ON TEMPERAMENT. 173 



thus -| added to ^ give f, the major third ; and these, added 

 to if, give |- the perfect fourth ; and so on , y>, if, f, ^, 

 J, if added together give f or the octave. 



In addition to these three relations, f less if gives 128:135, 

 or the chromatic semitone ; thus there are in the scale two 

 tones, | large and \ less, and two semitones, ^f diatonic 

 and iff chromatic. 



Besides these four relations, the three musical differences 

 of the scale are also to be found on this instrument, viz., ^f 

 less if = 24 : 25, the imperfect chromatic semitone f less 

 V = 80 : 81, the comma and the schisma 32,768 : 32,805, 

 which is also deduced from these relations. 



The comma is the difference between the large and less 

 tones or steps of the scale. 



The schisma is the difference between a sharp tone and a 

 flat one, say between D| and El?. 



The round added digital in each octave does not belong 

 to the series of the major scale ; it produces the major seventh 

 and sixth in the minor scale, and also introduces the im- 

 perfect chromatic semitone of 24 : 25, being *- less if. 

 The effect of this additional digital is very peculiar; the 

 tone seems to be too flat till it is heard in the chord. 



These seven are all the primary musical relations and 

 differences to be found on the natural fingerboard ; there 

 are many secondary keys to which I have paid no attention, 

 they may be very beautiful, but I do not enter into that 

 question for I have yet to learn, first, if they are true, and, 

 secondly, if they are necessary or useful. The introduction of 

 a round digital, placed on the white as well as on the coloured 

 digitals, would supply every secondary key that the most 

 exacting musician can demand ; but I was anxious to avoid 

 anything that would complicate or confuse the fingerboard. 



Musicians have greatly to complain of mathematicians for 

 carrying their formulae beyond their legitimate sphere into 

 the domain of music ; and mathematicians, on the other hand, 

 have to complain of musicians for demanding a number of 

 secondary keys which are not mathematically or musically in 

 true key relationship. It would be quite easy to make an 

 instrument with them all, and at no great cost of money ; 

 but what would be the advantage ? 



In constructing this instrument the principles I had to 

 consider were : a mathematical series of sounds on the one 



