INTERVAL. 



lefs may poflibly be conceived. It is true, with recprd to 

 praftice, there are limits, which are the greateft and leaft in- 

 tervals our ears are judgjes of, and which may be aftually 

 prod'iced by voice or inftrument. 



The degrees of tune are proportional to the number of 

 vibrations of tlie fonorous body in a given time, or the ve- 

 locity of their courfes and rccourfcs. Now thefe differences 

 in tune conftitute, as has been already faid, the intervals in 

 mufic ; thefe therefore mull be greater or lefs, as the differ- 

 ences are ; and it is the quantity of thefe which is the fub- 

 jeft of the mathematical part of mufic. Thofe intervals are 

 meafiired, not in the fimple differences, or in arithmetical 

 ratios of the numbers espreffing the lengths or vibrations, 

 but in their geometric ratios ; fo that the fame interval de- 

 pends on the fame geometrical ratio, and vies ivr/d. It is, 

 however, to be obferved, that in comparing the equality of 

 intervals, the ratios exprefllng them muft be all of one fpe- 

 cies ; otherwife this abfurdicy will follow, that the fame two 

 founds may make different intervals. To defcribe the par- 

 ticular methods of meafuring tlie inequality of intervals 

 would be too tedious : this one rule may be obferved, that, 

 to determine in general wliich of two or more intervals are 

 the greateft, take all the ratios as proper fradlions, and the 

 leafl fraction will be the greateft interval. 



The ancients were extremely divided about the manner of 

 meafuring intervals. Pythagoras and his followers meafured 

 them by the ratios of numbers. They fuppofed the differ- 

 ences of gravity and acutenefs to depend on tlie different ve- 

 locities of the motion which caufes found ; and therefore 

 concluded, that they could only be accurately meafured by 

 the ratios of thofe velocities. Which ratios are faid to have 

 been firft inveftigated by Pythagoras, on occafion of his 

 paffing by a fmith's ihop, and obierving a concordance be- 

 twixt the founds of harameis ftriking on the anvil. 



Ariftoxenus oppofed this. He thought reafon and ma- 

 thematics had nothing to do in this cafe, and that fcnfe was 

 the only judge in the dil'pute ; the other being too fubtile to 

 be of any ufe. He therefore determined the oftave, fifth, 

 and fourth, which are the nioft fimple concords, by the ear ; 

 and by the difference of the fourtli and fifth he found out 

 the tone ; which, once fettled as an interval the ear could 

 judge of, he pretended to meafure every interval by various 

 additions, and fubtrattions, made of thefe mentioned, one 

 with another : but this method is very inaccurate. 



Ptolemy keeps a middle courfe betwixt the two : he finds 

 fault with the one for defpifing reafon, and with the other 

 for excluding fenfe ; and (liews ho-.v thefe two may mutually 

 affift each other in this matter. Malcolm. 



Intervals are founded on certain ratios or proportions ex- 

 preffible in numbers, which may all be analyfed into the prime 

 numbers 2, 3, and j. And all intervals may be found from 

 the oftave, fiftlv and third major, which refpeftively corre- 

 fpond to thofe numbers. Thefe are the muflcian's elements, 

 from the various combinations of which all the agreeable va- 

 riety of relations of founds refults. 



This is the modern fyftem ; and a late author affures us, it 

 may be looked on as the ftandard of truth ; and that every in- 

 terval that occurs in mufic is good or bad, as it approaches to 

 or deviates from what it ought to be, on thefe principles. He 

 obferves, that the doctrine of fome of the ancients feems dif- 

 ferent. Ptolemy, for inttance, introduces not only the primes 

 2, 3, 4, J, but alfo 7 and 1 1, &c. Nay, he feems to think 

 »11 fourths good, provided their component intervals may be 

 expreffcd by fuper-particular ratios. But thefe are juftly 

 exploded conceits ; and it feems not improbable, that the 

 contradictions of different numerical h)-^»othefcs, iven in the 

 age of Ariftoxenus, and their iiiconfille:icy with < xperience, 

 



mio-ht lead him to rejeiS nurnbers altogether. Dr. Pej^ufc. 

 api'Phil. Tranf. N"48i. p. 267, 26S. 



M. Euler defines an interval, the meafure of the differ- 

 ence of an acute and grave found. Tentam. Nov. Thcor. 

 Mufic, p. 72 and p. 103. 



Suppofe three founds a, b, c, of which e is the acute, a 

 the moft grave, and b the intermediate /bund. From the 

 preceding definition it appears, that the interval between the 

 founds a and c is the aggregate of the intervals bctv.-een a 

 and*, and between i and r. Therefore, if the interval be- 

 tween a and b be equal to that between b and c, which hap- 

 pens wlien a:b:: c : d, the interval between a lo c will be 

 double the interval a to i, ovbtoc. This being coiifidered, 

 it will appear that intervals ought to be expreffcd by the mea- 

 fures of the ratios conftituting the founds forming thofe in- 

 tervals. But ratios are meafured by the logarithms of frac- 

 tions, the numeraters of which denote the acute founds, and 

 the denominators the grave. Hence the interval between 

 the founds a and b will be expreffed by the logarithm of 



the fradion -, which is ufually denoted by / — , or, which 

 <2 a 



comes to the fame, lb — la. The interval therefore of 

 cqnal founds, a to a, will be null, as /a — la==. o. The 

 interval called an oitave, or diapafon, wiU be expreffed by 

 the logarithm of 2 : and the interval of the fifth or diapente, 

 will be /3 — Iz. From whence it appears that'thefe inter- 

 vals are incommenfurable : fo that no intervals, liowever 

 fmall, can be an aliquot part, both of the oftave and fifth. 

 The like may be faid of the intervals / ;, and 1 1, and others 

 whofe logarithms are difiimilar. But intervals expounded 

 by logarithms of numbers, which are powers of the fame 

 root, may be compared. Thus, the interval of the founds 

 27 : 8, will be to the interval of the founds 9 : 4, as 3 is 

 to 2 : For / y = 3 / J , and 3 / f = 2/3. Euler, ibid. p. 74. 

 But though the logarithms of numbers, which are not 

 powers of the fame root, be incommenfurable, yet an ap- ' 

 proximating ratio of fuch may be found. Tluis the meafure 

 of the oftave is /l = 0.3010300, and tlie meafure of the 

 fifth is /3 — /2 = 0.1760913. Hence the interval of the 

 odave will be to that of the fifth, nearly as 3010300 te 

 1760913 ; vi'hich ratio being reduced to fmaller terms, in the 

 method explained under the head R.ATio, will give us thefe 

 fimple exprelTions for the ratio of the oiTtave and fifth : 2 : 

 1,3 : 2' 5.: 3> 7 : 4' 12 : 7. i" = 1°. 29 = 17. 4' = 24, 

 ^^ : 31, which lail is vci-y near the truth. Euler, ibid. 



In like manner intervals may be divided into anv number 

 of equal parts : for this purpofe we need only divide the 

 logarithm of the propofed interval into the fame number of 

 parts, and then find its correfpondent number by the tables. 

 The ratio of the number fo found, to unity, will give the 

 required ratio of the divided interval to its propofed part. 

 Thus let the third part of an oftave be required ; its loga- 

 rithm will be = 0.1003433 z= 'j 1 2. The ratio correfpond- 

 ing nearly to this v/ill be 6^ : 50, or lefs accurately, 29 : 

 23, or 5 : 4, which laft cxpredcs the third major ; and this 

 is by the lefs knowing taken for the third part of an oftave, 

 and feems to be fuch on our harpfichords and organs, where 

 from C to E is a third, from E to G «: another, and from 

 G «« or A '" to f another third. But the more intelligent 

 krow, that G * and A" ought not to be reputed the lame 

 found, fincc they differ by a diefis enharmonica, wliich is 

 nearly equal to two commai-:. 



M. Euler has intVrted a table of inter\'als in his " Tenta- 

 men N.iva- Theoria:^ Muficse :" he f ippofes the logarithm or 

 meafure of the ofteve to be i.oooooo, whence the logarithm 



