418 



Mr. A. J. Ellis on the Temperament [June 16, 



son now uses 40 tones to the octave on their justly intoned organs, the 

 condition of having twelve tones and no more, does not seem to be inevit- 

 able. It will therefore be necessary to determine what would be the best 

 system of temperament for the complete equally tempered scale of 27 tones, 

 and how great a sacrifice of musical effect is required by the use of the 

 Hemitonic system. 



In Table XV. I have calculated for each of the 59 (reducing to 51) 

 systems of equal temperament already named, the interval errors of the 

 Vths, Illrds, and 3rds, and the sums of the squares of the 23 interval 

 errors and the 6 beat meters of Table XII. I have then arranged the 

 temperaments in order according to each of these five results, and numbered 

 the order. Finally, I have added the five order numbers together and ar- 

 ranged the whole in the order of these sums. The smallest number would 

 therefore clearly give the best temperament, supposing that all the five 

 points of comparison were of equal value. Now the first and second tem- 

 perament on the list, or No. 26 and No. 2, only differ from each other in 

 the fifth, sixth, or seventh place of decimals with respect to these five re- 

 sults, a difference which no human ear, however finely constituted by nature 

 or assisted by art, could be taught to detect. As No. 2, or the Mesotonic 

 system, is determined in the simplest manner, I consider it as the real head 

 of the list. There is, however, little to choose between it and any one of 

 the ten or twelve systems which follow, except in simplicity of construction 

 and comparative ease in realization. The Hemitonic system, however, comes 

 35th in the list, and the old Pythagorean, recently defended by Drobisch 

 and Naumann (op. cit.)y and asserted to be the system actually used by 

 violinists, is the 45th. No one who has heard any harmonies played on 

 the Pythagorean system will dispute the correctness of the position here 

 assigned to it, which fully explains the absence of all feeling for harmony 

 among the nations which use it — the ancient and modern Greeks, the old 

 Chinese, the Gaels, the Arabs, Persians, and Turks. No modern quartett 

 players could be listened to who adopted it. 



The contest lies, therefore, between the Mesotonic and the Hemitonic 

 systems. The Mesotonic is that known as the old organ- tuning," or, since it 

 was generally used as a defective twelve-toned system, as the "unequal tem- 

 perament." Within the limits of the nine scales already named, the superiority 

 of the Mesotonic to the Hemitonic system has long been practically acknow- 

 ledged. But the extremely disagreeable effect of the wolves (more espe- 

 cially to the performer himself) has finally expelled the system from Ger- 

 many altogether, and from England in great measure. On the pianoforte 



rV., 5 tol4 ; v., 5 tol6 ; VL, 6 tol5 ; VII., 7 tol6 ; VIH., 7 to 17 j IX., 9 to 

 13 ; X., 9 to 13. Tones not forming part of any chord and reqmred chiefly by 

 the system of tuning: "[d? tt^l? t/'l^ tt/b tta> hb\> t&^b fcb. Complete 

 keys : F, C, G, D, ; B, F^. The keys of E7, had their synonymous, 

 and ^E, their relative minors perfect, 



