>■ Hi^ ' t mmmamie^mf \ \ .^■£i HARMONICS, O R THE PHILOSOPHY OF MUSICAL SOUNDS. B Y ROBERT SMITH, D. D, F. R. S, And Mafter of Trinity College In the Univerfity of Cambridge. O Deais Phcebi — JDidce lenimen. - 6 laborum THE SECOND EDITION, Much improved and augmented. ' — - ■ — - - ■ '■» LONDON: Printed for ^. and J.Merrill Bookfellers in Cambridge; Sold by B. D o D, J. W H I s T o N and B. W h i t e, J. NouRSE, and M. Cooper in London j J. Fletcher, and D. Prince in Oxford, MDCCLIX. Scimtis muficen^ mathefin^ atque adeo veram phyficam nojtris moribus n o n ahejfe a Principis perfona : ^ce quidem qmnia apud Gracos non laude foluni^ fed honore et gloria digna ducehantiir . Epaminondas, Imperator ilk infignis^ ne dicam fum- mus vir unus omnis Gr^ci^, philofophimn et muficam egregie didicit. Nam do5ius eji d Dionyfw, qui fuit eximia in muficis gloria. At philofophia praceptorem hahuit Lyjin Pythagoreum, neqiie prius euni d fe dimifj, qiiam doiirinis tanto anteceffit alios, ut facile intelligi pcjfet, pari modo fuperatiirum omnes in ceteris artibus. Corn. Nep. vit. Epam. fub initio. T O HIS ROYAL HIGHNESS WILLIAM DUKE OF CUMBERLAND^ This Philofophical Treatife, For a lafting Teftimony of Gratltudcj Is humbly offered and dedicated. By His ROYAL HIGHNESS'S moft devoted and mofl dutiful fervant Robert Smith 1G62315 THE PREFACE TO THE FIRST EDITION. TH E want of an elementary trea- tife of harmonics, fuch as might properly have been quoted in fupport of my demonftrations, has obliged me to begin the following work from the firft principles of the fcience. The antient theorifts conlidered no other confonances than fuch as are per- fed, and yet all their mufical fcales compofed of thefe confonances, have in pradlice been found difagreeable. The reafon is, they neceflarily contain fbme imperfedl concords, whofe imperfedlions are too grofs for the ear to bear with. The skill of the moderns has been chiefly employed in the bufinefs of tem- pering the antient Icalcs, that is, in di- ftributing thcfe groffcr imperfedions in fome of the concords, among all the reft or the greater part of them. By a 3 which VI THE PREFACE. which nteans, though the number of impeffed concords be greatly increaf- edj yet if their feveral imperfedlions be but as much diminifhed, the ear will be lefs offended than before. Becaufe it is the tranfition from a better harmony to a worfe, which chiefly gives the of- fence ; as is evident to any one that attends to a piece of mulic performed upon an inftrument badly tuned. It follows then that the inftrument would be better in tune, if all the confonances were made as equally harmonious as poiTible, though none of them were perfect. And if this be the true defign in tuning an inftrument, or tempering a fcale of founds, a theorift ought to be- gin with the fimpleft cafe; and inquire in the firft place, whether it be pofli- ble for two imperfed confonances to be made equally harmonious; and if {oy what muft be the proportion of their temperaments or imperfedions ; and . alfo whether different confonances re- quire different proportions, Thefe and the THE PREFACE. vU the like queftjons being rightly fettled, we may then determine in what pro- portion thofe grofler imperfedions in the antient fcales ought to be diftribut- ed, fo as to make all the concords equally harmonious in their kind, ei- ther exadlly or as near as poffible. But as none of the writers that I have feen, have attempted to give us the leaf!: notion of the nature and con- fiitution of imperfed confonances, nor of any one property or proportion of their effeds upon the ear, except a iin- gle conjedure whofe contrary is true (^), it was not poffible for them to deter- mine, from the principles of fcience, what diflribution of thofe groffer im- perfedions in the antient fyftems, would produce the moft harmonious fcale of mulical founds. As this is one of the moft difficult and important problems in harmonics, in order to a fcientific folution of it I found it neceffary to premife a Theory of Imperfed Confonances (^), wherein a 4 i (a) Prop. XIII. coroll. 8. {h) Se<51;. vi. viii THE PREFACE. I have demonftrated as many proper- ties of their Periods, Beats and Har- mony as I judged fufficient for fblving that problem, and probably any other that belongs to harmonics. This theory with its preliminaries and confcquences takes up a large part of the prefent treatife. As to the reft I chufe to refer the reader to the book it felf or the In- dex, rather than trouble him with a further account of it: a fhort one would be imperfect or obfcure, and a perfect one, too long for a preface. Having been asked more than once, whether an ear for mufic be neceflary to underftand harmonics, it may not be amifs to give this anfwer : That a mu- lical ear is not neceflary to underftand the philofophy of mufical founds ; no more than the eye, to underftand that of colours. Our late ProfeflTor of Ma- thematics was an inftance of the latter cafe, and the xx^^ propolition of this treatife affords an inftance of the for- mer. For by the folution of that pro- polition and a new way of tuning an viii'\ THE PREFACE. ix VI 11^^, defcribed in prop, xi, fchol. 2, art. 6, a perfon of no ear at all for mufic may foon learn to tune an or- gan according to any propofed tempe- rament of the fcale ; and to any defired degree of exadnefs, far beyond what the fined ear unafTifted by theory can poffibly attain to: and the fame perfon, if he pleafes, may alfo learn the reafon of the pradlice. But though an ear for mufic is not neceflary to underftand this treatife, yet thofe that are acquainted with mu- fical founds will more readily appre- hend many parts of it, and receive more pleafure from them. In the firft fcholium to prop, xx, I obferved that the winter feafon had prevented me from tuning an organ by the fecond table of beats, in order to try what efFed the fyftem of Equal Harmony might have upon the ear. But upon telling Mr. Turnery one of our organifts at Cambridge, how he might approach near enough to that fyftem, by flattening tlie major iii'^', till X THE PREFACE. till the beats of the v^^ and vi^^ major with the fame bafe, went equally flow, by his great dexterity and skill in tun- ing he prefently put my rule in execu- tion upon a ftop of his organ ; and af- firmed to me, he never heard fo fine harmony before, efpecially in the flat keys; but he added, that for want of more founds in every o£lave the falfe concords were more intolerable than ever : and no wonder, as their common difference from true concords was then increafed from one fifth to one fourth of the tone. Nor will it be improper to mention a like experiment made by the ac- curate hand of Mr. Harrifon^ well known to the curious in mechanics by his admirable inventions in watch- work and clock-work for keeping time ex- actly both at fea and land : which if duly encouraged and purfued will un- doubtedly prove of excellent ufe in na- vigation; by corre<^ing the fea-charts, with refped to longitude, as well as the reckonings of a fhip, to as great ex- THE PREFACE. XI exadlnefs, in all probability, as need be defired. But in regard to the experiment I was going to mention, he told me he took a thin ruler equal in length to the fmalleft firing of his Bale Viol, and di- vided the ruler as a monochord, by tak- ing the interval of the major iii*^, to that of the viii% as the diameter of a circle, to its circumference. Then by the divifions on the ruler applied to that firing, he adjufted the frets up- on the neck of the viol, and found the harmony of the confonances fo extreme- ly fine, that after a very fmall and gra- dual leno-thening of the other ftrino-s, at the nut, by reafon of their greater ftiffnefs, he perfectly acquiefced in that manner of placing the frets. It follows from Mr. HarrIfo;i\ af- fumption, that his iii'^ major is temper- ed flat by a full fifth of a comma. My iii^ determined by theory, upon the principle of making all the concords within the extent of every three odlaves as equally harmonious as poffible, is tern- xii THE PREFACE. tempered flat by one ninth of a com- ma ; or almoft one eighth, when no more concords are taken into the cal- culation than what are contained with- in one odlave. That theory is therefore fupported on one hand by Mr. Harri- fo?zs experiment, and on the other by the common pradice of muficians, who make the major iii*' either perfect or ge- nerally fharper than perfect. We may gather from the conftruc- tion of the Bafe Viol, that Mr. Harrifoit attended chiefly, if not folely to the harmony of the confonances contained within one odave ; in which cafe the differences between his and my tempe- raments of the major iii^, vf^ and v% and their ieveral dependents, are re- fpedively no greater than 4, 3 and i fiftieth parts of a comma. And confi- dering that any afligned differences in the temperaments of a fyftem, will have the leaft effe6l in altering the harmony of the whole when at the beft, I think a nearer agreement of that experiment with THE PREFACE. xid with the theory could not be realbnably expelled. Upon asking him why he took the interval of the major iii*^ to that of the viii''' as the diameter to the circumfe- rence of a circle, he anfwered, that a gentleman lately deceafed had told him it would bring out a very good divifiow of a monochord. Whoever was the au- thor of that hypothefis, for fo it muft be called, as having no connexion with any known property of founds, he took the hint, no doubt, from obferving that as the odave, confifting of five mean tones and two limmas, is a httle bigger than fix fuch tones, or three perfed major iii''', fo the circumference of a circle is a little bigger than three of its diameters. When the monochord was divided upon the principle of making the major III'* perfed, or but very little fharper, as in Mr. Huygens\ fyftem refultin^ from the odave divided into 31 equal intervals, Mr. Harrifon told me the major vi'^' were very bad, and much worfe xlv THE PREFACE. worfe than the v**''. In which he judged rightly, as 1 further fatisiied my felf by trying the experiment upon an organ; and being foHcitous to know the reafon of that effed, that is, why the v"^' and vi'^' major, when equally tempered, fhould differ fo in their harmony, after various attempts I fatisfied my curiofity. With a view to fome other inquiries I will conclude with the following ob- fervation. That, as almoft all forts of fubftances are perpetually fubjedb to very minute vibrating motions, and all our fenfes and faculties feem chiefly to de- pend upon fuch motions excited in the proper organs, either by outward ob- jeds or the power of the Will, there is reafon to exped:, that the theory of vi- brations here given will not prove ufe- lels in promoting the philofophy of other things befides mufical founds. Rob. Smith. Trinity College, Cambridge, Dec. 31. 1748, THE PREFACE TO THE SECOND EDITION. JN this fecond edition of thefe har- monics^ bejides many /mailer im- prove?nents^ the properties of the pe- riods^ heats aiid harmony of imper- feB confonances are more explicitly de- monfl rated [a) and confirmed by very eafy experiments [b). T'he idtimate ra- tios of the periods and heatSy which are generally more ufeful a?2d elegant than the exaEi ratios ^ are proved to be fuf- ficiently accurate for mofl purpofes in harm.onics [c). More methods are added for finding the pitch of an organ {d) and for tuning ity either by efiimation and judgment of the ear {e\ or more exaElly and readily by ifochronous beats of different concords [f\ as well as by complete {a) Lemma to prop, ix, and prop, ix, xi and corol- laries. {b) Prop. XI. fchol. 2. (<:) Prop. XI. fchol. i. \d) Prop. XVIII. and fchol. &<;« . ((?) Se6l. IX, art. i. (/) Prop. XX. fchol. 2. xvi THE PREFACE. complete tables of heats. An enquiry is made whether colncidejit pulfes be necef- faryy or only accidental to a perfeSi con- fonance {g ). And lajlly^ as the harpjtchord has neither firings nor keys for a?ty of thefe founds D^ A% E% B% F*^ A\ D\ G^, &^c^ which yet are fo often wanted that far the greater part of the befl compofitions cannot be perfor7ned with- out them^ except by fubflitutiitg for them E\ B\ F, C, G, G^ C^ F% Wc^ refpe&ively^ which by differing from them by near a fifth part of the i072e^ make very bad har?no?iy j and as the old expedient for introducing fome of thofe founds by infer ti?2g more keys in every oUave^ is quite laid afide by rea - fon of the difficulty ifi playing upon them ; / have therefore invented a bet-^ ter expedient-^ by caufing the feveral keys of thofe fubfiitutes^ E^, B^, F, C, G, G% C% F% &Cy to firike either Eb(?r D^ B^^r A*, F or E% C ^rB% G (,§■) P^op- ^^' fchol. 4. art. 7. &c. THE PREFACE. xvii G or F*^ C^ or A> y C* or DS F^ or G\ &€, For fines both the founds in any one of thofe couples arefeldom or nevsr tifed in any one piece of mufic^ the jnufcian by movijtg a few flops before he begins to play ity can immediately introduce, that found in each couple^ which he fore- fees is either always or oftenefi ufed in the piece before him. Two different conflruElion of thofe flops are here defer ibed ( h ), one of which is applicable at a f mall expenfe to aity harp- fichord ready made^ a7td the other to a new harpfchord^ a?id upon putti?ig them both in pra&ice^ they have perfcElly an-- fwered my expeBation, Several properties a?td advantages of this changeable fcale are defer ibed in the eighth SeBion, In a word^ the very worfl keys i?i the common defeUive fcale^ by cha?7ging a few founds are prefemly fnade as complete as the befl in thatfeaky a?ul more harmonious too^ becaufe the a chano'e- {h) SciTt, VIII. art. 18;, ig. xvHi T H E P R E F A C E; cha?7geahle fcale admits of the very hejl temper a7ne72t^ and^ which is another ad- 'ua7itage^ will therefore Jland loitger in tune than the co?nmon fcale which cannot ad?nit that temperame?2t, Thefe improvements of the harpfichord^ it is hoped^ may encourage others to ap- ply the like methods to the fcale of the organ^ which is equally capable of them and to greater advaiitages. Rob. Smith, Trinity Colle2;e, Cambridge, Oaob. 21, 1758. A T A B LE of the S E C T I O N S. Pag. (Sect. I . Philofcphical principles of harmonics, i Sect. II. Of the names and notation of confonances and their intervals, 9 Sect. III. Of perfeB confonances and the order of their fmpli city. 1 4 Sect. IV. Of the antient fyjiems of perfedi confo- nances. 2 3 Sect. V. Of the temperaments ofimperfeB inter- vals and their fynchronoiis variations. 35 Sect. VI. Of the periods^ beats^ and harmony of imperfeci confonances. 56 Sect. VII. Of a fy ft em of founds wherein as many concords as pqjftble^ at a medium of one with another^ fiall be equally and the moft harmonious. 123 Sect. VIII. The fcale of mufical founds is f idly explained and made changeable upon the harpfichord^ in order to play all the ft at and ftoarp founds that are ufed in ajiy piece of mufic^ upon no other keys than thofe in cominon ufe. i6q Sect. IX. Methods of tuning an organ and other inftrurnents, _ 187 f XX] Sect.X. Of occafional temperaments ufed in concerts well performed upon perfect tn^ Jirume?its. 225 Sect. XL Of the vibrating motion of a ?nufical chord. 230 An Appendix containing illuflrations and diffe- rent demonflrations of feme parts of the the- ory of imperfeB confonances. 244 AD VE R TIS E ME NT. A Complete Syjiem of Optics in four Books, viz. a popular, a mathematical, a mechanical and a philo- fophical treatife, by Dr. Smith : Cambridge 1738, 2 Vol. 4«o. Harmonia 'Menfararum, five analyfis et fyntheils ^per rationum et angulorum Menfuras promotse : ac- cedunt alia opufcula, n^rnpe de Limitibus errorum in mixta mathefi, de metbodo Differentiarum Nevv- toniana, de conflruclione Tabularum per dilTerentias, ^de defcenfu Gravium, de motu Pendulorum in cy- cloide et de motu Frojeftilium, per Rdgcrum Cote- fium : Edidit et auxit Robertus Smith : Cantabrigias 1722, 4^". Hydroftatical and Pneumatical Leolurcs by Mr. CoteSy publiflied by Dr. Smith: Cambridge 1 747, 23) elaftic bodies, which communicate the like vibrations to the air, and thefe the like again to our organs of hearing. Pliilofophers are agreed in this, becaufe found- ing bodies communicate tremors to diftant bo- dies. For inilance, the vibrating motion of a mulical firing puts others in motion, whofe tenfion and quantity of matter difpofe their vi- brations to keep time with the pulfes of air, propagated from the firing that was flruck. Galileo explains this phsenomenon by obferving, that a heavy pendulum may be put in motion by the leafl breath of the mouth, provided the blafls be often repeated and keep time exadly with the vibrations of the pendulum j and alfo by the like art in raifing a large bell 5 and pro- bably he was the firil that rightly explained that phasnomenon {a). 2. If {a) For he fays, in the perfon of another, il problema poi trito delle duecorde tefe all unifono, che al fuonodell' A una 2 HARMONICS. Seal. 2. If the vibrations be ifochronous the found is called Mufical, and is faid to continue at the fame Pitch ; and to be Acuter, Sharper or Higher than any other found whofe vibrations are flov/er j ' and Graver, Flatter or Lower [b) than any other whofe vibrations are quicker. For while a mufical Jflring vibrates, if its ten- fion be increafed or its length be diminifhed, its vibrations will be accelerated -, and experience fhews that its found is altered from what is called a graver to an acuter j and on the contrary. And the like alteration of the pitch of the found wiU follow, when the fame tenfion is given by a weight, firft to a thicker or a heavier ftring, and after that to a fmaller or a lighter of the fame length, as having lefs matter to be moved by the fame una I'altera fi muova et attualmente rlfuona, mi refta an- cora irrefoluto ; come anco non ben chiare le forme delle confonanze et altre particolarita. Dialogo i* attenente alia Mecanica^ towards the end. (Zi) As the ideas of acute and high, grave and loWj have in nature no neceflary connexion, it has happened accordingly, as Dr. Gregory has obferved in the preface to his edition of EudicTs vv^orks, that the more antient of the Greek Writers looked upon grave founds as high, and acute ones as lovf^ and that this connexion was afterwards changed to the contrary by the lefs antient Greeks, and has fnice prevailed univerfally. Probably this latter con- nexion took its rife from the formation of the voice in finging, which Ar'ijlidei ^intilianus thus defcribes. F/vs- '^a.K &\ yf /Jib /3a^UTnf, KarooGsv ava^sco/jtsvs t» "nrvsu- lj.cf\(^, n' J'' o|uT7)r, tTTiTroXris" wpo'iijAvis. p. 8. Et qui- dem gravitas fit, fi ex infcriore parte (gutturis) fpiritus fur- fum feratur, acumen vero, fi per fummam partem prorum- pat, as Alcibofnius tranflates it in his notes, pag. 208. Arts- HARMONICS. 3 fame force of tenlion. And thefe changes in the pitch of the found are found to be conftantly greater or leller, according as the length, tenfion, thicknefs or denfity of the firing is more or lefs altered (c). 3 . Therefore if feveral firings, however dif- ferent in length, thicknefs, denfity and tenfion, or other founding bodies vibrate all together in equal times, their founds will all have one and the fame pitch, however they may differ in loud- nefs or other qualities, and are therefore called Unifons : and on the contrary, the vibrations of unifons are ifochronous. This obfervation reduces the theory of all forts of mufical founds to that of the lx)unds of a fingle firing ; I mean with refped: to their gra- vity and acutenefs, which is die principal fub- jedi of Harmonics (d), 4. Con- (c) The Greek muficians rightly defcribe the difference between the manner of finging and talking. They con- fidered two motions in the voice, xivyicrsir d^uo ', the one Continued and ufed in talking, ^ /jtev cuvs^n? ts y.a.1 Xoyi- X)], the other difcrete and ufed in fmging, »] cP'k c/^jag-jj- ^xcdiii-A T£ y.ai jutsXaJc^otv). In the continued motion, the voice never refts at any certain pitch, but waves up and down by infenfible degrees; and in the difcrete motion it does the contrary ; frequently refting or flaying at certain places, and leaping from one to another by fenfible inter- vals : Euclid^ Introdudlio Harmonica, p. 2. I need not ob- ferve, that in the former cafe, the vibrations of the air are continually accelerated and retarded by turns and by very fmall degrees, and in the latter by large ones. {d) Ptolemy hys^ Ap/uiovtHyi /jtsv Wi c\viXiJ.lz y.c{}akn- A 2 Harmonics 4 HARMONICS, Sed. L 4. Confequently the wider and narrower vi- brations of a muiical ftring, or of any other body founding mulically, are all ifochronous very nearly. Othenvife, while the vibrations decreafe in breadth till they ceafe, the pitch of the found could not continue the fame ; as by the judg- ment of the ear we perceive it does, it the iirft vibrations be not too large : in which cafe the found is a little acuter at the beginning than af- terwards. 5. In like manner, fince the pitch of the found of a ftring or bell or other vibrating body, does not alter fenlibly while the hearer varies his diftance from it 5 it follows that the larger and leffer vibrations of the particles of air, at fmaller and greater diftances from the founding body, are all ifochronous : and confequently that the little Ipaces defcribed by the vibrating particles are every where proportional to the celerity and force of their motions, as in a pendulum (f). And this difference of force, at different diftances from the founding body, caufes a difference in the loud- nefs of the found, but not in its pitch. 6. It follows alio, that the harmony of two or more founds, according as it is perfect or im- perfed: when heard at any one diftance, will alfb be perfed or imperfed: at any other diflance : which Harmonics is a power apprehending the difFerences of founds, with refpe^l to gravity and acutenefs. ♦ (c) See'Newton's Princip. Lib. 11. Prop. 47. Art. 7- HARMONICS. 5 which being a known fad in a ring of bells for inilance, is mentioned here as a confirmation of thefe principles of Harmonics. 7. If two mulical firings have the fame thicknefs, denfity and tenfion, and differ in length only, (which for the future I fliall al- ways fuppofe,) mathematicians have demon- ftrated, that the times of their fingle vibrations are proportional to their lengths {f). 8. Hence if a firing of a mufical inflrument be ftopt in the middle, and the found of the half be compared with the found of the whole, we may acquire the idea of the interval of two founds, whofe lingle vibrations (always mean- ing the times) are in the ratio of i to 2 3 and by comparing the founds of 4, .|, \, ±, 4., I, t^, &c. of the firing with the found of the whole, we may acquire the ideas of the intervals of two founds, whofe fingle vibrations are in the ratio of 2 to 3, 3 to 4, 3 to 5, 4 to 5, 5 to 6, 8 to 9, 9 to 10, &c. 9. A Mufical Interval is a quantity of a cer- tain kind {g), terminated by a graver and an acuter found. In (f) As a clear and exaft demonflration of this curious Theorem depends upon one or two more, of no fmall ufe in Harmonics, and requires a little of the finer fort of geo- metry, which cannot well be applied in icw words, I have therefore referved it to the laft Se6lion of this Treatife ; which the reader may confult, or, taking it for granted at prefent, may proceed without interruption ; as he likes beft. {g) See Dr. JVallis's preface to Porphyry's comment on Ptolmfs Harmonics. Oper. Math. vol. iii. Euclid fays, A 3 'an 6 HARMONICS. Sed. I. In a ring of bells, for example, the founds of the firft and fecond bells, counting either from the biggeft or the leaft, terminate a cer- tain interval j thofe of the firft and third a greater interval ; thofe of the firft and fourth a greater ftdll j &c. So that the interval increafes by degrees, either as the graver of the two founds defcends, or as the acuter afcends j and within the interval of the founds of the biggeft and leaft bells, the intervals between the founds of all the reft are contained. 10. Mufical intervals are Meafures of the Ratios of the times of the fingle vibrations of the terminating founds, or, cceteris paribus, of the lengths of the founding ftrings {Jo). For it is obfei-vable in the experiments laft mentioned (?) and is univerfally allowed by mu- licians, that when the lengths of thofe ftrings have an Interval is to ttrspts^o/asvov utto o'xio ^Gofycov avco/jtofoov o|ut)t1{ y.aX (iixpuryfliy what is contained by two founds different in gravity and acutenefs. Introdu(Slio Harmonica p. I. Arijloxenm defines a mufical found thus, ^ovJt's' tB-lo)- ), may be reprefented by a feries of equal parts contained in a right line; as in 02, 03, 04, &c. Confequently when two founds are heard, two of thofe lines, as 02 and 03, will rightly repreient the two feries of equal times, if the magnitude of the equal parts in one line, be to the magnitude of thofe in the other, in the ratio of the fingle vibrations of the founds : or, the whole lines being fuppofed equal, if the numbers of aliquot parts in each, as 2 and 3, be feverally the fame as the leaft numbers of the vibra- {0) The old method of refolving concords Into their elements may be feen in Dr. IVallis's divifion of the mono- chord, or fecStion of the mufical Canon, as the antients called it. Philofoph. Tranfacl. N°. 238. or Abridg. by Lowthorp. vol. I. p. 698. firft edit. Art. 2. HARMONICS. ij vibrations of each found, made in the fame time reprefented by the line 02 or 03 {q). 2. And the founds being heard together, if we conceive the two equal and parallel lines that rightly reprefent them, as 02 and 03, to coin- cide throughout, the points that divide the fepa- rate lines, vv^ill fubdivide the combined lines into fmaller portions, as in Fig. 5, reprefenting a third feries or Cycle of times, in which the pulfes of both founds interchangeably fucceed one another in beating upon the ear. 3. Such a mixture of pulfes, flicceeding one another in a given cycle of Times, terminated at both ends by coincident pulfes, and fufficiently repeated, is the phyfical caufe that excites the fenfation of a given confonance : Efpecially when confidered as diftindt from any other confonance, whofe lingle vibrations having a diflierent ratio from that of the former, will conftitute a dif- ferent cycle, and excite a different fenfation. But if that ratio be the fame, though the abfo- lute times be different, the confonances are fimi- lar and may be looked upon as the fame in this refpecffc, that their cycles have the fame form ; the times in both having the fame order, and the fame proportions j and in this other alfo, that the interval of the founds is the fame (r). 4. This being premifed, one confonance may be confidered as more or lefs fimple than another, ac- {q) See Art. 12. following, (r) Art. 10. Sea. i. i6 HARMONICS. Sed. III. according as the cycle of times belonging to it, is more or lefs fimple than the cycle belonging to the other. And upon this principle all confo- nances may be ranged in due Order of fuch fimplicity, by the help of the following Rule. 5. One Confonance is Simpler than another in the fame Order, as the fum of the leaji terms, ex- prejjing the ratio of the Jingle vibrations, is fmaller than the like fum in the other confonance \ and when fever al fuch funis are the fame, thefe confo- nances are funpler in the fame order, as the lejfer terms of their ratios are fmaller. For the fimplicity of a confonance or cycle of times, confifts partly in the number of times con- tained in the cycle, and partly in the different proportions they bear to one another. Fig. 4. When the numbers of times in dif- ferent cycles are difterent, and die times in each cycle are equal to one another, as when we com- bine the founds 01 and 01, 01 and 02, 01 and 03, 01 and 04, 01 and 05, &c, the cycles of this fort may be ranged in the order of their fimpli- city above defined, 'by the order of the numbers of equal times in the cycles, or of the magni- tudes of the numbers i, 2, 3, 4, 5, 6, &c, or of 2, 3, 4, 5, 6, 7, 6cc, that is, of the fums of the terms of the ratios i to i, i to 2, i to 3, 1 to 4, I to 5, &c. In the other cafe, where the numbers of times in different cycles are the fame, and the times in each cycle bear different proportions to one an- other, as when we combine the founds 01 and 06, Art.6. HARMONICS. 17 06, 02 and 05, 03 and 04, that cycle is iimpler than another, in which the equal times between the pulfes of die acuter found, are lefs interrupted and fubdivided by the pulfes of the graver. Accordingly in the firft of thefe cycles compo-i fed of 01 and 06, not one of the 6 equal times between the pulfes of the acuter found 06, is fubdivided by any pulfe of the graver o i j but in die fecond cycle compofed of 02 and 05, one of the 5 equal times, between the pulfes of the acu- ter found 05, is fubdivided by one pulfe of the graver 02 ; and in the third cycle compofed of 03 and 04, two of the 4 equal times in the acu- ter found 04, are fubdivided by 2 pulfes of the graver 03. By which it appears, that the iiril: cy- cle is fimpler than the fecond, and the fecond Iim- pler than the third -, and that the order of fimpli- city of this fort of cycles, anfwers to the order of the magnitudes i, 2, 3 of the lefler terms of the ratios. 6. Now by the firfl: part of the rule above, the integers in the fecond column of the following table, are the feveral funis of tlie terms of the op- posite ratios in the firft, diminiflied by i , which alters not the order of their magnitudes, but only makes the fcries begin with i, anfwering to the fimplefl: confonance. By the fecond part of the rule, the ratios whofo terms have the fame fum, as i : 6, 2 : 5, 3 : 4, are ranged in the order of their lelTer terms 1,2, 3, or, which alters not die order, of thofe terms leverally diminifhed by i, as of o, i, 2, or of the B fra(5lions i8 HARMONICS. Secft. III. A table of the Order of the fimplkity of confo7^ances of two founds. Ratios of the vibra- tions. I : I I : 2 I ■ 3 Order of the fimpli- city. intervals of the founds. Continuation of the table. I : 15 15 16 i6|- 3VIII + VII I : 16 2: 15 5: 12 8: 9 4VIII 2VIII 4- VII VIII 4- 3d T I 2 3 4 41 VIII VIII + V 1 : 4 2 : 3 2VIII V 1:18 3:16 4: 15 9 : 10 18 18J i8| i8| 4VIII -f- T 2VIII 4- 4^*^ VIII -j- VII t I : 5 5 6 ^1 7 7^ 2VIII + III I : 6 2: 5 3: 4 2VIII + V VIII + III 4th VI I : 20 5:16 20 20i 4VIII 4- III VIII 4- 6th I • 7 3: 5 I : 22 3: 20 5:18 8:15 22 2VIII 4" VI VIII 4- 7'^ VII I : 8 4: 5 I : 9 8 8i 3VIH III 9 3VIII + T 3VIII -{• III 2VIII + T VIII -j- 4'^ 3^ I : 24 9 : 16 24 4V111 4- V 7th I : 10 2: 9 3: 8 5: 6 10 104. io| lOi: I :28 5 : 24 9 : 20 28 28i 28.^ 2VIII 4- 3^ VIII 4- t I : 12 3: 10 4: 9 6: 7 12 124 I2J: I2.| 12-^ 3VIII -|- V VIII -j- VI VIII -j- T 6th I : 30 15 : 16 30 3014^ 4VIII 4- VII H 32:45 45:64 76K- io8|-> IV 5th Art. 7- HARMONICS. 19 fradlons 4i T> T5 whofe common denominator 3 is the number of the ratios whofe terms have the fame fum 7. Thefe fractions either by them- felvesorthe mixt numbers 6, 64) 64) made by annexing them to the number 6, may therefore denote the order of the Modes of fimpHcity of fuch confonances as have the fame Degree of lim- plicity denoted by 6 or 7 — i . And thus die or- der of the fimplicity of all confonances whatever, is denoted by the order of the magnitudes of the integers and mixt numbers in the fecond column of the table. 7. This feries increafes from unity in feveral arithmetical progreffions, except that a term or two is here and there omitted, where ratios occur which being reducible to fimpler terms, have been confidered before, or elfe are not Perfect Ratios, which are fuch only whofe terms are i, 2, 3, 5, with their powers and products (j). For example, writing down all the ratios in due order, whofe terms make a given fum, as i to 8, 2 to 7, 3 to 6, 4 to 5, 1 rejed die two middlemoft for the reafons juft mentioned, and place the reft in the firft column of the table ; which may dius be continued with certainty and order as far as we pleafe. 8. Hence we may diftinguifli confonances into two forts. Pure and Interrupted ; pure, where none of the equal times between the pulfes of the acuter found, is fubdivided by any interme- B 2 diate (s) Sea. II. art. 5. 20 HARMONICS. Sed. HL diate pulfe of the graver ; and interrupted, when any of thofe equal times are interrupted by one or mere pulfes of the graver found. In the fecond cohimn of the table, the leaft fimple or loweft mode of each degree of inter- rupted confonancy, is every where placed above the next inferior degree of pure confonancy, as 4-I- above 5. For fhould we deprefs the mode 44. to a place next below the degree 5, why not even to a place next below 6 ? though not below 64, as being a more complex mode of a lefs fimple degree. But if that were allowable, by parity of reafon we ought to deprefs 44 j 64 > 64, next below 7, though not below 74, and like wife 44, 64, 64, 7^ , below 8, though not below 84 , and alfo' 44, 64, 64, 74 > 81 below 9 and 10, and fo forth to infinity : which, by deprefimg all the modes of interrupted confonancy, below all the degrees of pure confonancy, would render them heterogeneal, and incapable of any order or com- parifon with one another. The table is tlierefore rightly ordered. 9. Hitherto we have only confidered the num- ber and proportions of the times in the cycle by which a confonance is reprefented, without re- gard to the quality of the pulfes, as to magni- tude, duration, ftrength, weaknefs or other ac- cidents ; whereas the pulfes of graver founds are generally flronger, larger, obtufer, and of longer duration dian thofe of acuter founds, and affedt the ear diiferenriy. But ftill diis alters not the ra- tional idea of the confonance, as above defcribed, provided Art. 10. HARMONICS. 21 provided we take the middle inftant of each pulfe as we did in Art. i ; nor does the ear perceive any alteration in the kind of mixture, or in the interval upon foftening or fwelling either found, while the other retains the fame flrength. 10. It is well known in general, that fimpler confonances aifetft the ear with a fmoother and pleafanter fenfation, and the lefs fimple with a rougher and lefs plcafant one. And this analogy feems to hold true accordino^ to the order in the table, as far as the ear can judge with certainty. Thofe that are willing to try the experiment, may readily do it by the help of the third column of the table, fhewing the mufical intervals anfwer- ing to the relj3e6tive confonances. But the ana- logy will be plainer perceived by intermitting fe- veral confonances, and trying it, for example, in this feries of all the concords not exceeding the odave; VIII, v, 4*^ vi, iii, 3"*, 6^^; but then they fliould not be tempered as ufual, but tuned perfed:. And if die experimenter be fkilful in me- lody and compofition, he muft endeavour, as much as poffible, to diveft himfelf of all habitual prepofleffions in favour of this or that concord, or hiccellion of concords, acquired from the rules and practice of his art ; in order to an impartial judgment of the fimple perception of the fmooth- nefs and fweetnefs of each concord, and a fair comparifon of fuch perceptions only. 1 1 . Though nature has appointed no certain limit between concords and difcords, yet as mu- sicians diflinguifli confonances by thoie names for B 3 their It HARMONICS. Sea.IIL their own ufes, I may do the like for mine ; call- ing unilbns, iii"^', v'*"' and vi'^% and their com- plements to the viii'^ and compounds with viii*''% Concords, and all other confonances, Difcords. 1 2. If the times of the fingle vibrations of any two founds be V and i;, and if V : i; : : R : r, reprefenting the leafl integers in that ratio -, the length of the cycle of times between the fuccef- live coincidences of the pulfes of V and 'u, is r V = R'u. Becaufe thefe multiples of V and v are the leaft of any that can be equal. For the fame reafon, if V : a: : : S : 5 in the leaft integers, the cycle j V = S jv. 1 3 . Hence the length of the cycle of V and v, is to that of V and .v, as r to s-, that is, tlie cy- cles of confonances that have a common found or vibration V, are proportional to the Numera- tors of the fractions -J V = 'u, -J V = x, ex- preffing the times of the fingle vibrations of the other founds, as in Fig. 3, or to the lefler terms of the ratios in the firft column of the table of the order of the fimplicity of confonances. 14. Confequently were the degrees of fimpli- city of confonances to be eftimated by die fre- quency of tlie coincidences of their pulfes, or the fhortnefs of their cycles, as is commonly fuppo- fedi the unifons, viii^'^^ viii -f v*^', 2 viii*''*, 2 VIII -f iii*^', &c, whofe cycles are but i vibra- tion of the bafe, would be equally fimple j and the fame may be faid of the v''^', viii 4- 111*'% 2 VIII 4- T', &c, whofe feveral cycles are but z vibra- Art. I. HARMONICS. 23 vibrations of the bafe ; and the fame alfo of all confbnances having the fame number for the leifer term of their perfed: ratios 3 which flievvs that the frequency of coincidences is, of itfelf, too general a character of the fimplicity or fmoothnefs of a confonance, and therefore an imperfed one. SECTION IV. Of the antieju Syjiems of perfe& cofifonances. 1. 1" F no other primes but i, 2, 3 were admit- I ted to the compolition of perfeift ratios, a fyftem of founds thence refulting could have no perfed: thirds 3 nor any perfed confonance whofe vibrations are in any ratio having the number 5, or any multiple of it, for either of its terms, as 5 to 4, 6 to 5, I o to 9, 1 6 to 1 5, 6cc : it being im- poffible for any powers and produds of the given primes i, 2, 3 to compofe any other prime or multiple of it. 2. Fig. 3. The minor tones Z)£, Gv^ being thus excluded, and major tones being put in their places, every perfed major 11 1'^ will be increaf- ed by a comma, as being the difference of the tones {f) ; and eveiy hemitone and perfed minor 3*^ will be as much diminiflied j becaufe the 4*^^ B 4 and (/) Se6l. II. Art. 4. ^4 HARMONICS. Sed. IV, fend v^^% as CF and F<:, cG and (?C, are per- fect, whether 5 be admitted or not, as depending on the primes i, 2, 3, only. 3. Thefe diminiihed hemitones being called Limmas, the od;ave is now divided into 5 major tones and 2 limmas j as reprefented to the eye in Plate II. Fig. 6 j where the confonances whofe vibrations are exprefled by fuch high terms as the powers of 8 and 9, &c, muft needs be difagree- able to the ear, according to the foregoing ana- logy between the agreeable fmoothnefs of a con- fonance and die iimplicity of the numbers ex- preiiing the ratio of its vibrations (u) : and that in reality they are fo, any one will foon find if he pleafes to tiy the follov^ing experiment. 4. Fig. 6. Afcending by a perfedl v'^ and des- cending by a perfedl 4*** alternately, upon an or* gan or harpfichord tune the following founds, fromFtoC, CtoG, GtoD, D to A, Aio E, E to B, and the od:ave F/'will then be divided into 5 major tones and 2 limmas ; becaule the differences of thofe fucceflive v*^' and 4*''^ are ma- jor tones. Then having tuned perfect ocftaves to every one of thofe notes, try the confonances that would be perfect if the number 5 were admitted, as thirds major and minor, with their complements to the viii''' and compounds with viii'^^; and you will find tliem extremely difagreeable (x). 5. But (u) Se6l. III. Art. 10. (x) The v*Jis and 4*^$ being tuned by the judgment of the ear, if any one doubts whether their fingle vibrations be Art. 5- HARMONICS. 25 5. But if 5 be admitted among the mufical primes, the ratios 10 to 9 and 16 to 15, belong- ing to the minor tone and the hemitone, are alio admitted, and the elements that now compofe the ocflave, are 3 major tones, 2 minor and 2 hemi- tones, as in Fig. 3 . PROPOSITION I. Afyjletn of founds whofe elements orfmaU lejl intervals are t07ies 7najor and mi- nor and hemitoftes^ wilhtecejfarily con- tainfome imperfeEi concords, 6. Whatever be the order of thofe elements in any one od:ave, it muft be the fame in every one ; to the end that every found may have a per- fect odlave to it, as being the beft concord. And in order to have as many perfect v'^' as poffible, and confequently viii + v^''^ which concords are the fecond beft (_y), the elements muft be ranged in fuch order, that the contiguous couples fhall make as many perfed thirds as poffible, both ma- jor be as 3 to 2 and 3 to 4, let the Mufician compare the found of 4. of a mufical firing, and alfo |- of it with that of the whole, and he will acknowledge thefe concords and thofe which he tuned upon the inftrument to be the fame, and of confequence to have the fame ratios of their fmgle vi- brations. {y) See the table of the order of concords in Sed. in. Art. 5. 26 HARMONICS. Sea. IV. jor and minor ; thefe being the intervals which compofe the perfed; v'*"'. And that order being rightly determined, we fhall have the greateft number of perfedl concords of all forts. Becaufe the complements to the o6tave, of perfect thirds and v'^% will alfo be perfect, and fo will their compounds with any number of viii'^^ Now it is obfervable of the feven elements T, T, T, /, /, H, H, which compofe an odtave, that T and H, T and / are the only couples which make perfed: thirds (2;), all the reft, T and T, t and ^, t and H, H and H, making thirds im- perfed: by a comma, except H and H, which compofe an imperfect tone, bigger than the ma- jor tone by almoft a comma [a). Hence either T and H, or T and / muft be the outermoft elements in the odtave, as in the following table. For if the firft element in every odave in the fyftem be T and die feventh be H, the feventh in any o(5lave, combined with the firft in the next odave, will compofe the intei-val H 4" T of a perfed (z) Sc£l. II. Art. 5 and 7, {a) Putting H ~ log. _ zi o. 02803 Then 2H zz 2 x log. — =: o. 05606 And T n: log. -- z: 0.051 15 o Whence 2H — T zz o. 00491 81 And the Comma zz log. — - zz 0. 00540 oO Difference o. 00049 Art. 6. HARMONICS. 27 perfedl minor 3*^, and tlius the contiguous octaves will be joined in perfedt concord. T*ahle of the Eleme77ts, &c 7 I 2 3 4 5 6 7 I &c fH Caf.i. \ T+ t H+T-h t t+T+H T-f H T LH T + H t+T4- t T + H T r t 1 T+ t H + T-fH T4- t T Caf. 2. i L t T + H t+T+H H +T-I- t T+ t T Likewife if the iirft element in every odave be T and the feventh be /, here alfo the feventh in any o(5tavej together with the firft in the next oc- tave, will compofe the interval t-{-T of a perfedl major iii'', and thus the contiguous odaves will again be joined in perfe6t concord j and in no other cafe befides thofe two, as appears by the ob- fervation above. Caf. I . Now if the fecond element be t, the firft joined to it compofes the perfed: major iii*^, T-f^. And if the fixth element be T, die feventh joined to it will compofe the perfed minor third T-f H. Two of the feven elements in the odave being thus difpofed of at each end of it, the contiguous couples 28 HARMONICS. Sed. IV. couples of the remaining three cannot compofe perfed: thirds in any order different from this, H4-T-i-/, or its reverfe ^+T+H; both which being transferred into the interval between thofe extreme couples, fhew, that the elements in the fecond and third places, compofe either the im- perfed minor third /-{-H, or the imperfed: major third t-^t. If H be the fecond element, as in the third rank of the table, the firft couple does now com- pofe the perfed minor third T+H, and the laft being alfo T-fH, as before, the three remaining elements mufl have this order t-\-T-\-t^ to make perfe6t thirds of their contiguous couples j and being thus transferred into the interval between thofe extreme couples, they fhew, that the fecond and third elements do again compofe an imper- fe(fb minor third H -\-t. Caf. 2. Here alfo the lixth element mufi: be T, fmce no other joined to the feventh can make a perfed third, as T-\-t. Now if the fecond element be /, this joined to the firft makes the perfed: major third T -\-t. And two of the feven elements in the odave being thus joined at each end of it, the contiguous cou- ples of the remaining three, cannot compofe the intervals of perfed thirds in any order different from this, H-f-T^-H^ which being transferred into the interval between the extreme couples, Ihews, that the fecond and third elements do here alfo compofe the interval ^-f H of an imperfed minor third. If Art. 7- HARMONICS. 29 If H be the fecond element, as in the next lower ranks, then the firft couple compofe the in- terval T-f H of a perfed minor 3'', and the laft couple being T-f-/ as before, the three remaining elements mufl have this order, t-\-T-\-'H., or its reverfe, H-|-T-i-/, for the reafon above ; and be- ing thus transferred into the middle interval, tliey iliew, that the elements in the fecond and third places do again compofe an imperfect minor third, H-1-/, orelfe an imperfect tone H-f-H; which being joined to the major tone on either fide of it, compofes an imperfect major third, greater than /-f T by almofl two commas, as appears by the preliminary obfervation. Now any one of thofe imperfed: minor thirds, /-f H, together with the contiguous perfeift major II i*^, compofes a fifth equally imperfed:, and fb does the imperfedt major third t-]-f with the per- fedl minor third next to it. And the complements to the VI II''' of thefe imperfed: concords, as well as their compounds with viii*''^ are alfb equally imperfed, which proves the propofition. For hav- ing fhewn the neceffary defeds in thofe fix arrangements of the feven elements, we are freed from the trouble of confidering the reil (I?). Qj;. D. 7. Cor&il. Of thole fix arrangements of the ele- ments, the firil and fifth in the table are equally good (h) Mr. De Afoivris general corollary to the xvi pro- ■blem of his Dodlrine of Chances, gives 210 permutations cf thefe feven things, T, T, T, t, t, H, H. 30 HARMONICS. Sed. IV. good, and better than any one of the reft, as pro- ducing as many perfed: thirds, and a greater num- ber of perfedt v^^ PI. II. Fig. 7. In order to enumerate them with certainty and eafe, if die circumference of a cir- cle, be divided into feven arches, CD^ DE, E F, FG, GA, AB, BC, proportional to T, /, H, T, t, T, H, placed in the relpedtive angles at the center j they and their fums, whether fmaller or greater than the circumference, here confidered as a continued ipiral, will reprefent all the inter- vals in a fyftem compofed of any number of oc- taves, and the correfponding intervals in different odtaves will be denoted by the fame arch and let- ters : as appears by conceiving the bafe of the third Figure coiled round into the circumference of a circle, equal to the line Cc or cc &c. (r) In this notation then we have only three major ii\^\ CE, FA, GB, and they all perfedi and four minor thirds, DF, E G, AQ BD, the firft of which being compofed of /4-H, inftead of T+H, {c ) In this notation of intervals by circular arches, that the reader may not be at a lofs for a fuitablc notation of the lengths of the correfponding homogeneal firings ; let the radius OChe i and in OD, OE, OF, OG, OA, OB, OC^ from the center fet ofF-i, ^, ^, -f, i., ^L, ~ of the radius. Thefe are the fame lengths as thofe of the Monochord in Fig. 2, or Fig. 3 ; and as a regular curve drawn thro' the ends of the parallel ftrings in Fig. 3. is a Logiftic Line whofe Afymtote is the line Cc, fo a regular curve drawn thro' the ends of the diverging ftrings in Fig. 7. is an Equiangular Spiral whofe Pole is the center of the circle. See Sedl. I, Art. 10. and Mr. Cota'^ Harmonia Menfurarum, Prop. V and VI, Art. 8. HARMONICS. 31 T-f H, is too fmall by a comma -, and fix fifths, FAC, CEG, GBD, DFA, ACE, EGB, all perfedl but DFAy which being compofed of the defedive minor third DF and the perfedt major iii'^ FA, is too fmall by a comma. Thefe imperfedions being caufed by the conti- guity of t and H in the cycle of the elements, cannot be avoided while the hemitones are fepe- rated j there being but 3 major tones in the cy- cle ; and if they be joined, as in Fig. 1 2, the con- fequences will be worfe. The reil will appear by enumerating the thirds and fifths in the 8'\ 9^^ ^o'^ II'^ and 12**^ Fi- gures, made according to the other five arrange- ments in the Table of Elements (^). 8. Now if any one pleafes to try the following experiment, he will find what effed: thefe imper- fect fifths and fourths and their compounds with VIII**'', will have upon the ear ; that of the thirds and fixths having been tried before (e). In (d) Sir Ifaac Newton happily difcovered, (Optics Book i. Part 2, Prop. 3) that the breadths of the feven primary co- lours in the fun's image, produced by the refraffion of his rays through a prifm, are proportional to the feven difte- rences of the lengths of the eight mufical ftrings, £>, E, F, G, ^, By C, d, when the intervals of their founds are T, H, t, T, t,H, T : which order is remarkably regular ; but though it agrees beft with the prifmatic colours, it is not the propereft for a fyftem of concords, as producing one major third, two minor thirds and two fifths feverally imperfedl: by a comma. See Fig. 13. N°. 2. (^) Sea. IV. Art. 4. 32 HARMONICS. Se^IV. In Fig. 3 , tune upwards from C the two per- fed: v'^'CG, Gd, and the perfed xvii^^ or 2 VI 1 1 4- 1 1 1, Ce^ then downwards the v*'' ea, and the intermediate fifth ad will be too little by a comma, as including the imperfect minor third df. And by tuning an eighth below a we have the imperfect fourth Jld too large by a comma. 9. The difagreeable efFed: of this fifth da and fourth dA in every odave, and of their com- pounds with viii^''% and alfo of the third ^and and fixth fd' in every odtave and of their com- pounds with viii^^% and of many more fuchim- perfed concords, when the ufual flat and fharp founds are added to complete the fcale, has obli- ged pradlical muficians, long ago, to diftribute that comma, wanting in the fifth da^ equally among all the four V''', CG, Gd^ da^ ae^ con- tained in the xvii*^ Ce. And this interval Ce may be increafed or decreafed a little before it be divided into 4 equal v'^'. In any cafe fuch di- ftribution is therefore called the Participation or Temperament of the fyftem, and when rightly adjufted is undoubtedly the finefi improvement in harmonics. 10. If it be afked why no more primes than 1, 2, 3, 5 are admitted into mufical ratios j one rea- Ibn is, that confbnances whofe vibrations are in ratios whofe terms involve 7, 11, 13, &c, cceteris paribus would be lefs fimple and harmonious (y ) tjian (/) Sed. III. and Table pf the order of the fimplicity vf confonances. Art. II. HARMONICS. 33 than thofe whofe ratios involve the lefler primes only. Another reafon is this ; as perfedl fifths and other intervals refulting from the number 3, make the Schifm of a comma with the perfect thirds and other intervals refulting from the num- ber 5, fo llich intervals as refult from 7, 11, 13, Sec, w^ould make other fchifms with both thofe kinds of intervals. 11. The Greek muficians, after dividing an o(5lave into two 4*'^', with the diazeudlic or ma- jor tone in the middle between them, and ad- mitting many primes to the compolition of mu- fical ratios, fubdivided the 4^^ into three inter- vals of various magnitudes, placed in various orders, by which they diilinguifhed their Kinds of Tetrachords (g). Two of them have oc- curred in this Secflion. The firft, or l:=|.x|.x *4^, anfwering to the 4*'*=T + T4-L, in Fig. 6, is Ptolemy s Genus Diatonum ditofiiawiy and refults from that divifion of a Monochord which bears the name of Euclid'^ Sedion of the Canon } the fecond Kind, or ^--=:|.x ^x^^, anfwering to the 4'^=T4-/'-hH, in Fig. 3, is Ptokfnfs Diatonum intenfiim. 12. Since the invention of a temperament, all thofe antient fyflems have juflly been laid afide, as being unfit for the execution of mufi^ [g) Dr. JVallis has given a table of them in his Ap- pendix to Pto/emfs Harmonics. Oper. Math. yol. ill, pag. 166. C cal 34 HARMONICS. Sed. IV. aurcccj Xupat? cvap- U7rap^«v efHoc* »5 /^^toi y etf ^^^Tav ifacra, ■ZBrXar©' gyef. UcWaKic y' »v ri^fj-oS^ j^cxaVav apj'^a. Xu'/ogtv, eT5p^ [XH-iKO^ dy.^^Col; li|'yi^juiocraTO' wavla^^S '^ n ca^no-is iixh 0^ y.^TY]^ov, oJr -ar^^f ra; cv to? /3/ctj ^g«'a?. hoc eft, ^id mirufn, fi Eucrafiam m fath mnplam Jatittidine7n extendiint univcrfi\ qiiando et in lyris confonantiam ipfam qua fumma exaSiiJfimaqiie fit^ unicam atque infc^abikm ejje probahile fit^ et quce in ufus hoyninurn venit, certe latitudi- nem haheat. Sape tiamque^ \_qi{a?n'] percornmode temperajfe ly- ram videaris, alter fuperveniens tnuficiis exaSIius temperavit : fiqiddeni nobis ad omnia vita miincra feJifus ubiqiie judex e/l. Ex quibus Galeni verbis liquido conftat, confonantias, qui- bus Prop.II. HARMONICS. 35 alio becaufe we are afTured from hifloiy, that experience and neceffity did introduce fomething of a temperament before the reafon of it was diiccvered, and the method and meafure of it was reduced to a regular theory, as in the foU lowing proportion. SECTION V. Of the temperaments of imperfeB in- tervals and their fynchronoiis varia- tions, PROPOSITION II. To reduce the diatonic fy [I em of perfeSl confonances to a tempered fyflem of Mean Tones, Plate III. Fig. 13. When the elements are ranged in this order, T, t, H, T, t, T, H, or this, t, T, H, T, t, T, H, which two we fhew- ed to be the beft (/), and the arches C Dy C 2 -DE, bus in muficis utebantur inftrumentls, jam tunc imper- fe6tas effe, quin potius et fuifle femper et femper efle fu- turas. De Mufica lib. m. cap. 14. Be it fo ; but did they know, that all the concords cannot be tuned perfedl, and why they cannot ? (i) Sea. IV. Art. 7. 36 HARMONICS. Sed. V. DE, EF, FG, GA, AB, BC, are propor- tional to tliem, let the major iii'' CE, fituated between the two hemitones, be bifeded in d -, and let the other two major tones, FG, AB, be diminifhed at both ends by the intervals FJ] Gg, Aa^ Bby feverally equal to half Z)(^; and the o6tave will then be divided into five mean tones and two limmas, each limma being big- ger than the hemitone by a quarter of a comma. For the interval Dd being half the difference between the major and minor tones, CD, DE, is half a comma {k), and therefore the new tone C^or dE is an arithmetical mean between tliem. And each of the temperaments Ffy Gg, Aa, Bb, being made equal to half Dd or a quarter of a comma, it appears that every major tone is diminiflied by half a comma, and that every minor tone is as much increafed, which reduces all the tones to an equality. And by the conftrudiion the limmas bC, Ef exceed the hemitones by a quarter of a comma apiece. Q;E.D. CorolL In the fyftem of mean tones every perfed v^^ is diminiihed by a quarter of a comma: as will appear by going round the 13*'* figure, and comparing the tempered v*% faC^ CEg, gbd, dfa, aCE, Egb, with tlie perfed ones, by means of the notes T, t, H in the angles. Tliis (>f) Sea. 11. Art. 4- Prop. n. HARMONICS. 37 This is ufually called the vulgar temperament and might be proved feveral other ways indepen- dent of the firft and fecond propofitions ( / ). PRO- ( / ) Salinas tells us, that when he was at Rome, he found the muficians ufed a temperament there, though they underftood not the rcafon and true meafure of it, till he firft difcovered it, and Zarlino publifhed it foon af- ter ; firft in his Dimonftrationi Harmoniche, Ragiona- mento quinto, ^ropofta i^^, and after that, in his Infti- tutioni Harmoniche, part. 2. cap. 43. After his return into Spain, Salinas applied himfelf to the latin and greek languages, and caufed all the antient muficians to be read to him, for he was blind ; and in 1577 ^^ publifhed his learned work upon mufic of all forts ; where treating of three different temperaments of a fyftem, he prefers the diminution of the v'^» by a quarter of a comma to the other two, which he fays are peculiar to certain inftruments. De Mufica Lib. iii. cap. 22. Dechales fays, that Gu'ido Aretinus was the inventer of that temperament : Ipfe nulla habita ratione toni majoris et minoris, hunc unius quintae defectum aliis omnibus quintis communicat, et quafi dividit, jta ut nulla deficiat nifi quarta parte commatis. Hoc fyftema, quod valde commodum eft, dicitur Aretini. Curfus Mathem. Tom. i, pag. 62. De Progreffu mathefeos et muficae, cap. 7 ; et Tom. IV. pag. 15. cap. xi. But that opinion wants confirmation, efpecially as Dechales makes no mention of the claims of Zarlino and Salinas to that invention i for it feems they had a difpute about it, 38 HARMONICS. Sea.V, PROPOSITION III. If the Jive 7nean tones and the two Urn-' maSy that compofe a perfeB oUavCj he changed into jive other equal t07tes and two equal limmas, of any inde- terminate magnitudes ; the fynchro- nous Variations of the limma L, the mean tone M, and of every interval compofed of any numbers of thei7ty are all exhibited in the following tabky by the numbers and fgns of any f mall tndeter?ninate interval v : And are the fame quantities as the variations of the temperaments of the refpeUive imperfeB intervals. For 2d lid 3^ \\v 4th IVth L M L4-M 2M ' L + 2M 3M 5"^ — IV Q^V —41; V — 6v —5^ IV — 31; ^v — 1) 6v L-I-5M 2L+4M L+4M 2L+3M L-f3M 2L-I-2M Vllth ^th Vlth 6th Vth 5'^ Prop.III. HARMONICS. 39 For fince the perfect viii''^ = 2L+5M is in- variable, if the variation of L be put equal to 5'y, as in tlie table, that of 2L is i cu, and that of 5M, as being the complement of 2L to the viii^'^, is — icuj whence the variation of M is — 2V. Confequently the variation of the mean 3*^, L-fM, is 5'u— 2'u=3'u, and that of the iii^ 2M, is — 4'-j, and that of the mean 4''', L+2M, is ^v — 4'u=i;, and that of the mean iv*'', 3M, is — 6v. The variations of the intervals in the lower half of the table, are refpedively equal to thofe in the upper half, but have contrary figns j the correfponding intervals being complements to the perfed: odave. For which reafon the compounds of every one of thofe intervals with any number of odaves, have refpedively the fame variations both in quantity and quality. And if the fign of the variation of any one interval be changed, the figns of all the reft will alfo be changed ; becaufe their quantities will vanifh all together when v or any one mul- tiple of it vanishes. As to the fecond part of the proportion, it will appear in Fig, 1 3 , that any variation v of the mean interval CdEf is the fame in quantity as the variation of the temperament Ff of the faid interval CdEf: and the like is evident in any other inftance. Q^. D. C 4 CorolL 40 HARMONICS. Sed:. V, CoroU. I. It is obfervable in the table, that the variations of all the major mean intervals ii'^, iii*^, iv'^, V*'', vi'*^, vii'^, have the fame lign, and thofe of the minor intervals the con- trary fign. CcrolL 2. Having extended the circumference CdEfgabC of Fig. 13 into a right line, as in Fig. 1 4, at the points i E, g^ a^ b^ that termi- nate the major mean intervals 11^, iii**, v^'', vi^'', vii''', meafured from C, (and the minor too meafured from c the other extreme of the oc- tave Cc) place the refpedive tabular numbers 2, 4, I, 3, 5, denoting the proportions of their fynchronous variations; and in Fig. 15 divide any given line 6 into 6 equal parts, at the points I, 2, 3, 4, 5; then conceive the 14'** Fig. transferred to the 1 5''' five feveral times, into five parallel pofitions, fo that the feve- ral points I, 2, 3, 4, 5 in each Figure may co- incide. And it will be evident, by coroll. i, that any right line Ovvvv, draw^n from O, ter- minates the fynchronous variations, iv, 2'u, 31;, 41;, 51;, of thofe mean intervals, v*'', ii'', vi*^, iii'^, vii''', the variations being meafured from their refpedive origins i, 2, 3, 4, 53 and that thefe arc alfo the fynchronous variations of the temperaments of the refpedive imperfed: inter- vals, and of their complements to the viii'^ and compounds with viii'^', that is, of all the inter- vals in the fyllem. For as to the mean iv'*' fby Fig. 14, its con- temporary variation in Fig. 1 5, will be the line 6v Prop. III. HARMONICS. 41 61; in the fixth parallel F6B'F', when its tem- perament5'6 or B' /?' is taken equal to twice 5^ and placed the fame way from its origin b' or 6. Becaufe in Fig. 14 the temperament of the im- perfect iv'^fb is Ff-\-Bb=2Bb. As want of room in Fig. 1 5 will not permit the feveral intervals CG^ CD, &c. even lefs than one odave, to be reprefented in their due pro- portions to G I , the quarter of the comma, which is but the 223'' part of an odlave^ we muft conceive them continued far beyond the margin of the paper. Coroll. 3. When the iii'^ is perfect, the tem- peraments belonging to the v*^ and vi*'^ are fe- verally 4: of a comma, the former in defed:, the latter in excefs : and if either of them be made lefs, the other will be greater than ^ comma. PI. III. & IV. Fig. 15, 16. For when the Temperer Ovvv falls upon E, the iii'* CE is perfe6t, and the tempered v^'' Ci is lefs than the perfedl v'^ CG by G i, and the tempered vi '•* C3 is bigger than theperfed vi'^ CA by ^3 = G I = ■ comma. Hence when Av, any other temperament be- longing to the vi'^, is lefs than ^3 or ^ com- ma, Gv the correfponding temperament be- longing to the v'^, is greater than G i or i- com- ma: and on the contrary, when Gv is lefs than G I, the refpedlive Av is bigger than yf 3. And whatever be the magnitudes of thefe tem- peraments of the v^'' and vi'^, thofe of their com- 42 HARMONICS. Sea.V. complements to the viii*'* and compounds with VI 1 1'^* are the fame. CorolL 4. When the vi''' is perfed, the tem- peraments of the v'^ and iii*^ are feverally ^ comma, and are both negative. Fig. 16. For when the temperer Owv falls upon the line OHAI^ the temperament of the vi'^ vanifhes, and thofe of the v^'' and iii"^ are GH and EI, and are equal. For the equal lines Gi, y^3 and equal triangles GEO, AOE fliew, that the line GE is parallel to AO; whence GH is equal to £/, and the fimilar triangles lEO, ^30 give /£=^^3 = *xi,comma=4. comma. Coroli. 5. When the v*^ is perfed:, the tem- peraments of the vi^'^ and iii** are feverally equal to a comma in excefs. For when the temperer Owv falls upon the line OGKLy the temperament of the v*^ va- nifhes, and tliofe of the vi*^ and iii*^ are now AKand EL, which are equal, becaufe of the parallelograms A EGO, AELK; and ELk= 4 X G I or four quarters of a comma. Cor oil. 6. When the temperer Owv falls within the angle AOE, the temp^ v'^=temp*. vi^'^+temp^ III'', that is, theline G1;=^'u-l- £1;, or the lines Gi + i'u=^3 — 3'u + E'y, that is, putting the letter 'u for the line iv, ^c -^'v = ~c — 3 1; -1- 4^', which is evidently true. Coroli. 7. When the temperer Owv falls within the angle £0G, the temp^ vi''^ = temp*. v'^ -f temp*. 1 11^, that is, the line -^i;=G "J Hr Ev, Prop. III. HARMONICS. 43 E"J, or the line sAi, + 3 "J=^G 1 — 11; + Evy that is, putting v for the line ii;, Lc-\-2'v=^c^-^ i;+4'u, which is true. Coroll. 8. When the temperer falls any where out of the angle AOG^ the temp'. iii'^=temp^ v^'' -1- temp', vi'*^, that is, when it falls beyond the fide AO, the temp'. EI-fIv=GH-\-Hv ■^-Avj or putting the letter v for the line Hv, ^c-\- /\.v=^c-[-v-\- 2'Ui which is true: and when the temperer falls beyond the other fide OG, the faid temp'. EL-^Lv^Gv-VAK-^- Kv, that is, putting v for the line Gv, c-^^v =v + r -f 3 1;, which is true. Coroll. 9. The fum of the temperaments of the v'^ and vi'^ is 4: a comma when the iii*^ is perfed:; is lefs than 4 a comma by -i the tem- perament of the III'' when flattened -, and grea- ter than -i a comma by -i the temperament of the III'* when fharpened. For in the firft cafe the faid fum is Gi +^3 ; in the fecond, it is G i + 1 v-\-A2 — 3 'u =G i -\- A 2 — 2Vy and in the third, it is Gi — 1"^ + A 2 + 3 v=G I +^3 -\-2v, in which latter cales the temperament of the iii<^ is /\.v. Coroll. 10. Hence the fum of the tempera- ments of all the concords is lefs when the in"'' are flattened, than the like fum when the 1 1 1*'* are equally fharpened ; and the fum is the leaft of all when the 11 1''" are perfe(fr, as in the iy- flem of mean tones (m), (m) Prop. II. Scholium* 44 HARMONICS. Sea.V. Scholium, From the third and tenth corollaries I think we might juftly pronounce the fyflem of mean tones to be the beft poffible, were it evident that equal temperaments caufe different con- cords to be equally difagreeable to the ear («). But if it fhall appear, that the vi*^ and 3'* and their compounds with odlaves, are more difa- greeable in their kind, than the v'^ and 4^^ and their compounds with o6laves, all being equally tempered, as in that fyftem ; will it not follow, that the temperament of the former Parcel of concords lliould be fmaller than that of the lat- ter, to make them all as equally harmonious as poffible, without fpoiling the harmony of the III"* and 6^^ and their compounds with odtavesj which third parcel makes up the fum of all the concords in the fyflem. For («) Mr. Huygem has pronounced it the beft, in faying that the muficians in the other planets may know per- haps, cur optimum fit temperamentum in chordarum fy- ftemate, cum ex diapente quarta pars commatis ubique deciditur; Cofmotheoros pag. 76; but has given us no reafon for his aflertion, either in that incomparable book or in his Harmonic Cycle ; where he only appeals to the approbation and pra6lice of muficians and refers to the demonftrations of Zarlino and Salmas. But neither of thefe celebrated authors do any thing more, if I rightly remember, (for I have not the books now by me) than reduce the Diatonic fyftem of perfedl confonances to that of mean tones, by diftributing the fchifm of a whole comma into quarters; not at all confidering, whether thofe equal temperaments have the fame, or a different effect upon the feveral concords. Prop. III. HARMONICS. 4^ For if it be the immediate fucceflion of a worfe harmony to a better, as in inftruments badly tuned, which chiefly offends the ear ; it muft be allowed, that a fyftem would be the better, cceteris paribus^ for having all the con- cords as equally harmonious in their kinds, as the nature and properties of numbers will permit. In order to refolve thofe queftions upon phi- lofophical principles, and to determine the tem- perament of -a. given fyftem, that fhall caufe all the concords, at a medium of one with ano- ther, to be equally, and the moft harmonious in their feveral kinds, I found it neceffaiy to make a thorough fearch into the abftradl na- ture and properties of tempered confonances ; and thence to derive their effeds upon our or- gans of hearing: A large field of harmonics hi-- therto uncultivated. But before I enter upon it, it will be conve- nient to finifh this fedlion with a determination of the leaft fum of any three temperaments in different parcels, when any two of them have any given ratio. PRO- 46 HARMONICS. Sea.V. PROPOSITION IV. th To find a fet of temperaments of the v VI*'' and \\\^ upon thefe conditions \ that thofe of the v"^ and v^^ fhall have the given ratio of r to s, and the fum of all three fh all he the leafl poffible. PI. V. VI. Part of the 17*'^ and iS''^ Figures being conftruded like the 15^**, from A to- wards K take AM'. G\ :: s : r, and through the interfed:ion^of the lines AG^ Mi, draw the temperer Or j/^ J I fay Gr, As, Et are the temperaments required. For by the fimilar triangles Grp, Asp, and G ip, AMp, we have Gr : As : : (Gp : Ap : : G I : AM: :) r : s by conftrudtion, as required by the firft condition. Again, in the fame line MAC take AN= AM, and through the interfedion P of the lines AG, Ni produced, draw another tempe- rer ORST ; and by the fimilar triangles GRP, ASP, and G i P, ANP, we have GR : AS : i (GP :AP ::Gi: AN or AM: :) r : s by con- ftrudion, which likewife anfwers the firft con- dition ; and it is eafy to underftand, that no other temperers but thofe two can anfwer that condition. Now Prop. IV. HARMONICS. 47 Now whatever be the quantity and quality of the given ratio r to i, I fay the fum Gr-\-y4s-\- Et is lefs than GR-VAS^ET, Cafe I. Fig. 17. For when r is bigger than s, or the ratio of r to j, or of G i or yf 3 to AM or AN^ is a ratio of majority, the temperers Op, OP fall within the angles AOE, AOCxt- fpedllvely j as appears by the conflruftion. Whence, by coroll. 6 and 8. prop. lu, Gr = As-\-Ef, andET==GR-\-AS; and therefore Gr-^-As^Et : GR-\- AS -[-ET : : Gr : ET, which is a ratio of minority, becaufe Gr is lefs than GHov EI{o) and £/lefs than ET. Cafe 2. Fig, 1 8. When r is lefs than s, or the ratio of r to j, or of G i to AM or AN is a ratio of minority, the temperers Op, OP fall within the angles EOG, AOC refpe6tively ; as appears by the conflrudlion. Whence, by co- roll. 7 and 8. prop, iii, As=^Gr-\-Et, and ET = GR^AS, and therefore Gr-+AS-VEt : GR 4- AS-\-ET: : As : ET, which is a ratio of mi- nority ; becaufe Gr : As : : r : s : : GR: AS, whence, as Gr is lefs than GR, fo As is lefs than AS, which is lefs than iTj which is lefs than ET. Cafe 3. Fig. 17 and 18. When r to s, or Gi to AM or AN, is the ratio of equality, the temperer Orst coincides with the line OE, and Orst is parallel to GA-, whence it is plain, that the fum of the temperaments G 1+^^3+0, is lefs than GR-^AS-VET, as required. C^E. D. Coroll. {0) See coroll. 4. Prop. iii. 48 HARMONICS. Sed. V. Cor oil Putting c for the comma EL ov four G I, when the temp*, v : temp*, vi : : r : 5, the required temperaments of the v, vi and iii are, y + s ^ y-T s yrs And according as r is bigger or lefs than s, the temperer Orst falls within the angle AOE ot EOG. Fig. 17 and 18. For, As : Or :: s : r, and As : ^Gr or sK :: s : 2^, and As : As -\-sK or c (p) : : 5 : 5 + 3 r. Whence As = -j^r; ^y and Gr = *'- As = -^ r, and in the angle AOE, Et = Gr^As=~c, but in EOG, Et=As — Gr, by the equations in cafe 1,2. PROPOSITION V. To find afet of temperaments of the v***, vi'*" and ii\^ upo?t thefe conditions \ that thofe of the v*^ and in'* fhall have the given ratio of x to t^ and the fum of all three fhall be the leajl fojffible, PI. VII. VIII. Fig. 1 9, 20. If / to r be a ra- tio of minority, or of equality, or even of ma- jority lefs than i to --g — or o. 843070 &c, from E towards / take EM:Gi wt'.r, and through [p] See Dem. coroll. 5. prop. iii. Prop.V. HARMONICS. 49 through the interfedion /> of the lines Mi, GE produced, draw the temperer Orsf, and the re- quired temperaments will be Gr, As^ Et. But if the ratio of / to r be greater than i to o. 843070 &c, in Fig. 20, from E towards L take EN: Gi : : t : r, and through the inter- fedion P of the lines Niy G Ey draw the tem- perer ORSTy and the required temperaments will be Gi?,^*S,£r. And if ^ : r : : i : o. 843070 &c, the requi- red temperaments will be Gr, As, Et, or GB., AS, ET", their fums being equal. In the firil cafe, Fig. 19, take EN==EM, and in the fecond. Fig. 20, EM= EN; and through the interfed:ions P, p of the lines A^i, Mi with GE, draw two more temperers ORST, Orst. Then by the fimilar triangles G r p, E t p .indi G ip, E Mp, we have Gr\Et \: {Gp : Ep : : G I : E M : :) r : t by conftrudion, as re- quired by the firil condition. Again, by the fimilar triangles GRP,E TP and G I P, ENP, we have GP : ET: : {GP : EP :: G I : EN::) r : f by conftrudion, which alio anfwers the firil condition j and it is eafy to underfland that no other temperers but thofe can anfwer that condition. Cafe I. Fig. 19. Now when f is to r, and therefore EM or EN to G i in a ratio of mi- nority, tlie temperers Op, OP fall within the angles AO E, EOG refpedively by the con- D flrudion. so HARMONICS. Sed.V. ilru(flion. Whence, by coroll. 6 and 7. prop. 3, Gr=^s-\-Et and J[S=GR-\-Er. But Gr : Et ::r : ty and G r : i. £ / orri : : r : 1 /, and Gr : Gr — r i ox ~c : : r i r — \t : : Ar : ^r — t. Whence Gr=: — ^ / I / / 4/ — ];ZZxx — ^_^x—.^—0. Whence xx—:^x=^t ^"^ ^^ — ^^^+ T^T^ T^T + I: =tX-8-|- ~^ = g"g whofc fquare roots r_ - -■/ ^^3 . „,^__ _. !1 - Ir^v/33 -- 1^5. 744<;62 &c. are *■ — i — — ^ 3 whence ;e or - n::; g (r) For fince the root o. 843070 &c, when fubftkuted for X, will make the value of xx — ^x — '^., or of x — i — — zi o; a fmaller number fubftituted for x, will produce 2X a negative value of the latter, and confequently of the former quantity. D 2 52 HARMONICS. Sedt. V. cides with OHAI. Hence Gr-\-As-\-Et ht- comes=GI^+o-f £ J, and is to GR-^AS^-ET : : 5 : 6, a ratio of minority, produced by putting /=r in the terms of that ratio in cafe i or 2. CLE. D. Coroll. When the temp*, v : temp*, iii : : r : /, if -be bigger than 0.843070 &c, the re- quired temperaments of the V, VI and 11 1 are, Gr= -^- c, A5= -''-^ Cy Et= — ^ — c. And 4r — t 4r — t 4r — t the temperer Orst falls within the angle AOE or AOCy according as r is bigger or lefs than f. But if - be lefs than o. 843070 &c, they are GR = - - c, AS=!±Lc,Er^:=±', and ^r+t ^r-\-t Arr-\-t the temperer ORST falls within the angle EOG. Andif -== 0.S43070 &c, their fums are equal and either of them anfwers the pro- blem. P R O- Prop.VI. HARMONICS. 53 PROPOSITION VI. To find afet of temperaments of the v^^ vi^^ and 1 11^ upon thefe conditions that thofe of the vi^'' and iii'^ fl^all have the given ratio of s to t, and the [urn of all three pall he the leaft pofftble, PI. IX. X. Fig. 21 and 22. From E towards C take EM '. A^ : : t : s and through the in- terfedlion p of the lines M3, AE draw the temperer Orsf, and the required temperaments will be Gr, As, Et. For by the fimilar triangles A s p^ Et p and Aip, E Mp, v/e have As \ Et w {Ap : Ep : : A 2 ' E M ::) s : t hy conftru6tion, as re- quired by the firil: condition. Again, taking E N=E M, through the in- terfedtion P of the lines N^, AE produced, draw the temperer O R S'T, and by the fimilar triangles ASP, ETP ^ndAT^P^E NP, we h^v^AS :ET::{AP:EP : : A ^ : E N or E M) : : s : t by conftrudiion, which alfo an- fwers the firft condition ; and it is plain that thofe are all the temperers which can anfwer it. Now whatever be the ratio of j- to /, I fay that Gr-\-As^Et is lefs than Gi^+^5+£-r, D 3 Cap 54 HARMONICS. Sea.V. Cafe I. Fig. 21. When / is to s, or E M to ^ 3 in a ratio of minority, the temperers 0/, OP fall within the angles y^OE, EOG re- fpedively, as appears by the conjftrudion. Whence by coroll. 6 and 7 prop, iii, Gr=As -\rEtzneiAS=GR-VEr. But Et \ As \\t '. 5, and Et '. ^As ox If ::/:l5 and E^ : E/+7^or4-c (i) ::/:/+ *i 1:3 2^: 3/4-45. Whence £r=_^^r. And^j=EifX7=j7^/- And Gr=^j + £ /= -^ - ^, by the equation in the laft para- graph. Again, ET: A S :: t : s and ET : ^ A S or lT::t:^s and ET'.IT'^ET, or I E or l-c :: t : ^s — / : : 3 / : 4 i — 3 /. Whence ET=-j^^c2indAS = ET>^'- = -^^c, Therefore Gr-{-As-^Ef : GR^AS-VET : : Gr : AS : : _i±^ : l^Ill! : : ^ss-V^st-^ist — 3//, or ^ss-\-'7^st — 2st — 3// : 45^4-3 5/, which is evidently a ratio of minority. Cafe 2. Fig. 22. When / is to s, or E M to ^3 in a ratio of majority, the temperers Op, OP fall within the angles AGE, GOc re- Ipedively. Whence, by coroll. 6 and 8 prop, jii, Gr-=As-\Et and ET-=GR-\- AS, and Gr-{-As-^Et : GR^AS-^ET: : Gr : ET, which is plainly a ratio of minority. Cafe (j) See Dem. coroll, 4. prop, iii. Prop. VI. HARMONICS. ^^ Cafe 3. When t=s or EM or EN=:A'i^, the interfcdtion P vanifhes, and the temperer O R *S T' coincides with OGKL, as appears by the conftrudion. Whence by tlie conclufion of the fecond cafe, Gr-\-As-]rEt : o-\-AK-^ EL :: Gr : EL, a ratio of minority, as be- fore. QJE.D. Co?'oIL When the temp*, vi : temp*, iii : : s : /, the required temperaments of the v, vi and III are, Grzz — ; — c, As— — .— f, Etzu 4^+3^ ' . 4/+3' ' 4^4-3^ ' and the temperer Hes within the angle A O Ey whatever be the quantity and quality of the ra- tio of s to f. Scholi turn. Thefe three problems comprehend the fblu- tion of a more general one, namely. To find the temperament of a fyilem of founds upon thefe conditions ; that the odaves be perfed, that the ratio of the temperaments of any two given concords in different parcels be given, and that the fum of the temperaments of all the con- cords, be the leaft poffible. The reafon is, that the given ratio of the temperaments of any two concords, determines the pofition of the temperer of the fyflem, and this the three magnitudes of the temperaments of all the concords, whatever be their numiber. But if bodi the given concords be contained in any one of the three parcels above men- D 4 tioned; 56 HARMONICS. Sed. VL tioned (/), the given ratio of their temperaments can be no other than that of equality 3 and this datum is plainly infufficient. SECTION VL Of the Periods^ Beats and Harmony of itnperfeSi confonances, DEFINITIONS. I. Any two founds whofe fingle vibrations have any fmall given ratio, are called Imperfect Unifons : II. And the cycle of their pulfes is called Simple or Complex, according as the difference of the leaft terms of that ratio is an unit or units : III. And when a complex cycle is divided into as many equal parts as that difference con- tains units, each part is called a Period of the pulfes : IV. And the cycles of perfed conlbnances are often called Short cycles, to diftinguifh them from the long cycles of imperfed: unifons. PRO- (/) Schol. -prop. III. Prop. VII. HARMONICS. ^y PROPOSITION VII. In going from either end to the middle of any Jtmple cycle or period of the pulfes of imperfeEi unifons^ the Alternate Lejfer Intervals between the fuccejfive pulfes increafe uniformly^ and are proportional to their diftances from that end\ and at any difla?2ces from it lefs than half the fimple cycle or period^ are lefs than half the lejfer of the two vibrations of the imperfeEi unifons. Let the vibrations be V and v, and V : i; : : R : r, the integers i?, r being the leafl in that ratio 3 and putting d=R — r, we have the com- plex cycle rY-=Rv^rv-\-dv {ii)^ and the pe- riod ^ V== ^ i; 4- 1;, which when d= i,is a fimple cycle [x). PI. XI. Fig. 23, 24, 25, 26. To affiil the imagination, let the fucceffive vibrations V, V, V, &c, be reprefented by the equal lines AB, BC^ CDj Sec, and the middle inflants of dieir pulfes (u) Se6l. III. Art. 12. {x) Def. 11. 58 HARMONICS. Sed.VL pulfes (y) by the points ^, B, C, &c -, and the lucceffive vibrations v, v, a;, &c. by the equal lines a If, be, c d, &c, and the middle inftants of their pulfes by the points a, b, c, &c. Then beginning from two coincident pulfes at A or a, it is obfervable, that the fucceffive intervals of the pulfes are alternately bigger and lefs ; and that the Alternate Leffer Intervals Bh, Cc, D d, &c, or V — 'u, 2 V — 21;, 3 V — 3 f, &c, increafe uniformly, by the repeated addi- tion of the firfl leffer interval V — 'u, at every equal increment V or 1; of their dilfances from A. The alternate leller intervals are therefore proportional to their diftances from the coinci- dent pulfes A, a. Now any affigned diftance 3 V : r V : : 3 'u : rv :'. 3 V — 3 "J : rV — rv=dv, by the equa- tion y whence 3 V : j V : : 3 V — 3 "j : v -, con- fequently if tlie affigned diflance 3 V or y^D be lefs than half the fimple cycle or period ^ V, the adjoining interval 3 V — 31;, or D d is lefs than half v ; but if bigger, bigger than half v. And the argument is the fame in going back- wards from the two next coincident pulfes at U and w, U and x, &c, dieir larger and lefler al- ternate intervals being evidently of the fame mag- nitudes as in going forwards. Fig. 24. Now if the difference d=2, and the length of the complex cycle be the line AU or a x=}'V=rv-'r2 V, having divided it into two equal (;•) Se J=4; &C. CoroIL 3. Some of the alternate leller inter- vals of the pulles of imperfe(fl unifons, are the differences of equal numbers of their vibrations, counted from the nearefl coincident pulfes j and others are the differences of equal numbers of the fame part or parts of their iingle vibrations, counted from the neareft periodical point. CorolL 4. If the vibrations of two couples of imperfed: uniibns, or of any two confonances, be proportional, the periods and cycles of their pulfes, whether fimple or complex, will be in the ratio of the homologous vibrations. Let T and t be the vibrations of one couple, and V and v thofe of the other j and lince T : t ::Y '. V :: r-\-d : Ty the cycles of their pulfes {ire rT=r+^x t and r Y=.r-\-d x 1;, and the periods are ^ T= ^^ t and ^ V= ~- v -, and are in the ratio of T to V, or of / to v. Coroll. 5. The length of the period of the Leaft Imperfediions in any confonance of im- perfed: unifons, is the fame as that of the period of its pulfes. PI. XI. 64 HARMONICS. Sea. VI. PI. XL Fig. 23, 24, 25, 26. For unifons are perfect when their fucceffive pulfes are conftantly coincident (^), and imperfed: when the ratio of their vibrations is a little altered from the ratio of equality {i)) ; and then the pulfes are gradu- ally Separated by Alternated LefTer Intervals, which are the Imperfe?, are proportional to the vibrations y4Bj a by that is, whofe inter- val Qr^ is an alternate leller inteiTal of their pulfes (/). For fince the feveral lengths Q^ and ry, XA and y i, Ai^and gA, &c, of the fubfe- quent fhort cycles, are proportional to AB and a by the remaining diftances XZ and yn, AZ and g)j, KZ and Aw, &c, are alfo proportional to JIB and a 6 ; which Hiews that the diiloca- tions of the exterior pulfes X and j, A and e, K and A, &c, and of the interior too, are con^ ilantly fome of the alternate leffer inter\^als of the pulfes of AB and ab. And thus the period of the leaft diflocations of the pulfes of the im- perfe(5t conlbnance, or of die leaft imperfedtions in its fhoit cycles, is constantly the fame as that of the pulfes of AB and ab. Fig. 35 fliews the fame thing, when 2AB and ^aby or AC and ady are different mul- tiples of AB and ab, whole pulfes make long fimple cycles; and when they make periods, the like is evident by inipedion of the pulfes about the periodical points Xy Ty in Fig. 24, 25, 26, fuppoling the proper numbers of pulfes to be intermitted. And univerfally, the leaft terms of any per- fect ratio being m and «, the periods of im- E 2 perfect (/) Prop. VII. 68 riARMONlCS. Sed.VL perfed unifons whofe vibrations are AB and ^/^, will be changed into periods of the fame length of an imperfect fharp confonance whole vibrations are ?fiAB and ;z^^ by intermitting m-^—i pulfes of AB and «— i pulfcs oi ab-, or into equal periods of an imperfeft flat confo- nance whofe vibrations are mab and 7iAB, by intermitting w — i pulfes of ab and ;z — i pulfes of ABy fo as to leave equidiftant pulfes at larger intervals for the pulfes of the refulting confonance. For tho' fome of the alternate leifer intervals of the pulfes of the imperfed: unifons are de- flroyed by thoie intermiffions, yet the remaining pulfes continue in their own places and make periods of the fame length as the whole num- ber of pulfes did before. QJE. D. Coroll. I . With refped to the perfed: confo- nance whofe vibrations are mAB, zndftAB, the former iniperfed confonance of ??iAB and nab is tempered fharp by the tempering ratio nAB to nab (^ ), and the latter imperfed con- fonance of mab and nAB is tempered flat by the fame ratio 7nAB to mab of the vibrations A By ab of the imperfed unifons, whofe inter- val is therefore the temperament of both thofe imperfed confonances. And the fame might be faid with refped to this other perfed confonance of the vibrations mab and nab^ whofe interval is the fame as that of the former perfed confonance, the perfed ra- tio being the fame in both. Coroll. (^) Se(Sl. 11. Art 5 and 6. Prop. VIII. HARMONICS, 69 Coroli. 2. The lengths of the perfect cycles of thofe perfedl confonances are ?n?i AB and mnab-j ( becaufe f?iAB : iiAB : : m : n \ : mab : nab ; ) and mnAB being the greater of the two, is therefore the whole length of the imperfedl fhort cycle of either of the foregoing tempered con- fonances. Coroli. 3. Confequently the imperfeft (hort cycle of any imperfed: confonance contains equal numbers of the flower and quicker vibrations AB^ ab of the imperfed: unifons from whence it is derived. Coroli. 4. The fame multiples of the vibra- tions of imperfed unifons, will be the vibrations of other imperfed unifons, whofe period is the fame multiple of the period of the given uni- fons [h), and whofe interval is the fame too at a different pitch 3 becaufe the ratio of the vibra- tions is the fame (/). LEMMA. ^ P]. XIL Fig. 36. The logarithms of f mall ratios^ a o / are fuppofed to be very fmall in comparifbn to the terms themfelves. It appears then that the logarithm of the ra- tio a to bo is to the logarithm of co to do, as the area abfe to the area cdhg, or very nearly as the trapezium ablk to the trapezium cdnm, or (becaufe j / is the mean altitude of both) as the red angle under ab and st to the redangle under cd and st, or as ab to cd. Q^^E. D. Coroll. I. The logarithms of fmall ratios, a to bo, a to CO, which have a common term ao, are alfo very nearly proportional to the dif- ferences of their terms j but not fo nearly as if the terms had a common half fum. For the logarithms of the ratios ao to bo, a to CO are proportional to the areas abfe, acge^ or very nearly to the trapeziums ablk, acmk, or to the redangles under their bafes ab, ac and their mean altitudes, or nearly to the bafes them- felves : becaufe the ratio of the mean altitudes is veiy fmall in comparifon to that of the bafes. CorolL Prop.VIII. HARMONICS. 71 Cor oil. 2.. PI. XII. Fig. 37. The logarithms of any finall ratios ao \bo^ co \ do are very nearly in the ratio of - to - , or of ^' to ^ , compound- ao CO bo do ^ ed of the dired: ratio of the differences of tlie terms of the propofed ratios, and the inverfe ra- tio of their homologous terms. For fuppofing ao : eo -. : co : do^ thcfe ratios have the lame logarithm. Whence the loga- rithms of the propofed ratios, ao \ bo^ co\ do, / ^ 1 ab .^ ae or ao : eo. are as abX.o ae by cor. i, or as — to — ■' no ^n or - , becaufe ae'.ao: -. cd : co by the fuppolition. The fecond part may be proved in like manner by taking/i? : ao : : co : do. Coroll. 3. If ^ and b be the terms of any fmall ratio whofe logarithm is c and - ^ be any part or parts or it ; taking s-= and a=- , s ■\-- dto s — - d is the ratio whofe los^arithm is p P "-* - c very nearly. For the terms of both thofe ratios have a common half fum s, and fince s-\-d^=a and s — d=bi the difference of the terms a and b is 2 d. and that of the terms s -{■ ~ d and s — - d P P is -^ d. Whence by the lemma, the logarithm of ^ to ^ is to the logarithm of ^ + - ^ to s — - a : : 2d : ^- d ::i:- : : c :- c, and c bein? the lo^^a- p p P 00 E 4 ritlirn 72 HARMONICS. Sedl. VI. rithm of a to Lz ^ is the logarithm o£ s -^^-d to s — -d, P Cor oil. 4. Hence as mufical intervals are pro- portional to the logarithms of the ratios of the fingle vibrations of the terminating founds (^), if any part or parts of a comma c denoted by - r, be the interval of imperfedl unifons, the ratio of the times of their fingle vibrations will be I 6 I p-\-q to I 6 ip — q. For the comma c being the interval of two founds whofe lingle vibrations are as 8 1 to 80 (/), by fubilituting 81 for ^ and 80 for ^ in the laft corollary, we have 5 = — , J=~ and j+ -— ^, the ratio of the fingle vibrations be- longing to the interval ~Cj very nearly (m). This (k) Sea. I. Art. 11. (/) Sea. 11. Art. 4. (?«) And converfely, if the ratio of the times of the fingle vibrations of imperfea unifons be V to Vy their interval is Y y -- — xi6if. For fuppofing V : 2/ : : ibip+q: 161 p — a V-\~v p and q being indeterminate numbers; orV— i6i/)+? and vzz-ibip — q y then V — vzziq and V-j-i»— i6ix2/>, and — - — -i— . Whence =^r- — xi6i ^— ^ (. the in- V-f-y i6i/» V-f-^' p terval belonging to the ratio i6jp-{-q : 16 J p — ?, orViVy by coroll. 4. Prop. VIII. HARMONICS. 73 This ratio approaches furprifingly near to the truth, as will appear by an example. Let -c=^Cy or />=4 and ^=1, then i6i/>+^ ; 161/ — q : : as 645 : 643. Now by tlie Tables of Logarithms, The log. ^ =0.00539.50319 -^; of it =0.00134.87580 *^^^°g-^ = o» 00134-87417 the difference = o. 00000. 00163 and the logarithm 539-50319 divided by the difference 163 gives thequotient 330984, which {hews that ~ c deduced from the ratio 645 : 643 differs from the truth but by — ^—- part of a J 330984^ comma ; a degree of exa(5lnefs abundantly fuf- ficient for every purpofe in harmonics. CorolL 5. The times of the fingle vibrations of imperfe, -u, i;' in coroll. 5. to be variable and^ to increafe to in- finity in any finite time, the intervals - f , - c^ ■will decreafe and vanifh in the ratio of qto q' firfl: given; the ratio 161 p — q to i6\p^-q will alfo decreafe and vanifli in the ratio of equa- lity; and therefore the Ultimate Ratio with which the increafing periods ^——^ V and ^ - 7^ V' became infinite at the end of the given time, and vaniflied into innumerable fhort cycles of perfed: unifons, is - to -, , or, by a like argument, - to - . Coroll. 8. Hence if two confonances of im- perfed: unifons have a common found or vibra- tion V=V', or 'V=-v\ the ultimate ratio of their periods is q to q^ the inverfe ratio of their intervals ; (0) This agrees with Cor- 4. Prop, vir. Prop. IX. HARMONICS. 75 intervals ; and confequently is the inverfe ratio of the differences of their vibrations (/>). Cor oil. 9. If the two llov^er or the two quicker vibrations of two confonances of im- perfedl unifons have the ratio of their intervals, the periods of their pulfes are ultimately equal. V v For if V : V : : ^ : q\ then - : - : : i : i, which 1 1 is the ultimately ratio of the periods : and the like argument is applicable to the ratio v : v'. PROPOSITION IX. If the interval of two founds whofe per- feSi ratio is m to n, be ijicreafed or diminifded hy -^Cy and the times of the complete vibrations of the bafe and treble of either of thefe confonances be 7j and z and the period of its leafi imperfeEiions be P, then in Car. I, P = i^il=:-^ X ''- or i.^lP±5 x ^ , •J ^ zq m 2q n' t~t r Tj l6ip4-q Z 161 p — q z CaL 2, P = ^ ^ \ y.- or ■ ^— -? X - . -f ^ 2 q m 2 q n PI. XII. Fig. 38. For if AV and av, or V and V be the times of the complete vibrations of imperfed unifons whofe interval is the tempe- rament ^ c, then V : -J : : i6ip-\-q: 16 ip—q {q) and the period of their pulfes, or of their leaft imper-i (/») Coroll. I. and Se£l. i. Art. 11, {q) Coroll. 4. Lemma. 76 HARMONICS. Sed. VI. Imperfedions, is - ^ V= "^^ i; (r) and has the fame length as the period of the leaft imperfedtions in a fharp confonance whofe vi- brations are mW and nv^ or in a flat confonance whofe vibrations are jii'u and «V: both con-r fonances being derived from the perfedl one whofe vibrations are niN and ;zV, or mv and nv (i). Hence in Caf i . taking 7j=mV and z=nv, we have - =V and -='u ; which values being fubftituted for V and v in the period of imper- fect unifons, give P= ~^^ x -= ' '^"t ^ x - ^ And in Caf. 2. (Z and ;z being fuppofed indeter- minate) taking X^^mv and z=;zV, we have - =i; and-=Vi which being fubftituted for •u and V in the faid period of imperfed unifons, y» 161/14-0 Z 1616 <7 2; ^r^ -l^ -l-x. give P= — ^-^ X - = ^ ^ X -. Q, E. D. The value of P in Cafe 2 is deducible from its value in Cafe i , only by changing the fign of q 5 that is, by fuppofing - <; to be the nega- tive or flat temperament, as it really is when + I c is the fliarp one. And thus one expref- fion of the value of P might have ferved both cafes of the propofltion, but two are more difl:ind for future ule. CorolL (r) Sect. VI. Defin. iii. or Cor. 5; Lemma. {s) Pror. vixi. Corol. i. Prop.IX. HARMONICS, 77 Cm'oll. I . Hence if any two imperfe6t con- fonances have Z and Z'for the times of the complete vibrations of their bafes, z and 2' for thofe of their trebles, - c and - c for their tern- P P peraments, whether fliarp or fiat, or one of each fort, m and tn' for the major, and n and ;/ for the minor terms of the perfect ratios j the Ulti- mate ratio of their periods is — to -r^ , or ^ qm qm ^ to ^, '. The proof of which is the fame as qn qn ^ was given for the Ultimate ratio of the periods of imperfed unifons in Coroll. 7. to the Lemma. Coroll. 2. Hence when the temperaments are equal and the major terms the fame, the pe- riods of the leaft imperfections have ultimately the ratio of the fingle vibrations of the bafes. CorolL 3. When the baies are the fame, the periods have ultimately the inverfe ratio of the temperaments and major terms jointly. Coroll. 4. When the bafes and major terms are the fame, the periods have ultimately the inverfe ratio of the temperaments. CorolL 5. When the bafes and temperaments are the fame, the periods have ultimately the inverfe ratio of the major terms. Coroll. 6. All thofe corollaries are applicable to the trebles and the minor terms, only by reading trebles inftead of bafes and minor terms inftead of major ; and then, as before they had no dependence on the trebles and minor terms, fo 78 HARMONICS. Sed.VL fo now they have none upon the bafes and ma- jor terms. phtEnomena of beats. If a confoitance of two fouiids be uni~ for my without any beats or undulations^ the ti^nes of the f?tgle vibrations^ of its founds have a perfeSi ratio \ but if it beats or undulates y the ratio of the vibratio7is differs a little from a perfeB ratio^ more or lefs according as the beats are quicker or flower. Change the firft and fmalleft ftring of a vio- loncello for another about as thick as the fe- cond, that their founds having nearly the fame ilrength may beat ftronger and plainer. Then fkrew up the firft ftring; and while it ap- proaches gradually to an unifon with the fecond, the two founds will be heard to beat very quick at firft, then gradually flower and flower, till at laft they make an uniform confonance with- out any beats or undulations. At this juncture either of the ftrings ftruck alone, by the bow or finger, will excite large and regular vibra- tions in the other, plainly vifible to the eye ; which fliews that the times of their fingle vi- brations are equal {t). Alter the tenfion of either firing a very little, and their founds will beat again. But now the motion ( t ) Sea. I. Art. I. Prop. IX. HARMONICS. 7^ motion of one ftring flriick alone makes the other only ftart, but excites no regular vibra- tions ; a plain proof that they are not ifochro- nous. And while the founds of both are draw- ing out with an even bow, not only an audible but a viiible beating and irregularity is obfervable in the vibrations, which in the former cafe were free and uniform. Meafure the length of either firing between the nut and bridge, and, when they are perfect unifons, at the diilance of \. of that length from the nut mark that firing with a fpeck of ink. Then placing the edge of your nail on the fpeck, or very near it, and prefling it to the finger-board, upon founding the remaining ^ with the other firing open, you will hear an uniform confonance of V*^', whofe fingle vibra- tions have the perfed: ratio of 3 to 2 {u.) But upon moving your nail a little downwards or upwards, that ratio will be a little increafed or diminifhed; and in both cafes the imperfed: V'*^* will beat quicker or flower according as that perfect ratio is more or lefs altered. The Phsenomena are the fame when the parts of the firing have any other perfe6l ratio ; ex- cept that the beats of the fimpler concords are plainer than thofe of the lefs fimple and thefe plainer than thofe of the difcords, which being very quick are not eafiiy diflinguifhed from the uniform roughnefs of perfecl difcords. The (a) Sea. I. Art. 7. So HARMONICS. Seft. VL The founds of an organ being generally more uniform than any other, their beats are accord- ingly more diftindt, and are perfedlly ifochro- nous when the blaft of the bellows is fo uni- form as not to alter the vibrations of either found. Beats and undulations when every thing elfe is filent, are alfo pretty plain upon the harpfi- chord, efpecially while the founds are vanifh- ing. Quicker undulations are beats, and are re- markably difagreeable in a concert of ftrong, treble voices, when fome of them are out of tune ; or in a ring of bells ill tuned, the hearer being near the fteeple j or in a full organ badly tuned : nor can the beft tuning wholly prevent that difagreeable battering of the ears with a Gonftant rattling noife of beats, quite different from all mufical founds, and deftrudlive of them,- and chiefly caufed by the compound flops called the Cornet and Sefquialter, and by all other loud ftops of a high pitch, when mixed with the reft. But if we be content with compofi- tions of unifons and odlaves to the Diapaibn, whatever be the quality of their founds, the beft manner of tuning will render the noife of their beats inoffenfive if not imperceptible. Thefe are the general phcenomena of beats, whofe theory I am going to explain. PRO- Prop.X. HARMONICS. 8i PROPOSITION X. An imperfeB confona7we makes a heat in the middle of every period of its leaft imperfeBions-, a?jdfo the time between its fucceffive beats is equal to the pe- riodical time of its leaft imperfeBions, PI. XI. Fig. 23 to 27. 34, 35. Any /fimple cycle or any period of the pulfes of imperfe6l unifbns, contains one more of the quicker than of the flower vibrations (.v), and the fhort cycle of any imperfe(5l confonance contains equal numbers of the quicker and flower vibrations of the imperfed: unifons (^'). Confequently after taking away the greateft equal numbers of fhort cycles, that can be taken from both ends of the Ample cycle or the period of the imper- fecl unifons, fome part of another fliort cycle or two, as confiding of unequal numbers of the quicker and flower vibrations of the imper- fe(fl unifons, will always remain in the middle of the cycle or period. And this part, by in- terrupting the fucceffion of the Ihcrt cycles, wherein the harmony of the confonance con- fifts, interrupts its harmony and caufes the noife which is called a beat : efpecially as the inter- ruption is made where the (hort cycles on each {x) Prop. VII. coroll. i. \y) Prop. viir. coroll. 3. F fide 82 HARMONICS. Sed:. VI. fide of it are the moft imperfe6l and inharmo- nious. Therefore the time between the fuccef- five beats, m.ade in the middle of each period or Umple cycle of the pulfes of the imperfed: unifons, or of the leall: imperfections of the con- fonance (2), is equal to the time of this period. And the caufe of the beats of imperfedt uni- fons is a like interruption of the fuccellion of their fliort cycles, in the middle of every period or limple cycle of their pulfes, where they are moft imperfed: and inharmonious. QJ^. D. Coroll. The time between the fucceffive beats of an imperfect confonance is the fame as the periodical time of its Greateft Imperfedions. PROPOSITION XI. If the interval of two founds whofe per- fect ratio is m to n, be increafed or diminifhed by the temper a7nent -i c (a), and j8> be the number of beats made by either of thofe confo?mnces while its hafe is making N, and its treble M complete vibrations ; then in Caf. I, /2= ■x--'^— mN, or-^^ nM, J ' f^ 161 p—q ' 161 p + q ' Caf. 2, iS = -7^ V ^ N, or -/-^- nM. •J ' 161 p + q loi p—q For if the time between the fucceffive beats of either confonance be called P, and the time of (z) Prop. VIII. (rt) See Lemma cor. 4. p. 72. Prop.XI. HARMONICS. 83 of a complete vibration of its bafe be Z and that of its treble z ; the time of their beat- ing and vibrating will be conftantly meafured by /3P=NZ or Uz. Hence /3 = -^ or ^ and fince the time P is equal to the period of the leafl: imperfections of the confonance [b), by fubftituting its values in Prop, ix, wt have in Caf. I. /3=NZx /^" ^ = -^-"-^ , and fo of ' ibxp—qsL 161/ — q the other values of /2. Q^. D. Coroll. I. Hence if any two imperfecSt conr fonances have Z and Z' for the times of the iingle vibrations of their bafes, z and z for thofe of their trebles, i c and ^ (^ for their temr peraments, whether fiat or fharp, or one of each fort, m and rrl for the major, n and ri for the minor terms of the periedl ratios, N and N' for the numibers of complete vibrations made by the bafes, and M and M' for thofe made by the trebles in any given time; the ultimate ratio of the numbers of their beats, made in that time, will be ^ /^z N : qm^\ i^r qnyi\ q?2M\ ^^ qm am an an orV :%- oil- 'J—. The manner of proving the two firfl: ratios has been (hewn before (<:), and the given time being conftantly N Z=N' Z'=M2;=M' 2;', we have N : N' : : ^ : ^, , which ratios compound- ed with ^;« :^'/«' give ^;;/N : qmW : : ^ : ^'. F 2 Like- [h) Prop. X. {c) Lemma cor. 7. 84 HARMONICS. Seft. VI. Likewife M : M': : - : -, which ratios com- z z pounded with qn : g'n give q nM. : q'n'M' : : ? " . / "' X ' z' ' CorolL 2. Hence, when the temperaments are equal and the major terms the fame, the beats of the confonances, made in a given time, have ultimately the inverfe ratio of the iingle vibrations of the bafes. CorolL 3. When the bafes are the fame, the beats have ultimately the ratio of the tempera- ments and major terms jointly : And therefore when the bafes and beats are the fame, the tem- peraments have ultimately the inverfe ratio of the major terms. Coroll. 4. When the bafes and major terms are the fame, the beats have ultimately the ratio of the temperaments. CorolL 5. When the bafes and temperaments are the fame, the beats have ultimately the ratio of the major terms. CorolL 6. All thefe corollaries are applicable to the trebles and minor terms, by reading tre- bles inftead of bafes and minor terms inlliead of major : and then they have no dependence on the bafes and major terms, as in the former cafes they had none upon the trebles and minor terms: which abfent terms may therefore in both cafes have any magnitudes whatever without al- tering the ratio of the beats. CorolL Prop. XL HARMONICS. 85 Coroll. 7. Things remaining as in the propo- fition, we have in Caf. 1 . 7j : z w m -\-^\n \\ tn \7i — ^ . Caf. 2. X : z :: m — :^: n :: m : 7i -V y:^ > For by tlie Prop, in Caf. i . /3 = \l^p_ j whence ^: m '.'. 2q I 161 p — q and compojite ^ + xv : m :: i6ip-{-q : 161/ — ^, either of which ratios being the tempering ratio and m to ;; the perfed: one, the imperfed; ratio is plainly m -^-'t,'. n '.:7j '. z. An4 a like refolution of the other values of ^ in the proportion gives the other proportions. Scholium I. To Jhew that the ultimate ratio of the beats or the periods of imperfeB con- fonanceSy when ufed injlead of the exaB ratio y can produce no fenfble difference in the Harmony, I. The temperaments of apy two conib- nances being - c and - c. the difference between die exadl and the ultimate ratio of their beats, made in any given time, is the ratio 161 p^. q to 161 p:^q' ; where the fign of q or q' is nega- tive if the refpedtive temperament be fharp, or affirmative if flat. F 3 For 86 HARMONICS. Sedl.VI. For that ratio compounded with the exad ra- tio of the beats, which is -^J"- — to —^ — •, (^, makes their ultimate ratio ^ ;;z N to ^' m N' (f). 2. Now the magnitude of the ratio i6 ip'^q to i6ip:^q\ hke that of all ratios, being greateft or leaft according as the difference of its terms is greateft or leaft in proportion to the terms themfelves ; it will follow, that in the moft harmonious fyftem of founds hereafter de- termined (/), the ultimate ratio of the beats of any two concords cannot differ from their exad: ratio by any ratio greater than 3624 to 36 1^-, or lefs than 2901 to 2900. For the temperaments ^- c, - c of any two concords in that fyftem, have no other values than a couple of thefe, A^^, -^tC, and -r\c (f). Where p being 18, the greateft magnitude of the faid ratio i6i/'4r^toi6i^4r^'is 161x18+3 to 161K18 — 5, or 362 4. to 361I, and the leaft magnitude of it is 161x18 + 3 to 161x18 + 2, or 2901 to 2900. 3 . Hence the number of beats in either term of any ultimate ratio in that fyftem, cannot dif- fer from the number of them in the corre- fponding term of the exad ratio, by above t-tt part of that firft number : and therefore not by {d) Prop. XI. {e) Prop. XI. cor. 1. (/) Prop. XVI. Schol. 2, Art. lO and 13. Prop.XI. HARMONICS. 87 by a fingle beat when that number is lefs than 361. For let ^ to ^ be an ultimate ratio which ex- ceeds the correfponding exad ratio by the greateft difference 3624- to 36i-|-. Then by fubflrading this difference, and neglecting the fractions, the exa-il ratio is 361^ to 362^', that is, a to ^+-34t^, or a — -%^-za to e. 4. Now let two v^'^^ or any two concords of the fame name, near the middle of the fcale of a good organ, have the fame bafe and diffe- rent trebles ; and fuppofe them fo nicely tem- pered, that in a given time one of the v'''* fliall make 362 beats and the other 361. This indeed is extremely difficult to execute, the numbers of beats being fo large. But fuppo- fing it done, my opinion is (from my own ex- perience in fmaller numbers) that the mofl critical ear could not diftinguifli the leaft diffe- rence in the harmony of thofe v'^'% or in the rate of their beating : no not if the ratio of the beats were much greater than 362 to 361 : And if it could not, without doubt the theory of ultimate ratios is Efficiently accurate for de- termining and adjufling the Harmony of the beft fyftem of founds. Becaufe it will be fliewn hereafter, that the beft method of tuning any fyftem, is to adjuft every v^^ to the number of beats it fhould make in that fyftem. 5. In lefs harmonious fyftems, the difference between the exad and the ultimate ratio is fome- F 4 thing 88 HARMONICS. Sed. VI. thing greater than 362 to 361 ; as 322-^ to 3214 in the fyilem of mean tones (g) ; but fhll not fo great in any tolerable fyflem as to afted; the mofl critical ear : and what has been proved of beats holds true of Periods, the ratio of the periods being the inverfe ratio of the numbers of beats made in any given time. 6. Therefore the ultimate ratios of beats and periods ought to be ufed in harmonics, their terms being always limpler than thofe of the exa(5t ratios, as appears by comparing Prop, ix and XI with their corollaries. For inftance in the fyftem of equal harmony, the temperament of the v''' is ^ c, and of the vi*^ is — I Cy whence if their bafes be the fame I 9 the exa6t ratio of their beats, made in any given time, is 36 It to 362T by Prop, xi; but their ultimate ratio is that of equality by coroll.3, which is fimpler, and the harmony of the con- cords not fenfibly different [h). Scholi, turn 2. 'To Jhew that the theory of beats agrees with experiments, I. PI. I. Fig. 3. The exponent of the time of a fmgle vibration of any given found, as f, in {g) Prop. 2. coroll. {h) Art. 4. of thefe. Prop.XI. HARMONICS. 89 in a given fyftem of perfed: confonances may be changed into i by dividing every exponent by that of the given found, which changes them to thofe in PI. xii. Tab. i. without altering their proportions. Then if the found whofe exponent is i, be a little altered to y either higher or lower, the numbers of beats made in any given time by the feveral imperfed; confonances of y with every one of the other founds, will be pro- portional to the Denominators of their ex- ponents. For when y is flatter than <:, all the intervals above c are increafed by the common temperar mcnt cy = -c in Prop, xi, where in Cafe i tbe number of beats made by any given confb- nance yd, while its bafe y is making N vi- brations, is ,^^_ - m N. And all the perfe(ft intervals below c being diminifhed by c y, in Cafe 2 the number of beats of any given con- fonance y B, while its treble y is making M vibrations, is , ^^ n M. 161 p — f Here the numbers N, M of the vibrations of y made in any given time are equal, and ->— ^ being the fame in both cafes, the beats oi yd are to thofe of q/ B as »7 to «, that is as the ma- jor term 9 of the perfedl ratio 9 to 8 belonging to 90 HARMONICS. Sed. VL to y dy is to the minor term 1 5 of the perfed: ratio 15 to 16 belonging to ^ B j and thofe terms are the Denominators of the exponents of d and B, the treble of the former and the bafe of the latter confonance. And iince the beats of yd and y B are as 9 to 15, and by the fame demonflration thofe of yB and ^^ as 1 5 to 5, ex cequo the beats of y d and y e, having the fame bafe, are as 9 to 5 ; which terms are the Denominators of the expo- nents of the trebles dy e. And by the like proportions the beats of ^ B, yA, which have the fame treble, are as 15 to 5, the Denominators of the exponents of the bafes B, A. And when y is fliarper than r, the two theo- rems above are changed to thefe —, — ^ m N and ^ \o\ pr q - , ^: - 11 M, and the demonftration goes on as \6\p-\-q ' o before. Qj:. D. 2. In Tab. 2 and 3 each feries of fractions, being a geometrical progreffion in the ratio 2 to i, are the exponents of the lingle vibrations of fuc- ceilive viii^**', and are feverally deduced from the exponents of the bafes of as many given concords AC, AD, AF, AC;^, AE, AF;^. Hence in Tab. 2. the beats which the treble of any imperfed Minor confonance, AC, AD or AF, makes in a given time with its bafe and with every 8 ^^ below it and as many 8'^' above it Prop.XI. HARMONICS. 91 it as refult from a continual bifedion of the Numerator of its exponent, are all ifochro- nous. But the beats which that treble makes with the fucceffive 8'^' ftill higher are continu- ally doubled in any given time. T A B . 2. 12 1 6 ? I 3 5 1 10 20 A' A C a a d* 8 3 4 3 I 2 3 I 3 I 6 A' A D a d d' 7 8 5 I 4 5 2 5 t T A' A F a 1 a d' For the major term of the perfedl ratio of any Minor confonance is an Even number (/) and is the Numerator of the exponent of its bafe (that of its treble being reduced to i,) and when that numerator is reduced to an odd number by continual bife(Sions, this odd num- ber is the conftant numerator of the exponents of all the fuperior 8'^% whofe denominators muft therefore be continually doubled, which doubles the beats by Art. i . But the doubling the numerators of the exponents of the inferior 8'^' alters not their given denominator, as being an odd number, nor confcquently the beats. Tab. 3. (/■) Sect. 2. Art. I. Table. 92 HARMONICS. Sea.VI. Tab. 3. The beats which the treble of any imperfed: Major confonance, AC;^,AE orAF^^, makes with its bafe, in any given time, and with every S^^ above it and as many 8'^' below it as refult from a continual bifedion of the Deno- minator of its exponent, if an Even number, are continual proportionals in the ratio of 2 to I } and the beats of that treble with every 8'^ ftill lower are ifochronous. But if the De- nominator of the given bafe be an odd number, the beats which its treble makes with it and every 8^^ below it are all ifochronous, TAB. 3. 5. I 5. 2 5 4 I 5 s 5 16 A" A' A C;^ a a 6 I 3. I I 2 I I 4 3 8 A" A' A E a r/ 20 1 10 1 I 5 6 IZ A" A' A F)^ and x to be an indetermi- nate vibration, and V : x : : m : n, and the ra- tios of the indeterminate numbers m, n to ap- proach gradually to the given ratio of *y p to s/ q ; though the length nV, ^=^ m x^ of the indeterminate cycle of the pulfes of V and a*, increaies without bounds, neverthelefs the length -^ V, = -^ X, of the indeterminate period m — n m — « '■ of their pulfes tends gradually to a determinate limit -7^7- V = -~-^- 1;. And this is the vP—Vi VP-yq period of the pulfes of the incommenfurable vibrations V, ^', which excites the determinate ienfatlon of the imperfed: unifons, be the com- plex cycle of their pulfes ever fo long, infinite or impoffible. G 3 I or as 3 to 2 y^ 2, which are incommenfurable quanti- ties ', and that of a quarter note as / 9 ^^ '^ 8, which IS yet more incommenfurate ; and the hke for any other number of equal parts: which will therefore never fall in with the proportions of number to number. Upon the hnperfeaion of an Organ. Phil. Tranf. N°. 242, or Abridgm. vol. i. p. 705. edit. i. (z) Denique ob nullam fonorum rationem rationalem praeter oftavas, hoc genus [muficum] harmonise maxime contrarium eft cenfendum; etiamfi hebetiores aures dif- crepantiam vix percipiant. Tenta7nen riQVce Theoria muftcay cap. IX. fe6l. 17. Petropoli. 1739. I02 HARMONICS. Sea.VI. I fay determinate fenfation. For though the alternate leiTer intervals of the pulfes in the ieveral fucceffive periods of V and i;, even when commenfurate, are not precifely equal (^), yet it is highly probable that the ear could not diftinguifh a repetition of any one period from the fucceffion of them all, and feems agree- able to experience in obferving the identity of the tone of imperfect unifons held out upon an organ. 6. For further illuftration I will add an ex- ample or two. We fhewed above that the vi- brations V, V of the mean tone are as ^5:21:2. 23606796 &c : 2 : : m : n. Whence the length of the period of the pulfes of V and v^ is -^ V= ■ —-. — 77- = m — n o. 23606790 &c 8.47213 &c X V; which is a medium between 8 V and 9 V, the cycles of the pulfes of the major and minor tones, fomething lefs than the arithmetical, or even the geometrical mean, but not quite fo little as the harmonical mean be- tween them {b). Again, -when V and 1; are the vibrations of two founds whofe interval is a quarter of a comma, we found V : 1; : : 3 : 2 / 5 or 2. 99069756 &c \\ m \ n ', whence the pe- riod of the pulfes of V and v is - — V = 2-99'^ 97!; X V= -^2 1. 4060 &C x V. 0.00930244 &c ^ I • Or (n) Coroll. 2. Prop. vii. / (^) See Sed. vii. Def. 11. Prop. XI. HARMONICS. 103 Or thus. In approximating towards the ratio of V to "Jy or 3 to 2 / 5, or 3 to 2.990697, or 3000000 to 2990697 byfmall numbers (c), the ratios greater than V to 1; are 322 to 321, 967 to 964, 1612 to 1607, &c. Whence the cycle 3 2 1 V and the periods 321 4 V, 3 2 1 4 V, &c, are all too fhort. And the ratios lefs than V to f being 3 2 3 to 322, 645 to 643, &c, die cycle 322V and pe- riods 3214-V, &c, are all too long. There- fore the true period falls between the laft mentioned limits, agreeably to the former com- putation. From what has been faid of imperfed unl- fons the difficulty vanifhes in other imperfedl confonances, by obferving the redud:ion of the periods of their imperfedions to thofe of imper- fect unifons, as in Prop. viii. 7. If the ifocht'07ioiis '•oibratiom of contiguous parcels of air^ excited by different f rings, can- not be reduced to a fynchronifm by the mutual ac- tions of the particles, [as I think they cannot^ it will folhiv that coincideiit pulfes are not ne- cejfary but only accidental to a perfeB confo7iance. For while an imperfed: confonance is found- ing, if the ratio of the vibrations be made per- fed;, as in tuning a mufical inflrument, from the inftant of this change tlie diflocation of the pulfes, whatever it be, will continue unal- G 4 tered {c) See Mr. CoUi'i Harmonia Menfurarum, Pi op. i. Schol. 3. 104 HARMONICS. Sea. VL tered in all the fubfequent ihort cycles ; and thus the confonance is perfect without any coin- cident pulies, unlefs when the change of the ratio happens at the inftant of the coincidence of two pulfes. 8. This howe'ver ftems indifputabic, that coin- cident pulfes are not necejfary to fuch harmony as the ear judges to be perfeB. For if any long period of imperfed: unifons, in- tercepted between twobeats, be lengthened greatly and indeterminately, as in tuning an inftrument ; any given part of it, as long as any mulical note, will approach indefinitely near to perfe6l unifons ; certainly nearer than the ear can diftinguifh, as being often doubtful of their perfedtion. And yet throughout that part (fuppofed to be fmall in comparifon to the whole period) the pulfes of one found divide the intervals of the pulfes of the other very nearly in a given ratio, of any determinate quantity between infinitely great and infinitely fmall, in proportion to the di- .ftance of that part from the periodical point or point of coincidence. Neverthelefs the ear cannot diftinguifh any difiference in the har- mony of fuch different parts, as is evident by often repeating the fame confonance, which can hardly begin conftantly in the fame place of the long period. And the fame argument is applicable to any given confonance, as being formed by intermitting a proper number of pulfes of each found of the imperfe6l unifons : and the Prop.XI. HARMONICS, 105 the conclufion feems to be confirmed by the following experiment. 9. When any firing of a violin or violon- cello is moved by a gentle uniform bow, while its rriddle point being lightly touched by the finger, is kept at reft, but not prelTed to the fingerboard j the two halves of the firing will found perfect unifons, an eighth above the found of the whole ; and will keep moving conflantly oppofite ways. Becaufe the tenfion and flifFnefs of the parts of the firing on oppofite fides of the quiefcent point, compel them to oppofite and fynchro- nous motions, and thefe parts compel the next to the like motions, and fo on, to the ends of the firing. Hence, becaufe thefe oppofite mo- tions of the halves of the firing communicate and propagate the like motions to the contigu- ous particles of air and thefe to the next fuc- cefiively, it follows that different particles of air at the ear, placed any where in a perpendicu- lar that bifed:s the whole firing, will keep moving conflantly oppofite ways at the fame time ', thofe particles, which received their mo- tion from one half of the firing, going towards the ear, while others are returning from it, which received an antecedent motion from the other half of the firing : Or, in fewer words, the fucceflive pulfes of one found are conftantly bife(fling the intervals between the pulfes of the other : And yet the harmony of the unifons is perfedly io6 HARMONICS. Sea.VI. perfedly agreeable to the ear, as I have often experienced. 10. And in fo rare a fluid as air is, where the intervals of the particles are 8 or 9 times greater than their diameters (<^), there feems to be room enough for fuch oppoiite motions with- out impediment : eipecially as we fee the like motions are really performed in water, which in an equal fpace contains 8 or 9 hundred times as many fuch particles as air does (d). For when it rains upon flagnating water, the circu- lar waves propagated from different centers, appear to interfed: and pafs through or over each other, even in oppofite diredions, without any vilible alteration in their circular figure, and therefore without any fenfible alteration of their motions. 11. If it be objedted to the experiment above, that a conflant bifecSion of the intervals of the pulfes of one of the unifons by thofe of the other, if true, ought to excite a fenfation of a fingle found an eighth higher than the unifons, and as it does not, that of confe- quence there is no bifed:ion ; a fatisfad:ory an- fwer to the objection might eafily be drawn from die different duration and flrength of the fingle pulfes of different founds at a different pitch, were it neceffary to enter into that con- fideration. 12. But {d) Newt. Princip, Lib. 2. Prop. 50. Schol. and Prop. 23. Prop. XI. HARMONICS. 107 12. But after all, as abfolute certainty is dif- ficult to be had in this inquiry, I chofe to give the vulgar definition of a perfed: confo- nance in Sed:. iii. Art. 3, as a fimpler prin- ciple to build upon, and yet as fit for that pur- pofe as a more general one would be, even fup- pofing it were inconteftable. Scholium 5. Having obferved a very flrid: analogy be- tween the undulations of audible and vifible objeds, I will here delcribe it, as an illufiration of the foregoing theory of imperfed confo- nances. PI. XIII. Fig. 39. Let the points a^ by c, &c and a, /3, y^ &c reprefent tlie places of two parallel rows of equidiftant and parallel ob- jedls, fuch as pales, pallifadoes, &c, and let them be viewed from any large diftance by an eye at any point z. In a plane pafling through the eye and cutting the axes of the parallel objeds at right angles in the points, ^, by r, &c, a, /2, yy ScCy let lines drawn from z through ay (3, yy &c, cut the line of the other row in ^, By C, &c. Then by the fimilar triangles A B z and cc (3 Zy B C z and (B y z, C D z and y S" Zy &c, we have A B : a ^ : : {Bzi^z::) BC:(3y:: {C z : y z) ..CD : y S" : : &c. Therefore the antecedents A B, B Cy C Dy &c, which are to the equal confe- quents a jS, /3 9/, y S'y &c, in the fame ratio, are io8 HARMONICS. Sed. VI. are alfo equal to one another, and are the appa- rent projections of the confequents upon the line ab c oi the other row. Hence fuppoling m and n to reprefent the leafl whole numbers in the given ratio of AB to a b, we have a line m>^a b= nxAB, equal to the length of the cycle between the apparent coincidences of fome of the objecfls in one row with fome in the other ; as of ct and a at Ay of x. and m at K, &c : and if m — 71 be not an unit we have a fhorter line '!^=^ ah=-^AB = AXoy: XK, equal m m- — n ^ . to the length of the apparent period of their neareft approaches towards coincidences ; as on each fide of the point X, according to the de- monftration of the vii*^ propofition. But if the point z be fo iituated, that the lines A B and a b ov a fi, or B z and /3 Zj or Cz and y «, &c, which are all in the fame ratio, happen to be incommenfurable, it will be impoflible, mathematically fpeaking, for more than one couple of obje(5ts to appear coincident (f), and yet the periods of their ap- parent approaches will fubfiA in this cafe as well as in the other. Now if the objects be white, or of any co- lour that reflects more light to the eye than what comes to it from the ipaces between them, tind their breadth be confiderable as ufual, the rows will appear the leaft luminous about the coin- {e) See Prop. xi. Schol. 4. Art. 2. Prop. XL HARMONICS. 109 coincident ob)e(5ts and the periodical points, At Xy Ky ScCy where the objed:s of the nearer row hide the whole or fome part of thofe be- hind them In the remoter row ; and the rows will appear gradually more luminous towards the middle of the periods, where the objects will be feen diftind from one another If they be not too broad. And the contrary will hap- pen If the objects in the rows be lefs luminous than the Ipaces between them. Confequently if the fpediator ftands ftill and moves his eye from one end of the rows to the other, he will fee an alternate fucceffion of light and fhade; and while he moves for- wards in any tranfverfe direction z &>, and fixes his eye upon a given place of the rows, he will then fee an undulation of light and fhade, mov- ing forwards quicker or flower according to the celerity of his own motion. For then the apparent coincidences which were at A, K, &c, and confequently the in- termediate periodical points X, T, &c, will gradually fhift from A to B, &c, and from K to Ly &c, as is evident from the angular motion of the vifual rays about the fixt points or objedls a, (S, ^, &c, jc, A, ^, &c : And this is a known phaenomenon. If the fpedator recedes from the rows, the period -^ a b will grow longer, and upon his moving tranfverfely, the vifible undulations will be broader and flower than before, and at a very no HARMONICS. Sed.VI. very great diflance from the rows, will become imperceptible ; as being changed into an uni- form appearance of both rows in the place of one : quite analogous to the audible undulations of imperfed: unifons, as they grow flower and lefs perceptible while the unilbns are approaching to perfediion. The like phsenomenon refults from two rows of pales that meet in any angle. PROPOSITION XII. ImperfeU confonances of the fame Name are equally harmonious whe?t their fhort cycles are equally numerous in the periods of their imperfe&ions. As perfedl confonances of the fame Name are equally harmonious becaufe their cycles are fimilarly divided by the pulfes of their founds ; fo imperfe(ft confonances will be equally harmonious when their periods are fimilarly di- vided. Hence all imperfe(5t unifons whofe fingle vi- brations have the fame ratio, are equally har- monious, as having fimilar periods {/) ; and therefore all imperfed; confonances of the fame name whofe tempering ratios are the fame, are equally harmonious. For (/) Prop. VII. Cor. 4, Prop.Xir. HARMONICS. m For fince the vibrations of the correfponding perfect confonances have the fame given ratio, m to n, and the vibrations of the imperfedl ones are derived from thofe of the fimilar uniibns by intermitting m — i and n — i pulfes of their ho- mologous vibrations, (o as to leave equidiflant pulfes in every feries [g) ; the limilar periods of the unifons are thereby altered into fimilar pe- riods of imperfedt confonances ; and the equal intervals of the unifons into equal temperaments of the confonances (^6). And the lengths of theie limilar periods be- ing proportional to the lingle vibrations of their bafes or to equimultiples of them, that is, to the lengths of the fliort cycles of the perfed con- fonances, will contain equal numbers of imper- fed fhort cycles (/). q!j:. D. Coroll. Confonances of the fame name are equally harmonious when equally and fimilarly tempered. Scholium. After an organ had been well tuned by making all the tempered v*^' as equally harmo- nious as the ear could determine, I found that the numbers of their beats, made in equal times, were inverfely proportional to the times of {g) See Dem. Prop. viii. towards the end « And *' Univerfally, &c. {h) Prop. VIII. Cor. i, 2. (/) Ibid. Cor. 2. 112 HARMONICS. Sea. VI. of the iingle vibrations of their bafes or trebles, as nearly as could be expected : or that the times between their fucceffive beats, which are equal to the periods of their leaft imper- fections (/('), were diredtly proportional to thofe homologous vibrations, or to equimultiples of them, or to the lengths of the fhort cycles, which therefore were equally numerous in thofe periods. PROPOSITION xm. ImperfeSi confonances of all forts are "equally harmoniousy in their kindy when their Jhort cycles are equally numerous in the periods of their im- perfe&ions. PI. XII. Fig. 34. The times of the fingle vibrations of imperfed: unifons being repre- fented hy AB and ab, let AD and a Cj that is ^AB and 2a b be thofe of imperfect v'K And one length of their imperfed: fhort cycle being 2 A D = AG^ and the other being '}^ac = agi their difference G^ is the difloca- tion of the pulfes G, g at the end of the firfl fhort cycle AagG, meafured from the coincident pulfes Aa. And the greater of the two difloca- tions which terminate the feveral fucceeding cy- cles, is double, triple, &c of Gg (/). Again, (/J) Pro. X. (/) Prop. VII. Prop. XIII. HARMONICS. 113 Again, conceiving the pulfes f, g^ /, &c, to be now intermitted, let AD and a e^ that is 2,A.B and 4^^ be the fingle vibrations of im- perfedl ^^^\ And the two lengths of their firfl ihort cycle ANna being ^AD^=AN and 3 a e=a 77, their difference Nn is the difloca- tion of the pulfes N, n at the end of that cycle ; and in the feveral fucceeding cycles the greater of the two diflocations is double, triple, &c of Nt7, And the common period AZ or /? » of thole diflocations or imperfediions in the fhort cycles of the v'^^ and 4^^', is the fame as the period or fimple cycle of the pulfes of the vibrations AB^ ab oi the imperfect unifons (m). Now the two diflocations G g^ Nn^ in the iirft imperfe(fl cycles of the v**"^ and 4'^' in that period, are in the ratio oi AG to AN {n), the lengths of the cycles, that is of 2A D to 4. AD, or I to 2 : and the two greater diflocations Xj/y Qr, in the lafl: imperfedl cycles Xy g A, QVgA, in the fame period AZ, are in the ratio of their diflances Z X, Z Qj_ from this end of it : and this ratio is lefs than that of AX to A Q^or i to 2. But the two greater diflocations i^A, IIo- in the fubfequent cycles K A g A, n 0- g A, of the next period, are in the ratio of Z i^ to Z IT, which, on the con- trary, is greater than that of A iC to A n, or I to 2. H The (w) Prop. VIII, («) Prop, vii. ^ 114 HARMONICS. Sea.VI. The periods muft be conceived to contain a much greater number of Ihort cycles than can be well reprefented in a fcheme. And then, as the correfponding dillocations in the v*^^ and 4*^*' lie farther and farther from Z, the ratio of their diftances and magnitudes will approach nearer and nearer to i to 2. Therefore i to 2, or the ratio of the lengths of the fhort cycles of the v*''' and /\}^\ is either the exadt or the mean ratio both of the greater and the lefler dillocations in all their coixelpond- ing fliort cycles : becaufe the leller of the increaling diflocations in any fubfequent cycle, is the fame as the greater in the antecedent one. Now while the length ^G or ag remains unaltered, imagine the diflocation Gg of the •yths (Q ]^Q increafed in that ratio of i to 2, and then it will be equal to the former magnitude of the dillocation N?i of the 4'^", or to N ?t in Fig. 35, fuppofing the pulfes C, G, Z/, &c to be abfent. And the firft diflocation B b oi the pulfes B^ by of the imperfe(St unifons, being at the fame time increafed in the fame ratio, their period AZy which is alfo that of the difloca- tions in the v'^'(o), will be diminiflied very nearly in that ratio inverted (/>). And thus the pre- fent period of the imperfed; v^^* and the for- mer period of the 4^^^ are in the ratio of the lengths of their fliort cycles 3 wliich there- fore (c/) Prop. VIII [p) Cor. 7. Lemma to Prop. ix. Prop.Xni. HARMONICS. 115 fore are equally numerous in their reipedlive pe- riods. And fince the greater and lefler diflocations at the ends of the correlponding fubfequent fhort cycles of the v^^ and 4'^', are now refpec- tively equal, either exadly or at a medium of one with another, and equally numerous too, the whole periods compofed of thcfe fhort cy- cles, will be equally harmonious. Becaufe thole equal diflocations of the pulfes in the correfpond- ing fhort cycles, are the caufes that fpoil their harmony : and caufes conflantly equal will have equal effedls. The conclufion will be the fame if the diflo- cation Nn^ in the firfl cycle of the 4'*^= in ei- ther figure, be contracted to the magnitude of the diflocation Gg belonging to the v'^' in the other. For then the new period of the 4**'% being double of the old one \q), will be to the old one, or that of the v^^% as AN to AG, that is, in the ratio of the lengths of their fhort cycles, which therefore are equally numerous in thefe periods : and the diflocations at the ends of the feveral fubfequent fhort cycles of the 4'^% being likewife contrad:ed to the refpedtive mag- nitudes of thofe of the v^^\ the confonances are again made equally harmonious. And laftly, fince either of thofe confonances is equally harmonious to anodier of the lame name, at any other pitch, when their fhort H 2 cycles {q) Cor. 7. Lemma to Prop, ix. ii6 HARMONICS. Sea.VI. cycles are equally numerous in their periods (r), it appears that 4^^' and v'^' are equally harmo- nious at any pitches, when their fhort cycles are equally numerous in their periods. And the like proof is plainly applicable to any other cale of thefe or any other confonances : I mean when the common period of the imperfed: uni- fons is terminated at firft either by coincident pulfes or periodical points ; as will plainly ap- pear by conceiving a fhort cycle or two to refult from a proper intermiflion of the pulfes of im- perfed: unilbns on each lide of fuch points in iig. 24, 25. Q^. D. Coroll. I. Imperfect confonances are more harmonious in the fame order as tlieir fhort cycles are more numerous in the periods of their imperfedions. For if any two imperfect confonances be fup- pofed equally harmonious, their fliort cycles will be equally numerous in their periods, by the propofition. Then if either of the given periods be lengthened, the fhort cycles will be more numerous in it, and the leaft diflocation of their pulfes being fmaller than before, and the greateft much the fame (j), the diflocations will firfl increafe and then decreafe by fmaller and more degrees from one end of the period to the other. And thus the confonance will be more harmonious than it was at firfl, or than the other j^iven confonance. And (>) Piop, xir. (.») Prop. VII and viii. Prop.XIII. HARMONICS. 117 And on the contrary, if the period of either confonance be fhortened, the number of its fhort cycles will be diminifhed, and the dillocations of their pulfes will increafe and decreafe by larger and fewer degrees than before. And thus the confonance will be lefs harmonious than it was before, or than the other given con- fonance. Cor oil. 2. Imperfe6l confonances are more harmonious in the fame order, as their tempe- raments multiplied by both the terms of the ra- tios of the lingle vibrations of the correlpond- ing perfed: confonances, are fmaller j and are equally harmonious when thofe produds are equal. PI. XII. Fig. 34, 35. For the vibrations of im- perfed: unifons being A B and ^ ^^ and the terms of any perfed ratio of majority jn and 7?, ihe vi- brations of an imperfed confonance tempered /harp are ??2 AB and 7i a b, and thofe of the im- perfed confonance tempered flat are w a b and nAB; and tlie periods of tlie leaft imperfedions in both have the fame length as the period of the imperfed unifons (/) ; which length, fup- pofing A B : ab :: R : r in the leafl integers^ is ~AB'y call it p, K-—r -' Now the length of the imperfed fliort cycle of either of thofe imperfed confo- nances is ?nn AB (u) ; call it c. Then H 3 P_ c (/) Prop. viii. {u) Prop. VIII, Cor- 2. Ii8 HARMONICS. Sea.VI. '-= /- ^S X -i_ = -1 X /- = -i- by c K— r »z«AB mn R— r mnt ■' taking / = — , which being as the logarithm of the tempering ratio R \ r, or A B to ab [x) is very nearly as the temperament of both thofe confonances [y). Therefore in the fame order in which the values of - or — ^ are grreater, or the values of c mnt ^ mnt are fmaller, the correfponding confo- nances are more harmonious, by corol. i ; and are equally harmonious when the values of mnt are equal, by the prefent propofition. Coroll. 3. Confequently imperfed: confo- nances are equally harmonious when their tem- peraments have the inverfe ratio of the pro- ducts of the terms of the perfed ratios of the correlponding perfect confonances. For when the values of the product j?iny.t are equal, the values of / have the inverfe ra- tio of the values of m n, Coroll. 4. When the producfls mn of the terms of the perfed: ratios are equal, the tem- pered confonances are m.ore harmonious in the fame order as their temperaments are fmaller ; and are equally harmonious if their tempera- ments be equal. For (v) Cor. 2. Lemma to Prop, ix and Prop, viii cor. i» {y) Sed. I. Art. J I. Prop. XIII. HARMONICS. 119 For if the values oi mn or — be equal, the values of - = — ^ - are o:reater in the fame c 7nn t '-' order as thofe of - are greater, or as thofe of t are fmaller j and are equal when the values of / are equal. Cor oil.' ^, Therefore imperfecSt confonances of the fame Name are more harmonious in the fame order as their temperaments are fmaller ; and are equally harmonious when they are equal. Becaufe the terms of the perfect ratios of confonances of the fame name are the fame, and their product the fame. Coroll. 6. Imperfect confonances equally tem- pered are more harmonious in the fame order as the produds of the terms of the perfedt ra- tios belonging to the perfect confonances are fmaller J and are equally harmonious when thofe products are equal. For the values of / being fuppofed equal, thofe of - = — >; - are greater in the fame or- der as the values of - - are greater, o]- as thofe inn *-> of m n are frnaller ; and the former values are equal when the latter are fo. Coroll. J. Imperfed; confonances equally tem- pered are generally more harmonious in the fame order as they are fimpler, the pure ones chiefly excepted (2;), which are more harmo- H 4 * nious (z) Sea. III. Art. 8. I20 HARMONICS. Sea.VI. nious than fome others that are fimpler j though feparately conlidered they follow that order exactly. This will appear from the fixth corollaiy by a feries of the producls of the terms of the ra- tios in the firft column, compared with the fe- ries of numbers in the fecond column of the table in Se(5t. iii. Art. 5, fhewing the order of the limplicity of confonances. CoroIL 8. Confequently fimpler confonances will generally bear greater temperaments than the lefs fim.ple will ; or the lefs fimple ones ge- nerally ipeaking will not bear fo great tempera- ments as the fimpler will : contrary to the com- mon opinion {b). Cor oil. 9. The tempered concords in the fy- ftem of m.ean tones (<:) are not equally harmo- nious in their kinds. For by CoroU. 6, and by infpedlion of the terms of the perfect ratios annexed to the cha- raders of the concords in the iirft of the tables in {h) 06lav2e autem fiant exa£lae ; nam vel minimus oftavas defe6lus fit intolerabilis. Dechales Curfus math. Tom. IV. de Mufica, cap. xi. {b) 0£lavarum autem omnium unica eft fpecies, eaque perfe61:a ratione i ad 2 contenta. Hoc enim intervallum, propter perfcdtionem, vix aberrationem a ratione i ad 2 pati poitet, quin fimul auditus ingenti moleftia afficere- tur. Namque quo perfccSlius perceptuque facilius eft in- fcrvallum, co magis fcnfibilis fit error minimus ; minus autem fentitur exiqua aberratio in intervallis minus per- feflis. l^entamen naves Tlocorlcs inufxcs, cap. ix. fe<5l. 10. Petropoli lyic^. [c) Prop. 2. Prop. XIII. HARMONICS. 121 in the next fe(flion, it will appear, that the v*** and 4'^ and their compounds with viii*^% are more harmonious than the vi'^ and 3^^ and their compounds with equal numbers of viii*^% as being all equally tempered in that fyftem {d). Coroll. ID. The harmony of thofe concords is ftill more unequal in the Hugenian fyftem, refulting from a divifion of the od:ave into 3 1 equal intervals [e). Becaufe the common temperament of the vi*^ and 3*^ and their compounds with viii*^% which by Coroll. 4 and 9, fhould be fmallerthan that of the v*'' and 4^^ and their compounds with viii*^% to render them equally harmonious, is on the contrary fomething greater. CorolL 1 1 . Imperfedt confonances are more harmonious both as they beat flower, and as the cycles of the perfed: confonances are fliorter. For the quantities - will be greater on both accounts {f) and the harmony better (^). Coroll. 1 2 . Imperfed: confonances having the fame Bafe are more harmonious in the fame order as their Beats made in equal times and multiplied by the Minor terms of the perfed: ratios of the refpedive perfed conibnances are fmaller : and are equally harmonious when thofe produds are equal, that is, when the l^eats are inverfely as the minor terms {IS). For {d) Prop. HI. Coroll. 3. () and therefore more harmonious {q) than the harmonical mean among the fame. Q^E. D. (/.) Def. 2. CoroU. i. Sedl. vir. \) Prop. IX. coroll. 5. and Tab. i. at the end of the next Scholium. (f) Prop. IX. coroll. 4. 138 HARMONICS. Sea. VIL nexed to the fame charaders in Tab.i, according to the rule in prop. xiii. coroll. 3. 3. See whetiier all the temperaments in Tab. IV be rightly deduced from thofe ratios in Tab. ii"*, by the corollaries to prop, iv, v, vi ; and whether the numbers in the firfl: column of each table correlpond to the fame ratios and con- cords. 4. Examine whether the reciprocals in Tab. v, of the temperaments in Tab. iv be right, that is, whether the produd: of the quotient by the divifor, differs from the dividend by lefs than half the divifor. When a reciprocal is negative, as coming from a negative temperament of the iii'^ or vi''', which lies wholly out of the angle AOE^ I fubtra(5t it from o and place the re- mainder in the table inilead of the reciprocal itfelf Thus at N^ 10,— 6) 26 (= — 4. 33333 &c, which fubtrafted from o gives 5. 66667 to be transferred to Tab. ii*^ Part ii*^ and there ad- ded to the politive reciprocals, for the fake of unifoj-mity in the work j the integer 5 being only negative and the decimals .66667 affirma- tive. For 71 being any given integer, the number 71 — 4 — \ = ?i — 54-7. 5. See whether the reciprocals in Tab. v be rightly transferred into the refpedlive columns of Tab.!!"^ Parti 1'', which is readily done by means of the correfponding numbers in the iirfl: column of each table. 6. Caft up the feveral dozens of reciprocals in Tab. 11"^ Part u'', and transfer the fums to Tab. Ill and there cafl them up. 7. Tab, Prop. XVI. HARMONICS. 139 7. Tab. VI is thus deduced from Tab. iii. By the folution of the problem the fraction . — ^^ — , = G Z) in Fig. 44, is the harmonical 42. 72013 ^ mean among the temperaments Gr, Gr, &c j becaufe its reciprocal "*""' ^^°^^ is the arithmetical mean among their reciprocals, as being their fum divided by their number. The fame is to be iinderflood of all the other fractions : and as the value of the temperament E q, computed from coroll. 2. prop. XVI, comes out affirmative, by the coroll. to prop, iv, v, vr, it is part of the interval E C of the perfed; iii^, and therefore is a negative temperament of that concord, or an affirmative one of its complement to tlie oc- tave. This is the firft approximation towards the required temperament. 8. Tab. VII contains the calculation of Eq', the fecond approximation tow^ards the true tem- perament of the 1 11^, in a fyftem whofe extent is but one odlave, and is fufficiently evident from cor. I, 2, prop. XVI, and Tab. vi. And by a like calculation the values of E q\ in a fyflem of two and of three odtaves, will be found as put down under thofe of £ ^ in Tab. vi ; care being taken in the operations to continue the quotients in decimals as far as they are juft. 9. Therefore the refult of the whole is this. As all the parts of mulical compofitions in any given place (fetting afide double bafes) are ge- nerally contained within three o6taves, and as their harmony is ftronger and better within that compafs , I40 HARMONICS. Sed. VII. eompafs than it would be in a larger ; I chufe to make all the concords within eveiy three oc- taves equally harmonious and no more, be the extent of the fyftein ever fo great ; and confe- quently to diminifli the iii'^ by - comma, tliis being very nearly the value of the laft Eq = 0. 1 1024 in Tab. vi. 10. Hence in tlie iyflem of equal harmony the temperaments of the v^^, vi'^ and iii^ are ~^> "^"l and ~^- of a comma reipeclively (r) and are proportional to the muiical primes 5, 3 and 2. (5) 1 1 . In determining thefe temperaments of the diatonic fyflem, I have regarded no more confonances than the concords, i . Becaufe the difcords are feldomer ufed than the concords. 2. Becaufe the ear is generally lefs critical in the difcords than in the concords. 3 . Becaufe a mean temperament among thofe of the concords and difcords too, would differ from that of the concords alone, and therefore be lefs fuitable to them. 12. Laftly I have kept the odave perfect, 1. Becaufe it is the limpleft and moft harmo- nious {r) Prop. III. and its 2^ and 3d coroll. {s) But if any one chufes to have all the concords in 4. odiaves made equally harmonious, he will find by con- tinuing the tables, that the iii^ muft be diminiflied by Q87 of a comma, which being; nearly t't^ comma, the temperaments of the vth, wth and iiid will then be — , — and — of a comma refpectively. 40' 40 40 r y Prop. XVI. HARMONICS. 141 nious of all the concords, both in itfelf and its multiples. 2. Becaufe feme one interval muil: be kept perfect, in order to determine the variations of the temperaments of the reft (f). 3. Be- caufe upon fcveral trials of keeping other inter- vals perfedl inftead of the o6tave, many reafons have occurred to me for rejeding eveiy one of them. 13. Does it not follow then, that the fyftem of equal harmony, as above derived from the beft fyftem of perfed: intei-vals (u), is the beft tempered and moft harmonious fyftem that the nature of founds is capable of? {x). 14. It may not be amifs to obferve that in Fig. 44, 45, 46, Ec — E^y the difference of the Aritlimetical andHarmonical mean tempera- ments of the 1 1 1*^, computed for one odtave is — , for two is — ^ , for three is —,- of a comma. oH ' ,69 Hence in 3 octaves the arithmetical and harmo- nical mean temperaments of the v'*"' are as 76 to yy very nearly, and if the bafes of any v* in each lyftem be unilbns, their beats made in equal times are alfo as yG to yy (y) : wlience I judge that the harmony of the founds in the two fyftiems can fcarce be fenfibly different (z) . Neverthelefs it appears by the demonftration of the proportion, that an accurate folution of it required the hdp of Harmonical Means. (t) Prop. in. (u) Sea. iv. Art. 7. (x) See Scholium. Prop. in. [y) Prop.-xi. cor. 4. (z) Prop. XI. fciiol. I. art. 4. TAB, J42 HARMONICS. Sed. VII. TAB. 11 . PART I. N° Kacios of equal the temperaments for harmony of the 1 2 3 I V ana V 4"' 4.h VI as 3' VI 3^ 5 •• 2 5 ' ^ 5 ' 4 5 • 2 4 5 I 4 V V 4* 4'" III 6th III 6th lo : 3 20 : 3 5 • 3 lo : 3 6 7 I 6 VI VI 3^ III 6th III 6th 4 : 3 8 : 3 2 : 3 4 : 3 In one Odave I dozen 2 8 I 2 V and V 4* VI -hviii 3,*^ +VI1I VI +VIII 3*^ +viir 5 • I 10 : I 5 ' 2 5 : I I 9 10 5 V V 4.H 4"" III + VIII 6th +VIII III + VIII 6'^ + viii 5 • 3 40 •• 3 5 •' 6 20 : 3 I II 12 7 VI VI 3^ 3^ III +VIII 6^^ +VIII III + VIII 6'^ +VIII 2 : 3 16 : 3 I • 3 8:3 2 dozen TABLE I. facing p. 143, Contains the charafters and terms of the perfect ratios of all the concords. I'' Parcel. 2^ Parcel. 3^ Parcel. 2 ^- 1 VI. 3 5 -f 4*1 3^ 5 6 6-.| V + VIII. - 3 VI + VIII.-^ 10 III 4- VIII. - 4*+ VIII. 1 3^ + VIII.^ 6'^+ VIII. ~ V +2VIII. ^ VI + 2VIII.— 20 III +2VIII. I 5 4^^+ 2 VI 11.-^ 3^ + 2VIII. -^ 24 6^^+2VIII. -^ 32 V +3Viii.f^ VI + 3vni.^ I!I+3VIII.^^ 4*h+3Viii.l + 3Vin.| ^th+^vill.^ &c. &c. &c. &c. &c. 8cc. Prop. XVI. HARMONICS. TAB. II. PART II. H3 N° Reciprocals of the temperaments of the v,4th5.-Comp. 1 vi.3d& Comp. | iii,6th&Comp. I 2 3 I 3. 40000 3. 20000 3. 80000 3. 40000 8. 50000 16. 00000 4.75000 8. 50000 5. 66667 4. 00000 19. 00000 5. 66667 4 5 I 4 3. 70000 3. 85000 3. 40000 3. 70000 5.2^571 4.52941 8. 50000 5-28571 12.33333 25. 66667 5. 66667 12.33333 6 7 I 6 3-57H3 3.72727 3. 40000 3- 57143 6. 25000 5. 12500 8. 50000 6. 25000 8-33333 13. 66667 5. 66667 8-33333 Sums 42.72013 87.47583 126.33334 2 8 I 2 3. 20000 3. lOOOO 3. 40000 3. 20000 16. 00000 31. 00000 8. 50000 16. 00000 4. 00000 3-44444 5. 66667 4. 00000 I 9 lO 5 3. 40000 3.92500 5. 20000 3. 85000 8. 50000 4.24324 2. 36364 4.52941 5. 66667 52.33333 5. 66667 25. 66667 I II 12 7 3. 40000 3. 842 II 3. 25000 3.72727 8. 50000 4.56250 13. 00000 5. 12500 5. 66667 24-33333 4-33333 13. 66667 Sums 43- 49438 122.32379 144.44445 144 HARMONICS. Sed.Vlt TAB. II . PART I. N° Ratios of the temperaments for equal harmony of the 2 8 3 V 4-viii V 4-viii ^th 4.VIII 4'^ +VHI VI VI 3^ 5 • I 10 : I 5 '- 8 5 • 4 5 9 10 I v -f-viii V 4-viil 4**^ 4- VIII 4''^ 4-VIII III III 5th 20 : 3 40 : V 5 : 6 5 ' 3 1 6 12 I VI 4-VIII VI 4-VIII 3<^ 4-VIII 3"^ 4-VIII III 5th III 5th 2 ' 3 4 • 3 1 : 3 2 : 3 3 dozen 8 14 3 I V 4-VIII V 4-VIII 4''' 4-VIII 4^^^ 4-VIII VI 4- VIII 3"^ 4-VIII VI 4-VIII 3"^ 4-VIII 10 20 ' 5 4 I I • 4 2 4 16 4 V 4-VIII V 4-VIII ^th 4-VIII 4*^ 4-VIII III 4-VIII 6^^ 4-VIII III -hviii 6^^ 4-VIII ID 80 5 10 3 • 3 : 12 S 12 7 17 6 VI 4-VIII VI 4-viii 3*^ -fviii 3"^ -f viii III 4-VIII 6^^^ 4-VIII III 4-VIII 6'^ +VI11 I 8 I 4 ' 3 • 3 : 6 3 In two Odtaves 4 do zen Pfop.XVL HARM O N I C S. TAB. 11. PART II. H5 Reciprocals of the temperaments of the N* v,4th&Comp. 1 VI, 3d &Comp, 1 iu,6th&:Comp. 2 8 3 3. 20000 3. lOOOO 4. 60000 3. 80000 16. ocooo 31. oocoo 2. 87500 4. 75000 4 3 8 19 . ooooo • 44444 •33333 ooooc 5 9 lO I 3. 85000 3.92500 5. 2000c 3. 40000 4.52941 4.24324 2.36364 8. 50000 25 52 5 5- 66667 33333 6666y 66667 I 6 12 I 3. 40000 3- 57143 3. 25000 3. 40000 8. 50000 6. 25000 13. ooooo 8. 50000 c. 8 4. 5- 66667 33333 33333 66667 Sums 44. 69643 1 10. 51 129 122. 1 1 1 1 1 8 H 3 I 3. lOOCO 3.05000 3. 80000 3. 40000 3 I. oocoo 61. ooooo 4. 75000 8. 50000 3 3 19. - 5 44444 21053 ooooo 66667 4 16 4 3. 70000 3.96250 6. 40000 3. 70000 5.28571 4. 1 1688 1.88235 5-28571 12. 105 12 33333 66667 33333 33333 12 7 17 6 3. 25000 3.72727 3. 14286 3-57^^3 13. ooooo 5. 12500 22. OOOOO 6. 2^000 4- ^3 3 8 33333 66667 66667 33333 Sums 44. 80406 168. 19565 188 . 98830 K 14^ HARMONICS. Sea. VII. TAB. 11. PART I. N° Ratios of the temperaments for equal harmony of the 8 7 VI ~i~2VIII 10 : I H V 3*^ +2VIII 20 : I 2 r VI 4-2VIII 5 • I 8 4.^ 3^^ + 2VIII 10 : I lO V III 4- 2VIII 5 '- 6 M V 6^'^ -i- 2VIII 80 : 3 i6 4- III + 2VIII 5 : 12 9 4.1. 6-'^ 4- 2VIII 40 - 3 12 VI III 4- 2ViII I • 3 i8 VI 6^^ 4-2VIII 32 : 3 1 1 3^ III 4- 2VIII 6th j^ 2 VII I I : 6 16 : 3 5 dozen H V -i- VIII Vi 4- 27111 20 1 19 V -1- V I 1 1 3"^ -r2VIII 4.0 I I 4'^ + VIII VI 4- 2VIiI 5 : 2 2 I 4.'. + VIII -^^ -f2VIII 5 : I V 4- VI 11 III 4- 2VilI 5 3 20 V 4- VIII 6^^ 4- 2VIII 160 21 4.H + VIIJ III 4-2vni 5 24 5 4'^ -h VIII 6'^ 4-2VIII 20 3 17 VI -i- VIII III 4-2VIII I . 6 1 1 VI 4- VIII 0^^ 4- 2VIII 16 3 22 -i-VIII III 4- 2VIII I 12 7 3^ + V I i I 5th _j_ 2VIII 8 3 6 dozen 1 Prop. XVI. HARMONICS. TAB. IT. PART II. H7 N° Reciprocals of the temperaments of the v,4th&Comp. |vi,3d&Comp. | in,6th^^ Comp. 8 14 2 8 10 15 16 9 3. lOOOO 3. O5OCO 3. 20000 3. lOOOO 31. ooooc 61. ooooo 16. ooooo 31. ooooo 3.44444 3.21053 4. ooooo 3-44444 5. 66667 105. 66667 !>• 33333 52.-33333 5. 20000 3.96250 6, 40000 3.92500 2. 36364 4. II688 1.88235 4. 24324 12 18 17 1 1 3. 25000 3.91429 3. 14286 3. 84211 13. ooooo 4. 28125 22. OOOOO 4.56250 4-33333 45. 66667 3. 66667 24-33333 Sums 46. 08676 195.44986 243. 09941 H 19 I 2 3. 05000 3. 02500 3, 40000 3. 20000 61. ooooo 121. ooooo 8. 50000 16. ooooo 3.21053 3. 10256 5. 6666'j 4. ocooo I 20 21 5 3. 40000 3.98125 8. 80000 3. 85000 8. 50000 4.05732 I- 51724 4.52941 5. 66667 212.33333 2. ihhhj 25. 66667 17 1 1 22 7 Sums 3. 14286 3. 8421 1 3.07692 3.72727 22. ooooo 4.56250 40. ooooo 5- 12500 3. bbbby 24-33333 3-33333 13. 6G()6j 46.49541 296.79147 302. 813J0 K 148 HARMONICS. Sea. VII. TAB. 11. PART I. Ratios of the temperaments for N° equal harmony of the I 2 23 13 4 5 16 10 12 1 17 12 V -1-2VIII V 4-2VIII 4'^ -h 2VIII 4^'^ +2 VI 1 1 VI VI 3^ 5 • 5 • 5 • 5 • 2 I 16 8 V -1-2VI1I V 4-2VIII 4th _^ 2VIII 4_th _|_ 2VIII III 5th III 5ch 10 : 20 : 5 • 5 •• 3 3 12 6 VI -|-2vni VI +2VIII 3^ +2VIII 3"^ -I-2VIII III Ill 5th 1 ; 2 : I : I : 3 3 6 3 7 dozen 2 8 13 3 V +2v'lII V 4-2VIII 4'"^ + 2VIII 4'^ -1- 2VIII VI -j-VIII 3'* H-viii VI ^vjii 2^ -l-viii 5 • 10 : 5 • 5 • I I 8 4 I 9 21 I V +2VIII V 4- 2VIII j_th +2VIII ^|th _^ 2 VI 1 1 III 4- VIII 6^^' -hviii III 4-viii 6^^ -hviii 5 • 40 : 5 • 5 ' 3 3 24 3 17 6 22 I VI -1-2 VI 1 1 VI 4-2VIII 3^^ 4-2VIII 3^ 4-2VIII III 4- VI 1 1 6^'' 4- VIII III -j-viii 6^** 4- VI II I : 4 • 1 : 2 : 6 3 12 8 dozen Prop. XVI. HARMONICS. TAB. II. PART II. 149 N° Reciprocals ( V, 4th &Comp. jf the temperaments of the i VI, 3d&Comp. I iii,6th5fComp. I 2 23 13 4 5 16 10 3. 40000 3. 200G0 6. 20000 4. 60000 8. 50000 16. OOOOO 1.93750 2. 87500 5. 66667 4. OOOOO 3. 18182 8-33333 3. 70000 3. 85000 6. 40000 5. 20000 5.28571 4.52941 1.88235 2.36364 29. 66667 5.66667 12 I 12 3. 25000 3. 40000 3. 14286 3.25000 13. OOOOO 8. 50000 22. OOOOO 13. COOOO ■ 4.33333 5. 66667 3.66667 4.33333 Sums 49.59286 99. 87361 48. 18182 2 8 13 3 3. 20000 3. ICOOO 4. 60000 3. 80OCO lb. OOOOO 31. OOOOO 2. 87500 4. 750CO 4. OOOOO 3^44444 19. occoo I 9 21 I 3. 40000 3.92500 8. 80000 3. 40000 8. 50C00 4. 24324 1. 51724 8. 50c 00 5. 6666j 52.33333 2. 16667 5. 66667 17 6 22 I 3. 14286 3.07692 3. 40000 22. 0000© 6. 25000 40. OOOOO 8. 50000 3. 66667 ^•33333 3-33333 5. 66667 Sums 47.4:621 154. 13548 loi. 61 1 I I K ^JO HARMONICS. Sed.Vir. TAB- II. PART I. N° Ratios of the teinpcraments for equal harmony of the 8 14 3 , I 10 15 24 4 V 4- 2VI1I V 4- 2VIII ^th 4. 2 VI 1 1 ^th _|_ 2VIII VI 4- 2VIII 3d 4-2VIII VI 4-2VIII 3^^ 4- 2VIII 10 : 20 : 5 • 5 • I I 4 2 V 4- 2VIII V 4- 2VIII ^th -j- 2 VI 1 1 4'^ -f- 2VIII III 4-2VIII 6th -j_2VIII III 4- 2 VII I 6'*^ 4- 2VIII 5 • 80 : 5 • 10 : 6 3 48 3 22 7 25 6 VI 4- 2VIII VI 4- 2VIII 3^^ 4- 2VIII 3*^ 4-2VIII III 4- 2VIII 6'^ 4- 2VIII III 4- 2VIII 6'^^ 4- 2VIII I : 8 : I : 4 • 12 3 24 3 In three Odaves 9 dozen TAB. m. The numbers and fums of the Dozen | N8 | v, 4th & Comp. In I VIII^^ i^' 4'^ 12 12 12 12 42. 72013 43-49438 44. 69643 44. 80406 In 2 VII r". 6th /yth 48 12 12 12 12 12 175.71500 46. 08676 46.49541 49.59286 47. 41621 53. 22812 In 3 VI I r^*. 108 418.53436 > r n o :3 ft N V &c facing p. 150. VI ilprocals of all the different temperaments ^ Odaves. &c S"^ ^ Comp. ft) zL W fD •-1 <^3 I <^ en tr - 3 " s ^^ p p 3 ^ gcJQ ^< O n O n O 13- o 5 17 I 16 4 10 19 7 20 37 17 77 4 25 1 1 4 41 t In one Odaf 10 40 31 37 10 1 1 12 ^57 II 19 26 16 73 I 13 23 8. 50000 16. 00000 4. 75000 5.28571 4. 52941 6. 25000 5. 12500 Oaave 31. 00000 4.24324 2.36364 4.56250 13. 00000 2. 87500 61. 00000 4. 11688 1.88235 22. 00000 Octaves 4. 28125 121. 00000 4.05732 I. 51724 40. 00000 1.93750 I. 28302 76. 00000 O Slaves III, 6''' & Comp. 9 3 . 6 3 3 ■ 3 19 3 •12 6 3 39 3 .24 12 -II -48 24 ^7 16 19 37 77 25 AI 31 ^57 26 73 13 23 61 317 32 22 137 121 637 44 40 31 68 76 5. 66667 4. 00000 19. 00000 12.33333 25. 66667 8-33333 13. 66667 3 52 5 24 4 8 3 3 3 44444 33333 66667 33333 33333 33333 21053 66667 33333 66667 45. 66667 3.10256 212-33333 2. 16667 3-33333 3. 18182 2.58333 3. 16667 facing p. 150. Prop. XVI. HARMONICS. TAB. II. PART IT. 151 N° Reciprocals of the temperaments of the v,4th^cComp. I vi,3d&Comp. j iii,6th,SrComp. 8 14 3 I 3. lOOCO 3. 05000 3. 80000 3. 40000 31. ocooo 61. OOOOQ 4.75000 8. 500C0 3- 44444 3.21053 19. 00000 5. 66667 !0 '5 24 4 5. 20000 3.96250 13. 60000 3. 70000 2.36364 4. 1 1688 I. 28302 5.28571 5. 60667 IC5. 66667 2-58333 12.33333 22 7 25 6 3.07692 3.72727 3. 04000 3'57H3 40. 00000 5. 12500 76. 0000c 6. 25000 3-33333 13. 66667 3. 16667 ^'33333 Sums ^2- 22812 24^^.67425 172. 07164 1 TAB. III. reciprocals of the temperaments VI, 3d&Comp. 1 III, othi' Comp. 87.47583 122.32379 1 10. 51 129 168.19565 488.50656 195.44986 296.79147 99.87361 154.13548 245.67425 1480. 43123 ^26. 33334 144.44445 122. I I I II 188.98830 581. 87720 243.09941 302. 813IO 48.18182 lOI. 61 1 I I 172. 07164 1449.65428 152 HARMONICS. Sea.VII. TAB. VI. ^he values ofEqandEc( in Fig. 45 and 46, iviv^ds the temperament of the 11 1^, for equally and the moft harmonious. In I Od:ave. {^) 12 42. 72013 12 87-47583 = 0. 2808980 = GZ)= d = o. 1371808 =AH= h 12 -- =0.0949868 = EM^=m In 2 Oflaves. = o. 27^1606 = GD = d 175.71500 /^ y 488^50656 = ""' 0982587 =AH=b ^^-^^=o.o^2^gi6 = EM =m In 3 Odaves. — 2 = o- 258043 3 = GD = d 41^-53436 "^-^ — = O.0J2Q Kiy == AH = h 1480. 43123 / yj / ; ^7—75- = 0.074 coo c = EM=m 1449. 65428 z^-' -^ {a) See the laft Table, Prop.XVI. HARMONICS. 153 TAB. VI. being the firft and fecond approximations to^ making all the concords in i , 2 or 3 oBaves Hence Ef =4^—1 = o. 1235920 Ei = ^^ = o. 1504256 EM== m =0.0949868 3 ) o. 3690044 Eq = 0. 1230015 E^'= o. 122233 . In I Odlave, ^ J , Q See Tab, vii. Ef = 4« — I = o. 0926784 El = =0.2023217 EM== m =0.0824916 3)0-3774917 Eq = o. 1258306 Eq = o. iz^yig. in 2 Odtaves. E/ = /\.d — I 0. 0321732 Ei = I — 4^ 3 : 0. 2360634 £/«= m 0. 0745005 3 ) o- 3427371 Eq= o. 1 142457 £ / = 0. 1 1024 . . in 3 Odaves. 154 HARMONICS. Sed. VII. TAB. VII. The computation ofY^o^ in Fig. 46, being the ment of the 11 i^,for making all the concords 3. 56001 1 I I 42. 72013 _ ^=7=F =0.8628192= 3-476974 I 4 __ 4 __ r Gk ~ I \m~ 1.0949868 3-05^913 3) 10.689898 Arith. mean 3. 563299 GD'=d'== -^ = 0.2806^1 3.563299 -^ I I 87.47583 o , --— = = — == 6. '> ryo-J I Ae I — 3« o. 1573060 *^-'' -^ I 4 4 __ , Al I — d,w 0.7150396 5* 494 9 3 ) 19-240780 Arith. mean 6.41^593 AH' = h'=^-. ^ = 0.15)919 Prop. XVI. HARMONICS. 155 TAB. VII. Jecond approximation towards the tempera- in I oBa'ue equally and the moji harmonious, 1 I 126. 2^222 = 8.091138 Ef 4^—1 o. 1235920 -i.=-X_^ = ^= 6.647805 El I — 4^ 0.4512768 ^ -^ 3 ) 25. 266720 Arith. mean 8. 422240 EM' =^771 = = 0.1187^1 8. 422240 ' -^"^ Hence E/' =4/ — i = o. 122524 ^ ., I — 4^' El = — = o. 125441 Em'= HI = o. 118733 3) 0.366698 Eq' = o. 122233 See the values of Eq' in 2 and 3 oilaves in Tab. vi, part 2^, PRO- 156 HARMONICS. Se(fl.Vn. PROPOSITION XVII. Afyjlem of commenfur able intervals de^ duced fro7n dividing the oBave into 50 equal parts y and takiitg the Urn- ma L = ^ of them^ the tone T= 8 a7id confequently the lejfer 1^^ L -\-T = 13, the greater iii^ 2T= 16, the 4^ L^2r=^2iy the v'^ L + ^T = 29, &^Cy according to the table of elements {a)y will differ infenfbly from the fy ft em of equal harmony : I mean with regard to the harmony of the refpe&ive confonances in both. For fince L = 5, T = 8 and the iii'^ 2 T = 1 6 and the vi 1 1 = 5 T -h 2 L = 50, we have the 1 11"^ 2T : VIII : : 8 : 25; whence the iii"^ _ Q Q O lT ■= — VIII = — loo;. 2=— XO.'20I02. 25 25 o 25 •-' g()g^j = o. 09632. 95962, which fubtra(5ted from tlie perfed 111^^ = log. - = o. 09691. 001 30,leaves the temperament 0.00058.04168, which is to the comma <:=log. — = o. 00539. 50319 as 4 to 37 very nearly {b). Hence the tem- {a) Prop. III. {b) See an example of the like redudlion in the next Scholium. Prop. XVII. HARMONICS. 157 temperament of the 11 1** is — - r, and thofe of the v'*' and vi*'^ as in Tab. i, by prop. iii. cor. I. 2. 3. TABLE I. T:L::8:5 viii= 50 The fyftem of equal har- Ratios of the temperaments and of the beats made in any mony. given time (c). I I ^ ""4 ~37^ I I V c -c 4 36 J + r7^; + F6-369:370 1 I 3 4 37 4 36 I 31 3 4""37'4'"36'' ^^'' ^"^ 4 III — —c 37 4 III — -^c 36 -'- \''' 36: 37 37 36 ^ ^' Now though the concords of the fame name in this fyftem and that of equal harmony are not exadlly equally harmonious (^), and though the difference of an unit in the largeft number of beats made in a given time may be diftin- guifhed by counting them ; yet if the numbers be not fmaller than thofe in the table, the dif- ference in the harmony of the concords will be deemed infeniible by proper judges ; which are thofe only tliat have carefully attended to the beats of concords in tuning inftruments. But any one elle may be fatisfied experimentally, by caufing (f) Prop. XI. corol]. 4. {d) Prop. XIII. coroll. 5. ^ 158 HARMONICS. Sea. VII, caufing two concords to the fame bafe to beat as in the table. Q^. D. Scholium, In like manner if T= 5 and L= 3, then the odlave 5T+2Lis=3i and the temperament of this lyflem, which Hugenius has adopted (e), will be found as in the third column of the next table. TABLE II. T:L: VIII 2 : I 12 I I 3 V— - c-\- ~c 4 , 19 4 19 III , 12 19 T:L::3:2 VIII = 19 I 3 V c c 4 35 I I 9 VI 4- -c —c 4 35 III 12 35 T : L : : 5 : 3 VIII = 31 V c-\- — c 4 no VI 4- -c-\- ~c 4 no III no On the contrary, if from the given tempera- ment of a fyftem it be required to find the ratio of T to L, we may proceed as follows. Let it be propofed to approximate to tlie fyftem of equal harmony, where 2 T = in c [f) ; then [e) Cyclus harmonicus, at the end of his Works, or Hiftoire des Ouviages des S^avans, OcSlob. 1691, pag. 78. (/) Prop. XVI. Scholium 2, Art. 9. Prop. XVII. HARMONICS. 159 then fince 5T + 2 L = viii, we have 2 L = (viii — 5T=) VIII — -XIII — -Cj whence T:L:: iii — -c:viii — ^xiii — - c. 929 To find this ratio, we have the in = log. - = o. 09691. 00130 and the comma ^=log. — =0.00539. 50319 and - c=o. 00059. 94480* Whence 2T = iii — - c= o. 09631. 05650 and ^ X III — - c=o. 24077. 64125 and the 2 9 VIII =log. 2 = 0. 30102. 99957 ^^^ 2 L = VIII — -XIII — -(7=0.06025.35832, and 29 laftly T:L: : 963105650 : 602535832. Now the quotients of the greater term of this ratio divided by the lefler and of the lefler di- vided by the remainder and of the former re- mainder by the latter &c, are i, i, i, 2, 24, &c. Whence the ratios greater than the true one are 2 to I, 5 to 3, 8 to 5, &c, and the lefler are 3 to 2, II to 7, &c (g). Hence taking T to L fucceflively in thofe ra- tios, by the method ufed in the demonftration of the propofition, the temperaments of the ap- proximating rational fyftems will be found as in the tables. By which we fee how much and which (g) See Mr. G/tv's Harmonia Menfurarum, Schol. 3. prop. I. i6o HARMONICS. Sed. VIIL which way they differ from that of mean tones, as well as from that of equal harmony in Table i. SECTION VIIL The fcale of mujical founds is fully ex- plained and made changeable upon the harpfichord^ in order to play all the flat and Jharp founds^ that are ufed in any piece of mufic^ upon no other keys than thofe in common ufe, DEFINITIONS. I. The interval of a perfect odlave being di- vided, in any tempered fyftem, into 5 equal tones and 2 equal limmas [h)^ the excefs of the tone above the limma is called a Minor limma. II. The difference of the major and minor lim- ma is called a Diefis. III. If the difference of the intervals of two confonances to the fame bafe be a dielis, I fhall call either of them a Falfe confonance when ever, in playing on the organ or harpfichord, it is fub- flituted for the other which ought to be ufed -, as it often is for want of a complete fcale of founds in thofe inftruments. IV. The notes A-^^ B^, &c, fignify founds which are fliarper, and A^^ B^, &c, founds which {}}) See feci, iv art. 3, or the dem. of prop. 2, or prop. 3. ArM. HARMONICS. i6i which are flatter by a minor limma than the re- Ipedive primary founds^, B, &c : And ^**, Ji^^j &c, lignify founds whofe diftance from ^ is double the diftance oi A^ ov A^ from A and alike fituated. 1. PI. XVIII. Fig. 48 or 49. The interval of a perfedt o6tave being rcprefented by the circum- ference of any circle (/) and fuppofed to be di- vided by the founds A, By C, £), £, F, G into 5 tones and 2 limmas, towards the acuter founds take tlie interval AA^ equal to the minor lim- ma AB — J5C, and towards the graver take AA^ equal to AA^, and when the like flat and Iharp founds are placed at that diftance on each fide of the other primary founds J5, C, Z), E, F, G, every tone will be divided by a flat or a fharp found into a major and a minor limma, and by both into tv/o minor limmas with a diefis between them J and each primary limma, BC^ EFj will be divided by a flat or a fharp found into a mi- nor limma and a diefis, and by both into two diefes with an interval between them. 2. Fig. 48. In the Hugcnimi iyfi:em the odtave is divided into 3 1 equal parts, of which the tone is 5, the major limma 3, the minor 2 and thq diefis I {k). Fig. 49. In the fyfiem of Equal Harmony the odtave is divided into 50 equal parts, of which the tone is 8, the major limma 5, the minor 3 and the diefis 2 (/). Therefore the former tone is to the latter as L 5 31 (/) Sea. IV. art. 7. (i) Prop, xvii, fchol. (/} Prop. XVII. i62 HARMONICS. Sed.VIII. i VIII to - VIII, or as 12 c to 124, and the former dieiis is to the latter as -to -5 or 2 c to -3 1 ; and fince a quarter of a comma is about — viii (;«) 223 the former dieiis - viii contains above - , and 31 4 the latter almoft 5 of a comma. 4 3. Fig. 48 or 49. In either of thofe fyflems or any other of that kind, by going many times round the circle it v^ill appear, that in afcending from F continually by v'^^ the 7 primary notes will iiril: occur in diis order FCGDAEB, and then recur once fharpened in the fame order, and again twice (liapened &c : Likewife in defcending from jF by v*^% they will recur once flattened in that order thus inverted, B^ Rb A^ D^ G^ O F^, and again twice flattened &c : And thefc feveral cyclesjoined together make the following progref- fion afcending by v'^' ; E^b B^'o, Fb C^ Qb Db Ab Eb Bb.FCGDAEB, F'^ C^ G^ D^ A^ £* jB^, F^^ C^^ &c. 4. Hence a Table of the minor and major con- fonances to any number of Keys or bafe notes in that progreflion placed in the firfl: column (;z), is thus deduced. Oppoflte to any Key as Z) write the 1 2 trebles Eb, E, F, F^, &c of the minor and major confonances within the viii in the order of their marks, 2^ ii^ 3^ IIl^ &c at the top of the table, which trebles are found by going round the circle s then place die fame progreflion of v*^* above {m) Found by dividing the log. pf 2 by - log. — . («) Plate XIX. Prop.XVII. HARMONICS. 163 above and below the treble £^ in col. 2, as ftands above and below £^ in col. i ; and having done the like to the other trebles £, F, &c, the table is finiflied. For lince the interval D^ in col. i is equal to E^ Bl^ in col. 2, it follows that in col. i and 2 the interval AB^ equals D E^ j and the fame may be faid of the reft of the table. At the bot- tom of it the letters Z/, /, D fignify the major and minor limma and the dielis, as being the difte- rences of the inteiTals marked at the top. 5. As the organ or harplicord has but 1 2 founds in the odtave,whofe notes are F,C,G,Z),y^,£,5, widijp*, C*, G^, above, and E^, B^, below them in col. i j all the notes below E*^ in col. i and in thofe of the minor confonances, and all above G ^ in col. i and in thofe of the major confonances have no founds anfwering to them in thofe inftruments ; and are therefore excluded, or diftinguilLed from the notes that have founds, by circles round them, both in the table and in Fig. 48 and 49. Confequently when any of the excluded notes D^, A^, E^,B^, F*=», C^^, that are above G^y occur in a piece of mufic, as moft of them often do, the mufician is obliged to fubftitute for them the founds of E^ B^, F, C, G, D, re- fpedtively, which being higher by a diefis (0) make falfe confonances (p). L 2 Likewiie (0) As appears by Fig. 48, qr by the collateral notes in the columns of i\ ths and 5'hs in the table of confonances. (/>) Dtf. III. i64 HARMONICS. Sea.VIII. Likewife when any of the excluded notes AK Db, Gb, Cb, Fb, Bbb^ that are below EK occur, as fome of them often do, the mufician mufl fubflitute for them G'^, C^, JP^, 5, E, J, refpeftively, which being lower by a dielis make falfe confonances. Hence the two middlemofl; Keys Z), A have one falfe confonance in each, and the numbers of them in the fucceffive higher or lower Keys, in- creal'e in the arithmetical progreffion 2, 3, 4, 5, 6. Whence it is eafy to colled: that (tw^n twenty- fourths of the whole number of major and mi- nor confonances in the fcale of the organ or harp- fichord, are falfe j befides a larger proportion of falfe ones among the Superfluous andDiminiihed confonances hereafter mentioned. 6. The confonances to all the Keys above E have no flat notes; becaufe 5^ is the highefl flat note in every column of minor confonances, and is the higheft of all where it is the minor 5^*^ to the Key E^ Again, the confonances to all the keys below C have no iliarp notes ; becaufe F^ is the loweft fharp note in every column of ma- jor confonances, and is the lowefl: of all where it is the major iv^*' to the Key C. Therefore the confonances to thofe two and the intermediate Keys, CGDAEy have both flat and iharp notes among them- Hence it comes to pafs that the concords (q) to the 4 middlemolf keys GyD^Ay £, which are tlie {q) Seel. III. art. iith. Prop. XVII. HARMONICS. 165 the open firings of the violin, are all true, but not all the dilcords. 7. By adding a found for A^^ every one of the 6 lower keys £^ B^ F, C, G, D, will have one falfe confonance changed into a true one, as appears by infpeclion of the oblique diagonal rows of ^^ in the table. Likewife by adding ano- ther found for D^ every one of the 6 higher keys A, Ey B, F*, C"^, G'», will have one falfe confonance changed into a true one. Now in this inlarged fcale of 14 keys all the conlbnances to D and A, tlie two middlemoil, are true. And a like advantage will follow from giving founds to D^ and A^, the two next exterior keys, and fo forth. Therefore univerfally, the number of the mid- dlemoft keys to which all the minor and major confonances are true, is equal to the whole num- ber of keys or founds in the odiave diminillied by 12 ; fo that the 24 founds in col. i. would be neceffary to make all thefe confonances true in the 12 middlemoft keys. 8. But befides the major and minor confo- nances in the Table there are others in the fcale of Fig. 48 or 49, which I think are called Su- perfluous and Diminifl^ed confonances. The interval of a major confonance augmented by a minor limm^a makes the interval of a fjper- fluous confonance; and the interval of a minor confonance diminifhed by a minor limma makes the interval of a diminifhed confonance. L 1 Thus i66 HARMONICS. Sed. VIII. Thus the treble of a fuperfluous ii^, iii"^, i th B^,C^,D^,E^y refpedively; and the treble of the diminifhed 2^ 3^ 4*'', 5*^, 6'^, 7*^, &c to the key B, is C^, D^ EK F^, GK AK And the like is to be underftood in the other keys, where the trebles are often double fharp and double flat founds ; but are all omitted in the Table to avoid confulion by adding fo many notes to it. 9. I have heard of but one method of fupply- ing the organ or harpflchord with more founds in each odlave j which is by adding pipes or firings for A^^, D^, Sec, and dividing the keys of their fubfhitutes G '■^, C *, &;c, each into two keys J the longer of them for founding G^,C^^, &c as ufual, ^nd the fhorter for founding A^^D^^ &c : and by doing the like for D^, A^, &cc. But this method of fupplying the defecfls of the fcale is quite laid afide, on account of the great difficulty in playing upon fo many keys without extraordinary practice, and the following pallia- tive remedy is univerfally received. PI. XVIII. Fig. 48 or 49. The octave being al- ways divided into 5 tones and tv/o limmas ; by increailng the tones equally till each becomes double the diminiiliing limma BC or EF, the diefis, or difference between the major and mi- nor limma, will be contracted to nothing, which by Defin. iii annihilates all the falfe confonan- ces. But the harmony in this fyilem of 1 2 He- mJtones is extremely coarfe and difagreeable. For Prop. XVII. HARMONICS. 167 For the temperaments of the v*^ and 4*'', vi*'' and 3^ iii*^ and 6''' and their compounds with viii'^% are nearly -, ^ and -of a comma re- ' J 10' 10 10 ipe6llvely (r) and in the fyilem of equal har- mony they are - , ^ and - [s) ; by which fyilem, as being the moft harmonious, all other fyftems ought to be examined, as by a flandard. Now ~ being much lefs than ~ , makes the concords in the iirfl parcel {t) finer than they ought to be } and — and — being much greater than 2 and - , make the concords in the other two 6 9' parcels much coarfer than they ought to be, the two leaft of thofe temperaments being as great as thofe concords can properly bear. Now for want of another found to terminate each diefis in the fcale, it is necelTary in the tun- ing to diminifh the diefis till one found may ferve tolerably for the other, and thus to approach to- wards that inharmonious fyfi:em of 1 2 hemitones, till the harmony of the fcale becomes very coarfe before the falfe confbnances are barely to- lerable {u). L 4 9. That (?) Prop. XVII. Tab. ii^ col. i. \s) Prop. XVI. fchol. 2. art. lO and 13. (/) Prop. III. fchol. {//) This is done by fliarpening the major iiids more than the ear can well bear, which inlarges the tones and lellens the major limmas and diefes : or, becaule any 6 tones or 3 major iiids and a diefis (as A^C^CE-\. EG^ ^ G^ Ab) make up the odlave or circumfeienca in Fig. 48. i68 HARMONICS. Sefl. VIIl 9. That this is a bad expedient for fupplying the want of more founds, is farther evident from the Hugeniaji fyftem, wher-: the temperament common to the vi**^" and q^, &c being: - + — of a comma {x) is conliderably greater than it ouo^ht to be, that is, than 7 of a comma, as in the lyflem of equal harmony j and yet the Hu- genian dieiis is - of a comma (jy), which being Gonfidered as a temperament of the falfe confo- nances and being fb much greater than - + -^ of a comma mufl: needs make horrible diffonance. 10. Having therefore been long diffatisfied with the coarfenefs of the harmony even of the true confonances in the fcale of our prefent in- ftruments, which is fo defective too that not above a feventh or eighth part of the beft compolitions made fince Corelli% time, nor above a third or fourth of his can be played upon it without ufing many falfe confonances j and being flill more difgufled when thefe come into play, as they of- ten do in the remaining two thirds or three fourths o^ Corellts works, and fix fevenths orfe- ven eighths of all the refi: : I was c^lad to find out a better remedy for both thofe defects s at lead in a fcale of fmde founds. 1 1 . The ftrings of the fore unifon of the harp- fichord being tuned as ufual to the notes of the common fcale in the following lower line, let the founds [x) Prop, XVII. Tab. 2. co!. 3. {y) Se£h Viii. art. 2. Prop. XVII. HARMONICS. 169 founds of the back unifon be altered to the notes in the upper hne, each of which diiFers from the note under it by a diefis {z). G^A B^ B C C^D Eb E F F^G Now fince the jacks which flrike the firings of any of thefe couples of notes, as G'^ and A^y ftand both upon one key, by moving a ftop here- after defcribed, that key can ftrike either firing alone without founding the other : And fince both the founds in any couple are feldom or never ufed in any lingle piece of mufic, the mufician before he begins to play it, can put in, by the flop, that found which he fees mofl occafion for; and either of them being llruck by the fame key, tlie execution is always the fame as ufual. For example, if befides the founds F*, C"^, G^ in the common fcale, Z)*, A^, £^, 5*, F^^ fhould alfo occur in a piece of mufic (a) move their flops, and their firings will be flruck by the keys of E^ B^ F C G refpedively, Vv^hofe founds are ufually fubflituted for the founds required. 12. A mufician by cafling his eye over any piece of mufic, can foon fee what flat or fliarp founds are ufed in it which are not in the common fcalc J and to fave that trouble for the future, may write them down at the beginning of the piece. Now and then it may be proper to obferve whe- ther (z) As appears by Fig. 49. {a) As in Corelli's xi'*! folo and CarhmeVs ill^ 5cc. 170 HARMONICS. Sea.VIIL ther the outermoft of them in their progreflion by v^^', fhould be put in or not, left its fubftitute lliould occur oftener than the principal found it- felf. If both occur, that which recurs oftener muft be in the fcale. But as both occur very fel- dom the matter is fcarce worth notice. 13. To fliew by infpedion which are the falfe confonances in the Harpfichord after any flat or fliarp founds are put into it by the ftops j imagine the two middlemoft tranfverfe parallelograms in the Table {b) and alfo the circles furrounding the notes which are not in the common harpfichord, to be drawn with the point of a diamond upon a pane of glafs laid over them. Then if the founds of Z) ^ and A^ for inftance be put into the harpfichord, move the pane two lines higher till the uppermoft line of the two parallelograms juft takes in'thofe two notes in col. i, and in this pofltion the circles upon the pane will cover all thofe notes in the table which are not in the pre- fent fcale of the harpfichord, and point out the falfe confonances to every key. 14. Thus you fee how to make any given key as E or B, as free from falfe confonances as D or A is in the common fixed fcale ; namely by putting in by the ftops as many fliarp notes above G^ in the column of keys as fliall bring £ and B into the middle of the 1 2 keys then in inftru- ment. And the like may be done for any given key below D by putting in flat notes below £^. And {h) Plate XIX. Prop. XVII. HARMONICS. 171 And thus a mufician that tranfpofes mufic at fight can accompany a voice with the purefl and finefl harmony in the propereft key for the pitch of the voice. I fay the fineft harmony ; becaufe this changeable fcale may eafily be tuned to the moft harmonious fyftem {c) which is impradi- cable upon the common fixed fcale, becaufe the diefis would be fo large as to render the falfe con- fonances infufferably bad (d). 1 5. The famous Riickers and other muficians of a delicate ear, always valued the tone of a lingle firing for its diflindnefs and clearnefs, fpi- rit and duration, and preferred it to that of uni- fons and octaves. I mufl confefs I have long been of that opinion, even before I thought of this changeable fcale of fingle founds, which how- ever after fome years experience upon my own harpfichord has fully confirmed me in it. 16. Unifons by themfelves or with an od:avc are indeed an addition to the loudnefs of the tone, but not nearly in proportion to the number of ftrings. Firfl becaufe the opprefTion of the belly of the inflrument by the force of fo many fl^rings, hinders the facility and duration of its tremors j and fecondly becaufe in tuning unifons or odtaves, it is manifefl: that their tone is never clear, loud and flowing, like that of a fingle firing, except when they are precifely perfeft. But as this per- fedion continues but a very little time, efpecially after {c) Prop. XVI. fchol. 2. art. lo and 13. \d) Seft. VIII. art. 2, 272 HARMONICS. Sed:. VIIL after the room Is warmed by company, that clear linging tone is foon deftroyed. The compound tone of unifons by themfelves, or with an odtave, being of itfelf fo indillind:, what beating and jarring mufl refult from their compHcated mixtures in playing three or four parts of mufic? Efpecially as the imperfections of the unifons and odaves in the courfe of playing are frequently added to the temperaments of the other confonances, which if they were perfeifl could not bear thofe imperfedions fo well as the unifons and o6laves do when founded by them- felves (e). This confufed noile, like that of a dulcimer, is but too plainly perceived when the ear is held over the firings of the harplichord y and lince it refults from the multiplicity of firings, it appears that the heft way to improve thisinfhru- -ment is to find out methods for increafing the flrength and clearnefs of the tone of fingle firings. 1 7. To me, who feldom hear any other than the iingle firings of my own harplichord, die tone is as loud as I defire, not only for leifons and can- tatas but alfo concertos accompanied with inflru- ments in a large room. This indeed is more than a perfon could expert who has feldom or never attended to the tone of fingle firings except in the fnort pianos after the long continued y^r^^j- upon the full harplichord. The reafbn is that the very fame objeds affed: our fenfes very differently in different circumflances, as is very evident in at- tending to any other fenfation as well as that of founds. (e) Prop. XIII. corol!. S. Prop.XVII. HARMONICS. 173 founds. " For inftance, in coming out of a ftrong " light into a room with the window-fhutters al- " moft clofed, we immediately have a fenfation " of darknefs or a very little light, and this con- " tinues much longer than the pupil requires to " dilate and accommodate itfelf to that weak de- " gree of light, which is almojft inftantaneoufly " done. But after ftaying fome time in the fame " or a much darker place, the fame room which *' appeared dark before, will be fufficiently light." This obfervation is plainly applicable to founds, and more of them upon the other fenfes may be feen in T)v.Jm-in?> Effay on diflind: and indiftind: Vifion at the end of my Optics (/). 1 8. j^n expedient for changing the founds of any harpfechord ready made-^ where- by to experience the truth of the fore- going obfervations, PI. XXVI. The 66*'' figure reprefents the heads jf, II of two jacks {landing as ufual upon one key, with their pens pointing oppofite ways un- der the firings on each fide of them, as G* and A^^ the back unifon being raifed to A^. And abed reprefents a fmall brafs fquare of the fize in the figure, whofe fliorter leg ab h made very thin and placed between the jacks with its flat fides facing them; and the longer leg bced^ be- ing placed diredly over, and parallel to the next couple of firings that are clofefl together, is filed four (/} Art. 267. 174 HARMONICS. Sed. VIIL four fquare, and Hides lengthways in two fquare notches at c and d made in the parallel fides fcg^ hdi of a long brafs plate turned up like the fides of a long fhallow trough, which is fupport- ed a little above tlie firings by a row of fmall brafs pillars placed between the larger intervals of the ftrings, as at r, 5, &c, (but farther afun- der) and fkrewed faft into the pinboard of the harpfichord. Thefe pillars have long necks pafUng through the holes r, j, &c in the bottom of the trough, and the nuts r, i, &c are fkrewed upon die necks down to the bottom, to hold it faft upon the fhoulders of the pillars. And a brafs lid FGHI with oblong holes R S, &c correiponding to r, j, &c, being laid upon the troughy^/?/, the upper nuts Ry S, &c muft be fkrewed upon the fame necks, to keep the lid tightifh upon the longer leg of the fquare ^^c^ and others of the fame fize. A flit mfi is made in the lid for a fhort round pin e in the longer leg cd to come thro' it, and to move in it to and fro by a touch of the finger laid upon the pin. There mufl be as many fuch fquares as keys or couples of jacks, and the trough and lid may be each of one piece or confift of two or three pieces joined together at the necks of the pillars or any where elfe. While the jacks /, u are kept at their full height by holding down their key, with your fin- ger laid upon the pin e pufh the leg ai againft the far jack and mark the edge, or inner fide of it with a line drawn clofe by the upper edge of the Prop.XVn. HARMONICS. 175 the leg ^z^ J and after the fquare Is drawn back, make fuch another mark upon the edge of the near jack. Then from a fmall llender pin cut off a piece of a proper length meafured from the point, and taking hold of its thicker end with a pair of pliers, prefs the point into the inner edge of the jack, a little above the mark and far enough to ftick faft in it, and do the like to the oppolite jack. Let each pin projed: from its jack about a quarter of the Ipace between the two jacks, leav- ing about half of it void in the middle between the oppofite ends of the pins, as reprefented in the iigure. Now when the two jacks are again raifed by their key and kept at their full height, by draw- ing the fquare backwards with your finger laid upon the pin e in the longer leg, tlie fliorter leg nb will come under the pin in the near jack, and keep it fufjpended with its pen above the firing G^, which therefore will be filent while the far jack plays alone upon the ftring ^^j or, by pufh- ing the fquare forward with your finger at e, the leg ab will go under the pin in the far jack, and iiifpend its pen above the firing A^, while the near jack plays alone upon the firing G ^. When all the firings of the back unifbn are tuned to the notes in the upper line in art ii^'' all their jacks mufl be fufpended on the fliorter legs of the fquares; and then all die fore jacks will flrike the founds of the vulgar fcale ; and when other flat or fliarp founds are required in any piece of mufic, they mufl firfl be introduced by hold- ing 176 HARMONICS. Sea.VIIL ing down the keys of their ufual fubftitutes, one by one, and by drawing back the correiponding fquares with a finger laid upon their pins at e. So long as you choofe to play upon this changeable fcale, keep the knobs of the right-hand ftops ol a double harpfichord tyed together by a firing. When the firings are tuned unifons again, you may play upon them without removing this me- chanifm, provided you firfl draw every pin e to- wards the middle of the flit mn^ in the lid F/, till it be oppofite to the angular notch 0, and then draw the lid lengthways by the button /», till the notch embraces the pin e and keeps the fliorter leg abm the middle of the void fJ3ace between the ends of the pins in the oppofite jacks: other- wife thefe pins may fometimes ftrike againfl the fhorter legs of the fquares. If that middle fpace be too narrow, try whether it may not be widen- ed a little by feparating the Aiders with fome very thin wedges put between them : perhaps a little may be planed off from the back edges of the lliders without hurting them. I have defcribed this mechanifm {o fully, I think, that any man who works true in brafs may eafily apply it at a fmall expence to any harpfi- chord ready made, and take it quite away with- out the leaft damage to the inflrument. I have ufed it fome years in my own harpfichord with great pleafure and no other inconvenience than that of removing the mufic book in order to touch the pins in the brafs fquares behind it. But the following mechanifm for the reception of which Art. 1 9. HARMONICS. 177 a little preparation muft be made in the fabric of a new harplichord, is quite free from that in- convience, and changes any found together with all its ocflaves in an inllant, without putting down their keys. ig. To male a new harpjlchord wherein the founds bei?ig changeable at plea-- fure^ the ufualfet of keys pall imme- diately firike the proper fcale for any propofed piece of mufc, PI. XXVI. Fig. 67. Conceiving the pins, fbings and jacks which in every o6lave belong to the notes A^ B, Z), £, to be taken away from the fore uniibns of a common harpfichord, the re- maining pins, firings and jacks will be fufficient for the new harpfichord. Let the founds of thefe firings be altered to the notes here placed by the lides of their pins, and let thefe notes be written on the pin board of the new harpfichord j and that the tones of the firings founded by the jacks in each row, may be as like each other as pofTible, let the tongues of die new jacks be put as near as may be to their inner edges, and thefe oppofite edges be placed in the new Aider as near as may be to one another, as reprefented in the figure. Each of the keys, A^ B, £), £, that moves but one jack (which therefore muft be made as heavy with lead as two of the other jacks) flrikes always one and the fame firing. But each of the 8 remain- M ing i;^ HARMONICS. Sed.Vin. ing keys, BK Q C% EK F, F% G, G*, which moves a couple of jacks, is intended to ftrike ei- ther of their firings alone at pleafure ; that is, ji^ or BK B^ or C, C^ or DK D* or EK £* or F, F^ or G^ F'^'^ or G, G^ or ^K PI. XXVIII, Fig. 70. I think the beft way to do this would be to have eight flops or brafs knobs fKrewed as ufual on the flianks of eight draught irons made moveable in eight flits cut in the fore board of the new harpfichord : But to fave a quarter of the labour and expenfe I propofe to do it almofl as well with only fix, that is, three at each end of the fore board, as in the figure : where the notes of each couple of die changeable foufids are written on oppofite fides of each knob, to the intent that the found or firing fignified by this or that note to which the knob is piijhed, may be flruck alone by the key belonging to both the notes, while the other firing is filent. And fince eight founds are intended to be changed by fix knobs, each extreme knob is defigned to change two founds at one pufli of it towards either cou- ple of notes at the end of the flit, according to the fame rule as before. Hence by pufliing the two outermofl knobs at the bafe end of the fore board, towards the right hand, and all the reft towards the left, the keys will flrike the eight changeable founds in the vul- gar fcale, namely F^, C*, G^\ EK B^ F, C, G, to be occafionally changed by pufhing the knobs the contrary way : the other four, ^, Bj D, E, are fixt founds. The Art. 19. HARMONICS. 179 The notes of the changeable founds are placed in fuch an order, that the founds belonging to the notes on the fame fides .of die fucceffive knobs, continually afcend or defcend by v^^\ as in the Table of keys and confonances in Plate XIX. Becaufe diis order will be found much more convenient for altering and adapting the Scale to different pieces of mufic, than the alphabetical order of the fame notes. Now this delign may be executed as follows. PI. xxvi. Fig. by. When the pens for the back unifons are put under their firings, as denoted in the figure, andthofe for the fore unifons are drawn off from theirs by the flops of the common haip- fichord, the jack holes in the two parallel rows have the fame fituation with lefped to each other as they are intended to have in the new Aider, except as already obferved that the fpace between the two rows fhould be much narrower than in the common harpfichord. By thefe directions if an accurate draught of all the jack holes be made upon a long brafs plate, part of which draught is reprefented in the figure, it may ferve as a general pattern for mak- ing the new Aiders, or at leafl to give a clear con- ception of their dimenfions, which a workman may execute in what manner he pleafes. In or- der thereto let fix brafs plates well flatted by a mill be made equal to each other in all their di- menfions. Let the length of each be equal to, or rather longer at firft than that of a common fli- der, and the breadth of each be fufhcient for M 2 leaving i8o HARMONICS. Sed. VIII. leaving a pretty ftrong margin on the outlides of the jack holes, and then the thicknefs, after the work is finifhed and poliflied on both fides, need not exceed a twelfth of an inch. Fig. 67. In the 1" plate or Aider the oppofite holes for the jacks A^ and B^ and all their oc- taves, being made equal to thofe in the general pattern, let all the reft be made wider on each fide, than thofe in the pattern, by a twelfth of an inch, as reprefented at N"^ i, below fig. 6j. In tlie 2^ Aider the holes for the jacks B'^ andC and, to fave another Aider, for F^ ^ and G be- ing made equal to thofe in the pattern, let all the reft be made wider on each fide, than thofe in the pattern, by -of an inch, as at N*^ 2. In tlie 3"^ Aider the holes for the jacks C* and D^and, to fave another Aider, for F'* and G^ being made equal to thofe in the pattern, make all the reft wider on each fide by - of an inch. In the 4^^ Aider the holes for the jacks D * and E^, and in the 5*'' Aider for E^ and F, and in the 6^'^ Aider for G* and A^ being made equal to thofe in the pattern, let all the reft in each Ai- der be made wider on each fide, than thofe in the pattern, by - of an inch. PI. XXVII. Fig. 68. An under focket/^ being made of wood as uflial, that is, with jack holes diredily oppofite to one another but nearer toge- ther as already obferved, and being placed as near to the keys as may be, let an upper focket rj be made Art. 19- HARMONICS. i8i made of brafs, exadly equal to the pattern in fig. 67 but without the notches for the tongues to play in; in which ibcket let every jack hole ex- cept for A, B, D, Ej be made widei: on its left fide only by — of an inch. Leave - or half an inch in height above this upper focket r s, for the fix Aiders numbered I, 2, 3, 4, 5, 6 as above, to lie upon it, and let all he liapported as ufual by crofs pieces of boards fixed underneath. All the jacks being put thro' their holes, this 68''' figure reprefents a view of their edges and of die widening of the holes on each fide, as they would appear to a diftant eye placed at the fore end of the harpfichord, fuppofing the fore board and fore margin of the fockets and Aiders were taken away. When the two fockets /> 5' and rs are fb adjuft- ed in their places that the jacks ^, B, Z), E ftand upright and ftrike their firings, fix the fockets in that pofition at one end only, that the flirinking or fwelling of the harpfichord may not bend or ftrain them- Then pufli all the Aiders towards the right hand, till the pens on the right hand of the other jacks in the far row Aiall Arike their firings too. fee alfo fig. 67. Then if any Aider be drawn back again, which the widened holes will permit, it will draw back the jacks in its narrow holes only, without Airring th? reA, and bring the right hand pen of the A \r M 3 jack l85 HARMONICS. Sea.VIII. jack from under its firing on the right hand and put the left hand pen of the near jack under its firing on the left hand; and then this latter firing will be founded alone by the fame key while the Tormer is filent. The holes in the Aiders for the jacks ^, B, D, E, which have no motion lideways, need be wi- dened on their right fides only, as reprefented by the fliades, to make room for the fliders to move towards the left hand ; but if they be widened on both fides, according to the general diredlion above, no inconvenience will follow from it. According to the ufaal breadth of harpfichords tjie compafs of ourfcale may conveniently be from double G up to ^ in alf. PI. XXVIII. Fig. 69. When the fix fliders are laid upon one another in any order, provided they coincide in length and breadth (and keep fo by two pins put thro' two columns of holes at their ends) three round holes mufl be drilled through them all in the vacant places at k and /, oppofite to the jacks I? and d in ^//, and at m a little above e in alt ; and the hole at k in the i ^ Aider, at / in the 5'^ , as numbered above, and at m in the 2^ re- maining round, all the refl mufl be lengthened by — inch on the right hand and by as much on the left, to the end that a fteel pin put thro' the round hole in any of thofe fliders (g) may draw it {g) See thofe numbers in the lower line of Fig. 70. Art. 19- Hx^^RxMONICS. 183 it on either fide - of an inch, without moving any other Aider. Oppofite to the centers of the round holes at ky /, w, and at the diftance of about an inch and half from each center, are three other centers n, 0, p upon the pin board, where two concentric circles .are drawn about each, one with a radius about - , and the other about - or - of an inch. 4 . 4 10 Each larger circle reprefents a brafs plate having a cylindrical neck whofe bafe is the lelfer circle and height about a fixth of an inch. The up- per half of each neck is filed fquare and a fkrew hole is made in the middle of it. The round plates n, are flcrewed upon the furface of the pin board with flat headed iTcrews funk below the furface of the plates ; but the plate at p is firil let into the pin board as deep almoft as the plate is thick and then is fkrewed down. Three fteel pins made to fit the round holes ky /, m in the i^, 5^^ and 2^ Aiders, already men- tioned, are riveted to the far ends of three flat draught irons A^, O, P, and each pin is kept firm to each iron plate by a flioulder below and a col- lar above. Cut three flits in the fore board at r, j, /, di- reftly oppofite to the centers n, o.p, and having put the Ihank y of the fl:raitefl: draught iron P thro' the flit /, and its fl:eel pin into the column of holes in the Aiders at ?n, and the large cylin- drical hole P over the ncckp which jult fits it; by moving the fliank fideways the 2^ Aider whicli M 4 haij 184 HARMONICS. Sea. VIII. has the round hole, will be moved alone by the fteel pin. And to keep this Aider from being ftir- red by a like motion of thofe above or below it, a brafs waflier, or circular fpringing plate, whofe diameter it equal to that of the brafs circle below, is fitted tight upon the fquare part of the brafs neck and prefTed down upon the draught iron by a fteel fkrew fkrewed into the hole p in the middle of the neck. The other two draught irons O, N are made crooked to go round about die end of the bridge and the extreme pins in the pin board, and are placed in like manner as before upon the brafs circles 0, ?2 ; that upon having another large hole wide enough to receive the fkrew head at p and to give liberty to its own angular motion about the neck 0. The hole at N in the third iron plate being put upon the neck n and over tlie two former plates, has two other holes wide enough to receive the fkrew heads at and/*, and alfo to afford room for its own angular motion about the neck «, the planes of thefe two iron plates being fet off" upwards with cranked necks in order to move them above the other plates. Allowing — of an inch for the motion of any liider, or pen of a jack, the motion of any iron fhank at its flit is to — of an inch, in the given ratio of /> t to p m^ which motion is therefore de- tirmined; and the breadth of any fhank at the Hit added to its motion there, gives tlie length of tlie Art. 19. HARMONICS. 185 the flit, which length, if experience fhall require it, mufl either be augmented, or blocked up a little at either end or both, for adjufting the pro- per quantity of the Aider's mc*:ion. In the like vacant places at the bafe end of the Aiders, three other columns of holes mufl be drilled thro' them all, and the holes in the 3**, 6'^ and 4'^ Aiders remaining round {h) all the reft muft be lengthened on both fides, on pur- pofe to draw any of thofe Aiders feparately (with- out moving the reft) by fteel pins fixed as before in the ends of three other draught irons, of fimi- lar fhapes to thofe at the treble end : the ftraitefl being next the fide board and funk down a little, the 2** going over it, and the 3*^ over both. Note that the draught irons move under the ftrings of the harpfichord. Fig. 70. Two Aiders being faved, two founds be- longing to two extreme notes at each end of the two progreffions by v''^'', are changed both toge- ther by the extreme knobs, rather than any two belonging to the intermediate notes. For as the extreme founds G^ and D^yOr B^ and F^^^ are feldomer ufed than any of the means that are out of the vulgar fcale (/), if either of their ufual fub- ftitutes, F^ and C^, or Cand G, which are both excluded by the principals, G^ andZ)^, or B^ and F**, fhould chance to occur and interfere with them in the fame piece of mufic, it follows that (/;) See thofe numbers in the lower line of Fig. 70. (i) See the Table in Plate xix. i86 HARMONICS. Sed. VIII. that fuch accidents will happen feldomer in the former than in the latter caie. Upon communicating this method of changing the founds of a harpfichord at pleafure, to two of the mofl ingenious and learned gentlemen in this Univerfity, the Reverend Mr. Ludiam and the Reverend Mr. Michel, they encouraged me to put it in practice upon a harplichord made on pur- pofe by Mr. Kirkman; and were fo kind not only to direct and aiTifl the workmen, but alfo to im- prove the ufual method of drawing the Aiders in the accurate ileady manner above defcribed ; which anfwers the defign fo well, that a mufician even while he is playing, can without interrup- tion change any found for another which he per- ceives is coming into ufe : which however is fel- dom required it the fcale be properly adjufted be- fore he begins to play. It may not be amifs to obferve that a careful workman might fave fome time and labour, if inflead of widening each hole in the Aiders fepa- rately, he fliould pierce out fome long holes be- tween thofe couples of narrow ones which move the jacks ; leaving a few (lender crofs pieces here and there to add lufficient Hrength to the Aider v/hile he is working it. SECT. Sea. IX. HARMONICS, 187 SECTION IX. Methods of Tuning an organ and other i?ijlruments. Practical Principles. I. A confonance of any two mufical founds Is imperfed: if it beats or undulates, and is perfed; if it neither beats nor undulates. II. Any little alteration of the interval of a per- fed: confonance makes it beat or undulate, flower or quicker according as tlie alteration is fmaller or greater. III. If the inter\'al of a perfed: confonance be a little increafed, the imperfed: one is faid to Beat Sharp 'y if a little diminiflied, to Beat Flat [k). IV. An imperfed confonance will be difco- vered to beat fharp, if a very f mall diminution of its interval retards the beats ; or to beat flat if any diminution accelerates them. V. The harmony of a confonance is the finefl and fmoothefl when it neither beats nor undu- lates, and grows gradually coarfer and rougher while the beats are gradually accelerated by very fmall alterations of the interval. VI. A fmall alteration is fboner perceived in tlie rate of beating than in the harmony of a con- fonance, and both mufl: be attended to in tuning an {k) See Schol. 5. to Prop. xx. in the Appendix. i88 HARMONICS. Sedl. IX. an inftrument, efpecially the harpfichord, where the beats are weak and of (liort duration. VII. If any imperfed: confonance be founded immediately after another, an attentive ear can determine very nearly whether they beat equally quick, or elfe which of them beats quicker, even without counting the beats made in a given time; efpecially upon the organ, where the beats are ftrong and durable at pleafure. VIII. If feveral imperfedl confbnances of the fame name, as v**^' for inftance (by which the whole fcale is ufually tuned) beat equally quick, they are not equally harmonious ; to make them fo, the higher in the fcale ought to beat as much quicker than the lower as their bafes vibrate quicker ; that is, if a v*^ be a tone higher than a- nother, it fhould beat quicker in the ratio of i o to 9 or of 9 to 8 nearly ; if a iii'' higher, in the ra- tio of 5 to 4; if a v^^ higher, of 3 to 25 if an VI 11*^ higher, of 2 to i j &e. IX. PI. XX. Tab. IV. v. Beginning at any note as G of the undermofl progreffion by tones, if two afcending v^^' G d, da and the descending viii* a A and the two next afcending v^^^ Ae, eb be all made perfed:, the vi^^^^G^, db and the x*^ Gb will be found to beat fliarp and fo quick as to offend the ear by the coarfenefs of their har- mony compared with that of the v'^'. Which fhews that in order to make all thofe concords and their complements to, and compounds with VIII*''' more equally harmonious, the intervals of Art. I. HARMONICS. 189 of the v*''' and confequently of the vi*'^* and x*''' mufl be a little diminifhed in the following manner. PRECEPTS for tuning an organ or harpjichord by ejlifnation and judgment of the ear, I . PI. XX. Tab. V. Alter fucceflively the trebles of the afcending v**" G d,da till they beat flat, the lower v*'* very llowly and the higher a little quicker; and the defcending viii**» ^^ being made perfedt, alter the trebles of the next afcend- ing v'^* Ae, eb till they beat flat a very little quick- er than the two former refpedlively. Then if the x^*" Gb^ between the bafe of the firfl: and treble of the fourth v'*", beats fharp and about as quick as the v*^ Gd X.o the fame baie, thofe 6 notes are properly tuned for the defedlive fcale of Organs and Harpflchords in common ufe. But if that X*'' Gb beats fharp conflderably quicker than the v'^ G d^ every one of thofe four v'*"* G dy da^ Ae^ eb muft be made a very little flatter in order to beat a very little quicker than before. On the contrary, if the x*** Gb beats fharp, coniiderably flower than the w^^Gdy or not at all, or flat, thofe four v*^^ muft be made flaarper, to beat flat a very little flower in the flrfl cafe, and ftill riower in the fecond and third cafes ; till an equality J90 HARMONICS. Sea. IX. equality of beats of the faid x^^ and v*^ to the fame bafe G be nearly obtained. In like manner, the defcending viii^^ ^ B be- ing made perfed, let the two next afcending v^*"' Bf^, f^ f * be made to beat flat a '■eery little quicker than the v*^^y^^, eb refpedlively, fo as make the id^Ac^ beat fharp about as quick as the v^^ ^), and fo the number of complete vibrations {q) was 2x131 or 262 ) Seft. i. art. 3 and 7. {q) Sed. I, art. 12. 194 HARMONICS. Sed.IX. But upon a cold day in November I found by a like experiment, that the fame pipe gave but 254 complete vibrations in one feccnd^ fo that the pitch of its found was lov/er than in Septem- ber by fomething more than - of a mean tone. And upon a pretty hot day in Auguft I collec- ted from another experiment, that the fame pipe gave 268 complete vibrations in a fecond of time ; which fhews that its pitch was higher than in November by almoll half a mean tone. By fome obfervations made upon the contrac- tion and expanlion of air, from its greateft de- gree of cold in our climate to its greateft degree of heat (r), compared with Sir Ifaac Newton^ theory of the velocity of founds, I find alfo tliat the air in an organ pipe may vary the number of its vibrations made in a given time, in the ratio of 1 5 to 16) which anfwers to the major hemitone or about — of the mean tone and agrees very well with the foregoing experiments. CorolL In order to know when the pitch of an organ varies, and when it returns to the fame again, it is convenient to keep a thermometer con- flantly in the organ cafe. (r) See Mr. Cotei% xv^h Ledlure upon liydroHat. and Pneumat. towards the end, 'TEE Prop.XVIIL HAR MO N I C S. 195 THE SECOND METHOD of finding the pitch of an organ^ or the number of vibrations made in a given time by any given note. Let the notes C, Z), E, F, G, A, B, c, d, e,f, gy ciy by c\ be in the middle of the fcale, and from any Bafe note not higher than jD, tune up- wards three fucceffive perfecft v'^' DA^ Ae^ eby and downwards the perfed; yi^^ bdj then having counted the number of beats made in any given time by the imperfect; viii'^ Ddy the number of complete vibrations made in that time by its treble dy will be 8 1 times diat number of beats. For example fuppofe the viii*^Z)<^be found to beat65 times in 2ofeconds; then 8 1x65 or 5265 is the number of complete vibrations made by the treble ^ in 2ofeconds; and - — - or 26-2 is the 20 ■^ number made in one fecond. The time may be meafured either by a watch that fhews feconds, or a pendulum-clock, or a limple pendulum that vibrates forwards or back- wards in one fecond, whofe length from the point of fufpeniion to the center of the bullet is 39 inches and one eighth. And the perfon that ob- lerves the m.eafure of the time muft give a ftamp with his foot at the beginning and end of it, while another perfon counts the number of the beats made in that time j v/hich number dimi- N 2 niflied 196 HARMONICS. Sea. IX. nifhed by one, is the number of the intervals be- tween thofe fucceffive beats and properly fpeak- ing is the number required. For greater accu- racy the experiment Ihould be repeated feveral times and a medium taken among the fevera) refults. DEMONSTRATION. For if the notes Z), E, F, &c, ftand for the times of the fingle vibrations of their founds, wq have D :A:: 3 : 2, y^ : ^ : : 3:2,^:^: : 3:2 and 6 : d:: 2' 5> ^^^ ^7 compounding thofe ra- tios we have Did:: 81 : 40, which ratio being refolved into 81 : 80 and 80 : 40, ihews that the Yiii^ D d IS tempered fliarp by a comma. Whence in caf. i. Prop xi, putting /= i =§', /z=i wehave/3=--7-^ — r— =5- andoijS=M ' ' \blp-\-q 01 ' the number of complete vibrations of the treble J $1;* jB-y^ ^v^ /3^'* /3i;7 j8t;' ^v^ 4 8 8 16 16 32 32 In like manner, if the former feries of v*''* be continued downwards from dy as in Tab. 3, the numbers of their beats made in a given time will continually decreafe in the ratio of 1; to i or of I to-, and therefore will be, -' -4' ^> —' — > V V V *v^ v* v^ ^, i, ^, t, A. *U^ ^7 ^8 ^9 .yia Now (b) Prop. XI. corolla 2* Prop. XIX. HARMONICS. lot Now as often as thefe v^'^' are raifed by an oc- tave, as in the lower half of the fcale defcending from 4 in Tab. i or 2, fo often muft thefe beats be multiplied by 2 ; which produces this feries of beats ?> —5 — > ^y — > — ' — ' —5 i^> ^- ' 1; -y* i;J 1;^ ij^ V* V^ V^ IJ^ V'** Q^E.I. Scholium* For example, be it propofed to calculate the number of beats, in Tab. i, Plate xx, which every vth in the lyftem of mean tones will make in 1 5 feconds of time. Here the temperament of the v*^ is i of a comma (c), and fuppoiing the interval of the perfedl v^'' = log. - =0. 17609 1 3, we have the comma c = log. - = o. 0053950, and - r = 0.00134 88, and the v'^ — I c = o. 1747425, which is the logarithm of the number i. 4953 or lii21?j that is, of the ratio of 1.4953 ^° i> which in the iblution of the problem we repre- fented by -u to i . Now when the thermometer ftood at tempe- rate, the pitch of our organ at Trinity College, or the number of complete vibrations made in i fecond by the air in the pipe denoted by d in the middle of our table, was 262 (d). Hence (c) Prop. ii<^. cor. (^) Prop, xviii. 202 HARMONICS. Sea.IX. Hence to find the number of beats made in 15 feconds by the v'^ above d when tem.pered flat by - comma, in prop, xi we have ?7i : 72 : : 3 : 2 or z«= 3, ;z=2 and ^-c=-c, or ^==ij /'=4> and fince the bafe d makes 262 complete vibra- tions in I fecond, in the given time of 1 5 feconds it will make 15x262 fuch vibrations = iV, and the Abacus I. 0. 1747425 I. 5629841 I. 7317'^^^ I. v^zz Pjv' zz (iv^ zz 369 55^ 54, 667 81, 747 122, 24 182, 79 ^73^ 34 408, 73 611, 20 9^3^ 95 1366,7 N° of beats. 2 = 37 = 27 -^v'zz 4.1 4 31 g/3i'*=23 5^3ii6 43 "4 = 29 0. 6892716 ^ =4,8896 8/3 ^=39 o- 5145291 ;^ =3, 2699 83 ^ -T= 26 0. 3397866 ^ =2,1867 163 ^7-35 0. 1650441 -8 =1,4623 163 -^= 23 T. 9903016 3 :^ =o>9779 323 T. 815559^ ^^,, =0, 6540 323 ^^ ZZ 21 204 HARMONICS. Sea.IX. jS'u, jS-u*, jS'y^ Sec, as in the firft Abacus, and thence the correlponding numbers, which divi- ded by the proper powers of 2, as diredled in the folution of the problem, give the afcending half of the fet of beats oppolite to the pitch 262 in Tab. I. The log. of V fubtradted from o gives the log, of - , which log. continually added to the log. of /3, gives the loo;arithms of -,-,-, , 5cc as in the 2^ Abacus. And thefe logarithms give the num- bers tliemfelves, which multiplied by the proper powers of 2, as above dired:ed, give the defend- ing half of the fame fet of beats oppofite to the pitch 262 in Tab. i. The fuperior fets of beats are defigned for tuning the fame or different organs, when their pitch is higher than this by i, 2, 3 or 4 quarter- tones, as noted at the beginning of the table, and may be found by the continual addition of the logarithm of a quarter-tone to the logarithms in each Abacus ; and the iirft inferior fet of beats may be found by the fubtra(flion of the log. of a quarter-tone from the faid fet of logarithms, or by the addition of its arithmetical comple- ment : remembering to divide and multiply the correfponding numbers by the fame powers of 2 as before in each Abacus. And as the I tone is i of the 11 1^ its loga- 4 o rithm is i log. -= o. 0121138, which conti- 4 Dually Prop. XIX. HARMONICS. 205 nually added to the logarithm of 262, gives the fucceffive logarithms of the higher pitches 269, 277, &c, in the firfl: column of the table, over againfl: the correlponding fuperior fets of beats j and fubtrafted from the fame logarithm of 262, gives the log. of the loweft pitch 255, overagainft tlie lowefl: fet of beats. From the given temperament of the fyflem of equal harmony (e) the beats of all the v^'^' may be calculated by the fame method 3 and will be found as in Tab. ii*^. Plate xx. CoroII. I. Suppofing the niiddlemoft notes Jy in the firfl and fecond tables, to be unifons, the numbers of the beats in a given time, of any two correfponding v^'^'are very nearly in the given ratio of their temperaments ^ c and -^ ^ {f)i or as 9 to 10. For the beats would be in that ratio if their feveral bafe notes were exadlly uni- fons (g) ; and the difference of their pitches at the diftance of the tenth v*^ from the middle note ^, is but ten times the difference of the tem- peraments, or ^ Cj which produces the difference of but I beat in 290 in the extreme v*^^ in the tables, and lefs in the reft in proportion to their diflances from the note ^ (h). CoroJI. 2. In the fyftem of equal harmony the ratio of the numbers of beats of the iii*^ and v^^ to (e) Prop. XVI, Schol, 2. art. lo. (/) Prop. 2. cor. and prop. xvi. fchol. 2. art. lO. (g) Prop. XI. coroll. 4. and fchol. i. (h) Cor. 4. lemm. to prop. ix. and cor. 2. prop, xi. 2o6 HARMONICS. Seft.IX. to the fame bafe, is 2 to 3 in ^ given timej as being compounded of the ratio of their tem- peraments - and -^ Cj and of the major terms of their perfecfl ratios 5 : 4 and 3 : 2 (/). Coroll. 3 . For the fame reafon the beats of the v*'* and vi^'' to the fame bafe are ifochronous in the fyftem of equal harmony, whereas in that of mean tones they are in the ratio of 3 to 5 in a given time. PROPOSITION XX. To tune any given oi'gan by a given table of beats. Having found the pitch of the organ by any of the methods in Prop, xviii, look for the neareft to it in the firft column of Table i or 11, Plate xx, and overagainfl it is the Proper Set of Beats for tuning the given organ. If the weather be con- liderably hotter or colder at the time of tuning than it was when the pitch was found, allowance mufl be made in the number of vibrations deno- ting the pitch, by the fchol. to prop, xvi 11. Then flatten the treble a of the perfect v*^ a- bove d [k) more or lefs, till the number of its beats, made in 15 feconds (/), agrees with the tabular (/) Corvnll. 3. prop, xi and fchol. i. i^k) See the notation tab. iv. plate xx. (/) To be meaiured as direfted in the Second Method in prop. XVIII. But in this cafe count one beat more than the tabular number, as properly llgnifying the number of intervals between the luccefiive beats. Pi-op.XX. HARMONICS. 207 tabular number placed over that v^'' in the pro- per fet. From the treble of that v^^ da tune down- wards the odave a A, fo as to be quite free from beats, and repeat the like operation upon the next afcending v''^ Ae, and the like again upon the next till you have tuned all the Iharp notes in the fcale of your organ. Then going backwards from d, fharpen the bafe G of the perfed: v''^ below d more or lefs till die number of its beats, made in 15 feconds, agrees with the tabular number placed over this v'^ in the proper fet. From the bafe of the v^'' t/G, tune upwards the o(flave Gg and repeat the like operation upon the next defcending v^'^^^:, and the like again up- on the next, till you have tuned all the flat notes in the fcale of your organ. This being done, let all the other founds be made odtaves to thefe, and the fcale will be ex- actly tuned according to the temperament in the given table; that is, all the v'^' will be equally tempered, and confequently equally harmoni- ous {m)^ and fo will all the vi'^% and every other fet of concords of the fame name, which anfwers the deflgn of tuning hy a table of beats. If you chufe to tune the organ according to the Hugenian lyftem, the fet of beats in Tab. i, next below that which anfwers to the pitch, found by any of the foregoing methods, will ferve your purpofe. For [m) Prop. XII coroll. 2o8 HARMONICS. Sed. IX. For the Hiigenian v*'% having its temperament • — - r + — c fmaller than ~- c. in the ratio of 4 no 4 53 ^^ SS (^)> t>eats flower than the tabular v*'* in that proportion, which is but very httle flower than it would do, if its pitch were depreflTed by i of a mean tone. Q. E. F. Scholium I. 1. Since our organ at Trinity College was new voiced, and by altering the diipofltion of the keys was depreflTed a tone lower, and thereby re- duced to the Roman pitch, as I judge by its a- greementwith that of the pitch pipes made about the year 1720 ; by the help of fuch a pipe one may know by how many quarter-tones the pitch of any other organ is higher than that of ours, and thus (without any of the methods in Prop, xviii) determine the proper fet of beats for tuning it {0). 2. At a time when the thermometer fl:ood at Temperate, as it did alfo when the pitch of our organ was found to be 262, I aflifted at the tu- ning of the v'*^' of the open Diapafon hy the fet ' of beats oppoflte to that pitch in Tab. i, and up- on examining the iii'^' and x**'' I found them all perfed: : a manifefl: proof of the theory of beats and of the certainty of fuccefs in tuning by it. 3 . At that time the whole organ was tuned to the open diapafon, and is now univerfally allowed tg («) Prop. XVII. Schol. Tab. 2. {0) Sec column 2, Tab. i, 2, plate xx. Prop.XX. HARMONICS. 209 to be much more harmonious than before, when the major thirds were much fliarper than per- fedl ones ; and its harmony, I doubt not, is ftill improveable by making them flatter than perfed:, according to the fyftem of equal harmony. But at that time I had not finiflied the calculation of it, and to repeat the tuning of the organ over again would be troublefome and improper at the prefent feafon, when cold and damp wea- ther is coming; on very fall:. 4. For the properell: times for tuning the Dia- pafon of an organ feem to be, from the latter end of Auguft to the middle of Odober, when the air being dry, temperate and quiet, will keep nearer to the fame degree of elallicity for a given time. Becaufe a very fmall alteration in the warmth of moifl: air will fuddenly and feniibly alter its ela- ftic force and thereby the pitch of the pipes before the whole ilop can be accurately tuned. For that reafon conflant care mufl: be takea not to heat the pipes by touching them oftener than is needful 3 nor to flay too long at a time in the organ cafe j nor to tune earlyin the morning, but rather towards the evening, when the air is drier and its declining warmth is kept at a flay by the warmth of the perfons about the organ. But diefe and the like cautions may fooner be learned by a little prad:ice than by any defcrip- tion, and if not altogether necelTary, will how- ever contribute to the accuracy of tuning by fo nice a method which is plainly capable of any O defired 210 HARMONICS. Sedl. IX. delired degree of exadnefs provided the blaft of the bellows be uniform. 5. After tuning an organ according to any new fyftem whatever, we muft be cautious of judging too haftily of it. Some muiicians here who had conftantly been ufed to major thirds and confequently major lixths tuned very fharp, could not well relilh the finer harmony of perfe6l tliirds and better fixths in the organ newly tuned, till after a little ufe they became better fatisfied with it, and after a longer ufe they could not bear the coarfe harmony of other organs tuned in the ufual manner. It is therefore neceffary to have equal experi- ence in different objec^ts of fenfe, in order to judge impartially, which of the two is more grateful than the other, as is evident in almoft every thing to which we are more or lefs habituated. 6. If a machine were contrived, as it ealily might, to beat like a clock or watch, any given number of times in 15 feconds, between 20 and 56 or thereabouts 5 by fetting it to beat accord- ing to any given number in the table for tuning an organ, and by comparing its beats with thofe of the correfponding v^^, the ear would deter- mine immediately and exactly enough, whether they were ifochronous or not ; and thus a harpfi- chord might be tuned almoft to the fame exad:- nefs as an organ ; and the tuning of an organ might be performed much quicker by the help' of fuch a machine than by counting the beats as above. In HARMONICS. 211 Prop. XX. In the following method after 3 or 4 fifths are tuned by a little table of beats, the organ itfelf does the office of fuch a machine in all the reft. Scholium 2. *To tuns any given organ by ifochronous beats of different concords, I . The founds in the middle of the fcale being called CDEFGAB cdefgabc, make the VI 1 1^'^' Gg quite perfect, and let the v''^' cg^ Gdy ^<2 be made to beat flat 38, 28, 42 times refpec- tively in 1 5 feconds of time. NO of beats of the vith The Ha fyl>em of Equal rmony. Tab. i. A proper fyftem for defedive fcales. Tab. 2. The numbers of beats of the vths. c a ^) Prop. XIX. fchol. coroU. 3. Prop. XX. HARMONICS. 213 vi*^ differ from 40 by more than 3 and lefs dian 9, over or under, the next higher or lower fet refpedlively, is the proper fet. Becaufe an alte- ration of lix beats of the vi**" anfwers in time to an alteration of but one beat of the v''' to the fame bafe, as appears by the differences of the numbers in the firft and fecond columns. For which reafon, if in any experim.ent the beats of the VI ''^ ca fhould come out exadlly in the middle between any two numbers in the i^ co- lumn, you may take either of the fets oppoiite to them for the proper fet j the error from tiie truth being but half a beat, tliat is, half the interval of the fucceffive beats of the v^*" eg, which half cannot eafily be meafured. Yet the choice might be determined by altering the pitch of the found c a very little and repeating the experiment. 5. The proper let for a given organ being once found, the i'* experiment need never be repeated afterwards. For whatever be the proper fet in tem- perate weather, the next above it will be the pro- per fet in hot, and the next below it in cold wea- ther {q). At moft, the fet fo determined need only be verified as in Article 3 or 4 whenever the organ wants to be retuned. 6. After the v'^' eg, Gd, da had been made to beat 38, 28, 42 times in 15 feconds in the I ^ experiment, if the beats of die vi^'' ca had come out too quick, or too flow and indiflindl: to O 3 b^ {q) Prop. XVIII. fcholium, 214 HARMONICS. Sed.IX. be eafily countedj (which however cannot hap- pen unlefs the pitch of the organ be immoderately higher or lower than ufual,) in the iirft cafe ufe 13 feconds, and in the fecond 17 inftead of 153 and upon repeating the experiment the beats of the vi"^^ ca will come out within the limits 20 and 565 of the numbers in the iirft column and point to the proper fet. 7. Laftly, if the beats of any v''^ cannot foon be adjufted to the tabular number, which fome- times happens, and that number has the fign 4- after it, the excefs of a beat may be dilpenfed with as being lefs erroneous in that cafe than the defedt of a beat j and on the contrary, if the ta- bular number has the fign — after it. 8. PL XXV. Fig. 65. Now if you chufe to tune the organ tp the fyftem of equal harmony, which being the moft harmonious is tlie pro- pereft for a changeable fcale (r), in afcending from the founds, c^ G, ^, a tuned by the pro- per fet of beats, make the vV-"^ Ge beat fharp and juft as quick as the given v'^ Gd^ and do the like for every vi*'' in the order annexed. Likewife in defccnding from the faid given founds v VI VIII Gd, Ge da. db & tuning a A Ae, Af^ eh. ^c'^Sc tuning bB Bf, Bg^ &c, &c &c. (r) Scfl, viii art. ii^^ . Prop. XX. HARMONICS. 215 founds c, G, d^ a^ alter the common bafe F of the v^'' and vi''', Fc and Fd^ till v vi viii they beat equally Fc, F^& tuning Ff quick, the v^'^ flat, 5 ^/, B'o g andthevi^'^fharp; E^ B^, Eh Sc tuning Eh^ and repeat the like ^^£^, ^^f practice in the or- &c, &c &c. der annexed. If this fcale afcend fo high as to caufe a v^^ and vi'^, as el> and ec^ for inftance, to beat too quick, tune downwards the two viii^^^ eE and ifB and leaving out die uppermoft row of the /harp notes, proceed with the lower rows. And do the contrary in the defcending part of the fcale, leaving out the undermofl flat notes where you find they beat too flow and defcending by the higher notes. 9. But till Infl:ruments are made with a changeable fcale, it Is more proper to tune the de- fe(5tive fcale in prefent ufe by making every v *^ and iii'^, to the fame bafe, beat equally quick, the former flat and the latter fharp. Make the v^^' eg, Gd, da, Ae beat flat 31, 23, 35, 26, times refpedively, tliis being the proper fet found by Art. i, 2, for the given or- gan, and the \\\^ ce will beat fliarp equally with the v''' C7> O 4 PI. XXV, 2l6 HARMONICS. Sedr.IX. Plate XXV. Fig. 65. Then in afcending from the founds c, gy G, J, a, A^ e fo tur^d, make the next iii'' GB beat {harp and equally with the giv- en \^^Gd, and do the like for the reft in the order annexed. Likewife in defcend- ing from the fame given founds alter the B ^fy bafe F of the 1 1 1'^ FA till it beats fharp and equally vi^ith the given v^^ Fc to the fame bafe^ and do the like for the reft in the order annexed. v III VIII Gl G 5 and tuning £^ day df^ ACy Ac"^ eby eg^ FCy FAy & tuning Ff Bhfy Bbd EbBK EhG. DEMONS TR ATI 0*N. > PI.. XXIV. Fig. Ay which is defcribed in Prop. Ill and cor. 1.2. The beats of any given v^'' and vi^^ to the fame bafe will be ifochronous, as they ought to be in the fyftem of equal harmony, when their temperaments are as 5 to 3 [t). Whence by the coroll. to prop, iv, Gr is = -^^ and is the flat temperament of all the v'^^ {li) and As^= ■^ c is the fharp temperament of the refulting VI*''^ .. In Tab. 1 . column i the afcending numbers 35' 3^» '^7-> 3^' ^^ ^^^ alTumed for the beats to be made in a given time by the given v*^ eg in organs (/) Prop. XI. coroll. 3 and fchol. j. \u) Prop. XII. coroll. Prop. XX. HARMONICS, 217 organs of dliFercnt pitches (x) and from thefe and the given temperament ^ f , the beats of the other v'''' Gd^^da in col. 2. 3 of that table are ^computed by the method in Prop. xix. When the- v'^ and vi''' eg and ca, have any other temperaments, as Gp and A a, let a and h be the refpecftive numbers of their beats made in any known time, and /3 their common number made in that time when their temperaments were Gr ^= \c and^i= 4 <^ as before. Then fup- 18 18 ^ d poling the difference r^= -^r, by the limilar triangles Or^^ Ostr, we have S(t= -^c. And ^mce the numbers of beats made in equal times by a given confonance differently tempered are proportional to its temperaments (^), we have /3 : a : : Gr '.G^'.'.y. 5 — ^;, whence ^= IflZlf J likewife ^: b :: As : A a- : • 3 : 3 -1- 3.^, whence /3 = ^+ ' again ^== — . Therefore j/Sr— 54=^— /3 and r—a 6~* , . ^ • In the method above for finding; the proper fet of beats we affumed ^ = 38 and in vthe ex- ample we fuppofed ^ = 48, whence /S^^^^Sv^. ^^ZliL ■= '2Q- , the nearefl to which in column i tab.^l. being 4o,.^gives the proper fet, 40, jo-^, 45-. It. , ' ^ After ^''- ■ ■ ■ . (x) Prop. XI. Gor. 2. ly) Prop. XI. Gor. 4. and fchol. i. • ' 2i8 HARMONICS. Seft.IX. After this manner I found the proper fet for making the v'*^ and vi'^, cg^ ca^ beat equally quick in our oigan at Trinity College, to be 36, 27— , 40 +, in a warm feafon, after the pitch of it had been deprefled a tone lower, to the Ro- man pitch, only by changing the places of the keys. Confequently the proper fet for the former pitch would have been 40, 30—, 45— , becaufe the ratio of 36 to 40 or 9 to 10 belongs to the tone nearly. For this reafon in the method above I chofe to begin the experiment with the intenne- diate fet 38. 28 -f, 42 +, and adapted to it the column of beats of the vi^^ ^^ in the following manner. When ^ = 38, jS = 38 + i ^-38, whence it appears if ^ = 3 8, that /3= 3 8 3 if ^ exceeds 3 8, that /3 exceeds 3 8 by - of that excefs i if /^ be de- ficient from 3 8, that /3 is deficient from 38 by I of the defedt. So that the common difference of the values of jS or beats of the v*'' eg being unity in col. 1. Tab. i, the common difference of the correfpondent values of b or beats of the vi*'' ca muil: be fix, as in the column prefixed to the Table. If ^ = o, that is, if the vi^^ ca fliould come T O out perfe6l, we have jS = 38--6-=3i-, which in col. I. Tab. i continued downwards would give the pitch and the proper fet for an organ about a tone lower than ours, tho' lower than ordinary j Prop. XX. HARMONICS. 219 ordinary J becaufe the ratio of 36 to 3 1 - or 32, is 9 to 8 belonging to the tone major. Therefore in the experiment above for finding the proper fet, the VI''' ca cannot beat flat upon any organ now in ufe : and if its fharp beats come out too flow or too quick, a remedy has been given above in art. 6. The reafon of it is this ; if either of the temperaments G^, Acr^ of the v*^ and vi^'' be in- creafed, the other will decreafe, and accordingly the beats of the former confonance will be acce- lerated while thofe of the latter are retarded. So that when the beats of the vi*^ come out too quick to be eafily counted, they fliew that thofe of the v*^ were too flow at firfl: and therefore muft be accelerated, or the fame number of them mufl: be made in a lefs time : and on the contrary, when the beats of the vi'^ come out too flow. In Tab. 2, where the major 11 1^, in order to leflen the diefls {z\ is defigned to beat fliarp and as quick as the v'^ beats flat, the ratio of the temperaments Gp, Et, of the v'^ and iii*^ mufl: be 5 to 3 (^), and the temperer O ^ a- r mufl: be within the angle EOG. Hence G^= ^c(l?). But in Tab. i we had Gr= ^„ <:, and fo the ratio 18 of the beats made in a given time by the corre- ipondent (z) Sedt. VIII. art. 2. note u. (a) Prop. XI. cor. 3. Ichol. i. (b) Prop. V. cor. caf. 2, where - may ha\''e any value, becaufe the fecond condition of the propofition is not here required. ^^^ 220 HARMONICS. Sed. IX. ipondent v*''' in Tab. i and 2, is Gr to G^ or 2^ to 1 8 (c). In like manner a table of beats for any other fyftem might be computed and fubjoined to Tab I, in which the proper fet may be found as before in the i** experiment. Another lyftem might be tuned by ifochronous beats of the 1 1 1^ and VI*** , but it differs fo little from that of equal harmony that the mention of it is fufficient. Corollary. Hence we have the number of vi- brations made in a given time by any given found of a given organ. For the number of complete vibrations made in a given time by the found c is the produd of this conftant number 96, 7 - multiplied by the number of beats made in that time by the v'^ eg when it beats equally with the vi^^ ca, to be found by the method above de- fcribed. Thus if that number of beats be 36 in 15 Se- conds, in this time the found c makes 36x96,766 5cc = 3484 complete vibrations, that is 232 - vibrations in one fecond j and therefore the found ^, which is almoft a mean tone higher than c, ) makes 260 fuch vibrations in one fecond, which agrees with the experiment made with the brafs wire in prop, xvi 1 1 . For the temperament of the v^^ eg, when it beats flat and equally with the vi^^ ea, was found to be -o <^j = r <: in prop, xi, where in caf. 2, j8= lo P '{c) Prop. XI, coroU. 4, Prop. XX. HARMONICS. 221 iqmN ^^ ^r— ^^^P+^ p — 161x18 + 5 /?-_■ § .l6lp-j-q' 2qm 10x3 ^ ■■*9'6,7 -x/3, the number of vibrations of the found c. SchoUtim 3. If we could meafure any given part of a fe- cond of time more readily and exadlly by any Other means than by the beats themfelves, a fingle fet of beats for a given fyflem, as 38, 28, 42, &c would alone be fuflicient for tuning any given organ according to the given fyftem. For, by the method above, having found /S the proper number of beats wliich the v^'' eg ought %Q make in a gi\'en time as j 5 leconds, the time f , in which it will make the number 3 8 in the given fet, is to 15 feconds, as 38 is to ,/3; and that time fo determined is the Proper Time in which all the other numbers of beats in the given fet ought to be made by the other v'^* in the faid organ. But unlefs that time /, which will generally con- tain Ibme fraction of a fecond, could be readily and accurately meafured, this method will with equal expedition be leis accurate, or with equal accuracy v/ill be lefs expeditious than the former. For if inftead of the mixt number t we ufe the neareft whole number of feconds for the proper time, the limit of the error will be half a fe- cond ; whereas in uling the Proper Set for any given time, the error is but half the interval of the fucceffive beats of the v'^ cg^ which is tv^''o 05 222 HARMONICS. Sea.IX. or three times fmaller than half a fecond, be- caufe tlie number of its beats in col. i. Tab. i, is always between two and three times greater than 15, the number of feconds in which the beats are made. And the larger error cannot eafily be re- duced to an equality with the fmaller, unlefs by a fet of beats whofe numbers are between 2 and 3 times larger than thofe in the Tables, which would proportionally increafe the time and trou- ble of counting them. For inftance, inftead of counting 38, 28, 42 beats in 15 feconds, we muft count 96, 72, 107, in 38 leconds. Becaufe 15, 38, 96 are continual proportionals nearly. As the known method of tuning an inflru- ment by the help of a monochord is eafier than any other to lefs skilful ears, and pretty exadt too if the apparatus to the monochord be well contrived, it may not be amifs to fhew the man- ner of dividing it according to any propofed temperament of the fcale. PROPOSITION XXI. To find the parts of a given mo?tochorcly whofe vibrations fhall give all the foimds in an o&ave of any propofed tempered Jyjiem* Let the fyftem of equal harmony be propoled, and let the feveral parts of the monochord be meaiured from either end of it, and be to the whole. Prop. XXI. HARMONICS. 223 whole, in the ratios of the feveral numbers in the 3** column of the following table, to 1 00000 j I fay the vibrations of the parts fo found, and of the whole, will give all the founds in an odave of the propofed fyflem, as denoted in the firfl co- lumn of the table. Q^. I. For in the fcholium to prop, xvii we had 2T = o. 09631. 05650 and 2L = 0.06025. 358323 v/hence we have T= 0.04815.52825 L = 0.03012. 67916 T — L = /= o. 01802. 84909 L — / =^=0.01209.83007 From thefe logarithms of the tone, limma ma- jor and minor and the dielis, and from the lo- garithm 4.69897.00043 of the number 50000, the uppermofl in the table, all the logarithms be- low it will be found by the following additions : where the mulical notes in column i are fup- pofed' alfo to reprefent the logarithms over a- gainfl them, till you come down to C, which comes out 5. 00000. 00000 and fhews that the c + ^= B*, c + I =^cb , c + L = B B+/=:B^, B+L==A^, B + T=A A + /=A^ A + L^G'^, A+T=G G + /=G^ G + L=F*, G+T=F F+^=E=», F + / =F^ , F + L=E &c &c &c logarithms 5524 HARMONICS. Sed. IX. logarithms of the principal notes B, A, G, F, E, D are right j and thofe of the fecondary notes will be right too, if the operations in the addition be right. The correlponding numbers in column 3, which may be found by the tables of loga- rithms, fhew the required parts of the mono- chord J as a very little refledion will fatisfy any one that underftands the common properties of logarithms, and attends to the intervals of an oc- tave in Fig. 49 defcribed in Se(5l. viii, but not divided as there into 50 equal parts, which is only an approximation to the iyflem propofed* Q^E, D. Scholium, The numbers in the 4*^ and ^^ columns of the table (hew the parts of a monochord, whole vibrations will give the founds of the oppolite notes in the fyflem of mean tones and that of Mr. Hiiygejjs, who has fhewn how to find the laft column of numbers in his Harmonic Cycle. And as all the meafures in the 3 fyftems may be taken and marked upon the founding board of the fame monochord, the different effeds of thofe lyftems upon the ear, may be eafily tried and compared togedier, provided the tone of the mo- nochord be good and the divifions accurate, and the moveable bridge does not ilrain it in one place more than in another. SECTION facing p. 224. Tloe divifion of a Mo?tocbord, I c ci^ B B^ A* A A^ G Gh F« F F^ E E^ D=» D c« C 2 3 4 5 4. 69897. 00043 4. 71106. 83050 4.71699.84952 50000 51412 52119 50000 51200 52245 50000 51131 52278 4.72909.67959 4. 74712. 52868 4.75922.35875 53592 57441 53499 55902 57243 53469 55914 57179 4. 7772^. 20784 4.79527.05693 4. 80737. 88700 59876 62412 64177 59814 62500 64000 59794 62528 63942 4. 82540. 73609 4-84343-58518 4-85553-41525 66897 69733 71702 6687^1 69877 71554 66866 69924 71506 4.87356.26434 4. 88566. 09441 4.89159. 11343 74742 76853 77910 74767 76562 78125 74776 76467 78196 4.90368.94350 4.92171.79259 4. 93381.62266 80110 83506 85865 8000c 83593 85599 79964 83621 85512 4. 95184. 47175 4.96987. 32084 4.98197. 15091 89504 93298 95934 89443 93459 95702 89422 93512 95627 5. 00000. 00000 lOOOOO 1 00000 lOOOOO Syftem of equal harmony mean tones MwHuy- Prop.XXII. HARMONICS. 225 SECTION X. Of occajional temper a?nents ufed in con- certs well performed upon perfe& in- flruments. By a perfed: inftrument I mean a voice, vio- lin or violoncello, &c, with w^hlch a good mu- ficlan can perfectly exprels any found which his car requires. PROPOSITION XXII. T^e fever al Farts of a concert well per- formed upon perfeB injiruments^ do not move exa&ly by the given inter- vals of any one fy fern whatever^ but only pretty nearly^ and fo as to make perfeB harmony as near as pojjible. For inftance. If the bafe be fuppofed to move by the bcfl fyftem of perfedl intervals (^), the other part or parts cannot conflantly move by it too, without making ibme of the concords im- perfed: by a comma (f), which would grievoufly offend the muficlans (f). Confequently if they are pleafed, thofe intervals are occafionally tem- P pered {d) See Se), but a fcientific proof was ftill wanting till Dr. Taylor publiflicd his theory of the vibrating motion of a mufical chord (^)j which has lince been culti- vated by feveral able mathematicians (r), and being the principal foundation of Flarmonics, deferves to be further conlidered in the next fec- tion. SECTION XL Of the vibratt7ig motion of a mufical chord. PROPOSITION XXIII. TVhe?i a mufcal chord vibrates freely^ the force which urges any fmall arch of it towards the center of its curva^ tiircy ((?) StettI lungo tempo perpleffo intorno a quefte forme tlelle confonanzc, non mi parendo che la ragione, che communementefc n' adduce dagli autori, che {\ri qui hanno fcritto dottamcnte della mufica, foffe concludente a ba- fianza. Dicono effi ^^c. Difccrfi attenenti alia JHecanica, Dialogo i^, towards the end. {p) Ibidem. ( q) Methodus incrementorum, prop. 22, 23, and Philof. Traiif. N^ 337, IV. or Abrigd. by Jones Vol. 4. p. 391, (r) Commentarii Acad. Petropol. Tom. 1 11. Comment on the Principia Vol. 2. pag. 347* Mr. Maclaurin's Fluxions, art. 929. Prop. XXIII. HARMONICS. 231 ture^ is to the tenjion of the chord in the ultimate ratio of the le?'igth of that arch whe?t i?ifnitely dimi72ijhed^ to the radius of its curvature, I fuppofe the chord to be uniform, and very llender, or rather to be a mathematical line, flexible by tlie leail force and elafiic ; and its tenfion or quantity of elaftic force to be meafured by a weight, which if hung to one end of it, would diflend it to the fame lengtli which it has when it vibrates freely by the force of its ela- fticity. PL XXI. Fig. 5o.Let^CB reprefent fuch a chord fixed at the points A and B, CD any arch of it, CE and DE tangents at C and D j in either of which as EC produced if need be, take E F equal to ED, and draw FG perpendicular to FEj and D G to DE, and joining DF produce GE towards H. Then imagine the chord t6 keep its cur\'ature while a force applied at £ or i^ drav/ s the tan- gents EC, ED and thefe the points of contad: C, D, fo as to keep them in aqziiiibric. And fince the elallic forces at C and D are each equal to the force of tenfion, the direction of the third force at E will bifedt the angle CED un- der the other two diredions, and confequently will coincide with the line G EH, agreeably to the conilrudion of the equal triangles EDG,E FG^ P 4 Plea^ 232 HARMONICS. Sed-XI. Hence the three forces at H, C and D, which would keep the point E at reft, are proportional to the fides DF, FGy GD of the ifofcelar trian- gle DFG, to which their feveral directions E H, EC, ED are perpendicular 3 becaufe that tri- angle is fimilar to any other, as EDI, whofe fides are either parallel to, or in part coincident with thofe diredions, and therefore proportional to the forces ading in them, by the known theo- rem in Mechanics {s). Now fuppofe CL and Z)iW to be the radii of the curvatures of the chord at the points C, D, and the curve jL Af to be the locus of all the cen- ters of the curvatures at every point of the arch CD. Then conceiving the point D to move up to C, and confcquently iWand G up to L, the li- mit of the variable ratio DF to FG, of the faid forces, will be that of the evanefcent arch CD to CL the radius of its cuiTature. And a force conftantly equal and oppofite to the former of the two, is that which urges the vanifliing arch CD in the diredion E G, which ultimately coincides with CL ; and the latter was the force of ten- fion. Q^E. D. Coroll. When a mufical chord vibrates freely, the forces which accelerate its fmalleft equal arches, are conftantly proportional to their cur- vatures very nearly, provided the latitude of the vibrations be veryfmall in proportion to the length of the chord. For (0 See Theorem t,^ of Keill's Phyfics. Prop. XXIII. HARMONICS. 233 For the force of its tenfion being then very nearly invariable, the forces vs^hich accelerate its fmalleil equal arches are very nearly in the in- verfe ratio of the 7'adii of their curvatures (^), which is the fame as the dired ratio of the cur- vatures themfelves. DEFINITION OF THE HJRMONICJL CV RFE, Fig. 51. Let C be the common center of any two circles D F, EG, and C DEy CFG any two fe77ii diameters-, and of either of the included arches as DF, let FH be the fine., in which pf^ocluced both ways-, let the lines III and UK be fever ally equal to the other arch EG-, then while the fejnidiaiiieter CFG moves round the ceiuer C and carries with it the li7te IFHK, parallel to it felf and coiiflaittly equal to tiioice the arch EG J the extremities /, K will de' fcribe a curve whofe vertex is D a7jd axis D C, and whofe bafe ACB is equal to the femicircumference of the circle EG. (/) By the prefent Propofition. Coroll. 234 HARMONICS. Sed, XL CorolL I. PI. XXII. Fig. 52. Drawing FL per- pendicular to the bafe ACL^ a line KP perpen- dicular to the curve at Ky will be parallel to EL. For drawing KN perpendicular to the bafe, let the radius CFG go fonvards a little into the place Cfgy and carry the line KHFI into tlie place khfiy cutting KN'm O and FL in r. Then fince HK = EG by the definition, and alfo hk :^= Eg J their difference Ok is= Gg. Now by the fimilar triangles CLF and Frf, CFf and CGg, OKk and NPK, we have CL : CF iiFr: Ff, and CFiCG: : F/: G^, and ex aquo CL : CG or CE:: { Fr :Gg::OK: Ok:: ) NP : NK. Confequently the right ang- led Triangles CLE, NPK are equiangular, and the perpendicular KPM is parallel to the line EL. CorolL 2. At any point i^ the radius of curva- ture KM: LE::LE quad. : KNxCE. For drawing y/ parallel to FL ; another line kMy perpendicular to the curve at ky will be pa- rallel to Ely by coroll. i j confequently if the arch Kk be infinitely diminifhed, either of the coin- ciding perpendiculars KMj kM will be the ra- dius of the cui-vature at K. In the line El take Es = EL and joining Xj, the triangles jLE j and KMky CEL or CEl and sLly are ultimately equiangular. Now KNgvFL : LC :: fr : rF ox KO, and LC : L E :: P N : P K :: KO : Kk.znd ex aquo, KN: LE ::fr:Kk'j But Prop.XXIII. HARMONICS. 235 But C E : L E :: s L : LI or fr, therefore componendo^ KNxCE : LE quad. ::{sL: Kk : :) LE'.KM. Coroll. i^> Hence if the ratio of the circles CEG, CDF be vaftly great, the curvature at any point K will be extremely fmall, and its ra- dius KM :CE:: CE : KN very nearly ; be- caufe the lines LE and CE will be veiy nearly equal. Coro/I. 4. Upon the fame flippolition, the very fmall curvatures at any points D, K are very nearly in the ratio of their diflances DC, KN from the bafe AB. For when CE and confequently AB is given, the curvature at K, being reciprocally as its ra- dius KM, is diredtly as KN by coroll. 3. Coroll. 5. Fig. 53. While the greater circle remains let the lefler be diminiffied, and the curve AKDB will be changed into another AxS'B of the fame fpecies, and every ordinate to the common bafe will be diminifhed in the fame ratio, that is, NK : A^x : : CD : CcT. Fig. 52. For while any arch EG equal to HK or CN is p-iven in magnitude, let the other radius CD or CF be diminifhed, and becauie the triangle CF J/ retains its Jpecies, the line CH or NK is diminifhed in the fame ratio with CF or CD. Coroll. 6. Fig. 53. When the axes CD, Cj^of two curves are veiy fmall in comparifon to their common bafe AB, the curvatures at the tops of any 236 HARMONICS. Sed.XI. any two coincident ordinates NK, Ny,j are in the ratio of the ordinates. For if xfjt, be the radius of curvature at jc, by coroll. 3 we have KMxKN-== CE quad. = X /A X 3c A^; whence jc A^ : KN : : KM : k jx, that is, as the curvature at x to the curvature ati^. Coroll. 7. Hence, fuppofing the curve AKDB to have the elafllcity and tenlion of a mufical chord, it will vibrate to and fro in curves very nearly of the fame ipecles with the given curve AKDB, provided none of the vibrations be too large. For let the firfl: effort of the tenfion reduce that cur\T into fome other, as Ay, S" B, in the firft mo- ment of time y and lince the ordinates D C, KN are in proportion as the curvatures at D and K by coroll. 4, and thofe curvatures as the accelerating force at D and K {u), ading In the diredlons DC, KM or AW very nearly, and thefe forces as the velocities generated by them in that time, and the velocities as the nafcent ipaces D J^, Kk ; alfer- nando, we have DC -. DS" '. '. KN : Kx and di- 'videndo, DC : S'C : : KN : x N Confequently by coroll. 5 the curve Ay.S'B is very nearly of the fame fpecies with AKDB. And in the next moment it will be changed into another of the fame Ipecies, and fo on, till every point of the chord be reduced to the bafe AB at the fame in- ftant. And by the motion here acquired it will be carried towards the oppolite fide of the bafe, till by the oppofitlon of the tenfion, it fliall lofe all (u) Coroll. prop, xxiii. Prop. XXIII. HARMONICS. 237 all its motion by the fame degrees, and in the fame curves, by which it was acquired; and thus the chord will continually vibrate in curves of the fame Ipecies as the firft, neglecting the fmall difference in the directions KN, KM, and the reliilance of the air. CorolL 8 . The fmall vibrations of a given mu- fical chord are ifochronous. For if the chord at the limit of its vibratlort afllimes the form of the harmonical curve, it will vibrate to and fro in curves of that fpecies bycoroll. 7, and its feveral particles, being ac- celerated by forces conftantly proportional to tlieir diflances from the bafe ^B (x), will de- fcribe thofe unequal diflances in equal times, like a pendulum moving in a cycloid. If the chord at the limit of its vibration al^ fumes any other form, it will cut an harmonical curve, equal in length to it, in one or more points, as v^, K, L, B in Fig. 54 ; and the in- tercepted parts of the chord will be more or lefs incurvated towards ^B than the correfponding parts of the curve, according as they fall with- out or within them ; and will accordingly be ac- celerated by greater or fmaller forces than thofe of the correfponding parts of the curve (y). Therefore, fuppoling the chord and curve to dif- fer in nothing but their curvatures, the difference of the curvatures of the correfponding parts will be continually dimJniflied by the ditterence of their (x) Cor. prop. XXIII and cor. 6. Defin. curve, (y) Cor. prop, xxiii. 238 HARMONICS. Sea. XL their forces, till the parts coincide either before, or when they arrive at the bafe AB. And thus the times of tlie feveral vibrations of the chord will be the fame as thofe of the curve, and there- fore equal to one another. Co7'olL 9. The Figure contained under the har- monical curve and its bafe, is of the fame fpecies as the Figure of Sines. Fig. 52. For fuppoling the circle D F^to grow bigger till it becomes equal to EGR, the figured KDB will become a figure of fines. Be- caufe any ordinate KN to the ablcifs ^N or arch GRj being conftantly equal to FL, will then be equal to the fine of the arch OR ; and thus every ordinate as KN is incrcafed in the given ratio of CF to CO, or CD to CE. And on the contrary the feveral ordinates in the faid fio^ure of fines diminifhed in that confiiant ratio of CE to CDy are the ordinates in the figure AKDB of the harmonical curve. PROPSITION 'XXIV. ^The vibrations of a mujical chord Jlr etch' edby a weighty are ifochro7'ious to thofe of a pendulufn^ whofe length is to the length of the chords in a co7npoimd ra- tio of the weight of the chord to the weight that firetches ity a7id of the duplicate ratio of the diajiieter of a circle to its circumference. Fig. Prop. XXIV. HARMONICS. 239 PI. XXIII. Fig. 54. If P be the weight that ftretches the chord ADB, and D CM be the radius of its curvature at the vertex D, the force feat urges any fmall particle Dd towards Cis = ^xP by prop. XXIII. And fince Z)^ vibrates like a pendulum (2), if it were fufpended by a firing OP = Z)Cin a cycloid ^JPR = 2DC or DCF, and were ur- ged at the highefl points ^ Rhy 2. force ading downwards like that of gravity, but equal to the faid force jj-r^ x P, which urges Dd ot the limits Z), F of its vibrations ; the times of thofe ofcillations and of thefe vibrations would be equal to one another. Becaufe the forces being alio equal at all other equal diftances of the particle from P and from C, v/culd impel it through equal parts of the equal lines ^P, DC in equal times. Again putting /> for the weight of the chord ADB or ACB, the weight of its particle Dd Dd ^ '' = JB ^^ Hence if another firing L be to the firing OP or DCy as this latter weight •— x/> is to the former jr^ x P, equivalent to the force at Z) ; and the particle Dd he again fufpended by the firing L in another cycloid of the length 2 L ; fmce (z) Coroll. 8. Def. of the curve. 240 HARMONICS. Sed.XL liiice at the hlgheft points of this cycloid the particle is urged downwards by the whole force -— x/> of its own gravity, its ofcillations will be ilbchronous to thofe of the former pendulum {a). Becaufe we took their lengths in the ratio of the forces that aft upon them at the higheft points of the cycloids, that is, L : DC::pxDM:F 9c AB; which two ratios compounded with DC toAB, givQh: AB ::p ^DMx DC or p x CEq {b) : P X ABq, which was to be proved. For CEq is to ABq in the duplicate ratio of the diameter to the circumference, by the definition of the curve ; and we iliewed above that every particle of the curve vibrates in the fame time with the middlemoil. Q^. D. CoroU. I . The time of one femivibration, for- wards or backwards, of the chord AB meafured by inches and decimals, is ^ v^ ^ x .. and its reciprocal is the number of fuch vibra- tions made in one fecond. For the length of a pendulum that vibrates forwards or backwards in one fecond, is 39.126 inches in the latitude of London, and the dia- meter is to the circumference of a circle as 113 to 355 very nearly, and the times of the vibra- tions of pendulums are in the fubduplicate ratio of their lengths. Whence putting t for that of the {a) CoiolL Theor. 4 De Motu Pend. in Mr. CoU<'% Harmon. Mciifiirarum. {b) Coroli. 3. Dmiu Prop. XXIV. HARMONICS. 241 the pendulum L, we have /" : 1" : : ^ L = ^11^ P^ AB : ^ 39. 126, and t" == 111 3SS P ^ 355 y p AB 1 I -^55 /P 39-126 .1 ^ \/ -b X z 3 and - = ^^^ v^ - X ^^!j^ , the P 39. 126 / 113 /> yf5 ' number of femivibrations made in one fecond. CoroU. 2. Suppoling the laft number to be ;/, p we have the loo^arithm of ?f~ = lo^. — -— 4- 2, 58676. 52698, which gives 72 veiy expedi- tioufly. For the logarithm of ^^ x 39. 126 =: 2, 58676. 52698. CoroU. 3. If the lengths and tenfions of two chords be equal, the times of their iingle vi- brations are in tlie fubduplicate ratio of their weights, by coroll. i. CoroU. 4. If their lengths and weights be equal, tlie times of their iingle vibrations are re- ciprocally in the fubduplicate ratio of their ten- fions, by coroll. i. CoroU. 5. If their tenfions be in the ratio of their weights, the times of their fingle vi- brations are in the fubduplicate ratio of their lengths, by coroll. i. CoroU. 6. The weights of cylindrical chords are in a compound ratio of their fpeciic gravities, lengths and fquares of their diamet .ts, that is, p is as s X AB x d^ 5 when-e t is as AB x_d ^ p by coroll. i. Q_ CoroU, 242 HARMONICS. Sea.XI. Coroll. 7. Hence, if the tenfions and diame- ters of homogeneal chords be equal, the times of their fingle vibrations are in the ratio of their lengths. Coroll. 8. If the tenfions and lengths of ho- mogeneal chords be equal, the times of their fingle vibrations are in the ratio of their diame- ters. Coroll. 9. If the tenfions of fi.niilar chords be as their fpecific gravities, the times of their fingle vibrations are in the duplicate ratio of their lengths or of their diameters (<:). Scholium, I. Hence we may find the number of vibrations made in a given time by any mufical found, by comparing it with the found of a given chord firetched by a given weight. For example in the experiment abovemention- ed [d) I found the length of the vibrating chord y^ 5 = 35.55 inches and its weight ^=31 grains troy : And the found of it, when fiiretch- ed by the weight P = 7 pounds averdupois = 49000 grains troy, was two octaves below the found of the pipe d there mentioned. Hence by coroll. 2, we have n = 1 3 1. 04, the number of femivibrations made in one fecond by the wire [c) See GalhWs experiments on chords. Dialogo i^ at- tenente alia Mccanica, towards the end. (d) Prop, xviii. Prop. XXIV. HARMONICS. 243 AB, and 4^ = 524.16, the number of fe- mivibrations made hy-AB, by coroU. 7, or by 4 the pipe ^; which is double the number 262 of its whole vibrations. Before this experiment was made the orifice of the pipe was cut perfedly circular, and then the length of the cylindrical part was exadly 21.6 inches, and its diameter 1.9, which I mention becaufe the experiment, being accurate- ly made, is of ufe upon other occalions. 2. When -the thermometer is at Temperate, the latitude of a pulfe of the found of that pipe is to the length of the pipe, almoft as 2 - to i, by prop. L. Lib. 11. Princip. Philof. 0^2 AD' !244 APPENDIX. ADVERTISE MENT. THOUGH the theory of imperfeB confonances has been demonfirated pretty clearly^ I thinks in the fixth SeBio?!^ yet as I had confidered fome parts of it in d if went lights and fe arched a little further in- to fome others for my oivn diverfon^ I thought it not a?nifs to print my papers i?2 the form of the following Additions 'j that if the reader fhoidd def re any further infor7nation^ he may have recourfe to them whenever he pleafes. THE CONTENTS. Afcholiiim to prop, viii. An illufiration of prop, x, with a fcholium con- firming the theojy of the beats of impcrfeB con- finances. Another demonfiration cffihoUiim ^.prop. xi, [con- cerning the analogy between audible and vifible imdulatiofts) and of prop. vii. Another demorjiration of prop, xiii and its third corollary^ with an illufiration of the fir ft ^ and a fcholium or two confirming the theory of the har- mony of imperficB confonances^ and fiewing the abfolute times and numbers of their vibrations^ Jldort cylces and difiocations of their pidfes, con- tained in the periods between their beats. Schol. 4 to prop, xx^ containing tables and obfer- vations on the numbers of beats of the concords in the principal Jyfiems. SchoL APPENDIX. 245 SchoL 5 to prop, xx, fie'wing the methods of alter- ing the pitch of an organ pipe in order to tune it. Scholium to Prop, viii* Fig. 63 .T ¥ 7 H EN different multiples as 3 AB VV and 2ab of the vibrations AB, ah of imperfed: unifons, are the iingle. vibrations AD^ ac of an imperfe(^l confonance, the multi- pliers 3 and 2 are in the ratio of the Iingle vibra- tions T^AB and 2.AB, or ^ah and 2ab of the perfect confonance, and therefore fliould be ir- reducible to fmaller numbers. 1'he different mul- tiples of the vibrations of imperfedl unifons are therefore fuppofed in the propoiition to be the leall in the fame ratio. PI. XXV. Fig. 63, 64. But if different multi- ples of the vibrations AB^ ah^ as (iAB and \ab^ whofe multipliers 6 and 4 are reducible by a common divifor, be the fin2;le vibrations of an imperfed: confonance, (as they may by intermit- ting 6 — I pulfes of y^^ and 4 — i of ah^ fo as to leave iingle pulfes at firfl and between every intermiffion,) the period of the lmperfed:ions of this confonance will not be equal to tliat of tlie imperfed: unifons AB., ah^ but multiple of it by 2, the greateft common divifor of the multipli^ ers 6 and 4. For thofe multiple vibrations (^AB and A.ab are the fame as 3 x 7.AB and 2 ■-- ^ah, or ^AC ^nd 2ac, in v/hich the equimultiples 2AB and 2ab, or AC and ac in hg. 64, are the fingle vibrations of other imperfed unifons, reiulting Q_3 from 246 APPENDIX. from an intermiflion of every fecond pulie o£ AB and ab in fig. 63 ; and the period of their im- perfedions is equal to that of 3 AC and 2ac^ or AG and ae by this viij'^ propofition, and is the fame multiple of the period of AB and ab, as AC is of AB, or ac G^ ab by the coroU. 4 to this propofition, that is by 2, the greateft common divifor of the multipliers 6 and 4. An illujiratmt of Prop, x. THUS in Fig. 23, PI. xi, after taking away 9 fhort cycles from each end of the cycle ^{7of imperfed: unifons, there remains kKLiUy part of two m.ore; and in Fig. 25, after taking away 3 fhort cycles from each end of the period AX, there remains JD^, part of another 3 and in Fig. 27, after taking away 4 fliort cycles of imperfed o6taves from each end of the period A^iV, there remains iILm, part of 2 morej and laflly in Fig. 34, PI. xii, after taking away 2 fliort cycles of imperfed v'^' from each end of the cycle AZ of the imperfed; unifons, there re- mains 7iN^', part of another : which though not fituated exadly in the middle oi AZ, by rea- fon of the part A g Z of another fliort cycle, containing equimultiples oi AB and ab, is com- paratively very near it when the number of fliort cycles in the period is veiy large as ufual; in which cafe the beats will be made very nearly in the middle of every period. Scholimn. APPENDIX. :S47 SchoUu?n, It is very unreafonable to fuppofe with Mr. Sauveiir that the beats are made by the united force of the coincident pulfes of imperfed: uni- fons (e). For while the imperfecfl uniibns are made to approach gradually to perfedion, experience fhews diat they always beat flower and flower (f) and by theory (^) the periods of their pulfes grow longer and longer. Therefore in confe- quence of that gentleman's hypothefis, the uni- fons fhould alfo beat at the ends of the periods where the pulfes do not coincide : Becaufe it is very improbable that the cycle of unifons, fup- pofing it llmple at firft, while it lengthens gra- dually, will not fometimes be changed into pe- riods as well as into other limple cycles. Nor can it be allowed that the unifons will beat only at the ends of their complex cycles. For according as the numeral terms expreffing the ratios of the fingle vibrations of the feveral fucceffive unifons, happen to be reducible or not reducible or to be irrational, the cycles of the pulfes will fometimes be fhortened, fomedmes lengthened again, fometimes invariable and fome- times impollible, as iliall be explained by and by ; which accidents difagree with die condant gradual retardation of the beats in the prefent cafe. CL.4 If {e") Prop. XI. fcho]. 3. (/) See Phasnomena of beats placed before pr^p. x. (^} Cor. 5. lemma to prop. ix. 248 APPENDIX. If it be faid that the pulfes next to the perio- dical points fall fo clofe to one another, as to affed: the ear in the fame manner as if they were quite coincident ; it rnay be fo, and moft proba- bly is fo. And then it will follow that the bar-' mony of the fhort cycles terminated by fiich clofe pulfes, will there be much the fame as that of perfect unifons ; at leaft it will certainly be better about the periodical points and coincident pulfes than any where elfe in the periods. But the found of a beat has no harmony in it , on the contrary it rather refembles the common found of a beat or ftroke upon any grofs, irregular body : And this found refults from pulfes of air which re- bounding from different parts of the body, difpo- fed to vibrate in different times, will ftrike the ear one after another at irregular intervals, like the pulfes in the middle between the periodical points of imperfed: unifons. Therefore thefe are the only pulfes in each period, which can excite the fenlation of the beats of impeffed: unifons. And the like argument is applicable to any other im- perfed confonance by prop. viii. PI. XX IV. Fig. 55. As to thofe uncertain lengths abovementioned of the fimple and complex cy- cles of the pulfes of imperfed unifons, while their interval is continually diminished or increafedj let one of the founds be fixt and the time of its Imgle vibration be reprefented by any given line V and thofe of the variable found by the fuccef- five lines ^, 5, C, &c, all v/hich lines may con- ftitute any increaiing or decreafing progreffion ; and APPENDIX. 249 and fuppoling n to rcprefent any large given num- ber, let^ : V'. : n : a, B: F: '.n\b, C : F:i n : c, &c. Then will the cycles of the pulfes of F" and yf, rand B, Fund C, &c, be 72 F= a A, nF=bB, nF= cC, 8cc, provided every one of the num- bers a, d, Cj &c, be integers and primes w^ith refped: to the alTumed number 72. In which cafe the feveral cycles are equal to one another and to nF. But if the terms of all or any of thofe ratios have a common divifor, the correlponding cycles will be fliortened in proportion as the greateft common divifors are larger ; and therefore their lengths cannot increafe or decreafe fucceffively in regular order while the fucceffive intervals of the unifons continually decreafe or increafe, un- iefs the greateft common divifors decreafe or in- creafe in regular order too j which can happen but very rarely. And when the terms of the ratios of any of the vibrations happen to be incommenfurable, a fe- cond coincidence of their pulfes will be impoffi- ble : becaufe no muldple of one vibration can be equal to any multiple of the other. But in all cafes whatever, the periods of the pulfes of F and A F and B, F and C, &c, which are ^, JLK., ^, 5^c (Z)), will decreafe continually in the fame proportions with the frac- tions (/;) Def. 1:1. fed. vi. 250 APPENDIX. tions , — -5 whofe magnitudes can ne- n — a' n — b n — o ° ver be altered by any common divifors of their terms, whether integers fractions or fards. Another demonjlration of fcholimn 5. Prop, xiy and of Prep, vii. T H E breadth of the apparent Undulations of the lights and fhades feen at a diftance upon two rows of parallel objeds, may be alfo found by the following conflrudlion. PI. XXIV. Fig. 56. Let a plane palling through a diftant eye at z, cut the axes of the parallel ob- jed;s at right angles in the points a, b, c, &c, a, /8, y, &c, which are fuppofed equidiftant in both the parallel lines abc, a^y- From any objed: in one of thcfc lines to any fucceffive objects in the other, draw the lines a^, a/^, af, &c, and the lines z-vV, zxX^ ^jT^ &c, drawn parallel to them, will intercept the equal breadths of the apparent undulations. Becaufe while the eye is gradually dire(3:ed from the middle of any of the breadths VX^ XT^ &c, towards either of its extremities, the objedts will appear clofer together in couples, in propor- tion to their fmaller diftances from the next ex- tremity, which was fliewn in this fcholium to be the caufe of the undulations. For the lines ^cT, es^f^y ^jj, &c, being parallel to ^ct, are parallel to Vv by confliudion; and the lines /$, k i, /x, ///A, &c, being parallel to APPENDIX. 251 hoL, are parallel to Xx j and fo on. Let lines drawn from z through the objeds cT, g, ^, &c, of one row, cut the line of die other in JD, E, F, Sec. Then becaufe the rows are parallel, the ratio of D^T to £^2;, £g to ez, F^ to ^z, &c, is the fame as Fv to vzj or Xx to .x-z, &c (/). Whence alfo, becaufe of the parallels between the rows, we have Dd:dF::D: :Fv ivz-, Ik :kX III I : I z KI :l X::Ky: :x,z Lm : mX : : Lk : Xz &c. &c. That is, all thofe ratios are equal, and, alter- nately, the leffer apparent intervals Dd^Ee^ Ffy Ggj are proportional to their diftance dF, eF, fF, gFj from the next extremity F of the breadth F'X', and alfo Hi, Ik, KI, Lm, propordonal to zX, kX, IX, mX, their diftances from the next extremity X of the fame breadth FX, And the breadths FX, XT, &c, are equal, becaufe ab, be, &c are fo, and the triangles FzX ^jx^Laccb, XzY2sA boiCy 6cc, are iimilar by conflrudiion. Qj:.D. Coroll.i. The projeflions DE, EF, &c, of the equal intervals d^g, g^, &c, are to thefe intervals in (/) 2 VI. Euclid. 252 APPENDIX. in the conilant ratio of jD^j to S'z, or Ez to €Z, or Fz to vz, and confequently are equal to one another. Therefore fuppoling the lines DE and S'e ov de to reprefent the times of the iingle vi- brations of imperfed; unifons, the periods of the neareft approaches of their pulfes Z), £, &c. d, e, &c, are VX, XT, &c ; And in going from their extremities F, X to the middle, the alternate lef- fer intervals between the fucceffive pulfes, are proportional to their diftances from the next ex- tremity, as we fhewed jufl now : which is ano- ther proof of prop. vii. Cor oil. 2. If the eye be moved in a line paral- lel to the rows, the breadths of the apparent un- dulations will be conftantly the fame, and if it be moved uniformly in any other right line, their breadths will vary uniformly, and be conftantly proportional to the diftance of the eye from the rows. Becaufe the triangles FzX, FzT, &c, are conftantly ftmilar to accl^, aac, &c. And this conclulion feems to agree with what I have tran- fiently obferved of thefe undulations. But it is eafy to colled: from the conftrudion of the figure, and the different ratios o£ zF to zv exprelled by numbers, that the intervals be- tween the apparent conjundions of the objeds will increafe and decreafe very irregularly j and that no conjundions can happen except when the eye arrives at certain points of its courfe, and none at all, mathematically ipeaking, when its diftances from the two rows, meafured upon any right line, happen to be incommenfurable. Which APPENDIX. 253 Which conclufions being contrary to the conti- nual appearance of the undulations to the eye in all places, and to the regular increafe or de- creafe of their breadth, fhew, that their breadth is not equal to the interval between the apparent conjunctions, no more than the interval between the beats of imperfecft unifons is equal to the in- terval between their coincident pulfes. LEMMA. Ifi any period betiveen the fiiccejjive beats of an imperfeSi confonance, any given nu7nber ofJJjort cycles next to one fide of the leaf difiocation of the pulfes^ is more harmonious^ and the fame number of them next to the other fide is lefs har7noniom than the fame number of them next to either fide of the coincident pulfes : and thefe degrees of harmony diff.r more in thofe periods where the two leafi difiocations differ lefs, andmoft of all in the periods where thefe difiocatio?js are equal when pojjible. PI. XXV. Fig. 59. Let AB and ab reprefent the times of the lingle vibrations of imperfedl unilbnSj A and a their coincident pulfes, B, C, X), 8cc, b, Cj dy &c, their fucceffive pulfes on each fide of A, a ', Rr their leaft difiocation in any given period, and confequently the neareft to the periodical point 2;, which is here placed un- der A, for the convenience of feeing at one viewj 254 APPENDIX. view, the fhort cycles next to both fides of Rr and ^a. Firji I fay, the fhort cycles RS, ST, &c, which include z, are more harmonious, and R^^^P, &c, lefs harmonious than ^B, BC, &c, the numbers of them being the fame : and that the degrees of their harmony differ more in die periods where the two leaft diflocations i?r, sS differ lefs, and moft of all where Rr =sSy when poflible {k). For bB — {JB — Ab — RS—rs—)Rr+sS{l). And cC—{JC-^Jcz=iRr—rt=:)Rr-^tT. &c. &c. Hence the fucceflive diflocations sS, fT, &c, are refpedively fmaller than ^B, cC, &c, by i?r, as appears alio by their fmaller diflances from tz [m). But on the other fide of Rr, the difloca- tions ^j Pp, &e, are refpedively greater than bBj cC:, &c, by the fame Rr, for the like reafons. Now the fhort cycle RS which includes z, is more harmonious than ^B next to ^. For though the diflocations Rr+sS^rQ=l?B, yet thofe parts of l^B, as being fmaller than i>B, will give lefs offence to the ear than the whole : the whole may be perceived and give fome offence even when one or both its parts are impercepti- ble. And for the fame reafons the fhort cycle RS will be ftill more harmonious than j4B in other periods (i) See prop. VII. coroU. 2. (/) Sec prop. vji. coroll. I, {m) Prop. VJI, APPENDIX. 255 periods where Rr,sS are left unequal, and the moft harmonious where they are equal when poffible (n) : their fum being every where the fame. The next fhort cycle ^ST" is alfo more harmo- nious than5C; the diflocations sS^ ^'T being re- fpedtively fmaller than b By c C. Therefore the fliort cycles RS^ 5 T, taken together, are more harmonious than AB, BC taken together ; and flill more harmonious in otlier periods where sSy tT arc. fmaller, till sS he equal to Rr. But on the other fide of Rr and Aa, the fhort cycle R ^is lefs harmonious than AB, the dif- locations ^q, Rr being larger than Bb and o. The next fhort cycle ^P is alfo lefs harmonious than BC; the diflocations P/>, ^g being refpec- tively larger than Cc, Bb. Therefore R ^ ^P together are lefs harmonious than AB^ BC to- gether J and ftill lefs harmonious in other periods where Rr, ^, Pp are larger, till Pr be equal to sS. And the fame is evident in any larger equal numbers of fhort cycles throughout the period between the fuccefUve beats. Secondly J any imperfe(5t unifons will be chang- ed into imperfect o6taves whofe fingle vibrations are AC^nd. Ab^ ox Ac 2.ndAB, by conceiving every fecond pulfe of the feries Aj P, C, &c, or ^, b, c, &c, to be intermitted, which would de- prefs one of the unifons an odave lower. Now if that intermilTion fhould take away the alternate pulfes 5, U, &c, or s, «, &;c, the fliort cycles («) Prop, VII. coroll. 2. 256 APPENDIX. cycles of the ocftaves, next to one fide of Rr, will be R T, TJV, &c, and on the other, R P, PN, &c : I {^.y the former as including z are more, and the latter lefs harmonious than ACj CEy &c, the numbers of them being equal. For we had Rr -\- t'T= c Cy confequently the fliort cycle ^ T' is more harmonious than u4 C, for the fame reafon as in unifons, and be- caufe the intermediate dillocations sSy bB are vaniilied, one of their conftituent pulfes in each being taken away. And i^ 'T is ftill more harmo- nious than y^C in other periods where Rr and tT are lefs unequal. The next fhort cycle 'TW^h alio more har- monious than CEy the diflocations tTy wW be- ing refpedively fmaller than cCy eEy as in uni- fons ; and is llill more harmonious in other pe- riods where t%wJV are fmaller, that is where t T'and Rr are lefs unequal. But on the other fide of Rr and ^a, the fhort cycle RP IS lefs harmonious than y^C, and PN than CEy the dillocations Rr, Pp being relpec- tively bigger than o and Cc ; and Ppy Nn bigger than Ccy E e, refped^ively : and is Hill lefs harmo- nious in other periods where Rry Ppy Nn are larger, that is where Rr, t'Ts.re lefs unequal. Therefore the Ihort cycles RTy TlV, &c are more, and i^P, P A^, &c are lefs harmonious than ^C, CE, &c. Likewife if that alternate intermiffion ll:iould take away the pulfes P, T, JVy &c, or r, /, w, 6cc, then the lealt dillocation is s S, and the iliort cycles APPENDIX. 257 Cycles S ^ ^O, &c, as including z, will be more, and SU, UX, Sec lefs harmonious than ACy CEy &c, for the very fame reafons as before. Th'rMy, any imperfedt unifons will be changed into imperfed: v''^^ whofe vibrations are j^c and AD, (or AC and Ad) that is zalf and ^AB^ by intermitting 2 — i pulies of the feries a, If, Cy dy &c, which deprefles the acuter unifon an VI 11*'' lower, and 3 — i pulfes of the feries Ay B, Cy Dy &c leaving fingle ones between, which dcpreffes the graver unifon a xii'*^ or viii + v^^ lower ; and thus the interval of the new founds is an imperfed: v^'', as reprefented in the upper- moft parallel in the figure. Now in the period where thofe intermiffions leave the pulfes r, t, w, ^, &c, i?, U, T, &c, (as in the 4'^ parallel) the intermediate ones will be taken away, and then i?r being the lelTer of the two diilocations in the fhort cycle RT which includes z, is the leaft of all in this period. And the fhort cycles RTy &c, on this fide of Rr^ will be more harmonious than AG, &c (in the firfl parallel) j and on the other fide, the fhort cycles RLy &c, will be lefs harmonious than AGy &c : For the fame reafons ^s above. Likewife in the period where the pulfes ^, s, Uy Xy &c, ^, Ty Xy Scc Q-VC Icft (iu the ^^^ paral- lel), the intermediate ones will be abfent, and then ^q is the leafl diflocation in this period, and ^ greater difference than before will be found in the harmony of the fhort cycles on e^ch fide of 25^ APPENDIX. !^q and y4a ; the difference Xx — ^q being lefS than Ty — Rr in the former cafe. La/ify in the period where />, r, f, w, &c, P, S, IV, &c, are left (in the loweft parallel), the intermediate ones are intermitted, and then Pp is the leaft dillocation in this period, and a difference ftill greater will be found in the har- mony of the fliort cycles on each fide of Pp and A a, for the like reafon. And the greatefl diiference will be found where thefe diflocations are equal when poffible , that is, when a perio- dical point z bifedis a fliort cycle of any confb- nance, which confifts of any odd number of thofe of the unifons ; and alfo when either of the coincident pulfes at the ends of the complex or fimple cycles of the unifons, bifeds a fhort cycle of any ccnfonance confifting of any even number of thofe of the unifons as in Fig. 35. Plate XII. The like proof is plainly applicable to the vibrations AC, Ad, or to thofe of any other confonance. QJ^. D. Coroll. Hence any two imperfe.fl confonancea tvill be as equally harmonious as they poffibly can be, when the periods (betv/een their fuc- ceffive beats) which are bifeded by their coin- cident pulfes, are made equally harmonious j^ thefe periods having a mean degree of harmony among thofe of all the other periods in each con- fonance. All thofe degrees of harmony occur in prac- tical mufic, and whether feniibly different or not,. APPENDIX. 259 not (0), muft be ufed as if they were equal, and in theory we muft take the medium among them. As the proof of this conclufion has been pret- ty long, I avoided it in the Book by a paragraph in the dcmonftration of prop, xiii, which may now be proved fomewhat differently. Another demo7iJiration of Prop, xiii» and its third corollary, PL XXV. Fig. 60, 61. Let op and OF repre- fent the times of the fingle vibrations of imper- fect unifons ; ab and AB thofe of other imper- fed: unifons j and O, a and A their coincident puh'es ; and \i ab ^= op^ the period of the pulfes of the former unifons, will be to that of the latter, ultimately as bB to pP (p). I. Taking bB to pP as i to 2, this is now the ratio of the lengths of the periods of the unifons op and OP, ab and ABy and the latter is of the fame length as the period of the leaft imperfedions of od:aves, whofe fingle vibra- tions are ab and AC or lAB^ by intermitting 2 — I pulfes of the feries Ay B-, C, Z), Sec, by prop. VIII. Now the fhorter length of the fliort cycles of the unifons c/», OP, is op = ab, and that of the fhort cycles of the imperfect od:aves is ac or 2aby R 2 and {0) Prop. XI. fchol. 4. art, 5. laft paragr. (/>} Cor. 8. lem. \o prop, jx, and prpp. xi. fchol. i. :^6o APPENDIX. and the ratio of their lengths is i to 2, Which being the fame as that of the periods of the uni- fons and odaves, fliews that their fhort cycles are equally numerous in them. The longer length of the fhort cycle of the odtaves is ^-f Cor 2ABi and the difference of the lengths is 2AB — 2^^ = 2^i5 = <:C the dif- location of the pulfes at the end of the iirft fhort cycle, and is equal to pP, becaufe we took b B : pP :: i : 2 j therefore the feveral diflocations eE, 6cc, q^ &c, at the ends of the fubfequent fhort cycles of the oftaves and unifons, are equal refpedlively throughout their half periods, which are therefore equally harmonious : Becaufe thofe diflocations are the caufes that fpoil the harmony, more or lefs according as they are greater or fmaller; and caufes conflantly equal muft have equal effed:s : And becaufe the harmony of thefe half periods is the medium among the degrees of harmony of all the reft, by the coroU. to the lemma. 2. Fig. 60, 62. Again, taking bB to pP as I to 3, this is now the ratio of the periods of the imperfed: unifons op and OP^, ab and AB; and the latter period is equal to that of imperfe(5l xii*^% whofe vibrations will be ab and AD oz "^AB by intermitting 3 — i pulfes of the feries A, Bj Q -D, &c, fo as to leave fingle pulfes be- tween every intermifhon (g). And fince Pp = i^bB =) ciD, it appears that the feveral fubfe- quent diflocations q^ &c, gG, &c, of the uni- fons (q) Prop. VIII. APPENDIX. 261 fbns and xii^' are equally numerous and equal refpedively throughout the half periods on each lide oi oO and a A-, which render the confo- nances as equally harmonious as they poffibly can be, for the reafons above mentioned. 3. Fig. 60, 63. Laftly, taking bB to pP as I to 2x3, this is now the ratio of the lengths of the periods of the dillocations of the imper- fed: unifons op and OP^ ab and AB^ for the reafon above. And the latter period is of the fame length as that of imperfed: v'^^, whole fingle vibrations 2.ab and 'X,AB refult from in- termitting 2 — I pulfes of the feries ^, b^ r, d^ e, f, g, &c, and 3 — I pulfes of the feries A, B, C, Z), £, Fy G, &c, fo as to leave fingle pulfes at the beginning, and between every intermiflion, by prop. VIII. Now the fhorter length of the fhort cycle of the unifons op, OP is op = ab, and that of the fhort cycle of the imperfed: v'^' is 2x3^^ (be- caufe 2a b : 3^^:: 2 : 3) and the ratio of thefe lengths is i to 2 x 3 , the fame as that of the pe- riods of the imperfed unifons op, OP and the v*^% whofe fliort cycles op and ag are therefore equally numerous in them. The longer length of the imperfed Hiort cycle of the v^*^^ is 2 X 3 AB (becaufe zAB : 3 AB : : 2:3) and the difference of the longer and fhorter lengths is 2 X 3 AB — 2 x 3 ^^=2 x 3 x AB — Ab == 2 X 2bB=gG, the diflocation of the pulfes at the end of that fhort cycle, and is equal to />P, R 3 becaufe 262 APPENDIX, becaufe we took l?B :pP :: i : 2x 2- Therefore the feveral diilocations ?2N, &c, q^, &c at the ends of all the fubfequent fhort cycles of the v^^* and unifons, are refpedively equal in magnitude and number too, throughout the half periods on each fide of the coincident pulfes aA.oO; which equalities make thefe confonances as equally har- monious as they poiTibly can be, for the reaibns above. 4. Inftead of the terms 2 and 3 of the ratio of the vibrations of perfed: v'^', if we fubftitute thofe of any other perfect confonance, or m and n indeterminately for them, the method of de- monftration will be evidently the fame as in the laft example. Now thofe imperfed; confonances of viii*^% ^jjths^ yths^ ^^ ^j-g j^Q|. Qj^iy equally harmonious with the fame imperfed: unifons op, OP, but alfo v/ith one another j the diflocations/>P, cC, ^Z), g G, at the ends of their firft and fubfequent fhort cycles, being equal and equally numerous in their periods. And fince any one of them is equally harmonious to another of the fame name at any other pitch, when their fhort cycles are equally numerous in their periods (r), it appears that all forts of imperfect confonances are as equally harmonious as poflible, when their fhort cycles are equally numerous in the periods of their imperfedions. QJi. D, The equal harmony of flat confonances is de- monftrable in the fiime manner. CoroIL [r) Prop. XII. APPENDIX. 263 Coroll. Hence when imperfcd: conlbnances are equally harmonious, their temperaments have very nearly the invcrfe ratio of the produdts of the terms expreffing the ratios of the Ungle vi- brations of the perfedl confonances. This is the third corollary to prop, xiii and may be demonflrated in this other manner. The interval of the founds of imperfe(5l unit- fons is the temperament of the interval of any confonance \^'hofe iingle vibrations are different multiples of the vibrations of thofc unifons (i). Now in all the exam.ples of tempered con»- ibnances vve made the vibration ab conftant and AB variable. Confequently the feveral in- tervals of thefe imperfed: unifons, or the loga- rithms of the ratios qI ab to AB were very nearly proportional to the differences bB {f)^ which in the viii*^' and v^'^' v/ere made equal to -^ pP and — pP refpedively. Therefore 1x2 ■' 2x3 i i: J when theie confonances are equally harmonious, the ratio of their temperaments is — to — ^ 1x2 2x3 very nearly. And when either of them is equally harmo^- nious to another of the fame name at a diffe- rent pitch, their temperaments are eqml {u)y and the terms of the ratio of the vibrations of R 4 the (i) Prop. VIII. cor. i. [t] Cor. I. lem. to prop, IX. («} i^rop. XII. coroll. ■264. APPENDIX. the perfed confonances of that name are the fame. Confequently the dired ratio of the tempera- ments and the inverfe ratio of the produds of thofe termSj are very nearly the lame in all equally harmonious confonances. j^n illujlration of corolL i . Prop. xiii. PI. XXIV. Fig. ^y. Let the intervals of the equi- diftant points ^, I, II, III, &c be the longer or the fhorter lengths of the imperfedl fhort cycles of any given confonancej whofe half period is AP-, A and a its coincident pulfes i ab the lefTer of the vibrations of the imperfed: unifons w^hofe half period is alfo AP. Make the perpendicular P.^= -ab^ and draw ^ ^cutting the perpen- diculars at I, II, III, &c, in Z), Z), Z), &c. Then are thefe perpendiculars equal to the dillocations of the pulfes between the Hicceffive fhort cycles of the imperfed: confonance, by prop, vii and VIII. Fig. 58. Make the like conftrudion denoted by the greek letters for any other imperfed: con- fonance of the fame or a different name. And if it.be equally harmonious to the former, its half period clit will contain the fame number of fliort cycles as AP does (x) j fuppofe 6 in each. By leffening its temperament, let its half period be lengthened to a^, where ered:ing the perpen- dicular (-r) Prop. XII, XIII. APPENDIX. 265 dlcular/'^='7r)c join ctq cutting all the inter- mediate perpendiculars in ^,f,&c.Then the feve- ral new diilocations i ^, 2 ^j 3 ^, &c will be fmal- ler than iJ", 2j^, 3cr&c refpedively. Therefore the fhort cycles a. 6 £', contained in a part of the new half period a^, are not only more harmo- nious than the fhort cycles a6j^, contained in the old half period a7r>c, or than AN\D^ but thofe in the remaining part e6j%e continue the harmony in die new half period a/, after that of the old half period is quite extinguiffied by the beat at the end of it. Coroll. Since only the correiponding fhort cy- cles of imperfedl confonances can admit of a juft comparifon, one by one, in the order of their fucceffion, beginning from the coincident pulfes, or from their leaft diilocations next to the perio- dical points, (as explained in the demonftration of the lemma,) if the periods of two confo- nances contain unequal numbers of their fhort cycles, the comparifon will be imperfedij which is another argument a priori for the truth of the XI I ^'^ and xiii^'^propofitions. Scholium I, In any pure confonance i^y) the fliort cycle contains but one vibration of the bafe, as in Fig. 61, 62, and the equal times between the pulfes of the treble are never fubdivided by any pulfes of the bafe, except at the ends of the iliort (;') Se6l. III. art. 8. 266 APPENDIX. fhort cycles; and here the diflocations cC^ dD are confidered and adjufted widi the analogous ones in other pure confonances, bythexiii^^ propofition. But In any other confbnance whofe fhort cycle contains feveral vibrations of the bale, the equal times between the pulfes of the treble are Sub- divided by the pulfes of the bafe, not only at the ends of the fliort cycles, but between them, as at £), Ki &c, Fig. 63 ; where the confidera- tion of the inequalities of the intervals cD and De, /if and Kl, &c, feems to have been neglect- ed in the faid propofition, but in reality is implied in it. Fig. 63. For fuppofing the alternate pulfes Z), K, 6cc to be intermitted or taken away, thofe v'l^' will be changed into XII ^''^ an od:ave lower than the xii^^^ in Fig. 62 ; but will not be equal- ly harmonious with them, as the v'^' were fup- pofed to be before that intermiflion, till the dift locations, ^G, nN^ &c, in Fig. 63 be doubled; tliat the temperaments of both the xii*^' may be equal and their periods proportional to their vi- brations and fhort cycles [z). While the diflocations ^G, 72 N, &cc, remain doubled, reflore the pulfes D, K, &c, to their places, and now the intermediate inequalities .Dc — De, Ki — Kl, Sec are alfo the doubles of their former magnitudes and the new v*'^^ are lefs harmonious than the xii^"^' in Fig. 62, and will i\ot be equally harmonious with them till the diflocation, ^.v) Prop, XII. corolJ. APPENDIX. 267 diilocation, gGy nNy &c, and confequently tlie inequalities Dc — Di\ Ki — Kl^ &c be contrac- ted to dieir former magnitudes. Therefore thefe interrupted confonances are not confidered as pure ones in the xii*'^ and xiii^ proportions, but allowance is made on courie for the effed of the intermediate pulfes of the bafe, Scholium 2, PI. XXV. Fig. 63. Suppofing the letters d^ k to be reflored to the places of the ablent pulfes of the imperfed; unifons, that fall next to JJ and K^ I call the lines or times dD^ ^i^the Aberrations of the interior pulfes D, K^ from the places d^, k which they have in the perfed fhort cycles. Likewife in the upper part of the fame figure, if AE and adh^ the fmgle vibrations of an imper- fed: 4'^, then £^ is an aberration of one of the interior pulfes of the bafe in the firil fliort cycle. Now if the ratio of the times of the iingle vibrations of any perfed conlbnance be m to n in the leafl integers, and when it is tempered, if 2Z) be the fum of the exterior diflocations in any given fiiort cycle, the aberration or fum of the aberrations of the interior pulfes of the bafe, from the places they have when the confonance is perfed, will be 7z — i x D, The reafon of the theorem will fbon appear h^ drawing a fhort cycle or two of a 4^'' , iii^, &;c. ^68 APPENDIX. &c, and by obferving, that as ;/ is the numter of the vibrations of the bafe contained in any fhort cycle, fo n — i is the number of its pulfes exclulive of the extreams, and that the fum of the exterior diflocations is equal to the fum of any two interior aberrations equidiftant from them, or to double the aberration in the middle; as is plain from the arithmetical progreffion of the alternate lefler intervals of the imperfed: unifons, from which the given confonance is de- rived. Therefore in two equally harmonious conib- nances, as the fum of the exterior diflocations in any fhort cycle of the one, is to the fum of them in the correfponding fhort cycle of the other, in a certain conflant ratio [d)^ fo the interior aber- ration or the fum of the interior aberrations in the former fhort cycle, is to the fum of them in the latter in another conftant ratio ; and compo- nejjdoj the totals of the exterior diflocations and interior aberrations are alfo in another conflant ratio. But tlie temperaments and periods of the tAVo confonances muft be adjufled by the firft given ratio alone, without any regard to the fecond or third. I . Becaufe the exterior diflocations are of a different kind from the interior aberrations. For as in feeing fo in hearing, it is more difficult for the fenfe to perceive the quantity of a fmall inequa- lity in the larger fucceflive interval of the points or {a) By the foregoing illuflration. APPENDIX. 269 or pulfes Cy D, e^ Fig. 63, than to perceive the fame or a different fmall quantity when bound- ed by two vifible points or audible pulfes g^ G. And the difficulty is greater in more complex fhort cycles of imperfed /\}\ ilia's vi^% &c, where the fucceffive intervals between the points analogus to f, D, e, do not err from the fimplejft ratio of I to I, but from the more complex ones of I to 2, I to 3, &c ; as will eafily appear from the difpofition of the pulfes in fuch cycles in Fig. 5, Plate i. For which reafon the ratio of the fum of the interior aberrations ought not to be compared and compounded with that of the ex- terior dillocations. 2. Becaufe it appears from the corollary to the foregoing Illuflration, that no other regard can be had to the interior aberrations^ than what fol- lows on courfe from the given ratio of the exte- rior dillocations, determined by the equality of their numbers in the periods of the two confo- nances, as in prop, xii and xiii. Scholium 3. To give the reader more determinate ideas of the numbers of vibrations, fhort cycles and dif- locations contained in the long cycles and periods of imperfe(5t confonances, and of their abfolutc juration in practical mulic, I will add a com- putation of them in a confonance of v**"' tem- pered 17^ APPENDIX. pered by - comma, as it ufually is, more or lefs, in organs and harpfichords. PI. XII. Fig. 34. If AB '. ab '.', 322 : 321, the interval of the founds of thefe vibrations is i comma nearly (<^). Whence 321 AB^=^i22ab = AZ, is the length of the fimple cycle of the diflocations of the pulfes of the vibrations AB, a by or of the period of the imperfedions of any confbnances whofe vibrations are different mul- tiples of AB and ab {c) and whofe tempera- ment is the intei-val of the founds o£ AB and ab (d). Now the vibrations of imperfedt v*^** are AD and ac, or 2^^ ^^^ ^^^^ and the two conftant lengths of their imperfect fhort cycles are AG= 2AD=2\'2AB and ag='i^ac='2\2ab. Hence ^Z=32i^5 = ^'x 3^5 = loj AD-='^x6AB = s2>l^^' Likewife AZ == 022^/^ = — x 2^/^ =2 2 l6iac='^~x6ab = s?>i^g' And after the coincidence of the pulfes, their jErfl dillocation is ^G == -i. AD j and the limi^ of the g;reateft dillocation is - j^ 5= ^ AD. ^ a . 644. Foj; {b) Prop. XI. fchol.4. art. 61 (/) Prop. VIII. {d) Prop. vm. cgr. i. APPENDIX* 27> For AB : AB — M, or bB \i 322 : i, whence Bb=^ AB, and ^G = 6^5 = ^ ^22 '^ 322 ^l4S = -Ljn, and '.a6= I. ?il^B== 161 161 2 2 322 ix 11' X -AD = ^ AD, and is the limit of 2 322 3 644 the greateft diflocation, or alternate lefTer inter- val of the pulfes of AB, ab in any half period, by prop. VII. Now by an experiment mentioned in prop. XVIII, I found that the particles of air in an or- gan pipe called d or d-la-fol-re, in the middle of thefcaleof the open diapafon, made 262 com- plete vibrations or returns to the places they went from, and confequently propagated 262 pulies of air to the ear (e) in one fecond of time ; though the pitch of the organ was above half a tone lower than the prefent pitch at the Opera. And taking that found for the bale of our v^^, whofe vibration AD reprefents a certain quantity of time, we have 262 AD = 1 fecond and hence the abfolute times AZ, AG, Gg and ~ ab are the following fractions of i". For, 262^!): ^Z or 107^2) ::i":~-^ xi". and262^D :^Gor 2AD\\\" '.— ^\" 131 and 262^2) :Gf ox-^ AD'.\^'',-r^ ^ 161 262 X 42182 {/) Se£l. I. art. 12. 262 X i6i and 272 APPENDIX. and 262^2) : ^ ^^or iH7 ^jD :: i": -^i^ 2 644 262x644 ,// I ^n ?< I == X I . 1568 And the reciprocal of the periodical time AZ,= 122 X i", is the number of periods and alfo of 262 beats in i" (/) namely ^ = 2. 45 nearly in i'', or 245 in 100" nearly. And the leafl: dillocations in the fhort cycles, as AgAK, which include the fucceffive periodi- cal points Z, are 1, -> 1 of the dillocationG^ next to the coincident puUes. And thefe meafures are to thofe In any other v**" in the fcale of that organ, in the given ratio of the times of the fingle vibrations of their ba- fes. And the like meafures in any other given confonance, whofe temperament is given, may be computed in the like manner, or derived from thefe by the corollaries to prop. ix. In this example the v^*"* were tempered fharp, and when they are equally tempered flat, by tak- ing yld and AC for the fingle vibrations, the computation and the meafures vy^ill be but very little different. (/) Prop. X. Scholium APPENDIX. 273 Scholium 4 to prop, xx* tables and obfervations on the numbers of beats, of the concords in the principal fyjiems. The following table flicws the number of beats made in 1 5 feconds, by the feveral con- cords to the bafe note D, or D-fol-re, at the Roman Pitch, and likewife the proportions of the beats of the fame concords to any other bafe note ; with this defign, that perfons want- ing leifure or proper qualifications for examin- ing the principles and conclulions requifite to determine the fyftem of equal harmony, may yet form fome judgment of its advantages and difadvantages , when compared with the fy- ftem of mean tones and diat of Mr. Huygens; upon this allowed principle, that, a^eteris pari- bus^ concords to the fame bafe are more or lefs difagreeable in their kind for beating fafler or (lower refpedlively. To affift the reader's judgment I have added the following obfervations refulting from inipec- tion of the firft table. 1. The beats in column i differ lefs from one another than thofe in column 2 and 3 do, agreeably to the Name oi tlie fyiiem. 2. The beats of the V, v-f viii, v4-2viii in col. I, are a little quicker than thofe in col. 2, in the ratio of 10 to 9, and quicker than thofe in col. 3, in a ratio fomething greater. But the S beats 274 APPENDIX. beats of the vi, vi + viii, vi + 2viii in col. | are much flower than thofe in col. 2 and 3, in the ratio of 2 to 3 and more. 3. Thefe quick beats of the vi^^ in col. 2 and 3 have the difadvantage to be doubled in every . TAB. I. I'kc beats in 1 5" of all th /'—_-. VI +2VIII.— 20 80 120 ' '33 r; - V -f2VIII. 2 40 36 8 34-77 III +2VIII. J VI -f VIII. ^ 10 13 60 r 47 '1- 40 66^ II V + viii.i 3 20 18 17- •' II III -f VIII.- 5 13 I ^-4 71. i 5 20 30 3377 V. ^ 3 20 18 17- ' II ill. i 5 Syftem of ^3 I equal harm. mean tones 2 Column I 1 3 Appendix. 275 every fuperior o6tave, whereas the quick beats of the v''* in col. i. remain the fame in tlie fe- cond odtave and are only double in the third &c. 4. The beats of the iii*^ and compounds are quicker indeed in col. i than thofe in col. 2 and S 2 3; TAB. L - - - - - concords to the bafe D-fol-re. 3' 4-2VIII. _5_ 24 4^^+2VIII. 3 16 6^^+2VIII. _5 32 3^^ + VIII. 5. 12 4^^+ VIII. 3 8 6*^+ VIII. 5 16 3 • b 4th 3 ^ '4 ^ '8 Syftem of Column 48 72 SI 48 42 24 36 26^ 3 24 2oi 5 equal harm. mean tones I 2 80 46 14 40 23 7 gens. 276 APPENDIX. 3 ; but being flower than the beats of the other concords in all the fyftems, they can fcarce be fo oiFenfive as thefe will be. 5. Likewife the beats of the 6^^ and com- pounds in col. I, being (lower than thofe of the th TAB. II. 'The order of the harmony VI +2VIII V +2VIII III +2VIII VI+ VIII V + VIII III+ VIII VI V III Syftem of Column 240 40 13 120 20 26 60 40 equal harm. 360 399,7 36 8 34- I 180 »99^ i8 17- ' ir 8^ 2 90 99^ 36 8 34- 17 mean M. Huy- tones gens. 2 3 — ». I APPENDIX. 277 4'^ and 3^ and compounds with the fame num- ber of viii'^'in all the fyftems, can hardly of- fend the ear fo much as the quicker beats of thefe other concords will. 6. The fums of the beats of all the concords to TAB. IL of all the concords. 3 +2VIII 4*^+ 2 VI 1 1 6*^+2VllI 3^ + VIII 4^^+ VIII 6^^+ VIII th 4 5th Syftem of Column 480 321 240 2X0 120 80 104 equal harm. 720 288 360 144 800 279 400 138 70 180 200 72 69 35 mean M. Huy- tones gens. 2 3 27S APPENDIX. to the note D-fo]~re in both the col. i, 2, 3, ari! reipedively 759,702,805, whofe proportions with refped to the fame or any other bafe note are 38, 35, 40 veiy nearly. The fmall excefs of the firft fum above the fecond arifes chiefly from the beats of the iii"^ and 6*'' with their compounds, which in all probability are inoffen- five, as we faid before. But a completer rule for comparing the har- mony of imperfed: concords to a given bafe, appears in the fecond table; Hoat concord to which a fmaller nwnber correfponds^ being more harmonious, in its kind, than any other to which a larger nu?nber corresponds ; which affords two or three more obfervations. 7. In col. I and 3, the iii''^ become more harmonious by the addition of VII^*'^^ 8. In all the fyilems, thev-fviii is more harmonious than the v and v H- 2viii. 9. In col. I, the v'^ is more harmonious than the iii*^ and vi'"", the v + viii than the iii+viii and VI 4- VIII, and likewife the 4'^ and com- pounds than the 6''' and 3*^ and compounds with equal numbers of vIII*-*^^ 10. It may be objecfled to the fyflem of equal harmony that the beats of the v'*^^ are not only a little quicker, but fomething ftronger and di- ftindier than thofe of the other concords; which deferves to be confidered. On the other hand it fliould be confidered too, whether thofe very quick and lefs diftinct beats of the vi*'' and com- poundsj APPENDIX. 279 pounds, have not a worfe eiTe^t in dellroying the tleamefs of their harmony. Thefe are the principal advantages and dilad- yantages that occur in comparing thefe fyftems. For as to the falib concords being fomething worfc in the fyftem of equal harmony than in the otiier two (g), this is no objecflion to the fy- ftem, but only to the application of it to defec- tive inftruments j and I have fliewn above how to fupply their defects, without the leaft incon- venience to the performer [h). I fliall only obferve, tliat the firft table was calculated from the temperaments of the fy- ftems in prop, xvii and fcholium, by the corol- laries to prop. XI ', and that the numbers in that table multiplied by the numerators of the known fradlions, annexed to the chara ^^^- ^^' Puifes of air or founds confidered i; 8, art. 12; 20, art. 9. of imperfect unifons have cycles and periods ^6^ def. 2, 3, in which the alternate leffer inter- vals increafe and decreafe uniformly by turns 57^ VII. their coincidences not neceffary to the good har- mony of imperfed confonances loi, art. 5; 103, art. 7, 8. _ the frequency of their coincidences is but an im- perfed: charadter of the fimplicity and fmooth- nefs of perfect confonances 22, art. 14. Q. Quality of the puifes of founds, as to duration, Itrength, weaknefs, &Cj need not be confidered in harmonics 20, art. 9. R. Ratios perfeftwhat 10, table col.i ; 1 1, art. 5; queflion- ed by Galileo iig^ lin. ult. the INDEX. the proportion of their logarithms when fmall 69, lemma and cor. i, 2. Ratio of the times of the fmgle vibrations of two founds, whofe interval is any given part or parts of a comma, how expreffed 72, cor. 4. S. Salinas fays the antients ufed imperfedl confonances 34 (h), claims the firft explication of a mufical temperament 37 (/)'. Samenr's attempt to explain the reafon of the beats of imperfedt confonances 94, fchol ; confuted 96, art. 2; 247. Simplicity of confonances, what 15, art. 4; degrees of it reduced into order by a Rule 16, art. 5, and by a Table 1 8, and compared with the percep- tions of the ear 21, art. 10. "Scale of mufical founds, defective in organs and harpfichords 163, art. 5, which founds are moft wanted in it 165, art. 7, Plate XIX, how to fupply them wholly or in part without the incumbrance of more keys 1 6^j art. II ; 173, art. 18 ; 177, art. 19. of fmgle founds in the harpfichord preferable to unifons 171, art. 15, &c, loud enough if well penned and often heard 172, art. 17. Sound, how caufed i, when mufical 2, its gravity and acutenefs 2, its other qualities not conlidered in harmonics 3, art. 3 -,- 20, art. 9. Syftems of the antients '^^^ art. 1 1, unfit for mufic in feveral parts '^'^^ art. i2j 24 art. 4. Syfteni INDEX. Syftem ^ of major tones and limmas 23, art. 1,2,3, ^^^ ^^' perfeftions difagreeable to the ear 24. art. 4. of tones major and minor and hemitones 25, I, its leaft imperfecSlions 29, cor; difagreeable to , the ear 32, art. 9. of mean tones 35, II, how tempered 36, cor; pronounced to be the bell by Mr. Huygens 44 («), but is not equally harmonious 121, cor. 9. of Mr. Huygens, refulting from a fcale of 3 1 equal intervals in the o(5lave, how tempered 158, fchol ; not equally harmonious 121, cor. 10. of equal harmony, determined 130, xvi, is the moft harmonious fyftem taking one concord with another 141, art. 13, its temperament 140, art. 10, how computed by Tables 137, fchol, how to approximate to it, or any given tempered fyftem by fcales of equal intervals J 58, 159, differs infenfibly from a fyftem re- fulting from a fcale of 50 equal parts in the oc- tave 1565 XVII. T. Baylor the firft that determined a priori the time of a fingle vibration of a mufical chord 230, lin. 3. Temperament of a confonance, what 12, art. 6. Tem.peraments of a fyftem, what and whence occa- iioned 32, art. 9; no footfteps of them among the Grecian writers 34 (^), the invention of them claimed by Zarlino and Salinas 37 (/), the proportions of their fynqhronous variations 38, III, are fhewn alfo by a linear conftru(5tion 40, cor. 2. Tern- INDEX. Temperaments of the fyftem of mean tones 36, cor. of the fyftem of equal harmony, 140, art. 10. of fome of the concords cannot be lefs than a quarter of a comma 41, cor, 3. what the other temperaments are when that of the v'h, vi^'^, or iii"^ is nothing 41, cor. 3, 4, 5, in what cafes that of the v'^, vi^^, or iii*^ is equal to the fum of the other two 42, cor. 6, 7, 8, the fum of them all is lefs when the in* is flattened than when it is equally fharpened 43, cor. 10, when the fum of them all is the leaft pofrible43, cor. 10. their proportion requifite to make any two given confonances equally harmonious 118, cor. 3. Temperer of a fyftem, the limits of its pofition when the temperament of the v*'^, vi^^, or iii^ is equal to the fum of the other two 42, cor. 6, 7, 8, to find its pofition from the given ratio of the tempe- raments of any two concords indifferent parcels 46 IV, 48 V, 53 VI. Tenfion of a mufical chord how meafured 23 1 . lin. 5. Tuning by eftimation and judgment of the ear 1 87, 1 89, by a Table of beats 206, xx, is extremely accu- rate 209, art. 4. by a machine fuggefted 210, art. 6. by ifochronous beats of different concords 211. by a monochord 222, xxi. V. Variations of the temperaments of intervals fhewn by a Table 38, iii, and by a linear conftru6tion 40, cor. 2. Vibration INDEX. Vibration often fignifies the time of it 8, art. 12." a complete one, what 8, art. 1 2. Vibrations how communicated to diftant bodies i . whofe times are incommenfurable have determi- nate periods but no cycles, and yet afford good harmony loi, art. 5. of a mufical chord, wider and narrower, nearly ifochronons 4, art. 4, and alfo of the particles of air at different diftances from the founding body 4, art. 5, ^ their times determined from the length, weight andtenfion of given chords 238, xxiv and cor ; ceteris paribus are as the lengths of the chords 5, art. 7 ; 142, cor. 7. Voice, its motions in fmging and talking are different . 3 (0. in finging does not always move by perfect inter- vals 228, fchol; but makes occafional tempera- ments 225, feft. X, does not move by the given intervals of any one fyftem 225 xxii, but io as to make perfed harmony with the bafe 226. Voice part of an anthem ought not to be played on the organ 228, cor. 5. Undulations, audible and vifible compared 107, fchol i 250, lin. 6. Unifons perfedl what 2, imperfe6t what 56, defin. .1. the alternate leffer intervals of their pulfes increafe and decreafe uniformly by turns 57 vii. the periods of their pulfes what c^6^ defin. 3, are the fame as thole of their leaft imperfec- tions INDEX. tions 63, cor. 5, their proportions 6^, cor. 4, compared with thofe of imperfedl confonances 64 VIII, IX. in the harpfichord worfe than fingle founds 171, art. 16. W. Wdlis difapproves incommenfurable vibrations as imprafticable and inharmonious 100, art. 4. Zarlino the firft writer upon a mufical temperament 37 (0. FINIS, CorreBions. Page 6. note, Hn. 10, for J^uri^o^v, r. l^uTipyi, II. hn. 4, for ce^ r. cc. 51. note, hn. 2, ^^-/^ the firft minus—' 6gj 71, 73 in the running titles, for Prop. VIII. r. Lemma. 71. hn. penult, for — -a, r. -— ? d. 114, 115, notes (/>) and (q), for cor. 7. r. cor. 8. 168. lin. I. dele 9. 169. lin. 12. for fingle, r. one. ^^3> 1^5? 167, 169, 171, in the running titles, for Prop. XVII. r. Art. 5, 7, 9, 12, 1 5, refpeflively. 227. note {k) for fchol. r. fchol. 4. Pl.l 8 1 Fig. 2 +-H B c O Fig.l 4 *-> o o 9. 10 IS 16 8 d e J* A B C D E 3 3. 10 9. 10 R 9 IS f g' a i i R 10 /? c d' e' h;c 6 HCStefih cruijfc M i i &i ^'S'' ~ D E r GAB 9 4 5 i Fig.i A B C D E F J ^ C D E F G A Br t/ f .f !^ " /- r' -/• (->.;<■ Rff4 -I 1 1 1 \- \ 1 1 1 1 1- H 1 1 h- Fig-J —I 1 1 y- H 1 \ 1 1 H 1 H -^ Pl.TI. 'ig. 6. «i 729 i ^7 043 4 '--V'- d c \/Tig.i\y/ o \ V /Fig.i3\ V^ A" '1 4« c d D C 11 '': -' 7. c fA il r \T /'i' .5 J'' c E c ni 'j 4 61 c - //■ .r A B C ^D 1; F G~^ il I 5 X Z M N O P q~~^ R~^ S~^ T U ■2J AB C DE F GXH I KLMNYo PQRS fr ^5 ^ l;^ ( ^ 4 < ■( £ .^ ^ '. ^ , ■.'",'■'. '^ / f^ jc J/ z A B C ^~D £ F G X H I K I. >I n" OT P CI R 5 T U W 27e » 6 c d f S F h i k I m n e p g A B C 5 t V GH I KLlUNOPqRSTTT 1 A 1 1^ 8. 1 9 J C B E \/m. J^^ 6^ 33^ q A B C B E F G 37- 38- Tab ) . 9 6 ~i i ^ _8_ A ■5 y 2 CDEFGABfy,»,/'^<7 6 C' vm 7'* 6'-'' V 4'* 3'»' 2"' p' n m 4" v vi vn Z"' /' m i d o Vnr. .,, // ^ ^Y y (1 /. 1 1 /^"^"'^ -s- r D) O pi-Xiv: //^r % 44- G C \d 3''y;c c f^'hic E c yit t E D^ O vrKv. 4^l/c Fig- 45 G C o'U^C c r^ M E C V hr / 1) 1 O 4'^/r c il 1 f i Fig- 45- 'g f Whr / /h , .rOrr c -\' h/ / / '1' p / 3 r ra/iT / 7 M' 1 4 ^;";,v t.- ' y ,•/',. •'Cffn-foftafLcAf ^*> 24'^''y^ .,„„,/;'Kr f'/Af T^"-,. nM^ ^y-ify-m e/" 'y"" //t/rmtw// . 2 93 4 26 zs 29 43 33 4? 32 47 ^6 27 41 7,0 45 34 50 ■hs IS 42 32 47 35 S3 2fi5 3 ■26 38 2S 41 ■65 16 40 30 44 33 49 37 13 41 31 46 34 51 777 2 25 37 28 40 31 46 ■M 16 5p 29 43 32 4S 36 27 40 30 45 34 SO 269 1 24 36 27 39 30 45 34 ■25 37 18 42 31 47 31 26 39 29 44 33 48 2^2 23 3-; 26 3« 2? 44 33 24 36 27 4i 30 45 34 25 3R 28 42 32 47 255 1 23 34 25 37 25 42 32 24 35 26 40 3o 44 33 25 37 28 41 31 46 -Ve- ini kj- -fu- -l^ zkr -Ve- _kL 'e^ -Ve- Tab. I J'/Z-M.^A-^i^ y .TA, />i/i ■71/fya g/' A rii& / M N O r w X Y Ca BCD J g- / '" -^-r- 1 O 9 — V- R = 1 I J p] 1 ' — r— ' Y / N /' a ^ T ri- X _>/ m f z " - M V S w o Pl.xxvi. Fiff.66. •^ kB»Hili|IIHIUIl (§).' «|> o\e V <5> T X" I Ml- Ji 1 o o Bl- B 1 Dl- ^ D ; vv 1 o o Dl- D d O B« C* 1 Et E Y ! I 1 1 G": :~aI o d d d E* F* f| G* ■llilllllllll 1 II 1 nil ; A* j ■■■liiiiiiii -1 II 1 II fi A D E F G^ A^ Plxxvn. E * i m 1 i -'cT Li._il 11 ^ 7 ■ I. '■' y^rS./culf^. N A Bt B C dL D F.l- E F G^ G a1- G* 1 68. r V.70. B^ E* G C f-^ fA. 1 1 w w *jL.^Mram^ ??S5:S^rt ■V4 r^it^m^HS:i:l^^'BRAHvFACI[,r» - - ■-"".llllilillill A 000 035 738 University of California SOUTHERN REGIONAL LIBRARY FACILITY 305 De Neve Drive - Parking Lot 17 • Box 951388 LOS ANGELES, CALIFORNIA 90095-1388 Return this material to the library from which it was borrowed.