1864.] 



of Instruments with Fixed Tones. 



405 



composition of tones discovered by Prof. Helmholtz have enabled me to 

 present it in an entirely new form, and to determine with some degree of 

 certainty what is the best possible form of temperament. 



Let the compound tones P and Q, of which P is the sharper, form the 

 concordant interval^ ; q. Then P : Q=p : q, or qP=pQ, that is, the ^th 

 harmonic of P and the ^th harmonic of Q are conjunct. Now let P be 

 changed into P. (l + 0> where t is small, and rarely or never exceeds 

 ^= -0125. Then the ^th harmonic of P . (1 + will be ?P . (1 + 1) and 

 will no longer =pQ. The difference between the pitch of these harmonics 

 is qP . (\-\-t)—pQ=qt.P=pt. Q. Hence the number of beats in a 

 second produced by this change in P will be found by multiplying the 

 lower pitch Q by pt, which is therefore the beat factor, and will be positive 

 or negative according as the pitch of P is increased or diminished, or the 

 interval is sharpened or flattened. The other beats which existed between 

 the joint harmonics of the dyad P, Q may be increased or diminished by 

 this change, but in either case so slightly that they may be left out of 

 consideration in comparison with the beats thus introduced. But the dif- 

 ferential tone which was P—Q becomes Pt + P—Q, and is therefore a 

 tone which is entirely unrelated to the original chord, and which may be- 

 come prominently dissonant. This is an evil which cannot be avoided by 

 any system of temperament, and is about equally objectionable in all 

 systems. It may therefore be also left out of consideration in selecting 

 a temperament. 



The melody will also suffer from the alteration in the perfect ratios. 

 An interval is best measured by the difference of the tabular logarithms of 

 the pitches of the two tones which form it. Hence the interval error 

 e=log[P.(l + 0-^Q]-log [P: Q]=log(l+O = j"^, if the square and 

 higher powers of t be neglected, and fi be the modulus. Hence the beat 

 factor which =pt, will =^e-f-^, or (xpe. I call pe the beat meter, and 

 represent it by /3. 



We may assume that the dissonance created by temperament a /j^ Hence 

 for the same just interval p : q, variously represented in different tempera- 

 ments, the dissonance a e^. That is, the harmony varies inversely as 

 and the melody varies inversely as e^. Hence for the same interval the 

 harmony and melody both vary inversely as e^. The general harmony and 

 melody may be assumed to be best when and Se^ are minima, which will 

 not happen simultaneously. 



The following contractions for the names of the principal intervals will 



der k. Sachsischen Gesellschaft der Wissenschaften, vol. iv. Naclitrage ziir 

 Theorie der musikalischen Tonverhaltnisse, ibid. vol. v. Ueber die wissen- 

 schaftliche Bestimmung der musikalischen Temperatur, Poggendorn's Annalen^ 

 Yol. xe. p. 353. Naumann, Ueber die verschiedene Bestimmung der Tonver- 

 haltnisse und die Bedeutung des Pythagoreischen oder reinen Quinten-Systems 

 fur unsere heutige Musik, 1858. Helmholtz, Die Lehre von den Tonenipfin- 

 dungen, 1863. I am most indebted to Smithy Drobisch^ and Helnilioltz. 



2 G 2 



