PY Til AC OR AS. 



Such was the ancient philofophy of founds, of wliich 

 Pytliagoras is recorded as tlie lirlt teacher, lint iiow much 

 (it tiiis tlieory was founded on experiment and demonllration, 

 and liow much of it upon hypothefis ; how much of it was 

 known, and how much taken for granted, cannot certainly 

 be determined. The ilory juft now difcuded is too much 

 emharradcd with abfurdities and impolTibilities to guide us to 

 any probable conjefture, as to tlie method by wiiich Pytha- 

 . goras ailually arrived at his conclufions. 



The difcovcry, as far as it relates to the length of llrings, 

 waseafdy made, becaufe it depended upon an obvious expe- 

 riment. It was, likevvife, cafily perceived, that a fltort 

 Itring vibrated with more velocity than a long one ; but be- 

 tween the certainty of this general fa6t, and the certainty 

 that the vibrations were in a ratio exaftly the inverfe of the 

 lengths, there is a confiderablc gulph. (See Smith's Har- 

 monics, feft i. art. 7, and note f. ) We have no account 

 of the bridge upon which Pythagoras got fafely over. Ex- 

 periment, here, is out of the queftion ; for the flowed vi- 

 brations that produce mufical found, arc far too quick to be 

 counted or diitinguiflied. The inference, however, was na- 

 tural, though it does not appear that the ancients were able 

 • to fupport it by ilridl and fcientiiic proof. . 



Indeed it was fo late as the beginning of the prefeut cen- 

 tury, (17 14. See Phil. Tranf. and Methodus incremen- 

 tormn direfta et inverfa, by Dr. Brook Taylor,) before this 

 ancient theory of found was fully confirmed, and the laws 

 of vibrations, and the whole doctrine of mufical ilrings, 

 eltabliflied upon the folid bafis of mathematical dernonftra- 

 tion. 



The fecond mufical improvement attributed to Pythago- 

 ras, was the addition of an eighth ftring to the lyre, which, 

 before his time, had only feven, and was thence called a 

 heptachord. It is fuppofed by feveral ancient writers, that 

 the fcale of this inltrument, which was that of Terpander, 

 confiited of two conjoint tetrachords, EFGABbCD; 

 and that Pythagoras, by adding an eighth found, at the top, 

 and altering the tuning of the fifth, formed this fcale : 

 EFGA, BCDe, or a fimilar fcale, confiding of two 

 disjunft tetrachords. 



How this fcale was generated by the triple progreffion, 

 or ferics ,of perfect Jths, the abbe Roufiier has lately very 

 well difcuffed, in his " Memoire fur la Mufique des An- 

 ciens." We fliall endeavour to explain what is meant by 

 the triple progrefiion in mufic, which is the bafis of this in- 

 genious hypothefis ; referring the reader to the Memoire 

 itfelf for his proofs, as inferting them here woidd require too 

 much lime and fpace for a work of this nature. 



Let any found be repreientnd by unity, or the number i ; 

 and as the 3d part of a firing has been found to produce the 

 I2tli, or oftave of the 5th above the whole ftring, 11 feries 

 of 5ths.may be reprefented by a triple geometric progreffion 

 of numbers, continually multiplied by 3, as i .3 9 27 81 

 243 729 ; and thefe terms may be equally fuppofed to re- 

 prefent I2ths, or ^ths, either afcending or defcending. For 

 whether we divide by 3, or multiply by 3, the terms will be 

 in the proportion of a 1 2th, or odiave to the 5th, either way. 

 The abbe Rouffier, imagining that the ancients fung their 

 fcale-backv\-nrds, as we iliould call it, by defcending, an- 

 nexes to his numbers -the founds following : 



Term I II III IV V VI VII 

 139 27 Si 243 729 

 B E A D G C F . 

 out of which feries of Jths, by arranging the founds in dia- 

 tonic order, may be formed the heptachord, or 7th, 

 B C D E F G A ; and to thefe, adding the duple of the 

 highefi. found, in the proportion of 2 to i, the abbe fup- 



pofcs that Pythagoras acquired the odtave, or proflambano-" 

 menos. This is throwing a mite into the charity-box of poor 

 Pythagoras, without, however, telling us in what reign the 

 obolum was coined ; for we have met with no ancient author 

 who bellows the invention of proflambanomenos upon this 

 philofopher. The abbe does not let him or his followers ilop 

 here, but fuppoies an Sth term, 2187, added to tlie pro- 

 gredion given above, by which a B D was obtained, wliich 

 turniflied tiie minor femitone below B tj. Tlie fyftein of 

 Pythagoras, according to the abbe, was bounded by this 

 Sth term, and the principle upon which it was built being 

 lofl, the Greeks penetrated no farther into the regions of 

 modulation, where tliey might have enriched their mufic, but 

 contented themfelves, in after-times, with tranfpofition-j of 

 this feries of found. 



The abbe Rouflicr imagines, however, that though Py- 

 thagoras went no farther than the eighth term in triple pro- 

 grellion, yet tlie Egyptians, in very high antiquity, extended 

 the feries to twelve terms, which would give every poffible 

 mode and genus perfeft. A curious circumllance is obferved 

 by the fame •author, p. 28, ^ -|7, with refpedl to the mufi- 

 cal fylleni of the Chinefe, which well deferves mention here. 

 " In collecting," fays he, " what has already been advanced 

 concerning the original formation of the Cliinefe fyftem, it 

 appears to begin precifely where the Greek left off, that is, 

 at the Vlllth term of the triple progreffion, which is purfucd 

 as far as the Xllth term, by which feries, arranged dia- 

 touically, the Chinefe acquire their fcale, e [3, D b> B [3, 

 A b, G bj £ b, in defcending: or, as Rameau exprefleS 

 the fame intervals, in fliarps, afcending, G*, A*, C*, D*, 

 E«, g;X." — It is obfervable that both thefe fcales, which 

 are wholly without femitones, are Scottifli, and correfpond 

 with the natural fcale of the old fimple enharmonic, given 

 p. 34. M. Jamard, a late French writer on mufic, pulhing 

 calculation (Hll further than either the Egyptians or Chinefe, 

 has,obtained, by purfuing the harmonic feries, i, 2, 3, 4, &c. 

 &c. not only the enharmonic diefis, but even the mmute in- 

 tervals in the warbling of birds ; it is wonderful he did not 

 apply his ratios to human fpeech. 



After mufical ratios were difcovered and reduced to 

 numbers, they were made by Pythagoras and his followers, 

 the type of order and juft proportion in all things : hence 

 virtue, friendfliip, good government, celeftial motion, the 

 human foul, and God himlelf, were harmony. 



This difcovery gave birth to various fpecies of mufic, far 

 more ftrange and inconceivable than chromatic and enhar- 

 monic : fuch as divine mufic, mundane mufic, elementary 

 mufic, and many other divifions and fubdivifions, upon 

 which Zarlino, Kircher, and almoll all the old writers, 

 never fail to expatiate with wonderful complacence. It is', 

 perhaps, equally to the credit and advantage of mufic and 

 philofophy, that they have long defcended from thefe 

 heights, and taken their proper and feparate llations upon 

 ectrth : that we no longer admit of mufic tliat cannot be 

 heard, or of philofophy that cannot be underjiood. 



Ariilides Quintilianus afl'ures us, that mufic comprehends 

 arithmetic, geometry, phyfics, and metaphyfics, and teaclics 

 everything, hom folfuing the fcale, to the nature and con- 

 ftruftion of the foul of man and the foul of the univerfe. 

 To confirm this, he quotes, as a divine faying, a moft curious 

 account of the end and li.Jinefs of mufic, from one mailer 

 Panacmus, which informs us that the province of mufic is 

 not only to arrange mufical founds, and to regulate the voice, 

 but to unite and harmonize every thing in nature. This 

 writer, p. 102, in folving the aueilion, whence it is that the 

 foul is fo eafily afFedted by inllrumcntal mufic, acquaints us, 

 in the Pythagorean way, how the foul, frilking about, and 

 12 playing 



