though there is confiderable merit in this attempt, the work is more commendable for the matter than the form, which, as we have already more than once obferved, is not fo inviting as the nature of the fubject might give us reafon to expect, when treated by a writer of abilities.

ART. IV.

Memoire fur la Mufique des Anciens, &c.-An Effay on the Mufic of the Ancients, explaining the Principle on which the authentic Proportions afcribed to Pythagoras are founded; as well as the various mufical Systems of the Greeks, Chinese, and Egyptians : Together with a Comparison drawn between the Syftem of the Egyptians and that of the Moderns. By the Abbé Rouffier, 4to. Paris.

TH

AHE learned and very ingenious Author of this curious and profound effay attempts to prove and explain, by means of one fimple principle, the true nature and generation of the most ancient fcales of mufical founds; and particularly the mufical proportions known under the title of Pythagorean. His intention indeed is to fhew, not only that these ancient fyftems were founded on this principle, but likewife that all those which depart from it are falfe and defective. He undertakes to prove the first part of this pofition, both from the nature of the thing, and from the remains of antiquity; and appeals to the ear for the truth of the latter part of it. We shall endeavour to give the outlines of his fyftem in as clear a manner as the nature of the subject, and the limits to which we are confined, will admit.

The notes of the common fcale, or octave, as we have lately had occafion to obferve*, however natural that divifion may appear to be, are undoubtedly artificial, and the refult of much and profound thought. According to the Author, nothing can be more natural to fuppofe than that a fcale of founds was originally formed, by taking a certain perfect, concordant interval as a model or rule; by the fucceffive application of which, a series of founds would be produced, which being all brought down to, or raised up to the fame octave, according as the progreffion was taken upwards or downwards, would give all the requifite notes contained within the compafs of an octave. The concordant interval which he fuppofes to have been employed for this purpose by Pythagoras, and the Egyptians his mafters, is the fifth, taken in a defcending, or its equivalent the fourth, in an afcending progreffion: and as a series of numbers in a geometrical triplicate ratio to each other, will exprefs a fucceffion of perfect fifths (or rather of perfect twelfths, their octaves)

In our Review of The Principles and Power of Harmony, November 1771, page 374.

Nn 3

affuming

affuming to denote the fundamental note, he proceeds in a defcending triplicate progreffion, and thus procures a ferics of numbers, expreffing the increafing lengths of a fuppofed mufical chord, and denoting the different founds which it would produce. On this particular progreffion, according to him, as on a fundamental and inalterable principle, the genuine fyltem of the ancient Greeks was conftructed. Many of the fucceeding fyflems, naturally, and as it were, of their own accord, arrange themfelves under this fimple and luminous principle, to the difcovery of which the Author acknowledges himself indebted for the knowledge of an infinite number of particulars, which throw light on many queftions that have long divided the mufical world on this fubject. We have faid, the fyftem of the ancient Greeks; for, according to the Abbé, the knowledge of the principle which he here explains was very early loft; as it was unknown even in the time of Ptolomy, whofe errors have been adopted by all fuccceding writers.

Before we proceed further, we shall give, in one line, the first eight terms of this triplicate progreffion, in a feries of defcending fifths (or twelfths) formed by multiplying each preceding number by three; together with the names of the notes expreffed by them.. We fcarce need to add that the lower numbers are to be elevated, in a duplicate progreffion, in order to bring them up into the fame octave with any particular note with which they are to be compared. To fave the trouble of calculation, tables are given at the end of the work, in which are contained all the neceffary feries of thefe numbers, in duplicate and triplicate progreffion.

Ift term. II.. III.

1.

IV. V. VI. VII. VIII.

3. 9. 27.
81. 243.
729. 2187.
E. A D G. C. F. B flat.

B. Here, according to the Author, Pythagoras and the ancient Greeks clofed the progreffion; probably from an apprehenfion that the chromatic genus, which would be introduced by a further extenfion of the feries, might, from its effeminate nature, prove dangerous to manners: for it is well known that, long after the time of Pythagoras, the Lacedemonians punished Timotheus in an exemplary manner, for attempting to introduce that genus among them, by adding four ftrings to the ancient heptachord; as appears from the remarkable Senatus confultum iffued on that occafion, and which may be found in Bocthius, lib. 10.

The Author proceeds afterwards to fhew that the Egyptians added four more terms to this progreffion. He endeavours to prove likewife, that the Chinese musical scale of fix founds, of which he treats particularly in an article apart, commences with the laft term of the preceding progreffion; and draws

from

from thence conclufions favourable to his hypothefis. The four added terms are these :

E flat.

19683. 59049. 177147.
A flat. D flat. G flat.

The Reader has now before him a feries of twelve numbers, which are faid to express the value affigned by the Egyptians to the notes of their fcale. They carried on the progreffion no farther than the twelfth term, for an obvious reafon. The thirteenth, 531441, he obferves, which anfwers to C flat, in a manner excludes itself from the feries: as this C flat would be lower than the B natural (which is the fundamental note of this progreffion) raifed up to the nineteenth octave, and which is expreffed by the number 524288. The difference between thefe two numbers, it is well known, conftitutes the mufical interval known by the name of the Comma of Pythagoras, but hitherto fuppofed to be produced by an afcending progreffion.

8

8

5

9

243

768

To fave our mufical Readers the trouble of calculation, we fhall fubjoin a regular scale of founds founded on the preceding defcending progreffion, but here given in an afcending feries, and reduced, we believe, to the lowest terms in which the ratios can be expreffed without fractions. We fhall likewise place below them the numbers which correfpond to the fame notes in the modern diatonic scale; in order that the difference may be feen at one view: we fhall likewife add, between every two notes, the ratios expreffing the interval between them : Ancient scale 8 C D E F G & A B C 384. 432. 486 512 576 648. 729 Diatonic. • 38443210 480 18 512 3 576 640 € 720 1768 The Author having, by a variety of arguments and authorities, taken pains to eftablifh the preceding feries, as the genuine fcale ufed by the ancients, proceeds to fhew that this is the only just and natural method of dividing the octave; that Ptolemy and all the fubfequent writers of mufic lott fight of this juft and original principle, that of forming a mufical fcale by a feries of perfect fifths fucceeding each other; and that all the errors and imperfections of the prefent or diatonic fyftem, and the numberlefs difquifitions and difputes to which this fubject has given birth, proceed from our not having known and adopted this fimple principle, both in theory and practice. We fhall now proceed to offer a few obfervations that prefent themfelves on a confideration of this icale; firft briefly obferving, in general, that it not only differs, in many parts of it, from the diatonic fyftem, but that it is inconfitent likewise with many of the principles deduced from the experiments made with the ftring trumpet, the harmonical funds naturally produced by founding bodies, and other phyfical phenomena.

N 04

In the first place, we shall obferve that, on calculating the ratios of the numbers given in the fuppofed ancient fcale formed by a triplicate progreffion, it will be found, that the fifths are all perfect; that there is only one kind of tone in this fcale, and that the major, in the proportion 8:9; that the major thirds, in every part of it, confift of two fuch major tones, and confequently conftitute an interval larger than that in the diatonic fyftem, which is expreffed by the ratio 4:5; that the femitone, on the other hand, is every where less than the diatonic; that the minor thirds are likewife every wh re the fame throughout this fcale, and form an interval ́ smaller than the diatonic of 5:6. To fhew these differences at one view, and in the fmalleft numbers:-The ratios expreffing the prefent diatonic femitone, major third, and minor third are 15:16 (or 240: 256) 4:5; and 5:6. In the ancient system the fame intervals are expreffed by the ratios 243: 256; 4:5πi and 56. We fearce need to add, as it will appear on the bare inspection of the preceding fcale, that the minor tone of the moderns, 9: 10, is not admitted into this fyftem. This, as well as many other devices, tending to perplex the theory of mufic, and to disfigure genuine harmony, are here faid to be the invention of the modern Greeks.

The inalterability and indivifibility of the tone is strongly and frequently infifted upon by the Author; who affirms that there is not, nor can be, any other tone than the major; which is formed by the two extremes of any three fucceeding terms in the triplicate progreffion, given at the beginning of this article; the firft, taken in any part of the feries, being railed up into the fame octave with the third as B1, elevated, by a duplicate progreffion, to 8, and forming with A9, the ratio 8: 9.

It follows, as a neceffary corrollary, from this inalterability of the tone, that the interval of the major third in this fyftem mult be larger than the modern interval of the fame denomination, which, as is well known, confifts of a major and a minor tone, producing the interval 4:5; for X 2 = 1 = "÷=. But the major third of this fyftem, the true Diton of the ancient Greeks, is produced by taking the two extremes of any five fucceeding terms in the above mentioned feries, and raifing the lowest, B1, for example, fix octaves, that is, into the fame octave with G 81, which gives the ratio 64: 814:51%, and greater than the former interval by a comma. In fhort, to give a more familiar inftance, it is the interval produced by the extremes of four perfect fifths; between G, the open fourth string of a violin, and B, the perfect fifth of E, the open first ftring, After these two examples, we need not proceed farther to exemplify in what manner the minor third, and the femitone, are deduced from this progreffion. They are both contracted

by

by this operation. The former which, in the diatonic fcalė, is expreffed by the ratio 80: 96, or 5: 6, is here reduced, by an operation fimilar to thofe above given, to 81: 96, or 52:6; and the latter, 240:256 (or 15: 16) to 243:256. We need not mention the remaining intervals, which depend upon thefe. Such, according to the Author, was the fcale of founds, by which the ancient Greeks fung and executed their divine compofitions, at a time when mufic was among them the science of poets and philofophers: nay fuch, he affirms, are the tones which Nature forces even the modern European to produce, provided his ears have not been debauched to a certain degree, by our arbitrary, fictitious, and falfe proportions; or by having been long accustomed to the difcordant intervals of tempered inftruments. In the ancient fcale, founded on the defcending progreffion of perfect fifths, no fuch temperament was neceflary: and had Didymus and Ptolemy known or attended to that fimple principle, the mufical world would not have had their heads confounded with endless difputes and calculations, undertaken and inftituted in defence of complicated and erroneous fyftems; nor their ears wounded by falfe and difcordant intervals, the natural offspring of their reveries.

The felection and adoption of our prefent fyftem, which is no other than the Diatonicon fyntonon of Ptolomy, out of a great many others prefented by that writer (who feems to have taken a pleasure in fplitting of tones) according to the Author's ac count, arose from hence: it found favour, it seems, with Zare lin; has been adopted by all fucceeding theorifts, and acquired the epithet of a natural scale, merely because its concordant intervals happened to correfpond with the natural feries of the numbers 1, 2, 3, 4, 5, 6, in arithmetical progreffion. It is true, fays the Abbé, that there is real harmony between the numbers 1 and 2, as they represent the octave, between 2 and 3, which give the fifth; and between 3 and 4, which truly exprefs the fourth: but it does not follow from hence that harmony must be produced from the numbers 4 and 5, or 5 and 6, if they do not actually prefent fuch harmony. What reason, he adds, can be given for not carrying this progreffion further *? There are the fame grounds to expect harmony from the numbers 6 and 7, 7 and 8, &c. I mean, fays the Abbé, mufical harmony, harmony of founds, in fine harmony for the ear; and not a harmony of numbers, or of quantities proceeding in arithmetical progreffion.

This has been lately done, certainly to a very extravagant extent, by M. Jamard. A particular account of his arithmetical operations on mufic may be feen in the Appendix to our 44th volume, page 554.

The