chord of the added sixth), is often substituted for the chord
of the subdominant. And it should be borne in mind that in employ-
ing this chord the seventh of the secondary root should be prepared as
if it icere a dissonance of suspension.
.cl.
_C2_
-iS>-
_Ql
i:
-Q-
-(S*-
-Gh-
-(S*-
T^
■■(^-
hS^
jCi.-
i^^:
-<^-
W^
hS>-
-<s>-
-<s>-
-e>-
"?rr
-e>-
-s»-
i^z:
-QQ-
Q_
Si
-QQ-
-^— P^
-<s»-
'Z2.
-<s>-
22:
:P2
-o-
-<s>-
-(S>-
-e>-
^2:
6
5
-(f^-
-<S>-
-O-
THE PRINCIPLES OF HARMONY.
159
Sometimes tlie supertonic seventh is altogether
omitted, and then two curious results follow : first, we
get a chord exactly like the first inversion of the
minor triad of the supertonic; but in reality it is
only a dominant chord of the added ninth in its
third inversion : and secondly we avail ourselves of the
licence, peculiar to this form of the cadence, to intro-
duce without preparation not only the dissonance of the
sixth, but also that of the fourth.
Examples —
^
-<s»-
--^
-Gh-
Root.
9
7
5
C* '
^2:
^2:
-^-
1fS>-
'-^-
?2:
22
Z2:
Z2
iS>-
cj :
Roots
-&■
&—^
-&■
C G
G
&c., &c., &c.
Note. — It occurs frequently that a common chord, the hass of which is a third below
the tonic of our key, is interpolated between the tonic and subdominant (or substituted
chord). If we are in a minor key, this choi*d will be a major chord. If we are in
a major key, the contraiy will be the case. Examples —
-^
i^
-G>-
:r2
fls
1?*=^
-tSt-
-Gf-
-zj-
IGO
THE PRTNCTPLER OF HARMONY.
0. Ill order still further to avoid monotony, other modifications and
curtailments are sometimes introduced :
i. The subdominant or other substituted chord is omitted, the ? or
^ being retained upon the dominant, thus —
t=^-
-^-
s
^-
-(S>-
s
r
^
&-
-(S»-
:c2:
ii. Or the subdominant or substituted chord retained, while the
dominant V or V is omitted, thus —
^2:
^^^
^^^i
-(S»S>-
221
Z2:
-o-
fS> —
-o-
iii. The same, omitting the seventh of the supertonic, thus —
^
M
m
-iS>-
:^
-^
-o-
THE PRINCIPLES OF HARMONY.
IGl
iv. Introducing the chord of the dominant seventh on the super-
tonic root, thus —
i
fSt-
^p^
-/S*-
^=^
-e>-
s
-Gt-
g
z^zz=z?:±
-<s>-
b7
V. Introducing the first inversion of the same chord-
-^
-^-
f±
:^
1^
^<s^
r j r j
"O"
4 3
vi. Introducing the second inversion of the same chord-
J 3 I
^
SI «s-
Mr
S=l
-^2_
-<S|
:^
:g=^^
-<s>-
-s-
^2:
fS>-
b7
-<S^-
vii. Or the chord of the minor ninth on the supertonic root —
g_^^i^-
Z2:
-s(-
- g:3 rj
Gt-
~Z2~-
W^^^^
«
-&-
3
:z2:
102
THE PRINCIPLES OF HARMONY.
viii. Or the first inversion of the same chorcl-
±
-m
-G>r
m
#
-iS»-
i^:
m
-^
-w
a
22
-<S>-
Ix. Or tlie second inversion of the same chord-
-I-
:^=^:
:g=^^^
iq:
f^
1^21
^
e
lS>-
g
b5
D6
4
221
X. Or the chord of the augmented sixth —
-^ n
--gr
W^
'■^
lis
^
ZS2:
-<s-
• r-> r ^
-^*'
Z2:
di
xi. Or the inversion of the same chord-
i^
IS
S^'' —
1221
-^
^Gk
-^j- -7:y
o-
^21
^21
fe:
D2
THE PRINCIPLES OF HARMONY.
163
xii. Or, lastly, a new chord, which shall be explained in the fol-
lowing section —
XT
-^
-9&-
W=^
1^3:
t?:^-
icz:
m^.
:^-
m
fSK-
1)6
:^t
:^2:
7. On comparing our new chord in No. xii. with that in No. iii.,
it will be seen that the only difference is the accidental depression of
one interval, viz. the fifth of the dominant root G.
But this interval so depressed forms no part of the harmonic
series of G.
All the rest of the chord, however, belongs to it.
From what root then is this obtruded D b to be derived ''.
Clearly from C, of which it is the minor ninth.
But judging by the analogy of the double root-chords explained
above, one might think that such a derivation of the chord from
the two roots ^ would lead us to consider it as belonging rather to
the key of F, of which those roots are respectively the dommant and
supertonic.
And to a certain extent this is true ; and thus we see a connection
between this cadence and the plagal cadence, where a transient modu-
lation into the key of the subdominant actually takes place.
This new chord is then originally, though remotely, a subdominant
chord, though actually built on the tonic and dominant ; and whatever
Y 2
1(34 THE PRINCIPLES OF HARMONY.
elements of modulation it may possess are neutralized by the essentially
dominant character of the rest of the cadence.
Such then is the analysis of this remarkable and very beautiful
chord.
It is peculiarly suited to the minor mode, although occasionally it
may be followed by a major conclusion.
To distinguish it, it will be advisable to call it the " pathetic
cadence " : a name given to it by some authors on account of its peculiar
character. Dr. Crotch, Callcott, and most English theorists, have named
it the " Neapolitan sixth ^^ but such a name is very unmeaning, as it
certainly was at no time pecuhar to the Neapolitan composers.
It is susceptible of two mversions ; but they are difficult to mani-
pulate, and of doubtful advantage. Therefore they may be passed over
in this place.
CHAPTER XIV.
1. Cadences which lead from the key, may be fitly termed
"cadences of modulation." Of these there are many kinds.
a. Irregular Cadences, or perfect cadences out of the key, designed
not for final closes, but for variety. Examples —
From C to A minor.
ip
m
■3=t
-©
"^^
-^
n — sg-
Z2C2:
-^-
p:
:q:
From F minor to A flat.
M^S
f
o-
q:
^^-^=5g
(S>-
(S>-
b7
b5
te:
p6
bo
6
b4
b7
^22
=l2^
1G(>
THE PKlNCirLES OF HAILMOXY.
i
From 1) minor to A minor
^a^s:
5
22
3
5^
=S-
:^
-G>-
T2--
lOii.
(3. Incomplete Cadences, which close not on the tonic triad, but on
its first inversion, introduced by the third inversion of the
preceding dominant chord, thus —
cz:
±t=g
2±
'.^21
-(S"
-o-
^?=p-
iq:
-^-
-(S»-
y. False Cadences, where the cadence, after proceeding regularly as
far as the chord of the dominant seventh, is suddenly broken
in upon by the intrusion of a foreign chord ; which must,
however, be such as to allow all the intervals of the pre-
ceding chord to be regularly resolved, thus — ■
n.
i
231
22:
2?
22:
£
Z2:
=s
3
H _g__a
32
•<S»-
:?2:
:?z2i
^
6
4
S
:22:
THE PRINCIPLES OF HARMONY.
167
111.
IV.
^
-<si
ISZ
ffi
^s
-^
VS
s=^
-s
^2:
cz:
z:^:
-e^*^
e
fS-
:t=e
-lir^ '
±fc
-s*
-s>-
ICZ
V.
-A-
VI.
^ — ^-^-
22
-(SI-
#1
^^
:qq:
Z2:
=to:
-(S>-
-<S'-
-«s>-
^
-^-
■&-
^21
i^z:
bo
7
"^:?"
:c2:
^^-
In No. i. the dominant bass, instead of descending to tlie tonic,
rises a whole tone.
In No. ii. the same progression is accompanied by dissonances by
suspension.
In No. iii. the dominant bass rises a semitone.
In No. iv. the same is shewn in the minor key. And it must be
observed here that in the minor key the bass can only rise a semitone,
never a whole tone ; as, if it did, the intruded chord would produce
too violent a transition for the ear to bear.
In No. V. the same in another minor key, but in a different position,
and succeeding a pathetic cadence.
In No. vi. the intermediate notes are entirely omitted, making the
progression short and sudden.
IGR
TIIK PRINCIPLES OF HARMONY
It is evident that no real modulation is effected in any of these
cases. They are merely false cadences disappointing the ear, and
requiring a subsequent cadence to restore a feeling of rest.
If then we really wish to remain in the key to which the false
cadence has led us, we must make a regular cadence in that key for
the purpose, thus —
:-i
X2:
JQ-
22
1^3^
-<S
-<st
:q:
g=^
:i^
-^
^^£
C2:
?2:
-g--
-&■
22
1221
\y
u.
^
-(S>-
-st
^
S=^-g
zt
^
-(Sf-
4
fS>-
1221
/^
:22:
-eS"-
j:±
-s>-
: r - > ^^
-^<sJ
^
?2:
^1/
And it will be seen that, by combining these two examples, we
can modidate from key to key, as far as we please — ■
-8:
s»-
m
T^
-f^ . -P-
-fS^
io:
^fS»-
"fX^
=^ q =g5=
4^
fe
t^
THE PRINCIPLES OF HARMONY.
1G9
icz:
:c2:
-Gh
'JOL
-s*-
^
-o-
-G^
-o-
In the foregoing example (bar 5) an enharmonic substitution of BG
for C b has been employed, to avoid extreme keys ; at bar 7 we return
to our original key. And at the end of the example a plagal cadence
with dissonances has been added, in order to shew the use of such
cadences to intensify the force of a perfect cadence, and to induce
a feeling of final repose in the tonic.
If, however, we do not wish to modulate after the false cadence,
we can always return to the original key by means of the chords
of either the augmented sixth, the diminished seventh, or the seventh
on the supertonic ; thus —
/r\
n\
r^.
r\
ITS
fr\
rs
|g=:ttj
iq:
-^
P^^a^s^
22:
fS>-
Eg
:^
\u
b7
b6
-&-
Z2:
T21
fS>-
T2L
170
THE PRINCIPLES OF HARMONY.
Of course utiier inversions may be used, and dissonances introduced,
wherever it can be done without violating any of tlie preceding rules.
d. Interrupted, or Broken Cadence. Where a rest is placed where
the tonic should be ; as in the foUowmg example from Logier
(p. 214)-
Allegro.
■M
/TS
l^
'&.
f
/
d:
^^
/Tn
P
--m--
^
:^=E=
~-^-
^
^.
=|:
fag— g=:l
■z=a
iJ4»-j^-^w
Cf^^.
/
^=^
<s-
*^-
e^
nc
^^
J r^
5S
:^
s=e
ff
:i=F
i
=^
^=^:
r r 1-"
r J F
=1:
THE PRINCIPLES OF HARMONY.
171
€. Irregular False Cadence. Where the chord of the dominant
seventh, instead of being resolved regularly into its tonic,
passes into another fundamental discord, all its intervals, how-
ever, being regularly resolved. In this case the bass descends
a semitone, and the root of the new discord is the supertonic ;
thus —
^
(S>-
-<s
s
:z2:
:2:2
22
G)-
^<S>-
^s
P2:
:Sq:
z^.
m
■iSt-
:^±
z±
u^-
22
-^ ^
6
jr2L
hi
or wdth any other inversion of the minor ninth, or of the
chord of the dominant seventh, on D.
The chord of the added ninth may also be similarly em-
ployed.
^ Susjxmded Cadence. Where, instead of allowing the dominant
to proceed at once to its resolution, a few modulations are
interposed ; thus —
-/S
:^
t.
iq:
Z2
-Gh
i2^
S
-<sf
WEB
2^^
is:
fe^n
^.
If
ij
8^-g:
p£
^
m
-f^
221
-Gf^
Z2
fS^
T2L
rrj
221
b7 b7
b?
b7
b7
4
-<S>-
z2
CHAPTER XV.
1 . It has been already sliewn (Chapter VI, sect. 3, and Chapter XI,
sect. 9) how wide a field of varied moduhition is opened to the com-
poser by the equivocal character of the chord of the diminished seventh
and its inversions, as well as of the chord of the augmented sixth, when
these are treated enharmonically. Also it was shewn, in Chapter X,
section 1, how to modulate progressively, or by way of sequence, by
means of the chord of the dominant seventh and its inversions.
As a supplement to these observations it will be weU to point out
three other irregular or deceptive ways of modulating by means of this
last-named chord, which are often of great service to the composer.
2. The minor seventh in the dominant chord, instead of descending,
may ascend chromatically to a note of its own name, provided it thus
becomes a leading note, and ascends afterwards to a new tonic, thus —
_C2
i
fe:
C2:
-^-
THE rniNOlPLE.S OF HAiLMONY.
173
or it may be followed by a false cadence, thus, in the case of this
example, modulating to E minor, or to E flat major. Only provided
always that the progression of the leading note be as in the example.
Either of these chords of the seventh may be used in any of their
inversions, if the above condition be strictly observed. The leading
note of the first chord here is allowed to descend.
3. Another variety of resolution takes place when the chord of the
dominant seventh is succeeded by the first inversion of the dominant
seventh of a root a minor third below its own root —
ih-
-^-
-^^-
-^— H
ffi^ h' \^ f^ \
"k ' 1 '
J
/m^:
rzi
nn^
_ . f^ J
(if. j ■"! 1
v_^ 1 1 1
Roots. G E A
For the second of these chords a diminished seventh on the Gjf, arising
from the same root E, may be substituted.
4. The third irregular resolution of this chord is when the interval
of the minor seventh falls to the fifth of the next root, while the
leading note ascends a whole tone instead of a semitone. It is usually
confined to the first inversion of the chord, as in the annexed example —
^
^F=^
S
-<s«-
:t^
G
D
174
THE PlUNCirLES OF HARMONY.
5. Hitherto every device we have examined has had for its object
the attainment of variety in modulation. But very often there may be
a transition or progression from one key to another, either for a chord
or two, or permanently, without any modulation properly so called.
In such cases dominant harmony must be altogether laid aside, and the
progress must be from one tonic triad to another. All that is wanted
for this is some note in common between two adjoining chords, in the
same part, as in the following example —
^
T2:
-Gh-
-^
-i«5>-
^s>-
BEEB
-S*-
-JUt.
=F
:^
.C2.
e
Z2:
4^
-&
-&
z^:
:^
-Gt-
-^
.CL.
^■=8:
ts>-
m^
--g— g:
:p=:
--Bi
-is»-
:S
q:
T^
T^
T^
:^
22:
Z2
-^ ^ ' d:
This example has been simply constructed by taking the ascending
scale as a treble melody, and harmonizing each note with the three
triads of which it formed a part : treating it alternately as an octave,
a third, and a fifth.
It is obvious that every note is susceptible of three such harmonies,
and that only such of them really belong to the natural scale, as coin-
cide with the chords found for it in Chapter lY. Those which consist
THE PKTNCIPLES OF HARMONY.
175
of the imperfect or diminished triad are ambiguous and comparatively
useless. Still they Imve their place, and are therefore reckoned here
with the rest.
6. The first note of the scale is most naturally accompanied by its
lower octave, or tonic, as a bass, with its proper triad, major or minor.
It may also have the subdominant for its bass, of which it is the fifth ;
as for instance in a plagal cadence, when that note is in the treble —
1=1
:z2i
-&-
:z^z
But we may also take the third heloiv (either major, if we are in
a minor key, or minor, if we are in a major key), and thus get a new
bass, or, as it is called, a modified bass, which in the key of C would
be A B or A b —
m
B-
or, if in the minor-
:z2:
zi:
1^-
B
/S^
^
^
:Z2:
2:^=1
which gives variety without modulation.
7. The second of the scale has only one natural bass, namely, the
dominant, of which it is the fifth. There are, however, the two fol-
lowing modified basses —
17G
THE PRTNCTPLRS OF TTARMONV
I
-<s>-
S
or,
-o-
i
Z2:
#
-<s-
Of these the former, i. e. the octave, is good ; while the latter, the
third below, is to be avoided, because of the diminished fifth it contains,
which is a doubtful and unsatisfactory interval, suggesting the first
inversion of the chord of the dominant seventh, but without the octave
of the root; or else it might be an imperfect chord of the minor ninth
on the third of the scale, omitting both the root and the leading note,
and thus being vague, equivocal, and unsatisfactory to the ear.
8. The third of the scale has only one natural bass, which is the
tonic itself. But it may be accompanied by two modified basses, i.e. one
a fifth, and the other an octave below it, of which the former is to be
preferred, being the relative minor —
i
S
s>-
T^
23
9. The fourth of the scale has only one natural bass, namely, the sub-
dominant, of which it is the octave. It may, however, be accompanied
by two modified basses ; one a third, and the other a diminished fifth
below it. The last is of course undesirable, being the leading note,
and bearing the imperfect triad —
THE PRINCIPLES OF HARMOXY,
177
i
X^
^=^
-Gh
10. The fifth of the scale has two natural basses, the tonic and
the dominant. It may also be accompanied by one modified bass, the
third below —
B:
22:
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T^
11. The sixth of the scale is naturally accompanied by the sub-
dominant. It has two available modified basses, which are the fifth
and the octave below —
a:
e
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2^2:
12. The seventh of the scale has only one natural bass ; it has con-
sequently two available modified basses. But one of them should be
used with caution, as it is the seventh or leading note, and bears the
imperfect triad —
A a
178
THE PrtlNOTPLES OF HARMONY.
1=^
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13. Tlie following example will shew the use of these modified
basses, to secure variety, and to save continual change of key at the
same time —
III
2^
Z3 7Z K-
-(S>-
aa^:
:&
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m=m
-Gh -Gt-
«-
-zi-
m&
(S»-
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; rj CA-
-«s>-
i^i:
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izrzn.
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^
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ipzuc
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22:
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22ZZCZ
fS-
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2:i
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Here no fundamental discords, nor suspensions, nor retardations have
been vised, yet sufficient variety has been secured solely by the use of
modified basses.
CHAPTER XVI.
1. It has already been explained how to produce sequences of
various kinds.* In each case the result was a perpetual modulation
round and round a cycle of keys. Now as it is obvious that such
modulation, unless sparingly employed, must become very wearisome to
the ear, it is desirable to find some way of combining the advantages
of a regular sequence with the power of remaining in one key. And
this can be done by treating dominant sevenths and ninths in a manner
somewhat analogous to that by which the modified basses and triads
were obtained.
2. If a chord of the dominant seventh —
i
-Gh-
ZX
be only 'partially resolved, that is to say, if the seventh fall while tlie
leading note is suspended, the result will be as follows —
* See Chapter VI, section J), and Chapter XI, section 9, and Chapter XV, section 2.
A a 2
180
THE PRINCIPLES OF HARMONY.
fS-
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s>-
s>
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t^
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where the ear remains unsatisfied, and craves some further resokition.
This can be obtained by allowing the B to fall, as though it were the
dissonance of the ninth on the root A, in its first inversion; thus —
-9—^—
r^
A rD
ff ^ ^ -
' r^ f-^
l^ J ^
— 1 I
r*\' ^^
i^' ■-'-■-
V^ 1 rj
Roots. G A
where we see that it is a sort of false cadence, and that the B is
really the dissonance of the ninth by suspension.
But suppose that, with a view to establish a regular progression
of the basses from fifth to fifth, we change the bass from C to F, at
the same moment that the B descends to A, we shall then introduce
another dissonance —
I
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-CL
icz:
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THE PRINCIPLES OF HARMONY.
181
and this may be regarded as the dissonance of the ninth on the root D,
necessitating the descent of the dissonant note E to D —
&-
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22
m
m
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^
I^
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m
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In order, however, to keep up the progression of basses as before, it
will be necessary to let the basses change to B at the same time, thus
producing a new dissonance of the ninth on the root G, each dissonance
being in the first inversion. But instead of resolving this by allowing
the A to descend to G on the same bass, we change the bass to E,
thus turning the note D into a dissonance, and once more necessitating
its descent to C.
Without analyzing this process further, we will give the whole
sequence, and afterwards remark on some of its chords —
is-
:Z2:
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S
CZ]
r ' V
3. Now it will be observed that in this last example the tenor, or
third part, has been varied, so as to lack the symmetry of the other
parts. This has been done to avoid the hidden octaves which appear
in the previous short example between the bass and tenor in each bar.
182
THE PRINCIPLES OF HARMONY.
Indeed such a sequence of sevenths as this is most perfect and pure
when the tenor is omitted, and when the harmony is consequently in
three parts.
We will now give the same sequence in three -part harmony, and
then assign the proper roots to each chord —
fSK-
122:
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T2=^&-
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7 7
D G
-^
GAD
7 7
G C
F
7
D
G
7 7
G A
7
c
7
D
G
7
G
C
Here it will be perceived that the roots proceed regularly from A
to F, when an interruption occurs, in the shape of a double-root chord,
succeeded by an ordinary dominant seventh on G.
The chords marked * might have been treated as first inversions
of the triad of B flat, with the dissonance of the ninth, C ; but then
w^e should have been forced to modulate out of the key, which woidd
liave destroyed the symmetry and character of our sequence.
It was preferable therefore to look on these chords as double-root
chords of the minor seventh and minor third (commonly called the
" seventh on the second degree " ), leading back to the same dominant
harmony with which we started, and with which the sequence after-
wards closes.
4. By adding notes a third above the treble of the above sequence
of sevenths, the second chord of every bar is converted into a chord
of the ninth —
TTTE PRTN0IPLE8 OF HARMONY
183
S:
S
fS>-
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-^
orzr.
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Again, by taking tlie original sequence of sevenths, and adding
a tenor part progressing in sixths below the alto or second part, the
first chord of every bar is converted into a chord of the ninth —
^1 — r j-
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Or again, by combining these two sequences, a complete sequence
of ninths and sevenths in five-part harmony is produced —
s
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184
THE PRINCIPLES OE HARMONY.
5. By using the inversions of the chord of the dominant seventh,
a great variety of sequences can be obtained —
fe^^m
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r=B=rr-^
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THE PRTNX'IPLES OF HARMONY.
18:
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&C.
* N.B. This is not veiy good, as it contains hidden eights.
?=:
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&C.,
&c., &c.
Bb
1
6
5
6
186
THE PIUNCIPLES OF HARMONY.
From these specimens tlie student will see his way to the elabora-
tion of other forms of sequence according to the method here indicated.
The last seventh must be a real chord of the dominant seventh,
either in its fundamental position, or inverted, so as to lead to the
close on the tonic. All the other sevenths which occur in sequence are
of course different from the fundamental seventh, and only derive their
force from the dissonances of suspension from which they arise ; and
from the symmetrical progression of the basses, which gives an impulse
as it were to the whole sequence, and invests it all through with
a quasi dominant character. Of course it is possible to substitute a
real dominant seventh at any stage of a sequence, and thus modulate
into any key which that seventh leads to. This is a very useful way
of availing oneself of these sequences in the course of a piece.
The annexed example is intended as an exercise for the student to
harmonize —
IS
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9 8
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t 3
THE PKINCIPLES OF HAKMOXY.
187
6. Towards the close of this example occur a series of ascending
notes, each figured 7 6. It will be well to sav a few words about
these.
Of course the simplest way of preparing these dissonances is —
:c±
gfc g^E^dsJg^
-&■
-^
Z2:
m
22:
-s>-
s
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which produces an ascending sequence. But this leads to the con-
sideration of what this would be without the dissonances.
It would then become an ascending sequence of sixths accompanied
bv thirds ; thus —
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W-
s
:q:
-«s^
^z:
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B.
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Roots. C
6
G
6
D
6
E
] &c.
And these may either be derived from the tonic and dominant roots,
or from modified basses, such as were described in the last Chapter.
B b 2
188 THE PJUNCrrLES OF HARMONY.
Or the same may bo written in a descending sequence-
4
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122
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la
Roots.
6
D
6
B
6
A
6
G
'A
D
-0-
G
And this descending sequence of sixths admits of course of dis-
sonances to each note —
a
w
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Roots. G
6
E
7 6 7 6
D C
7 6 7 6
D A
7 6 7 6
3 -
G
D
G
There is always considerable risk and difficulty in adding a fourth
part in harmonizing a sequence of sixths, as there is danger of making
hidden consecutive octaves or fifths.
The following are some of the best methods of doing it —
-& — I
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22:
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22:
22:
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— ^— ^ Tr~- o ~~- ~—^ =g:^= &-^:r =^
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THE PRINCIPLES OF HAKMONY.
180
1^21
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The sixths may always alteniatc witli tifths, provided the iiftlis do
not produce the effect of consecutlves, Avhich tliey must do, more or
less, if they are on the accented beat, or (as it is called) the down-
beat —
-s"-
:r2
-(S>-
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Not so good.
Good.
This sequence of fifths and sixths, liowever, is not easily practicable
in descending —
^-r-jXJ — J j K^J ^ 1
6 5
6 5
^^
6 5
6 5
r^-r
The hidden fifths are here too offensive.
CHAPTEK XVII.
1. When the tonic or dominant root is sounded and held on, while
harmonies belonging to the supertonic, dominant, tonic, and subdomi-
nant, are made use of in the upper parts, it is called a " pedal," or, as
the French say, " le point d'Orgue." Any of the above harmonies may
be freely used on a pedal, and many cases occur where this resource
enables a composer easily to introduce discords which otherwise would
seem crude and harsh.
The dominant pedal is generally introduced whenever it is wished
to excite a special craving for the tonic of the original key of a piece
of music, after protracted modulations; or towards the close of a fugue,
for the same purpose.
The tonic pedal is only used by way of protracting the final cadence,
introducing the plagal cadence gradually, or intensifying the concluding
feeling of repose in the main key.'
2. The way to use a pedal can be best explained by example, and
a specimen is therefore given ; of course it is supposed to come not
near the commencement, but towards the end of a piece in the key
of C. It is introduced by a few prefatory chords, and succeeded by
a regular conclusion in the key of the piece —
THE PRINCIPLES OF HARMONY.
191
-^. ^=P:
1 h
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TIfK I'lUNOIPLES OK irAHMONV.
Dissonances and sequences may Ije introduced freely on a pedal
bass, to vary the effect,*
Note. — Many persons prefer neglecting the pedal altogether in the thorough-bass
figuring, and ])lacing the figures under the next lowest part, just as they would if no
pedal were there. Either plan is good. If the pedal passage in this example were so
figured, it would appear as follows : —
&c., &c., &,c.
3. Sometimes, though rarely, a long note like a pedal is intro-
duced in an upper or inner part of the harmony. It is necessary in
that case to be very careful not to let it clash with dissonances which
resolve upon it, especially fundamental ninths, major and minor. It is
very rarely that an inner or upper part will bear this treatment, though
many instances may be quoted, from the works of some of the greatest
masters, where such a contrivance produces a surprisingly good effect.
The proper name for it is an " Inverted Pedal."
* See Examples at the end of tlie work, Nos. 3. 3, 9, 10, 11.
CHAPTER XVIII.
1. A HARMONY is Said to be "broken," when the intervals of which
it consists are not heard simultaneously, but successively, yet so as to
produce on the ear the same harmonic effect as though they were
sounded together. Thus the following chords are shewn first unbroken,
and then broken in various ways —
I
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E
CJ^
s
-<s>-
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m
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194
THE PRINCIPLES OF HARMONY.
22:
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m
i^aa:
^
m
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-<S»-
When chords are broken, care must be taken, first, that every note
requiring any fixed progression or resolution, shall proceed correctly, and
be resolved according to rule ; and secondly, that no conseciitive fifths
or octaves arise in consequence of the new forms taken by the chords
wlien broken.
2. Before going further, it may be mentioned that when a whole
phrase or passage is played simultaneously in two or more difierent
octaves, such octaves are not regarded as faulty consecutions, but the
part so reinforced is said to be "doubled in octaves." Any number of
THE PRINCIPLES OF HARMONY.
195
real parts may be so doubled in octaves, provided always that no con-
secutive fifths ensue —
1.
I
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This last example (iii.) however must be regarded as faulty, on
account of the consecutive fifths between the second and third parts of
the harmony. At the same time, it must be admitted that Beethoven
has made use of just such a consecution as this in the first movement
C C 2
196 THE PRINCIPLES OF HARMONY.
of his "Sonata for Pianoforte and Violoncello, or Horn, in F" (Op. 17),
as well as in one or two other places.
Thus any of these doublings in octaves would be allowed, especially
in instrumental music. And still more would it be correct to double
the bass in octaves, which indeed is almost universally done in piano-
forte and organ music, and in that for a full orchestra.
3. The permission to double in octaves will account for the apparent
consecutives in the example of broken harmony given above, in the
last two bars especially. And it is just in such cases as this that the
student has the greatest need of care and discretion, so as to know
when and how to avail himself of this resource without infringing the
rules of counterpoint.
Above all he should remember that consecutive fifths can never be
excused under plea of " doubling," although consecutive octaves can.
He should also remember that the whole phrase, and not one or two
notes of it, must be doubled in octaves; otherwise the rule against
consecutive octaves will still be infringed upon.
4. From the consideration of broken chords the transition is easy to
that of what are called " passing notes."
These " passing notes " should be regarded as nothing more than
embellishments, and as in no respect affecting the harmony.
They form no part of the essential chords belonging to the melody,
and serve only as connecting links between successive notes when such
notes are more than a minor second apart.
For example, take the melody of the common chord ascending and
descending, as at i. in the following example.
THE PllINCirLES OF HARMONY.
197
It is allowable to interpolate passing notes between these essential
notes, which do not affect the bass or harmony. These passing notes
are indicated by dots at ii., and are written in the usual way at iii.,
where the essential notes are distinguished by a horizontal line over
them —
j=±
^
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-<S'-
u.
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'IZSl
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-s^
Z2:
-fS*-
^
It will be observed that in this example the passing notes are
placed on the ZM2accented parts of the measure. When this is the case
they are called "unaccented passing notes."
198
THE PRINCIPLES OF HARMONY.
5. When passing notes occur on the accented part of the measure,
they are called "accented passing notes." Example —
*-
s^^
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These are more dissonant than unaccented passing notes, because
they are heard at the same time that the bass and the rest of the
harmony are struck, while the unaccented are not heard till afterwards.
The following example from Logier will shew this —
s
p=^q^
(S>-
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6 6 6
6 6 6
E^E^EEi^
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— ^ ^^
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THE PRINCIPLES OF HARMONY.
199
A=;h=^r^-^-ih-'='-ir^iS^.U^^=-^
I
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= i
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J..^^:^ A.IA ^ J^l.J^
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It will be remarked that in these examples passing notes are intro-
duced not only into the melody, but also into the bass and inner
parts ; and that in the three last tlie inversions of the chord are
varied.
6. A chromatic scale may be easily produced by interposing passing
notes between the lohole tones of a diatonic scale ; only it is not usual
or desirable to insert one between the sixth and seventh degree, as the
effect is harsh — 1 i^ I K
G>
^
^
:q^
J2:
'B^
4:^
^
6 -
5 -
Roots. C
G
C F
c
F
6 7
G
C
200
THE PRINCIPLES OF HARMONY.
Eveiy one of these inserted semitones might also be harmonized as
essential notes, which would involve various modulations. Thus—
^
-^
fc
g
W\
^ 2 ^
4z=f
^=^
;if^=f:
4t-
2
6 - «5
^
^i
it 6
41-
-ly \
3^
s^
^i^"""!"^^
^^^
«^ <i^
:f=*
44-
2
i 6
^<h
Note. — When the parts proceed by contrary motion, whole chords may be introduced as
"passing chords," intervening between fimdamental discords and their resolutions; thus —
M^^^^^^^kE3E^
Z±
■^-
\ <S>-
^KHJL
r^^^^^,
F
1^
-^'
? -' % ^ 9 I e IT i_i §
ria
r^ '
-<&
-&-
7
t 5
i&i
Passing chords.
7. When no passing notes can be introduced, use is made of what
are called ^Wtiixiliory votpn."
THE PRINCIPLES OF HARMONY.
2U1
They may be written either above or below the melody, and may
be either accented or unaccented.
If helovj the melody, they should be only half a tone below.
If above the melody, they may be either a half or a whole tone
above, as the case requires.
Simple Melody.
s
w=^^
-0-
d S d -^
^
^2:
?
■# — m-
ES?
6 6 ^666
Auxiliary Notes added.
T^
^^=m
6 6
4
4 6 G 6
2 i i
3
zz
^
S
m
^
sf
fefe
^ f
:c2:
^
5=^
Simple Bass.
Dd
202
THE PRINCIPLES OF HARMONY.
m
^
-<s-
^
22:
-C3-
^^^^
Auxiliary Notes in the Bass.
8. When the notes of a melody move by a skijD, i.e. by intervals
greater than a third, passing notes may skip also to a semitone below
an ascending mterval —
-^ g^ P—
or to a semitone or whole tone above a descending interval —
^m
§
^
or
i
i=t±
1(S>-
Auxiliary notes, under similar circumstances, skip beyond the interval
to return —
-^ — 1
p —
1 —
^— I
II
X
/
^
■^^ 1
ff\^ *
•
■^
vU
>
11
eJ
9. Chromatic passing notes and auxiliary notes, if accented, exercise
a very great influence on the harmony ; so great, indeed, that it is
often difficult to determine whether they should be treated as such,
or as real and essential notes —
THE PRINCIPLES OF HARMONY.
203
SE^^lE^
iz=i
-G^
'jr:il
'W=^
J=i
-<SH
-z±
-Gi
-e»-
:«^-
:<S>-
-<S>-
"C?"
In this example the AJf in the second bar is treated here as a
chromatic passing note leading from AG to B. But it is more than
doubtful whether this is not wrong (although it is the usual way of
writing it), for the other notes of the chord indicate a change of root,
and if the note be wiitten B'? instead of AJf, it at once shews itself
as part of the chord of the minor ninth on root A, superposed on the
tonic pedal G, of which A is the su]3ertonic. On the whole it appears
therefore preferable to consider this note as an essential one, and to
write it B flat, in spite of custom.
In the third bar, on the other hand, the F sharp must be treated
as an auxiliary note, as it forms no essential part of the harmony of
the root A, from which root the chord is evidently derived.
Note. — Accented auxiliary notes are usually called '^ appnggiaturas,'' as they are sup-
posed to be a kind of buttress or leaning support to the note before which they are
placed.
10. This last example, in its third bar, introduces another rather
doubtful case —
^
^'
^
-s^
-G^
D d 2
204 THE PRINCIPLES OF HARMONY.
For the treble part might be Avritten, with good effect, thus —
■±
A^.
^
<^-
-G>-
:q:
@5
1t^
-^
o-
in whicli case tlie question arises, " What is the derivation of the
F sharp?"
It cannot be a passing or auxiliary note, as it moves to D by
a skip of a major third.
May it not be an instance of a double-root chord on the roots D
and A?
If so, it would be a sort of anticipation of the succeeding chord
of D, and would be an analogous case to that remarkable passage in
the quartett " When the ear heard her " in Handel's Funeral Anthem —
^^^,
s;s
6 7
4 5
where the treble goes up to G on the bass Ab, and where the chord
is an imperfect " chord of the added sixth " (as it is erroneously called),
THE PRINCIPLES OF HARMONY. 205
which we have proved to be derived from the dominant and sub-
dominant roots, which here would be -nu- This explanation is not
altogether conclusive, however, as the major ninth of the F ougJit to
be resolved, of course, by descending a whole tone ; or, if suspended,
it ought to be suspended in the same part and in the same octave,
neither of which conditions is observed in the present case.
It is probably an isolated example of licence, peculiar to Handel.
11. In the works of old Church writers of the English school,
especially Purcell, Blow, and Croft, it is no uncommon thing to meet
with a combination of the minor sixth with the major third, which is
treated as a fundamental discord —
If it is, in truth, a fundamental discord, it can be no other than
a fundamental minor thirteenth, and of course must be resolved into
the key of F.
The thirteenth division in the harmonic series certainly does give
a sound which approximates more or less closely to this interval, at
least it is not more different from it than the true harmonic funda-
mental minor seventh is from the tempered minor seventh which we
are obliged to use in music ; only, whereas the natural minor seventh
is too jiat, the natural minor thirteenth is too sharp.
But we may perhaps safely regard the above chord as a funda-
mental chord of the minor thirteenth, and use it, if need be, with the
rest of the harmonic series ; thus —
206
THE PRINCIPLES OF HARMONY.
\^^^^^?^
It produces a peculiarly plaintive and pathetic effect when judiciously-
introduced. But the student should not be too lavish of such extreme
discords.
Very often it has more the appearance of an auxiliary note or
appoggiatura ; for instance —
^
:«!2:
z±^
&-= — •
22^2:
22:
icz:
^m--
3g
-^G-
— (S>-
,^
22:
-O-
-<s»-
nzn
:r±
ir2L
And at other times it may be looked upon as an ordinary suspension.
Of course this can only be when it is regularly prepared in the pre-
ceding chord.
There is some difficulty in considering the minor thirteenth as an
available portion of the harmonic series of the dominant root, since we
have rejected the fundamental eleventh.
But in the first place the harmonic sound which represents the
latter interval is nearer to an augmented than to a perfect eleventh,
whereas the harmonic No. 13 is fairly near to the minor thirteenth.
THE PRINCIPLES OF HARMONY.
207
And in the next place there are really no cases in which the sup-
posed fundamental eleventh may not be equally well regarded either
as an inner (or upper) tonic pedal, or as a suspension, or as the minor
seventh of the supertonic, as has been already shewn.
It appears therefore unnecessary and unphilosophical to have re-
course to the "augmented foiu-th in the third octave" in the harmonic
series, that sound being very much out of tune, more so indeed than any
other natural harmonic sound in the first four octaves firom the root.
12. Under some circumstances it admits of a doubt whether the
chord of the fundamental minor thirteenth may not be more correctly
written as an augmented fifth —
i
i
for (as has been shewn above, Chapter XI, section 2) nature does give
us that interval in perfect tune.
And indeed if we invert the minor thirteenth, so as to put it in
the bass, it does actually produce a chord of the augmented fifth "'" —
:^
or
Roots. E
A
13. The major thirteenth is correct and in tune as found in the
series of natiu-al harmonic sounds. It is represented by the cube of
3, or 27. For 3 is the fifth ; 3 x 3 = 9 is the ninth or supertonic ;
3 X 3 X 3 = 27 is a fifth again above that, i.e. the major thirteenth.
* See Short Examples at the end of this book. No, 13.
208 THE PRINCIPLES OF HARMONY.
Hence are probably derived such cliords as —
'r^-
^=^-
Z3:
-^rx
ic^^
cz
Z2CZ
-Gh
^:
3
z±
3
:^2
(S»-
C2:
which indeed cannot well be explained in any other consistent way.
14. The province of the science of harmony does not extend further
than those various points which have been explained in this treatise.
If the student wishes to become a composer, he must add to these
a diligent study of counterpoint, fugue, form, and instrumentation.
Having thus counselled him, we will now leave him to apply what
has here been taught, both by analyzing the works of great masters,
and by harmonizing melodies according to the rules given in this
work.
EXERCISES
ON THE PRECEDING CHAPTERS.
No. 1. — Exercise on the Chord of the Dominant Seventh,
(Chapter II, 7, 8.) .
Fill in the omitted notes in the incomplete chords.
ZZ!2:
-«s>-
:z2:
i
* C J'
-Gh-
- r^ b
E
^
-nr
TU-
fe
-S»-
Z2_
J— rj
,CJ -
"< : ^ q:
:z2:
-(S>-
zz:
-<s>-
-s»-
4W
-s>-
-<s>-
:z2:
:if^
=S=
^
Z2:
"cr
-s>-
-o-
icz:
M:
E e
210
THE PRINCIPLES OF HARMONY.
No. 2. — Exercise on the Chord of the Added Ninth.
(Chapter II, 9.)
Fill in the omitted notes in the incomplete chords.
-C^_
g^fjE
:q:
1^
-<s» —
E
-^&-
-s>-
J2Q.
-<5>-
-<S>-
rf
ZZ
:r j r w> _
-<s>-
t:?-
-(S»-
i
i:
:z2:
-H<S>H-
^a^
Z2:
-«s>-
-o-
rj < ^-
-<s-
1W
No. 3. — Exercise on the same.
Take the two preceding exercises and transpose them into all the
major keys.
THE PRINCIPLES OF HARMONY.
211
No. 4. — Exercise on the Inversions of the Common Chord and their
Thorough Bass figuring.
(Cliapter III, 2, 3.)
Fill in the omitted notes according to the figuring.
^^
2i
3
:^ — ^ — ^
:^=
E^=g— g
zi
^
"O
"Cr
^1
g^E^
T^
o-
P2:
i^
Z2:
i^
-<s?
la
S' Gi
6 5
4 3
6 6 6 6
^^
:z2
-«^
^ ^j ^
i^
q:
i^i:
P2=fe=i:^
SEfe^E^
^ -Ts. r-- ^
1^2:
^2:
zd:
iq:
^ ^ J
-&■
z±
j:^
"cr
» 6
be
'^^-
i^
Z2:
-&
^-
T^
T^
^-
-&
m
t^
-^
4
-T^ I r^
-^-
D 6 t
:^
q:
-r±
<s>-
:^
Z2:
E e 2
212
THE PRINCIPLES OF HARMONY.
No. 5. — Exercise on the Chord of the Dominant Seventh mid its
Inversions, to he filled in like the last.
(Chapter III, 3—6.)
f^ d
-S ^
^
t^
^^=t^
r=^
g=?F=F
i
21
:P2:
J^
:r±
tt
5 6 6
f
l?P3=3^=
~S — T
1 1
r-r:r-^^
-^ ^
n — •
i(^ * r
__ •
[
y> ) \
1
(©f# \ — r
^=^
1
1 — r~'¥~\
^ "^ -
tLLi_J_J_
— S* —
\ ^_!
"^^^—^
r>
— 1 i —
07
kM^: — r
— ^ —
— ^* ■» —
\—
^f f"
— m —
-^ — S t-
J- ' ^
— & —
=3t:
— «sJ —
•J
T^r^g — ^
^~if^ — r~"
1 1 1 !
%^
j#
^ — U—
-J-J-^
d '
•
1 ^ ! U
No. 6. — Exercise 07i the same.
Transpose the last exercise into all the major keys.
THE rniNCIPLES OF HARMONY.
213
No. 7. — Exercise on the Inversions of the Chord of the Added Ninth.
(Chapter III, 9—15.)
i
4^
rs>-
m
s
1^=1^
1^-
-^-
-&^
:r±
:r2
^t^ES
3
:^=^
^s>-
nipz
ir:r
-<s^
_gj r :;^
i^i
f^-
g^
P r> -
hS*
-(S*^
22:
fS>-
-s>-
<s>-
:^:
Z2
1^2
^ r:^
5 6 6
3 4)-
3
:^^=R
1^2L
(S>-
gzip j
e
q:
z:^:
^
n d r ^
-^^ — h
:^
c^
32
a
b5
-b— f^-
e
^EE^3E^
-lS>-
C6
X2:
-<s»-
inz
(S>-
:^
-^- ^ ^
b7 !57
r^ g p :
1, ^^
4 R 7
-<S^
211
THE riUNCirLE.S OF HAUMONY.
No. 8. — Exercise on the Ascending Major Scale.
(Chapter IV, 4.)
:M
^^^
^^=^=?^
i*^
:^=^
iS^zfe
&?
^
^-^
6 6 6 6
C5
iA
P
1^-
^=*
^^i:
^=^
^— ^
^=^r
^&^E^
:p::^:^
^^3^
^
6 6 6 6
4 6
No. 9. — Exercise on the Descending Major Scale.
(Chapter IV, 5, 6.)
m>
is
FEE^
-^—0-
1^
3^
6 6 ^ fi :^-
6 6
THE PRINCIPLES OF HARMONY
215
rr—w
a
?
-q:
±izjL
:^=^
« ^r:i
I
<^-
<«=i^
fefc
g^^
?:^
I' f
6 (J 5
4 3
No. 10. — Exercise on the Ascendmg Minor Scale.
(Chapter V, 6.)
i
53
:«
^
s;
=ip=i^
-1
s--*— *
^E5
^
:^=qc
^S3EfS
« t
fi (j
6 6 6
i
-N
i^-
-<s>-
-s>-
E^3
^ — #■
6 e
t^-
3
"C?'
'O'
^^
216
THE PRINCIPLES OF HARMONY.
No. 11. — Exercise on the Chord of the Minor Ninth and its Inversions.
(Chapter VI, 1 — 8.)
C2:
w-
T2L
r^L
^ r ^-
v:2:
-^h-^
-Gl-
(S ^
^
-<s>-
T2L
^:
22:
fe
22
hS*-
-O-!-
-1^
?2:
fS>-
22:
i^:
e
-jit.
-Gi-
S
:r2~:
fS>-
@5
4:2:
(S>-
?2:
1^-
:^:
-si^^
^
Z2:
f
:p2=zi^
-iS>-
22:
-i(S»-
nfe
?2:
-iSt-
-tt:?-'
IQ
i^
C2_
-(S>-
22:
«P=
Q .
g5
Q ^
:?!2:
p?
s
d7
6
T^~~~^
:r2
2:i=^
22
-^
-^ rj-
-s>-
gTT~T"^=p
:q:
^ — ^
r^
I
X2:
-f5
D5
-<S>-
'I
THE PRINCIPLES OF HARMONY.
217
-& r ^ G-
-^
2±
22
'G* <5^ Gf-
Z2:
i=p p ^
-JiS'-
221
:i^
:fe±
-(S*-
b7
5
n7
C5
d5
C7
C5
I
XT
g^ C^
:1f^
iq;
-^
g
J^
e* Gt Gh
-&
W-
s
ff rJ rJ — rJ ^-
b7 d6
5 4
6 P7
4 tl5
6 6
igr^ ? r^— ^-
-f^ fS> .<s>-
-^ r^ io — &p-
br^ .
1
^
-1^ «s> <s-
«5
-tf(^ r-^
D7
D5
^.^-
b5
ip2=:f2i
"vT
b5
D
^^2:
g g p p p-
-C2-
i^
T21
-&-
-(S>-
:g^:
^P=^P==^
:q:
^s*-
.6
&4
3
:g^
B7
-S'-
22:
-(S>-
Ff
~t:7
Do
218
THE PRINCIPLES OF HARMONY.
T^
T^-
e
"r^ &-
-i^>-
-«5t-
i^-
-Gt-
-j:2L
-<s>-
^^-
Z2:
&i
a?
C5
fii
D5
g ^^ P ^-
:P2:
P
z:^:
-(S-
i^z
-o-
^
:S=2:
-S»-
22:
:g
-(S>-
1221
,6 D7
Go —
No. 12. — Exercise on the Descending Minor Scale.
(Chapter VII, 1, 2.)
^ p I rj P p
i^
3
i^
T2L
■&-
-(S»-
?2iz:
^
-P-^-
&-
^
r-^ Q
^
-G>-
-1^
6
6 ,6
b5
THE PRINCIPLES OF HARMONY.
219
i
^^-
hc^j^
~f^—r
-Gf-
'fSh-if,
;»p=^^
-(S>
=5^-
-1^-
m
^=X^-
1^I±
1 ^-
^= ^
6
b
6 «
■c^f-J?:;/— ^-
b D6
qi=q==r-
■^
-^ — ^^
Z2:
1^21
-s^
22:
P-p-tf^
-iS*
^i=^:
^^^:
(S>-
' ^ P f^T- g:
-<s>-
'^—rzr-
'^ ^ ^
^
b7
B5
Q »
i ^ b^
b
:g> ^ t^
-iS"-
3=^
:?^
iq:
-&-
-zJ-
~CJ
i^
_C2_
.Q-
P2=^=^
|S^
#
1^21
5
(S>-
^:^i4^±^^
22
» 6
6 t
a P5 7
6 )t
:^z:z
-^±
:^=P:
e
Z2:
-fS?
-^
"C?"
^
-^-
2i
-G>-
-^±
^=^^-^--
-&
iq:
5 6
i
b
6 6
b5
4^
2
^ I
\ t
F f 2
220
THE PRINCIPLES OF HARMONY.
i
No. 13. — Exercise on Dissonances by Suspension.
(Chapter VIII, 3—7.)
rzi
Sfi
ifS-
321
O-
:^
i^z:
-r2L
fS>-
-CZ.
*y
^TZTTZgZi:
T^
-^-
s
z:±
:c2:
:q:
-Si'
Z2
« t 3 ^ ^ t 3 '
:?2=
^ 3 ^ M 3
-»4^ <S>-
22:
JUL
-(SI-
f^^^a
-fs-
z^:
-p— P— r±
-is>-
2:2:
-(S»-
-^^-^
^:=1
2bir
^" ^ <^ cJ ; vd~f^
6 4
-5 - 5 6 9 69865
4 3 4 5 M -
I^
-l^-
:^
-^
.Q-
^^-
X2:
1^2:
zz
^tat
:^
i^
-&^
-(S>-
Z2:
-is»-
frt
1^21
-iS*-
T^
1% % %\ ^%
5 Go ce 6
3
1^:2:
6 (i 6
^-^
-s»-
7 6 5
THE PRINCIPLES OF HARMONY.
221
No. 14. — Exercise on Retardations.
(Chapter IX, 1—4.)
-G^
:-t=*
:<^ ^
. g * =:
«S^^^^
i3
^^
:^
^Ei
z:±
3
^
5 6 7 8
I »'S
M
:^^=f^
:^i=^
?:^
& — "
@^
t=-
?n=^=f:
*=#=
:^:
^21
:fe
I % I
i
4^ 6
2
^^
^51
i^^^
" Q eJ-
^ — rj
P^
SZ
6 7 6 7 8
-f5h-
rj -r^
Z3:
222
THE PRINCIPLES OF HARMONY.
No. 15. — Exercise on the same.
S^^
5^fe
t3tX3
7 G 6 7 6
y -
76 6 56 56 76
^^
6 7 6 6 5 -
4 5 4 3
^E^^
3
3
3^
^— ^
56 7 6 6 41- 56 76 7
^ ^
7 « ^
65-6 5 - 78
4 3 4 3 5-
4 3
No. 16. — Exercises on the Harmonization of Melodies.
(Chapter X, 4.)
1.
*t
»
.^
y ^ *\
m ^
^
— <i.
.^ 1^ 1 1
A '* A *
W \
^^
^
■^
^ 1
f(\\ V - *
1
1 i * " 1
V/ 4- • I
1
1 ' 1
J
=3F
H* ^
-^ ^ -
:4:
S
i
i
^^ee£
I ^1
:ifat
-«s^-
THE PRINCIPLES OF HARMONY.
223
^^^^
r p
i> J r
q^=p:
^:^ -ni^r_r
^ -9- '-^ '^ -^
^
]^
i^
u.
E
s
-^i:^
3
X2:
— \ \ — ^ rd — ^ ^
fe^-j^
-<s-
22:
5^
-tS^
^^ j=f=
i^
W
iV— ^— ^
22_
P g fS^-r& f^
1^2:
^^ig
tip
?2=^
^^^-
2±
"or
fe
224
THE PRINCIPLES OF HARMONY.
111.
^4?=1=1^
^ r ^ ^
g
S^i3
:^=e=^=^r+^^
-^ J > J B<J ^^
7^
T^-p — ^ 1 h ^ ^ ^ ^ > r- T !? r •
^)_2 ^'__^ — #L. ^ — .^ W . ^ k- ?- — I
^^^
^^
^
^Ee
J ^
Z2
No. 17. — Exercise on the Cliord of the Augmented Sixth.
(Chapter XI, 7, 8.)
.^L^ P~rP rj I ^ ~ r ^ n ~^^ x
= RP~^ J ^- J:
-o-
s»-
^
-<s>-
v::^-^
i^
^^tr]--^
S^
:^
e
:^
-(^
-(S»-
221
22
-&^
Lir^
-e^
I 6
t ?
6 7
^666 J
THE PRINCIPLES OF HARMONY.
225
-if— ^^, —
«s>-
^2Z issz:
lO-
-<s/
<s>-
^
fcj:^
(S»-
2:±
t::;^
Z2
-(S/-
"?:3"
^=^:
:ftc^:
11?
^^-
f
«5 6
122
-S^
-iS>
^2:
-(S> —
i 6
b
§ 7
4 5
No. 18. — Exercise on the same ivith Enharmonic Changes and Inverdons.
(Chai^ter XI, 9, 10.)
Z2:
¥^
m^^.
^^-
nf2±^
22
3
is^
D6 D7 6 6 - 7 6 ^B t 6 6
4»- #5 _ 4t- - b5 * 4)- 6
:b q b 2 (, 5
3
Z2:
-o-
6 5
4 «
P==P=^
be 6
^•#^=q:
22:
:^
tS'-
iS>-
-<s-
HS-
b 2
'2 I
4 3
9 8
bt 3
b7
Q5
bb
bS
b7
IS
p^
*^
-<s^
^ b c ^ ,. rJ
is:±
CJ
-o-
b6
4
"? 9 8 5 -
H 4 3
D
b7
b5
9 8
@S^
Sg
^2:
M
-<S>-
1^2:
i^
Z2:
-^
f- tt I
b7
b5
b5
be
D4
b7 e
CI5 5
9 8
7 6
4 -
t?
og
22G
THE PRINCIPLES OF HARMONY.
No. 19.— Exercise on the Chord of the Minor Seventh and Minor
Third and its Inversions.
(Chapter XII, 3.)
i^=r
T^~
221
6 7 6 5 1 5 i
4 3
No. Id.— Exercise on Cadences of all hinds.
(Chapter XIII, 2—7.)
1^
^
Z2
2-21
O-
Z2:-
^
zt^
^21
r^-
r:?
:^
:P2=^
:z2:
B6 6 6 6 5
9 § ^ 2
7 6 4 6
D6
b5
^
2i:
^=?=i
hS^
1^2
-«S>—
Z2:
fS>-
-?':M^-
221
3^
fS>-
q:
6 6
^iS
m
f 1-8
6 6 7 r: _
5 jj t D
4*- 6 P6
2 4
ae 6
b5
06
b5
M
_rJ g^
z±
?2:
--^-
■T±
m
6 D6
n
THE rUINCIPLER OF HARMONY.
227
No. 21. — Exercise on the same.
(Chapter XIV, 1.)
fS»-
:c2:
'-T^-
5S
Z2
- ?rr
til
6 7
IZZC2
fS>-
:ip
4^ — t^-
-^--
rj rJ
-^ —
Z2.
:^J
^^
6 7
4J-
2
7 B
n
T^-
m
:^
i^^zfet
J2Z
12^
-<s'^
-^
-&
b7
b 6 b
S
ip2:
i==:
-(S>-
rzjr-r_
(S>-
:i^
:^
fS>-
221
-eS*-
b7
(55
n
1
b7
b7 7 6
m^
1^
z:?:^:
■is>-
^:
^2:
:?:±
<^J rJ
^
b7
b5
6 7
/Ov
:^
I^ZJt^
i?:^:
:^=^
^s>-
}G^-
Tzti?dz
D6
^ i
6
[t^-
-C'-
-^:
1^1
•^-^ 6
:?:±
"O"
Gg2
228
THE PRINCIPLES OF HARMONY.
No. 22. — Exercise on Irregular and Deceptive Cadences.
(Chapter XV, 2—4.)
^^^.
T-
-G>-
-&■
22
im
-G*-
j?g:±= -^ fl< - y -S -
7 6 6 5-
4 » -
-7
#5
»
T2:
^i
:^2z=:tf?^
-«s-
q:
^21
2i
SE^
:i^
-c^-
-«
7 6 D7 6 5
5 4 J
D5 6 6
^^^^
T^
V^-=^^^=^
-^--
6 6
4
5
7 4 3-
5
^===^-^r ^^^
-^ f^-n
m
:a:
:z3:
-<s^
22
6 6 D
5 5
4 1(
^
^^^e
221
:5^2=r^
-o-
-^ flg>
221
6 6
Q5
7 6
D7
D7
5
5 6
7=^
m^B
-o-
t=^
hS^
2ZZ=^
e
22:
-.s--
l^-
6 5 - 44^
4 5 - 3
7 ~6 ^^ 5 -
4 7 4 «
6
THE nilNCIPLES OF HARMONY.
229
No. 23. — Exercise on Modified Basses.
(Chapter XV, 5—13.)
(S(h"^r] r
^
Z2:
^21
-f^-^
^-
:^
22
"C7
-S^
22
^2=^
_Q.
iS>- ^
#2=^
^^
ICZ
q:
Q5
6 7 7
li
No. 24. — Exercise on Sequences.
(Chapter XVI, 2—6.)
B
'JL ^ f
f^ ^ ^
^
6 6
f-
D6
^r ^ M f — f \ f -v-^—f-
-T^-n — r^—f-
v.=L4: •_! — * \ J J 1_
-t— -1- — J ^
6 7 7 7 7 7 7
7 6
:^=i
i^
i^
^
ti
-^ ^
=^=r=P^
=t:
6 6 6 6 6
6 7 6 7 6
5656 56 56Q76
rf=i^
'^
iffz^g^-:
8 7
6 5 6 8 7
5
6 6 7 6 7
i ^ i ^
D7
5
4t- 6 B
6 6
5
^
3
iczf:
^
at:^
-^"^.i-
2z:
,6 9 6
b5
6 7 8 7 5 -
5 4 3
230
THE PRINCIPLES OF HARMONY.
No. 25. — Exercise on the Introduction oj Dominant and Tonic Pedal.
(Chapter XVII, 1—3.)
^
t--^=^
p_j}fi-^
-^ — P :
-Q-
-^
T2L
e^^^
£
^^
-^ I f^ 1^ 1^ p ^p
-C2.
iS-
^f=Ptt«:
:^
44-
2
6 B 6
4 4 5
r-p— p
=^
1^=^
-:^
1 — I — p-
tztzit
Add the two inner parts without figures.
W,
22:
22:
22:
:q:
I
#-^
^2:
:^
^
^
e
^2:
:^
:z2z
@3.
-(S*-
-^S'
Z2:
22:
1^21
1^2:
S
-fS>-
:a:
S
icz:
-(Sl-
:^
la:
zi:
hS'-
3
?2=:^:
THE PRINCIPLES OF HARMONY.
231
i
t=^=^
t^-
<^-
1:2
S
j:±
IQ
Cr^:
2:2
-o-
Z2
"O
^^^
■^::^
^
^^
^2:
22
■c^
iS>-
-^
Z2:
^21
g> -^ ,^-gL,-g_^^
cz:
"C?"
"C2"
'^ ^ —
"^"
221
-<S>-
221
P-i — <s^-^
ff r j> p p:
-<s>-
irr
S
-<S'-
:z2
-<s>-
22:
IQ
22:
■^5"
"C?'
.C2.
L^I
(S^
fiS-
fS"-
rS^
-«S-=-
-e*-:-
-(S*-
:^±
^
2:i
-|S>-
^
ict
"22"
Q •
-C?- . "^2"
232
THE PRINCIPLES OF HARMONY.
No. 26. — Exercise on Breaking of Chorda.
(Chapter XVIII, 1.)
Break this harmony in as many ways as possible.
m^
icz:
-o-
:2z:
-<s>-
"TT
m
-^
-<SH
zz
-G>-
-rjL
-<s>-
-^
7 6
No. 27. — Exercise on Passing Notes and Auxiliary Notes.
(Chapter XVIII, 4—8.)
Adorn this melody with passing notes, and with auxiliary notes.
The bass and inner parts also.
u
■P^
y 1 7 * i
rj •
,^:j H" 1 ■■
yl r> • ' -^
— ' --^
rj .
r'
ffn * '
-^
—^
1
M^\) Z
_
J
r:i .
rmY W *'
r-j •
h-"
rj
l^'l ^ t'
V^k * 1
1
^ z
7 6 6
5
6 Be
i
J^_
:^
i^-
iS>-
^
-^s>-
221
T2L
221
hS^-
r:> »
i=^
-<S)-
r J Q
:C2IZZZZZ2
^ i
THE PRINCIPLES OF HARMONY.
233
^^ -1^
i^
/S>-
jQ.
icz:
-^2.
-<s>-
-<s^
/S>-
-G>-^
-o •
icz:
;^2:
221
-(S*-
H^
-^^
6 D
e t
5 -
4 3
No. 28. — Exercise on the Chord of the Fundamental Thirteenth and
its Inversions.
fm\'
<Si 1
1 %r^ 1
1 — —
1 — ^ — 1
r^
iP' (
V
rj
rj
v^ I
J
D7 6
5 -
t 3
5 6
6
8 7
6 5
b -
@3^
22zzr
-s>-
iq:
-«s>-
Z2:
-<s>-
1^2:
-<s^
76 76 76 5-
4 - 44^ 3
6 7
5 -
7 6
^
Z23:
:q:
:tt^
-<s»-
221
1^
Z2:
I 6
7 6
6 ,6
b5
9 8
9 8
D7 6
5 —
6 5
#
: Q
'JZZ.
7
-(S»-
icz:
:i^
Hh
:^
-7"^
« -
n\
- rj I rj — H
234
THE PRINCIPLES OF HARMONY.
List of Fundamental Chords.
Vll. Vlll.
zz
:^:
1^21
^a
22:
:?!2:
221
-«s*-
-s»-
-1^-
-^-
22:
-(S^-
22:
22:
-s>-
:cz:
-JZL
22:
-(S*-
— <s^
-s*-
Roots.
m
22:
22:
22:
22:
22:
-<S'-
-s>-
-<S'-
-<s-
-«S'-
XVI.
-<s»-
XVU, XVlll.
-G-
XIX.
-«s-
-<S'-
-G>-
-Gf-
-G>-
g:
:g
-e^-
-(S-
:5g
^S'-
-(S>-
-s»-
'(S*-
.<s>-
^
-^-
-o-
=<s>^
J@-
^<s-
-^
-G>-
-G>-
-G^
Df
-<S>-
hS^-
-«S^
Roots.
^^
tI9
7
tJ5
(lt\'
(<?;•
V.-'
& — ^
1 — Gt '
1 — ^ 1
— 1
— — I
9
D9
d9
d9
69
'/
7
7
7
7
t
%
%
t
i
THE PRINCIPLES OF HARMONY.
235
xxii. xxm. XXIV. XXV. xxvi. xxvii. xxviii. xxix.
-O-
-<S>-
-<s»-
-&-
C7
5
-<S^
-(S*-
-'S*-
-^» —
Roots.
C5
6 Jj
f 'i
-&-
-G)
-^
-<S»-
-<S»-
-<S^
-SH
t9
7
P9
7
09 EC
XXXI. xxxu.
XXXIU. XXXIV. XXXV.
XXXVll. XXXVUl. XXXIX.
Ma
-^^ — n
321
=s=
:zz:
-TTT
i^
-7^^
-<^-
-^
=8=^=3
jcz:
"C^
Tg:
-^
KZ
-^-
le-
1^
-^
'TTj'
-ys*-
Roots. 7
C5
7
*1
7
9
-(S*-
-s*-
-<s>-
-•s*-
-«s>-
-s>-
-<s*-
-<s>— -
-<s>-
-o-
-s>-
-o-
b9
'?
&9
7
Q9
7
1)9
b9
H li 2
236
THE PRINCIPLES OF HARMONY.
xl.
Z2:
xli
5S
xHi.
1^
xliii.
xliv.
xlv.
xlvi.
xlvii.
IgzziiS^fe
:feS:
xlviii.
^
?:22:
~o~
^^
<s>-
^^-
^'S^
^-^
Roots. ]'
b9
1^
-S»-
I^IT
Z2:
bb
b6
44-
2
6
t>5
be
B5
bb
b9
7
b9
t
b9
b9
b9
b9
b9
b9
1^2:
ib
b9
-<s>-
b6
b
b9
7
b9
r^\'
rj>
V^- r^
rT>
i*^"!
K^ c>
r:^
' «S
1 (S> — 1
— & — '
1 — o —
— <s» — -J
L — (S> — '
— <s» — 1
1 ^ 1
xlix.
1.
li.
lii.
:g:
liii.
liv.
-<s-
Iv.
icz:
Ivi.
221
Ivii.
:?2:
Iviii,
-<S>-
-<S>-
-(S»-
-fS'-
-G)-
IQ!
iS>-
-<S»-
122
:<S^
ic^:
:z2:
jQ_
-©•-
-O-
-G>-
-f^-
-«^
-iS>-
-G>-
Z2:
B5
Roots.
d5
&
m-
-<s> — >-
1
L- s" —
L- o — I
-
J s^
— 'S>
— s>—
— *s> — Ij
09
B9
7
V
¥
13
7
13
7
THE PRINCIPLES OF HARMONY.
237
lix.
"C?"
zz:
Roots.
Ix.
Ixi.
Ixii.
22:
-o^
fe
^
?:><^
1^
-<s>-
7
Ixiii.
S
Ixiv.
-^
-Q-
Ixv,
zz:
■^>-
di3
7
B13
7
«
di3
7
a\3
Ixvi.
Ixvii. Ixviii.
Ixix.
Ixx.
Ixxi.
Ixxii. Ixxiii. Ixxi^
122:
zz
TO"
22:
22:
zz
Z2:
-?:^
S!3-
^
-<s^-
:ig:
^
22:
-o-
tt^
:^f^
22:
_c2_
::?=:
22:
Z2:
22:
-^
Roots.
CJ^3
Be
1:7
D5
D4
CIS
D9
7
13
13
13
6
24
B13 D13
%
/^*
(^.
V-^
J — <s> '
1 — t^ — 1
1 (S» — I
:i3
For an explanation of the chords in the above list, the following
references are given to the body of the work : —
238
THE PRINCIPLES OF HARMONY.
Chap. II.
Chap. Ill, 2, 3.
Chap. Ill, 9 — 12.
Chap. Ill, 13—15.
Chap. VI, 2—8.
Chap. XII, 1—3.
Chap. XII, 3.
y Chap. XI, 4—10.
No. xxxvii.
No. xxxviii.
No. xxxix.
No. xl.
No. xli.
No. xlii.
No. xliii.
No. xliv.
No. xlv.
No. xlvi.
No. xlvii.
No. xlviii. . Chap. XIII, 7.
No. xlix.
No. 1.
No. li.
No. Hi.
No. liii.
No. liv.
No. Iv.
No. Ivi.
No. Ivii.
No. Iviii.
No. lix.
No. Ix.
No. Ixi.
No. Ixii.
No. Ixiii.
No. Ixiv.
No. Ixv.
No. Ixvi.
No. Ixvii.
No. Ixviii.
No. Ixix.
No. Ixx.
No. Ixxi.
No. Ixxii.
No. Ixxiii.
No. Ixxiv.
Chap. Ill, 6, and Chap.
XV, 5, 7, 9, 12, 13.
Chap. VI, 8 (note), and
Chap. XV, 5, 7, 9, 12, 13.
> Chap. XVIII, 11—13.
SHORT EXAMPLES
FROM THE
WORKS OF VARIOUS GREAT MASTERS.
No. 1.
?Ee
'■^r-m-
SE
J=d:
321
e
i
- fj*^^-— r
J 14^
-Q
z:±
^t^
^I
be
4
Z b-
5*=^=*:
=F
be o I
4 8 In
Koots.
izi
^
SEE
^^=b:
-c^
-s^
^
-Gh
From Weldon's Anthem " Hear my crying," referred to in the note
to Chapter XI, section 10.
This is supposed to be the earHest example of an inversion of the
chord of the augmented sixth. Here * it is analysed.
240
THE PRINCIPLES OF HARMONY.
No. 2.
i
^J-3:
et g]
ce,
f
^ "^s
^lA
3E^
^ *-w
i^55^
S
-^ -^
f
Roots.
i
^21
s:
1
/^ s b 1 *
^
-^ —
1 —
^ —
— ^ —
fr)^ — ^
— f.
gra
L: ^ J^
--i-.-
=fcj
r 1^
» —
^^^^^
Nt^^ —
Roots.
1*
^
^^^ \
1 ^ V-
■
1
THE PRINCIPLES OF HARMONY.
241
i
t=^P
^
&^^
^
^ gra
- ee
^=^^
pour
moi.
—7^
i
-^^
Roots.
r r 1
n
^■wi^-w
m
r==^
-o-
pp
^ r ^ h r
^ — ^
^
gra - ce.
I
Mi
gra - ce,
^.■.:
>b->^ !^
•^
;^
^B^^
■^^
I 5
^^^^
Roots.
-e( ^
I 1
242
THE PRINCIPLES OF HARMONY.
/
/r\
ei^fn*
:^=t=5S^1
/tN
^^ss
w^^
ffra
SS
ce pour moi.
4
i^=W^£:
f
-M=^-
ce
Roots.
m
^
3
-F-F-
From the conclusion of " Eobert, toi que j'aime," from Meyerbeer's
Opera of " Robert le Diable," where a most pathetic effect is produced
by the introduction of the inversion of the chord of the augmented
sixth at *, where the D b is admirably sustained by the second
bassoon.
THE PRINCIPLES OF HARMONY.
243
No. 3.
i5i
IS
^
fjgg
s
^t=g
Som - - bre fo - ret.
de-sert, triste
et sau - va
-^
ge.
1 I 3 ^ II '^3 .-0-
^
kj-
S:
5=t^
i^-^
1=1^
Roots.
iS^Ei
j-
j-
j-^
f^
f^
From Rossini's Opera " Guillaume Tell." At "'* the chord of the
augmented sixth inverted, and of course based upon a dominant and
supertonic root, is introduced on a tonic pedal, thus involving three
simultaneous roots — a rare case.
r 1 2
244
THE rniNClPLEfc) OF HARMONY.
No. 4.
From the opening of Haydn's " Creation," where another inversion
of the chord of the augmented sixth is seen at '". This is a very
peculiar combination of dominant and supertonic harmonies, as both the
roots bear their respective major thirds, and the minor seventh of the
supertonic is omitted. The chord marked f is to be regarded as a dis-
sonance by suspension and retardation ; but as it is prolonged and
emphatic, both the roots are here assigned : for in every case of sus-
pension or retardation two roots must overlap of necessity, though it
has not been thought requisite always to record them both in the
pages of this book. In the following examples it has been thought
needless to write the roots in a separate stave.
THE PRINCIPLES OF HARMONY.
245
No. 5.
f=fc5
^S
-b-
-e^
-!S»-
4S>-
^S>-
-Y^h-
m
From a Minuet from Haydn's Symphony in I), where at * we see
a chord of the augmented sixth on roots E and B introduced on the
pedal A, thus involving three simultaneous roots, as in No. 3.
No. 6.
i
s
yjzBzzE^
j l ^ JJ -
d?
221
-^
His mighty griefs . re - dress.
5=5=5=5^=^=^^9=^
h±:
^^
1^=^=^=^
at3t
H 1 (-
^t:^t=W=it
^
-^=^^
From Handels " Samson." At * there is a very good example of
the employment of an enharmonic change, to modulate suddenly from
D to Eb/
246
THE PRINCIPLES OF HARMONY.
No. 7.
±t±=^
^5
3^3=?=^
Thy re-buke hath bro - ken his heart.
1^-
E
:1^
-Gh
^-
— ^^
From Handel's " Messiah." At * there is a beautiful enharmonic
change to modulate suddenly from G minor to E minor. Here the
enharmonic change is between two different parts of the harmony,
instead of the same part, as is usual.
No. 8.
From the chorus "Let Sinai tell," in Crotch's Oratorio "Palestine,"
where at * there is a chord of the augmented sixth resolved en-
harmonically to introduce a sudden modulation from DiJ minor into
D d major. The effect of this is truly magnificent and sublime.
THE PRINCIPLES OF HARMONY.
247
No. 9.
^' ^ ^^
-eh -&r
^- z' N-
^
i=-- r^3:
i
^^
22:
22
z:^:
f7
-<s»-
.J
ST
«^^
^=?E^
J^
*=]
lEt
^^^^^^g
4'
-<^
^=^^^3
&
^
^1
1^
4-^- g^T7>^ -«-g-^->
^31
r:]l.r-^ r^
fe&l^
^
Example of an inverted dominant pedal from Beethoven's Pastoral
Symphony (No. 6).
248
THE PRINCIPLES OF HARMONY.
No. 10.
Y^=fC=T^
Z2:
T^~
T^-
m
^
s:
i2:
i^
I
r 1
r:^^
J^
?
rrri I
f^^
f
^^^-
r llJ 'y-Lj ' jj I jii
:^t;^-
2i
z^t
:^=2i
2±
:?:±
s: r 1
~^
Example of tonic, dominant, and supertonic pedals combined, from
the same. Here, in reality, there is only a double pedal, for the super-
tonic G does not hear any harmony of its own.
THE PRINCIPLES OF HARMONY.
249
No. 11.
^sm
g.
> 1 ^
-^1 =1-
toi
! 1 I 1
-^-^
-^ ^
=^ 11
?=
to:at
?i
J^
:^
i2^3ti^
s
*=t
^
i^^^^^
r rr
From Beethoven's Symphony No. 5, in C minor, where we see a
curious and original upper pedal held on by the Clarinet, and singularly
resolved by rising chromatically to the leading note of another key, for
a transient modulation, in order to introduce the perfect cadence.
Kk
230
THE PRINCIPLES OF HARMONY.
No. 12.
t-r^^-
m
" g r • ^ -r^^
My faith shall cling un - - sha
I
he^i
^g^ -
&;&
a?:
&«;— j—^^^qfcjzg
I I
I I
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l^
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b»^
i
s-^^
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Si
:Jf=^5Fi:^
iii:
i^:
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ken to Thee, my Sa
s^#
dJ
S^
ilt
viour, to
^-^^nit
-j^
iiSifj^s
^^^l=Sa^^%
^ff
SO'
THE PRINCIPLES OF HARMONY.
251
-^-=^M=^
^
P=P
— ^ r i
M—^
fe
Thee, to
Thee, my Sa
viour.
::fetagi|5
--«*:
I-
tJ^-
:i;
^=^#^J: i^°^gfeg
F^fr-W--^
^3^
Ktt^
$<^ •
fflE.
*
N
^
From Spohr's " Crucifixion." At * there is a very clever enhar-
monic modidation from B b minor to E Q major. At t Spohr has
written E Q, for convenience of reading, but the note ought io be
written Fb, being the minor ninth of the root Eb.
From the Overture to Mozart's " Zauberflote." At * the B C ought,
theoretically, to be Cb, and it would then be an imperfect inversion
of the chord of the minor thirteenth on the root Eb. As B 2 it could
only form part of the chord of G, which is not possible.
K k 2
252
THE PRINCIPLES OF HARMONY.
No. 14.
li
-C2-
^ (S — I — (S>
2Z
^
g
-S>-
fire that wastes
her
lo - ver's heart
S
^-
-TJ
^
-«S^
fire,
Hnt^Sizji
make
-O-
her feel
the
-^-
^¥ if==f^
4-^
fire,
make
her
E:-*gzzti
^21
9
s
is-
iS>-
cz:
:^
-fS>-
fire that wastes
/S^
her
lo
vers
Wi^
-o-
<s>-
fire
that
wastes
her lo - ver's heart
■i^-
K3
1 rJ
-G>-
-s>-
fire,
W¥^?=^
te
fire,
fire.
-O-
make
her
feel
iS"-
S>-
that
wastes
her lo - - ver's
^ ftr?-^^ =^
-s^
-(S>-
-<s>-
/^
THE PRINCIPLES OF HARMONY.
253
-^
lS»-
ICZ.
-n
-&^
3E!:
way
that wastes
z:±
22
:^±
-Gh
fire
that wastes her lo
heart
w
-^
4^
7^
=v
T^
-Gh-
-7-=
-(S>-
feel
the fire
m
-eh
heart.
i
-<s»-
^ — ^
^ ^
-fw^
MfS
way
and make her feel the self - same fire
-S»-
1=^=^
the self - same
fire,
w.
-<s>-
f l f ^
-^W^
heart,
and make her feel the self - same fire
^
*=£
122:
:P2:
W
and make her feel the self - same
fire
%-
^^3^
:r±.
-Gh
izz:
±=
I
254
THE FIUNCIFLES OF HARMONY.
This passage is from De PearsalTs splendid iriadrigal " Great God
of love." At * the seven notes of the diatonic scale are introduced
simultaneously, by means of a combination of the chord of the added
ninth with all the practicable suspensions and retardations. The effect
is admirable.
No. 15.
i
I J -J- 4 J^ !
m
^=a
E
icz:
=F=F^±1^
-e>-
t=
p
T=^=^
'^-
iEE^Ei^&d^E^EU
f
^21
-p -^
From the " Cum Sancto Spiritu " in Beethoven's Mass in C. At *
there is the fourth inversion of the added ninth irregularly resolved by
contrary motion of all the parts.
No. 16.
ii
J-
3
P^E^
^&-
^
^^E^
' -^1 Vi
uP^
jEgr^r^fM^gg^FFf^g
THE PRINCIPLES OF HARMONY.
255
From Mozart's Quartett in C (No. 6), where in eight bars tliere
are no less than six false relations, in defiance of the general rule.
Here they are introduced on purpose, in order to produce a vague,
sombre, and mysterious effect — which they do most admirably.
No. 17.
1*-^-^
-r^-r
3p:
frf- ,. -
^^
-^— K
^155:
1 1
^,^
^
^
' u I I ^ ■■ — I
135!:
&c.
25G
THE PRINCIPLES OF HARMONY.
From Bach's Fugue in C minor (No. 2 of the " Forty-eight Preludes
and Fugues"), where there are four false relations in three bars. Here
they are merely tolerated for the sake of the " Imitation " in the
counterpoint.
From Mendelssohn's " St. Paul." Example of curtailed resolution of
the first inversion of the chord of the minor seventh and minor third.
APPENDIX
Ll
EXPLANATION
OF THE
DIAGRAMS OF THE MUSICAL SCALE
AND ITS COMPONENT INTERVALS.
By WILLIAM POLE, F.R.S., Mus. Doc, Oxon.
The object of these diagrams is to present to the eye a representation of the relative
magnitudes of the various musical inten'als, analogous to the impression which they make
upon the ear.
The idea of doing this is not new. Mr. HuUah has long adopted the symbol of
a ladder representing the diatonic scale, the intervals between the third and fourth, and
between the seventh and eighth steps being only one half the length of the other degrees ;
and I believe somewhat similar figures have been used by other wi-iters ; but I am not
aware that any attempt has heretofore been made to represent the intervals graphically,
with such accuracy as to render appreciable to the eye the minute differences on which
the more delicate ajipreciation of the pitch of the sounds must depend.
I think the diagi-ams here given will fulfil this condition, as they have been very
carefully drawn, and are on a very large scale. The octave is above thirty inches long,
giving about two inches and a half for each semitone; and I presume that, on such
a scale, the minutest differences of pitch which can be distinguished by a sensitive ear,
will be rendered distinct to the eye. For example, the difference between the true fifth,
Ll2
260 THE PRINCIPLES OF HARMONY.
and the fifth of equal temperament, is only about one-fiftieth of a semitone, which few
ears could detect except by its beats ; yet its existence is made perfectly evident upon
the diagi-am, amounting to a space of about one-twentieth of an inch, or half one of the
subdivisions.
A few remarks will explain the principle on which the intervals have been laid down.
The pitch of a note depends on its number of vibrations per second ; and it might at
first be supposed that by laying down on paper (or, as it is technically termed, " plotting ")
the vibrations of two given notes by a scale of equal parts, a sufficient representation
of the interval between them might be obtained.
But this would not do, for the reason that it would give the same interval different
values according to the place in the scale where it was taken, which is at variance with
the idea conveyed to the ear.
It is manifest that the idea foi-med through the ear of the magnitude of any interval —
say, for example, of an octave — is the same, whether it be taken at a low pitch, thus —
Z2:
or at a higher pitch, thus —
"TTJ-
But if these were "plotted" on paper by the number of their double vibrations per
second, we should have the former represented by
132 — 66 = 66 divisions of the scale,
and the latter by
1056 — 528 = 528 di\nsions,
which would clearly be inconsistent with the idea sought to be conveyed.
Any representation of spaces derived from the harmonic divisions of the monochord
WDuld also fail, for a similar reason.
APPENDIX. 261
The mm*e correct method of representation is based on the principle that a musical
inteiTal is expressed not by the difference, but by the ratio of the vibrations of its two
sounds ; this ratio being always the same, at whatever pitch the interval be taken. We
only therefore require a convenient mode of laying doAvn this ratio on paper ; and this is
obtained, according to a well-known mathematical principle, by the application of logarithms.
If two numbers have a definite ratio, the difference between their logarithms will always
be equal, whatever the absolute magnitude of the numbers themselves may be. Thus, in
the two cases above cited, we have for the example at low pitch
For the example at high pitch
log. 132 =
2-12057
log. 66 =
1-81954
difference
•30103
log. 1056 =
3-02366
log. 528 =
2-72263
difference
-30103
To apply this principle, tlierefore, it is only necessary to take out, from an ordinary
table, the logarithms of the numbers of vibrations of any given notes, and the difference
of these logarithms will be the logarithm of their ratio. And if this latter logarithm be
laid down or " plotted " to any convenient scale of equal parts, the intei-val between the
two notes will be thereby represented to the eye, in a manner exactly conformable to the
idea conveyed by this interval to the ear.
On this principle the two accompanying diagrams, Plates I and II, have been con-
sti'ucted.
Plate I is a diagram of the extent of an octave, containing all the most usual inter-
mediate intervals. To form this, a scale of inches, subdivided into tenths, having been
fii-st drawn on the paper, the logarithmic difference representing the ratio of the octave
(•30103), magnified one hundred times, was laid down so as to occupy 30-103 of these
divisions, and within this space were also marked the other intervals, calculated in like
262 THE PRINCIPLES OF HAllMONY.
manner. Tluis 17"609 inches, corresponding to the log. of H, formed the interval of the
perfect fifth; and 9-691 inches, corresponding to the log. of f, formed the major third,
and so on.
The centre, or widest column of the figure, gives the diatonic scale ; the chromatic
notes being placed in narrower columns on each side, the sharps to the left, and the flats
to the right.
It will be seen that many of the notes have been given two values ; these are the
equivocal notes, the exact pitch of which may be deduced in two ways, giving two
different values for them. For example, the second of the scale, D, may either be deduced
as the fifth above G (= 5* 11 5), or as the fourth below A (= 4 5 76), the former making
it a major tone above the tonic, the latter a minor tone, Similarly the minor seventh B?
may be deduced either as a minor third above G (= 25'527), or as a perfect fourth
above F (= 24"988), and so on for others.
On each side of the true scale is placed a column giving the positions of the various
notes as determined by the system of equal temperament, each semitone interval being
equal, and = 2"5086 inches of the logarithmic scale. The nature of the errors induced by
this mode of temperament will be obvious by simple inspection.
To the right of the before-mentioned scales are five columns, designed to shew the
natural harmonic notes to the fundamental tonic, up to the thirty-second, which is pro-
bably higher than they can ever be actually distinguished in practice, even with artificial
aids. They are laid down by the logarithmic values of their intei'vals from the funda-
mental, in the same way as the notes of the scale, except that in the second, third,
fourth, and fifth columns the equivalents for one, two, three, and four octaves respectively,
are subtracted from those of the time intervals, in order to bring the whole of the notes
within the compass of one octave. This will be clearly explained by Tables B and C.
The names of the notes marked on the diagram are arranged for the scale of C,
i. e. C being the fundamental or tonic, this being the simplest scale to select for an
example ; but the same arrangement of lines would answer for any other fundamental by
merely chandne the names attached.
i
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DIAGRAM OE VARIOITS S]^IALL Iin:^ERVALS
T
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APPENDIX. 2G3
Ou the left of the cliayrum is a " Scale of Proportionate Vibrations," which shews the
number of vibrations made by any of the notes marked, in proportion to those of the
fundamental, the latter being taken as 100. Tliis is in fact the scale of natural numbers,
to which the divisions of the diagram are the logarithms.
The smaller diagram, Plate II, contains representations of several small intei-vals,
drawn on the same principle and to the same scale as those in the larger diagi'am.
The Table A gives a list of the various intervals (except the harmonics) contained in
Diagrams I and II, with full particulars respecting their derivation and composition, their
exa^t ratios, and the logarithmic numbers by which their places in the diagrams have
been determined.
Tables B and C are devoted to the harmonics only, as already explained.
The intervals ou the diagram may be measured and compared either with a pair of
compasses, or by marking the distances down ou a strip of paper.
London,
March. 18G8.
A.
TABLE OF THE VARIOUS MUSICAL INTERVALS CONTAINED IN DIAGRAMS I AND II.
The number in the last column determines the magnitude of the interval as laid down in the diagram.
Limiting
Notes.
Name of tlie Interval.
1
j
Derivation of the Higher Note.
1
Composition of tlie Interval (logaritlimic)
Ratio of Vi-
brations of tlie
two Limiting
.Sounds.
Logarithm
of this
Ratio.
c— c
C-B;j
Octave ....
2 : 1
125 : Gi
•30103
•29073
Major third above Gjf
Three major thirds ....
C— Cl7
Diminished octave
Minor sixth above EI?
f Minor third + minor sixth ; or t
L octave — s '
48 : 25
•28330
i
C— B
Major seventh . .
\ r Equal temperament
1 True interval . . .
Eleven mean semitones . . .
Fifth + major third
15 : 8
•27595
•27300
i f Minor third above G .
Fifth + minor third
9 : 5
.25527
C— Bb
Minor seventh . .
■{ By equal temperament
Ten mean semitones ....
•25086
t_ Fourth above F . . .
Two perfect fourths ....
16 : 9
•24988
C-AJt
Augmented sixth .
r Major third above F jf .
1 do. do. . . .
Major seventh — -S"
Major sixth + 8
225 :128
125 : 72
•24497
•23958
r Pythagorean ....
Three fifths — one octave . . .
27 : 16
•22724
C— A
Major sixth . . .
J By equal temperament
Nine mean semitones ....
••2-2577
LTrue
r Minor sixth + s ; or fifth + t; or i
fourth + major third . . J
5 : -i
•22185
C-At>
Minor sixth . . .
jTrue
I By equal temperament
Fifth + S; or fourth + minor third
Eight mean semitones . .
8 : 5
•20412
•20069
C— GJt
Augmented fifth .
1 Major third to E . .
Two major thirds
25 : 16
•19382
C— G
Perfect fifth . .
if True
I By equal temperament
3 : 2
•17609
•17560
Seven mean semitones . . .
f Major third below BP .
Fifth-s
30 : 25
•15836
C— Gb
Diminished fifth .
■i do. do. . .
Fourth + 6'
64 : 45
•15297
[^By equal temperament
Six mean semitones ....
•15052
TABLE OF INTERVALS (continued).
265
Limiting
Notes.
C— F
C— E
C— Eb
C-D#
C— D
C— Db
c— c#
(CJ— Db)
c— c
Name of the Interval.
Tritone or augment-
ed fourth . . .
Perfect fourth .
Major third . . .
Minor third . . .
Augmented second
Major .second . .
Minor second . .
Augmented unison
Enharmonic Diesis
Pythagorean comma
Ordinary comma (c)
Unison ....
Derivation of tlie Higher Note.
r Major third above D .
L Minor third below A .
r By equal temperament
LTrue
r Pythagorean , . . .
■< By equal temperament
[True ,
rTrue ......
I By equal temperament
Minor sixth below B .
f Fifth above G . . .
By equal temperament
Fifth below A . . .
■ Major third below F .
. By equal temperament
Minor sixth below A .
Composition of the Interval (logarithmic).
Fifth-S; or T+T+t .
Fourth + 8 ; or T+t + t .
Five mean semitones . .
Major third + S . . . .
Four fifths — two octaves
Four mean semitones . .
T+t; or minor third + s .
T+S
Three mean semitones
r Major seventh— minor sixth
1 T + s
Tone major (T) = « + c
Two mean semitones . ,
Tone minor {t) = S+s . .
Diatonic semitone (S) . .
One mean semitone . .
Chromatic semitone («) .
S-s
Twelve fifths— seven octaves
T-t
Ratio of Vi
b rations of the
two Limiting
Sounds
45 : 32
25 : 18
4 : 3
81 : 64
5 : 4
6 : 5
75 : 64
10
9
16
15
25
24
128
125
3'^
219
81
80
1
1
Logarithm
of this
Ratio.
14806
14267
12543
12494
10230
10034
09691
07918
07526
06888
05115
05017
04576
02803
02509
01773
01030
00590
00540
00000
M m
B.
TABLE OF THE NATURAL HARMONIC NOTES TO C.
The sign + signifies that the harmonic note is sharper than the one named, the sign — flatter.
Number
in rank.
1
2
Upper note,
tbe lower
one being
CCC.
Octave.
Ratio of
Vibrations.
Logaritlim of
this ratio.
ccc
cc
i First Octave -j
> Second Octave J.
1 : 1
2 : 1
■00000
•30103
3
GG
3 : 1
•47712
4
c
J I
4 : 1
•60206
5
E
5 : 1
•69897
6
G
» Third Octave J
J
i
6 : 1
•77815
7
Bb-
7 : 1
■84510
8
c
8 : 1
•90309
9
d
9 : 1
•95424
10
e
10 : 1
1-00000
11
ftf-
11 : 1
1-04139
12
g
' Fourth Octave .
12 : 1
1-07918
13
ab +
13 : 1
1-11394
14
bb-
14 : 1
1-14613
15
b
15 : 1
1-17609
16
17
c
db-
i
= 16 : 1
17 : 1
1-20412
1-23045
18
d
18 : 1
1-25527
19
5b
19 : 1
1-27875
20
e
20 : 1
1-30103
21
f-
21 : 1
1^32222
22
ft-
22 : 1
1^34242
23
gb
23 : 1
1-36173
24
g
> Fifth Octave .
24 : 1
1-38021
25
H
25 : 1
1-39794
26
ab +
26 : 1
1-41497
27
a
27 : 1
1-43136
28
bb-
28 : 1
1-44716
29
bb +
29 : 1
1-46240
30
b
30 : 1
1-47712
31
cb +
31 : 1
1-49136
32
a
C
.
32 : 1
1-50515
TABLE OF CONTENTS.
CHAPTER I.
1, 2. The Stave; Clefs. 3. Notes and Rests. 4. Time-table. 5. Sharps, Flats, and
Naturals; Table of Signatures. 6. Classification and Table of Intervals. 7. Chords;
Ti'iads. 8. Motion ; Consecutive Fifths and Octaves ; Other Consecutives. ... 1
CHAPTER II.
1. Harmony as derived from Natural Phenomena. 2. Origin of Tonic and Dominant
Triads. 3. Dominant Chord ; Connection of Dominant Chord and Tonic Triad.
4. Chai'acteristics of Tonic and Dominant Harmony respectively. 5. The same
further explained. 6, 7. Force of the Minor Seventh and Leading Note in Dominant
Harmony. 8. Resolution of Discordant Notes. 9. Chord of the Added Ninth. . . 14
CHAPTER III.
1, Different positions of the Tonic Triad, and of the Chord of the Dominant Seventh.
2. Inversions of the Tonic Triad. 3. Thorough Bass Figuring. 4. Inversions of
the Chord of the Dominant Seventh. 5. Their Characteristics and Resolutions.
6. Allowable omissions of Notes. 7. Allowable doublings of Notes. 8. Imperfect
Resolution on the |. 2. Inversions of the Chord of the Added Ninth. 10. Reso-
lution of Major Ninth first. 11. Awkwardness of Fourth Inversion accounted for.
12. Figuring. 13. Allowable omissions of Notes. 14. How to discriminate
between certain Chords. 15. Additional Thorough Bass Rules 26
CHAPTER IV.
1. General Paradigm of Harmonics. 2. Origin of the Diatonic Scale and its Harmonies ;
Origin of Modulation ; Logier's Rule. 3. Completion of Ascending Scale ; Re-
marks on the Sixth and Seventh Degrees. 4. Varied accompaniments by means of
Inversions. 5. The Descending Diatonic Scale. 6. Other ways of harmonizing it. . 56
b
CONTENTS.
CHAPTER V. PAGE
1. The Minor Mode deducible from Nature. 2. Comparative perfection of Harmonic
Intervals ; Derivation of Minor Third, and proofs. 3. The same subject concluded.
4. The Minor Triad essentially Tonic — why so. 5. Relative Majors and Minors.
6. Derivation of Minor Ascending Scale. 7. Remarks of its Sixth and Seventh
Degrees. 8. Why the consideration of the Descending Minor Scale is deferred. . 66
CHAPTER VI.
1. Natural origin of the Chord of the Minor Ninth; Its Invei*sions and Resolution.
2. Allowable omissions. 3. Chord of the Diminished Seventh ; Its Inversions ;
The Enharmonic Change. 4. Resolution into Major by licence. 5. Enharmonic
Modulations. 6. Rule for discovery of Roots. 7. The same. 8. Application of
this rule to Diminished Sevenths 74
CHAPTER VII.
1. Descending Minor Scale. 2. Alterations of Minor Scale. 3. Various Harmonizations
of the Minor Scale. 91
CHAPTER VITI.
1. New Discords. 2. Definition of Terms. 3. Dissonances by Suspension. 4. Prepa-
ration and Resolution. 5. Rules. 6. Dissonances by Suspension in Inverted
Chords ; Rules. 7. Double Dissonances by Suspension, 8. Additional Rules for
certain cases - ... 95
CHAPTER IX.
1. Retardations. 2. Their Preparation and Resolution, 3. Double Retardations.
4. Retardations and Suspensions combined 112
CHAPTER X.
1. Sequences of Dominant Sevenths. 2. Sequences of Diminished Sevenths. 3. Re-
versed Sequences of Dominant Sevenths, and of Diminished Sevenths. 4. Rules
for the harmonization of Melodies 116
CONTENTS. xi
CHAPTER XI. PAGE
1. Origin of Secondary Roots, in Nature. 2. The same. 3. Application of the principle.
4. Chord of the Augmented Sixth ; The Italian, French, and German Sixths.
5. Derivation of this Chord. 6. The same. 7. The same. 8. Rules for its
Resolution, and Examples, 9. Treatment of it enharmonically. 10. Its In-
versions 123
CHAPTER XII.
1. Chord of the Minor Seventh and Minor Third. 2. Chord of the "Added Sixth."
3. Inversions and Resolutions of this Chord 143
CHAPTER XIII.
1. Cadences. 2. Perfect Cadence. 3. Plagal Cadence ; Tierce de Picardie. 4. Im-
perfect Cadence. 5. How to introduce a Cadence. 6. Modifications and curtail-
ments. 7. Neapolitan Sixth 151
CHAPTER XIV.
Cadences of Modulation ; Irregular Cadences ; Incomplete Cadences ; False Cadences ;
Interrupted Cadences j In-egular False Cadences ; Suspended Cadences 165
CHAPTER XV.
1, 2. In-egular Modulations by the Chord of the Dominant Seventh. 3, 4. The same.
5. Progression by Triads. 6 — 12. Modified Basses 172
CHAPTER XVI.
1. Formation of Sequences of Sevenths and Ninths, without Modulation. 2. Derivation
of Sequences from Suspensions. 3. Example in Three-part Harmony, with the
Roots found. 4. Same with Ninths also. 5. Sequences of Inverted Sevenths.
6. Sequences of Sixths 1^9
CHAPTER XVII.
1. The Pedal or "Point d'Orgue." 2. Figured Example explained. 3. Upper and
Inner Pedals ^^^
xii CONTENTS.
CHAPTER XVIII. PAGE
1. Broken Harmony. 2. Doubling in Octaves, when admissible. 3. Application of
same. 4. Unaccented Passing Notes. 5. Accented Passing Notes. 6. Chromatic
Passing Notes. 7. Auxiliary Notes. 8. Passing Notes and Auxiliary Notes by
Skip. 9. Accented Auxiliary Notes ; Doubtful Orthography and Derivation
of Resulting Chords ; Appogiaturas. 10. Discords by anticipation ; Licence.
11. Chord of the Fundamental Minor Thirteenth. 12. Augmented Ti'iad ; Inver-
sion of the Chord of the Fundamental Minor Thirteenth. 13. Chord of the
Fundamental Major Thirteenth. 14. Conclusion 193
Exercises on the Preceding Chapters 209
Short Examples from the Works of Various Great Mastei's ......... 239
c.
TABLE OF HARMONIC NOTES TO C, COMPRESSED INTO ONE OCTAVE.
The number in the last column determines the position of the note in the diagram.
Number in rank.
Upper note,
the lower one
being C.
Ratio of
Vibrations.
Logaritlira of
this ratio.
2, 4, 8, 16, 32
C
2 : 1
•30103
31
Ct> +
31 : 16
•28724
15,30
B
15 : 8
•27300
29
Bb +
29 : 16
•25828
7, 14, 28
Bb-
7 : 4
•24304
27
A
27 : 16
•22724
13,26
Ab +
13 : 8
•21085
25
Gt
25 : 16
•19382
3, 6, 12, 24
G
3 : 2
•17609
23
G\>
23 : 16
•15761
11,22
n-
11 : 8
•13830
21
F-
21 : 16
■11810
5, 10, 20
E
5 : 4
•09691
19
Eb
19 : 16
•07463
9,18
D
9 : -8
•05115
17
Db
17 : 16
•02633
2, 4, 8, 16
c
1 : 1
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