TRANSACTIONS OF THE SECTIONS. 47 



vals ns rapidly decrease : the same number of vibrations which between the Ist and 

 2nd steps of the harmonic series produce an octave, between 2ud and 3rd step a 

 fifth, between the 4th and 5th steps a major third, between the 15th and 10th steps 

 produce only a diatonic semitone, and so onwards beyond the range of musical 

 computation. 



In contrast with this harmonic series of sounds, which is simple, arithmetical, 

 and perfectly ref^ular, we have the series of the musical scale, which is compound, 

 geometrical, and so irregular that two tones or steps of equal vibrations cannot 

 musically succeed each otlier. Of the 48 sounds in the harmonic series, 22 are 

 coincident with the musical series, and 26 ai-e not coincident. 



Of these 22 coincidences, the root, or lowest sound of the harmonic series, 

 occurs as — 



The 4th sound of the musical scale times. 



The 1st sound, or Tonic, occurs 5 „ 



The Gth „ of the scale „ 4 „ 



The 5th „ „ „ 3 „ 



The 3rd „ „ „ 2 „ 



The 2nd „ „ „ 1 „ 



The 7th „ „ „ 1 „ 



In all 22 „ 



Of these 22 coincidences between the harmonic series and the musical series, the 

 last are the numbers 24, 27, 30, 32, 36, 40, 45, and 48, which form the relations of 

 the musical scale. 



This full harmonic series can onhj be built upon Fa, or the 4th of the musical 

 scale, as its root ; and the first power of Fa, 10§ (as it appears in the lowest series 

 of the musical scale 8, 9, 10, 10§, 12, 13J, 15, 16), is the common multiplier and 

 divisor of the vibrations of all the sounds of the musical scale. Thus in the octave 

 from tenor C upwards the vibrations are : — 



CDE P a ABC 



256, 288, 320, 341^, 384, 426§, 480, 512. 



These, divided by the first power of Fa, or the 4th of the musical scale (say 10§), 

 give 24, 27, 30, 32, 36, 40, 45, 48, being the figui-es of the musical scale with 

 which the harmonic series closes. 



In this harmonic series the 8lh, 9th, and 10th tones or steps following in diatonic 

 succession are the 1st, 2ud, and 3rd tones of the musical scale, and the 15th and 16th 

 are the 7th and 8th of the musical scale. 



These figm-es give us the first or lowest relations of the musical scale, 8 : 9, 9 : 10, 

 and 15 : 16 : — 



The large step or tone of 8 : 9 occurs 3 times. 



The less „ „ 9:10 „ 2 „ 



The small „ „ 15:16 „ 2 „ 



Within the octave, in all 7 steps or tones. 



These relations of the tones or steps of the scale are always the same in every key. 

 C, =512 vibrations, is common to the keys of Bj;, F, C, and G; and the 7th or 

 diatonic semitone below, =480 vibrations, is common to the keys of C, G, and D ; 

 so with every musical tone. Each of these is represented by a digital upon the 

 natural finger-board of the author's voice-harmonium. 



For distinction the digitals representing tones common to 4 keys are white, those to 

 3 keys are coloured ; the 1st, 2ud, 4th, and 5th tones of the scale in every key are 

 white, and the 3rd, 6th, and 7th are coloured. 



In every key, looking along the fingerboard, the progression of the scale is the 

 same :— 8 : 9, 9 : 10, 15 : 10, 8:9, 9 : 10, 8:9, 15 : 16. From white digital to 

 white, or from coloured to coloured, there is always the large step or tone of the 

 scale 8:9; from white to coloured always the less tone of the scale 9 : 10 ; and 

 from coloured to white always the small step or tone 15 : 16, the diatonic semitone. 

 Looking across the finger-board at the digitals endwise, from the end of each white 



