1864] 



on Instruments with Fixed Tones, 



99 



the major scale of D are played as GBD, BF^ fA, and AXC^E, instead 

 of 0# fjE'. The dominant chord is therefore too flat by a comma, and 

 in passing from the chord of A to that of D, as in the ordinary cadence, 

 the note A has to be changed into -fA. If A is the highest or lowest note 

 in the chords, the eifect is decidedly bad. The flatness of the "leading 

 note'' t^^i ill place of C^, although only a comma in extent, is felt as 

 annoying in the succession Jcfl, d. The result is such that it would not be 

 worth while to invent new instruments with such a defect in common scales. 

 On the same instrument I have the three chords of the major scale of A 

 tuned as BF^-^Ay AXC'^Fy EtG^B, in which the subdominant chord is 

 now a comma too sharp. As the subdominant is a much less important 

 chord than the dominant, the effect is better, but trouble arises from having 

 occasionally to alter the tonic note A itself. Even the dissonance of the 

 dominant seventh, when played as FtG^Bd is perceptibly harsher than 

 the correct ^ J 5 (both forms lie on the instrument), although the 

 added seventh d now forms a true minor third with the fifth B, whereas 

 the correct note td forms a dissonant Pythagorean minor third with the 

 same note B. When, however, the first E is omitted, the chord of the di- 

 minished fifth I^G'^Btd is not so pleasant as J ^ d. Again, on the same 

 instrument, instead of having XBXF^A^ as the synonymous major of 

 tdfa in the scale of a minor, I have only B F^fA, which is a comma too 

 sharp. The rarity of the chord, however, renders the bad effect of less 

 importance. Again, I am obliged to modulate from B major to td minor 

 instead of d minor. Even here the error of a comma is perceptible. The 

 general result, therefore, is that commatic substitution, even within the same 

 melody or succession of chords, is inadmissible in just intonation. 



Professor Helmholtz {op. cit. pp. 433 & 484) has suggested what may 

 be termed schismatic substitution, or the use of one note for another which 

 only differs from it by a schisma, the eleventh part of a comma. Having 

 one concertina tuned to equal temperament, and another to just intervals, 

 the equation ^g=-%g has enabled me to test this suggestion by practice. 

 I rind that in slow chords, the altered fifth c ^g, the altered major third 

 ^gb, and the altered minor third e ^g are all decidedly, though only 

 slightly, dissonant. In rapid chords the eff'ect would be necessarily much 

 less perceptible. Such chords as OF ]^G, e \^g b are far superior either to 

 the Pythagorean CfEG, fegfb (of which I can produce the counterparts 

 FfA O, df-fa), or the still worse tempered chords G\\E\\ G, \\e \\g \\b. If 

 we modified Professor Helmholtz' s suggestion, and, where practicable, used 

 only entire chords which are too flat or too sharp by a schisma, so that the 

 schismatic errors would only occur in harmonies where a note was pro- 

 longed from a chord to which it belonged into another for which it was too 

 sharp or too flat by a schisma, then there could be no objection whatever 

 to schismatic substitution, which would be quite inappreciable in melody. 



Now schismatic substitution will materially reduce the number of dif- 

 ferent tones required. By referring to Table IV. it will be seen that all the 



VOL. XIIT. I 



