1864.] 



of Instruments with Fixed Tones, 



407 



Hence all intervals and pitches can be expressed in terms of v. This 

 further appears from arranging the 27 different tones required in tempered 

 scales, in order of Vths, thus 



«bb, ebb, 6bb,/b, cb, ^b, «b, eb, 6b, 



f, c> 9> dy a, e, I, 



c#, ^/fc d^, a% 6#,/X, cX, ^^X . 



It will be obvious from Table V. (Proceedings, vol. xiii. facing p. 108), 

 when the signs are omitted, that these 27 tones suffice for all keys from 

 Cb to (7#. This also appears from observing that the complete key of C 

 requires 7 naturals, 3 flats and 3 sharps, or 13 tones, and that one flat or 

 sharp is introduced for each additional flat or sharp in the signature of the 

 key. Hence for 7 flats and 7 sharps in the signature 14 additional tones 

 are required, making 27 in all. The rarity of the modulations into d\), g? 

 or cb minor enables us generally to dispense with the three tones «bb, ebb, 

 ibb, and thus to reduce all music to 24 tones. The system of writing 

 music usually adopted is only suitable to such a tempered scale, and there- 

 fore requires the addition of the acute and grave signs (tj) to adapt it for 

 a representation of the just scale founded on the numbers 1, 2, 3, 5. 



To calculate the value which must be assigned to v so as to fulfil the 

 conditions supposed to produce the least disagreeable system of tem- 

 perament, it will be most convenient to use logarithms, and to put log 

 ij=log|-—^=*l 760913— ^. The above arrangement of the requisite 27 

 tones in order of Yths, therefore, enables us to calculate the logarithms of 

 the ratios of the pitches of all the tones to the pitch of c in terms of oo, by 

 continual additions and subtractions of log v, rejecting or adding log 2 

 = '3010300, when necessary, to keep all the tones in the same Vlllve. 

 The result is tabulated in Table XII., column T. From this we imme- 

 diately deduce 



log m= log 6? —log c = '0511526— 2m 

 log ^=log/-loge = -0226335+ bx 

 log;g:=log/#-log/ = -0285191- 7x 

 log a =log^l7 -log/;g= — -0058851 + 12cV. 



To find the interval errors, the just intervals must be taken for the 

 commonest modulations into the subdominant and dominant keys, as ex- 

 plained in my paper on a Perfect Musical Scale (Proceedings, vol. xiii. 

 p. 97). As the method of determining temperament here supposed makes 

 the errors the same for the same intervals in all keys, that is, makes 

 the temperament equaly it is sufficient to determine the interval errors for 

 a single key. Hence the just intervals are calculated in Table XII., 

 column J, for the key of (7, and the interval error is given in column e, in 

 terms of x and /c=:log the interval of a comma. From these interval 

 errors the beat meters for the six concordant dyads are calculated in 

 column /3. To these are added the values of S and 2 /3^, also in terms 



