INTERVAL. 



of the fifth will be 0.554962, and the logarithm of the 

 third major will be 0.321928 : from thefe the meafures of 

 all other intervals may be found. But as it has been cufto- 

 niary for muficiansto racafurc their intervals by commas, we 

 ftiall here infert a table of intervals, v/ith their meafures in 

 commas ; where we fuppofe the logarithm or meafure of the 

 comma 1^ to be i. 00000: hence the logarithm of the oc- 

 tave A will be 55.79763, that of the fifth 32.63952, and 

 laftly, that of the third major 17.96282. From thefe all 

 the other intervals may be found in the manner exprefled in 

 the table ; where the firll column fliews the names of the 

 feveral intervals ; the fecond, the proportions of founds 

 forming thefe intervals; the third, the compofition of thefe 

 proportions fram the primes 2, 3, and 5. Thefmaller figures 

 marked above, and fomewhat to the right of the larger, in- 

 dicate the power to which the number exprefled by the larger 



figures is raifed. Thus "■ ^ - fhews that the feventeenth 



power of 1 multiplied by 3, and divided by the eighth 

 power of 5, will produce -j^ij-^ in the fecond column, 

 and that this is the proportion exprefling the interval called 

 efchalon in the firll column. The fourth column of the table 

 contains fome fimple figns of fonie of the intervals, as h for 

 h/p€roche, d for d'lsf.s, &c. and the fifth column fliews how the 

 intervals arife from others : thus over againft femitone major, 

 I find in the fourth column S, which is here only an arbitrary 

 mark for this femitone ; and in the fifth column I find j -|- 

 </ = IV — III, which fignifies that the femitone major is 

 equal to the lum of the femitone minor and diefis, or to the 

 difference between the fourth and the third major. Obferve, 

 that the comma is marked by a dot () ; when this is placed 

 over the letter or other fymbol, it fignifies that the interval 



is fuppofed to be heightened by a comma ; and on the con- 

 trary, when the point is placed below, it fignifies that the 

 interval muft be diminifhed by a comma ; thus / =• T figni- 

 fies that the tone minor increaied by a comma is equal to a 

 tone-major, and vice •vcrfd: "T — t (liews that the tone-major 



diminifhed by a comma is equal to the tone-minor. The 

 figns ->-,—, =, arc here taken in the fame fenfe as in al- 

 gebra, to fignify addition, fubtraftion, and equality. So 

 likewife the dot placed between two numbers, or between a 

 number and the fymbol of an interval, fignifies that the in- 

 terval is to be multiplied by the number. Thus 2. IV 

 fhews that the fourth is doubkd; and thus 7*" = VI -f S 

 = 2. IV = VIII - T, fliews, that the leflbr flat fcventh is 

 equal to the fixth major and femitone-major, or alfo to two 

 fourths, or to the odave. when. the tone-major has been taken 

 from it. Laftly, the fixth column of the table fliews the 

 meafures, or logarithms of the ratios in the fecond column. 

 Thefe are not the ccmmon logarithms of the tables where 

 1. 0000000 is the logarithm of 10. But here i.cocoo is 

 afTumed as the logarithm of |-J-, or of the comma, as before 

 mentioned. Thefe logarithms are eafily derived from the 

 common, of the large tables of Vlacq, or Briggs : thus 

 the logarithm of 2, or the oftave = 0.3010299957; the 

 logarithm of {, or of the fifth — 0.1760912590 : and laftly, 

 the logarithm of i, or of the third major = c 0969100130. 

 Now thefe logarithms being feverally divided by the loga- 

 rithm of -jJ, or the comma = 0.C053950139; the quotients 

 will give the number of commas in an odave = 55-79763 ; 

 in a filth = 32.63952 ; and in a third major = 17.962R2. 

 Hence all the reit may be found by addition and fubtrac- 

 tion only. Here follows the Table. 



A Table of the Mufical Intervals with their Meafu 



