1864] 



of Instruments with Fixed Tones. 



417 



2/i+2/5+2/9=2/2+2/6+2/io=2/3+2/7+2/n=2/4+2/8+2/i2='OlO3O00 {a) 



A system of unequal temperament may therefore be determined by arbi- 

 trarily selecting eleven different Vths, or else eight different Illrds and 

 three Vths. The equations («) show that if the temperament is not equal 

 (in which case all the y's are equal, and the interval error of the Ilird is 

 '0103000 = '0034333, as in the Hemitonic temperament), at least four 

 Illrds must have their interval errors greater than ^ X '0103000, that is, 

 there must be at least four Illrds in every unequal temperament which are 

 inferior to the very bad Illrds of the Hemitonic system. Kirnberger, 

 Dr. T. Young, and Lord Stanhope *, in the unequal systems they propose, 

 have each seven Illrds sharper, and therefore worse than the Hemitonic 

 Illrds. In one of Prof, de Morgan's unequal temperaments, six Illrds 

 are sharper than the Hemitonic ; in another four are sharper and four the 

 same ; in a third all are the same, but the Vths differ f . Hence nothing 

 is gained over the Hemitonic system in the way of harmony, while the 

 uniformity in the representation of the uniformity of just intonation is 

 entirely lost. 



In selecting a temperament, therefore, we may dismiss all unequal tem- 

 peraments, as they must be inferior to the Hemitonic in both harmony and 

 melody, and will have no advantage over it in the relations of chords or the 

 number of tones required. Also, if it is considered necessary to play in all 

 keys with only twelve tones, any system of defective equal temperament 

 will be inferior to the Hemitonic, on account of the various and distressing 

 wolf intervals which occur when the music is not confined to the six major 

 scales of jP, (7, G, D, A, and the three minor scales of g, d, a. Hence 

 the two conditions of having only twelve tones (exclusive of octaves) and 

 of playing in all keys, at once exclude all temperaments but the Hemitonic. 

 As, however, organs, harpsichords, and pianofortes with 14, 16, 17, 19, 21, 

 22, and 24 tones to the octave have been actually constructed and used Xi 

 as Mr. Liston used 59 §, Mr. Poole used 50, and Gen. T. Perronet Thomp- 



* Kirnberger, Kunst des reinen Satzes in der Musik. Dr. T. Young, loc. cit. 

 Charles Earl Stanliope, Principles of the Science of Tuning^ 1806. 

 t De Morgan, loc. cit. p. 129, temperaments Q, JR, S. 



t Mr. Farey (Phil. Mag. vol. xxxix. p. 416) gives the particulars of their scales, 

 builders, and localities. 



§ The following account of Mr. Liston's organ is deduced from the data of 

 Mr. Farey (Phil. Mag. vol. xxxix. p. 418). Scale : c -fc lc$ cfl </C> fcj fd9 XeX 



Xdcx d id xd^ d^ teb tt^b e /b \e t/b tt/b X4 4 f ]4 t/ Xftfi 

 ^ t^' l/x Xg fx g ig M cb g% tab tt«b a t« h\h f^bb t4 4 6b 

 \a% tSb %hh ^ \h tcb Xh% |c 1%. Chords : Table V. col. IH., lines 4 to 13 ; 



^6 ~ ^2 + 2/3» 



^7=^3+2/3— 2/4> 



^8 = ^4 + 2/4-2/5. 



^9 =^i + 2/lO-2/9> 

 ^10 = ^2 + 2/11 2/lO> 



^11=^3+2/12 — 2/ii» 

 ^12=^4+2/1 -2/i2- 



