180 LECTURES TO SCIENCE TEACHERS. 



From white digital to white, say from the 

 1st to 2nd, and 4th to 5th of the scale, 

 and from coloured to coloured, or from 

 the 6th to the 7th of the scale, the 



relation is always 8:9 



From white to coloured, being from the 

 2nd to the 3rd, and from the 5th to the 



6th of the scale 9:10 



From coloured to white, being from the 

 3rd to the 4th, and from the 7th to the 



8th of the scale 15 : 16 



From white to white, or coloured to coloured, is always the 

 large step. 



From white to coloured is always the less step. 

 From coloured to ivhite, the diatonic semitone or the 

 small step. 



The round digital is related to the coloured which succeeds 

 it as 15 : 16, and to the white which precedes it as 25 : 24, 

 being the imperfect chromatic semitone. 



Looking across the fingerboard at the digitals endwise, from 

 the end of each white digital to the end of each coloured 

 immediately above it, in direct line, the relation is always 

 128 : 135, or the chromatic semitone ; and from the end of 

 each coloured digital to the white immediately above it, in 

 direct line, the comma is found 80 : 81. 



Between all enharmonic changes, such as between A fe 

 404|4 to G | 405, the interval of the schisma always occurs, 

 32,768 : 32,805, the difference being 37. 



These simple intervals and differences, 8 : 9 9 : 10 15 : 16 

 24 : 2580 : 81128 : 135 and 32,768 : 32,805, comprise 

 all the mathematical and musical relations of the scale. The 

 larger intervals of the scale are composed of so many of 

 8 : 99 : 10 and 15 : 16 added together. 



The digitals rise to higher levels at each end, differing by 

 chroma and comma, or comma and chroma alternately ; this 

 causes separate levels on the fingerboard at each change of 

 colour; though these are not essential, they will be found 

 very useful in manipulation, and serve readily to distinguish 

 the different keys. 



The two long digitals in each key are touched with 

 great convenience by the thumb. The lower end of each 

 coloured digital always represents the 7th in its own key, and 



