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



the Mesotonic stands 2nd in order of melody, inappreciably different from 

 the 1st, and the Hemitonic 39th. 



If the 3rd harmonic only is developed in the qualities of tone combined, 

 the beats of the Vth are heard, but those of the other intervals are not 

 perceived. The beats of the Ilird and Vlth, which are so faulty on the 

 Hemitonic system, will not be perceived at all unless the 5th harmonic be 

 developed, and will not be much perceived unless it be strongly developed. 

 Now the 5th harmonic is comparatively weak on all organ pipes and on 

 pianofortes, and hence the errors are not so violently offensive on these 

 instruments. If, however, the * mixture stops,' which strengthen the upper 

 harmonics by additional pipes, are employed on the organ, the effect is un- 

 mistakeably bad, unless drowned by din or dimmed by distance. On the 

 pianoforte, however, these intervals, and even the still worse 3rd and 6th, 

 depending on the 6th and 8th harmonics, which are undeveloped on piano- 

 forte strings, are quite endurable. 



Hence the Hemitonic system, except as regards melody, will not be 

 greatly inferior to the Mesotonic on a pianoforte and on soft stops of organs, 

 but will only become offensive on loud stops. But for harmoniums and 

 concertinas, violins and voices, where harmonics up to the 8th, and even 

 higher, are well developed, the Hemitonic temperament is offensive. The 

 roughness of harmoniums is almost entirely due to this mode of tuning. 

 The beats of the Vlth, Ilird, and 3rd are distinctly heard, and the develop- 

 ment of differential tones is so strong as frequently to form an unintelligibly 

 inharmonious accompaniment*. Concertinas having 14 tones to the octave 

 are indeed generally tuned mesotonically (or intentionally so), thus c c% 

 d d% e e\^, ff^, g a a\>, h b\) . They are, however, occasionally tuned hemi- 

 tonically (or intentionally so) to accompany pianofortes, thus c cj$, d d% 



* The three recognized forms of the common major triad 4, 5^, 6 ; 5, 6, 8 ; 3, 4^ 

 or C E Gy JE G c, Gee, have the pitches of their tones as 4w, Qn ; ^n, Qn, 8n, and 

 Sn, 4:92, 5n respectively. They produce, therefore, the diiferential tones n, n, 2n j 

 n, 2n, Sn, and n, n, 2n respectively. If the chords are tempered, the altered 

 unisons n, n become pulsative, and the other tones disjunct. Now if in Table XIT 

 we put x=log (1+0 neglect t^, we shall have very nearly E=f^ . (l — 4t) .C; 

 G=^.(l-t).C; c=2C,e=^.(l-4:t).C; ff=3.(l-t).a The pairs of pul- 

 sative differential tones are therefore U- C=(^l-Ut) .0, G- C=(}^+{lt) . C, 

 ?ind e~ G=(^+j;t) . C, e — e= — -^i^) . C. The numbers of beats are the abso- 

 lute value of the differences of these pairs of numbers, or of ( — ■ and 

 ( — -1-2^0 . C. The squares of these expressions, and the sum of their squares, 

 will be minima respectively for 1=-^^-^, ^^=-00167070, log ^J= T745206, which is 

 nearly No. 88 (22) ; t=^, ^=-0011658, log ?; = T 749255, which is nearly No. 24 

 (20) ; and t^-^, ^ = '0013096, log -1747817, which is nearly No. 34 (8). 

 These beats, though perfectly distinct in some octaves, do not appear to be suffi- 

 ciently prominent to serve as a criterion of the relative value of different systems 

 of temperament, or to form the basis of a system, and they have consequently 

 not been introduced into the text. They were noticed and used by H. Scheibler 

 (t)er physikalische und musikalische Tonmesser, p. 15). 



