74 On Perfect Musical Intonation. 
exg. that Lx2x4xix4=—4 or £494. Again it is supposed 
that a similar series of tivelve ees wi end upon the octave: 
&, that LxeX4x% $x2x4xX2Xi=1 or 
213424. oF 524288 = IAL a ‘mathematics cannot be 
musical scale, which is founded upon them. ‘The result of this 
mutilation is, (as might. poerele be supposed, ) the destruction of 
pee harmony and melody in the tempered music. 
The question has been vie whether ratios which con- 
tain the prime seven should be considered harmonic. A standard 
elementary treatise before us contains the following: “ Higher 
primes than 5 enter into no harmonic ratios: such co ombina- 
tions for instance as 1:7, 5:7, or 6:7, are altogether discord- 
nt. The ear will not endure them, and cannot rest 
upon them.’’* 
The most certain method of determining oo eyed of hess | 
harmonic combination, is by an appeal to the The 
binations must first be heard, and the ear eat “decide cae 
them. Although combinations which contain the prime 7 are 
continually occurring in the performances of good singers and 
violin players, yet it might es difficult for one unfamiliar with 
them to know when they occur, If it be proposed to try them 
referred to. It is probable that the writer quoted above, never 
eard (knowing when he heard them) the combinations he con- 
demns. The writer of this paper has sysapr facilities for the 
experiment in question, inasmuch as he has nd an instru- 
ment of perfect intonation, upon which the ae of these, or any 
other combinations can be tried. On the evidence of his own . 
ears and those of every musician who has heard them, he must 
pronounce them altogether harmonious and pleasing.t 
* Prof. Benj. Peirce, “On Sound,” 
+ We have admitted in our system ee prime age than seven. The question 
may be asked, why the higher ey as 11, 18, 17, 19, 23, &e, should be exclu- 
ng g e mg a ne that they produce ratios too com lated for the ear 
appreciate. e primes, and the combinations produced by them, are ul- 
Dea peal nar 8 belong to the extended science of i isn would be 
vit were our ears sufficiently delicate e appreciate them. But “ig, natvels 
3 are illimitable, there is a limit to human bore ion of them. If any on 
is aetois rte o investigate this matter, ten can satisfy himself by atempting to sie 
one of these remote primes, as the 11th, for instance, "which Il be the easiest of 
the whole. As this cannot be obtained from the Athes ie (octaves, 
fifths, thirds, or sevenths,) it must be tuned as an eleventh at once e in a chord as fol- 
lows, 8:11, 9:11 or 10:11, de If it be — impossible to tune it, it cer- 
be impossible to use it in harmony o imposible 
y of of th oe 
d 
ears, and find that he can apprec nie i, (ie cikee voce 
tune it, or how when its tune) we wil agree that it may be used by himself 
and those w ho possess equally delicate ear 
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