or the Division of the Octave. 167 



have adopted this as a rule for tuning seems curious. Smith 

 discusses only systems with one class of whole tones (i. e. those 

 called by Mr. Ellis "commatic "); and practically he confines 

 himself to negative systems (t. e. systems with fifths flatter than 

 equal-temperament fifths). He regards perfect-concord systems 

 as quite unmanageable for practical purposes; and, admitting 

 the restriction to 12 sounds per octave, there is no doubt he 

 was right. His fundamental theorem is as follows : — The 

 octave, a constant interval, being made up of five tones and two 

 semitones, it' the tone and semitone be changed in any manner, 

 the change of the tone is to that of the semitone as 2 : 5. Now 

 in all such systems the tones are " two-fifths tones," i. e. are 

 got by tuning two fifths up, and the semitones are five-fifths 

 semitones, i. e. are got by tuning five fifths down, whence Smith's 

 result follows independently. 



Smith's criterion for systems consists in the consideration of 

 the dissonances of the tempered fifth, sixth, and third. He as- 

 cribes to each interval a measure of dissonance, i. e. supposes 

 that tempering by the same interval produces different disso- 

 nances in different concords. It is for the purposes of this mea- 

 sure that his investigation of the theory of beats is undertaken. 

 The work of Helmholtz has superseded his theory of beats for 

 practical purposes. Of the three temperaments Smith arrives at, 

 the one which he calls equal harmony is that which he prefers. 

 It makes the filths flatter than in the mean-tone temperament, 

 and the thirds fiat ; the fifths and sixths beat equally in oppo- 

 site directions. The mean tone he regards as preferable in the 

 next degree; he calls it the u vulgar temperament.'''' But as 

 both of these are very bad on the 1 2-key ed board, he proposes a 

 third system in which the thirds and fifths beat equally fast, the 

 fifths flat and the thirds sharp. This system is intermediate 

 between the mean-tone and the equal-temperament; and the 

 interest of it is that it, or something very like it, appears to have 

 been generally adopted in practice. The more recent form of 

 the unequal temperament is substantially this system, though it 

 may be doubtful whether its employment was derived from 

 Smith*. The properties of Smith's equal harmony are closely 

 represented by the cycle of 50 (negative of the second order), as 

 he himself points out ; and his third system (that above alluded 

 to) has very nearly the properties of the system of 43 (negative 

 of the first order). Smith describes an arrangement of stops 

 for introducing additional notes into the harpsichord for the 

 purposes of the mean- tone system or of his " equal harmony ;" 

 he possessed and used a harpsichord thus constructed. Mr. 

 Ellis has recently employed the same method ; but my opinion 

 * See ' Hopkins on the Organ,' p. 1S3. 



