i6o 



NA TURE 



{Dec. 21, 1876 



, 9 5 45 3 27 15 2 

 ' 8' i' i^' V 76' 8' ' 



and if we built it upon the subdominant - the vibration 

 numbers will be — 



, 10 5 4 3 5 



94323 



16 



Or, more generally, if C, D, E, F, G, A, B, 2C, be taken 

 to represent the vibration numbers of the so-named notes 

 in the scale of C, then the vibration numbers of the scales 

 on G and F will be — 



C, D, E, aF, G, j9A, B, 2C, 

 C, ^D, E,;F, G, A, ;B, 2C, 



where 



^ 



= 135 



128' 



, ;8 



80' 



a and /3 being respectively the chromatic semitone and 

 the comma. Here the law of the formation of the relative 

 scales is so obvious that they can be written down suc- 

 cessively at sight. 



Now let the fifteen scales be written down from C to 

 C|i in the one direction, and to Ct> in the other, when it 

 will be immediately apparent that the symbols are 

 arranged in groups of threes and fours, and if we draw 

 straight lines horizontal and vertical so as to enclose 

 these groups each in a rectangle, we have at once the 

 properly so-called " Natural Fingerboard," the rectangles 

 being the digitals, of which the larger are white and the 

 smaller coloured 



Here all the scales are true and adjacent according to 

 their relationship, the fingering obviously the same for 

 all. The relative minors are provided for by a round 



digital in the corner of each coloured digital, bearing to it 

 the vibration ratio of 15 : 16. 



The intervals on the board as seen above express them- 

 selves : moving from flat towards sharp keys the interval 

 from a white to a coloured digital is a, and from coloured 

 to white ^. 



Between digitals related as aG (Gi) and -A (Ajj), that 



is, between every pair similarly related in mutual azimuth 

 (to borrow a term) and distance, the schisma occurs. The 

 keyboard gives us the value of the schisma by inspection; 

 we may either take the route — 



lAto A 



A to G = -?- 



ID 



G to aG 



giving schisma = -^a^ Or thus- 



La to A = a : A to /3A = ;3 : M to «G 



IS 

 l6' 



This expression shows that the 



giving schisma = -~a^. 

 10 



schisma also exists between any round digital, and the 



coloured digital next below. Or we may get an entirely 



numerical value thus : — Since from white to white is 8 : 9, 



then from -- D diagonally to /3A is (8 : 9)* and from ^A 

 back horizontally to aC, a minor sixth, is 8 : 5, the interval 



of aC and -— D is 

 aj3 



Schisma 



8\8/ 



Again, the "comma of Pythagoras" being the excess of 

 twelve fifths over seven octaves is expressed by — 



that is, it is the excess of six major tones over one octave. 



The keyboardshows this immediatelv; from -— D diagonally 



to 2a^C is six major tones, thence back horizontally to a/3C 



is one octave, therefore afiC differs from --D by the 



" comma of Pythagoras." And every pair of white 



digitals (or coloured, as aD and — E) similarly related in 



ap 

 azimuth and distance have the same interval. Obviously, 

 by mere inspection of the board, the " comma of Pytha- 

 goras " is equal to comma -|- schisma. In fact this key- 

 board will well repay a very careful study. 



Turning to the practical aspect of the subject, the har- 

 monium on this principle must be considered "un fait 

 accompli," judging by the highly appreciative interest 

 shown by the South Kensington audience, who remained 

 long after the close of the lecture to listen to the instru- 

 ment which was exhibited, and played on. 



When we count the number of wires that would be 

 necessary for a piano, the prospect would be somewhat 

 alarming, did we not remember that on account of the 

 number of harmonics and sub-harmonics, or combination 

 tones that would be called into play, probably one wire 

 to each digital instead of two or three would suffice. 



The original account of this key-board is given in the 

 second of two small pamphlets entitled " Music in Common 

 Things," by Mr. Colin Brown. In these the numerical 

 basis of the diatonic scale as derived from harmonics is 

 laid down with remarkable perspicuity, and amongst other 

 things the reader will see, for the first time probably, 

 that although the fourth and sixth of the scale are want- 

 ing as harmonics to the tonic, yet they come out in the 

 diatonic scale as harmonics to the fourth of the scale. 

 Indeed the seven notes of the scale come out successively 

 as harmonics to Fa m the fifth and sixth ociaves. 



A. R. C. 



