Thomson — On Numerical Relations of fhe Musical Scale. 153 



that the discord is least in the case of the sub-dominant and domi- 

 nant, which are the principal notes after the tonic in any scale. 

 The third point to be noticed is, that all the numbers in this column 

 are approximate multiples of one constant : {a) = "0004904 or 

 •009587 of a tone (f)/ This is shown in column YIII.. Eeturn- 

 ing to Column lY., it will be noticed that the last three figures of 

 the logarithms follow a system. This points to the fact that there 

 is a relation between the distances apart of successive notes. In 

 every group of four the two outer intervals are equal to 11a, and 

 the inner one to 10a. We will call the former P, and the latter 

 Q. The interval P is one that constantly recurs in the mathema- 

 tical treatment of the musical scale. It is, in fact, the major 

 Comma (|^), and is the difEerence between the Major Tone (f) and 

 the Minor (Y)- The interval Q is called the Minor Comma (Hff). 

 It is the difference between the Enharmonic Diesis (yf|-) and the 

 Major Comma, the Enharmonic Diesis, being the difference between 

 the Major Semitone (|f) and the Minor (ff). The intervals be- 

 tween successive groups of four involve a third quantity B{ \ltl ^), 

 which may be called the Diminished Comma. This interval is the 

 difference between three minor semitones and a major tone. The 

 three constants P, Q, R give a convenient measure of the intervals 

 from C, of all the notes in the above scale ; and, moreover, any 

 musical interval which can be formed from the diatonic scale may 

 be expressed in simple terms of P, Q, R. 



NOTE ADDED IN THE PEESS. 



Since the fractions representing musical intervals must be multiplied 

 together in order to obtain the fraction representing the sum of those 

 intervals, therefore the quantity 34P + 19 Q + 12R (the octave) is really 



equivalent to (^J' x (^)" x i^^^^ This fraction when mul- 



1 This relation depends on the coincidence that 2^^^ is nearly equal to 10^^ x 3^* ; for 

 if • . 21" =: 1012 X 384, 



then 173 log 2 = 12 + 84 log 3 



4 log 2 + 49 log 2 + 84 log 2 +24 log 2 + 12 log 2 = 84 log 3+ 12, 

 ^ log 2 + tf log 2+ 7 log 2 + log 4.+ log 2 = 7 log 3 + 1, 

 A log 2- 1+log 2 + log 4 = 7 (log 3 -log 2 -1^2 log 2), 

 A log 2 - log f = - 7 (A log 2 - log t), 



SCIEN. PEOC, E. D. S. VOL. IT. PT. III. P 



