176 Mr. R. H. M. Bosanquet on Temperament, 



sevenths of the principal systems. The actual values in deci- 

 mals are easily deduced ; but those here given exhibit the rela- 

 tions better. (For a Table of the actual values I must refer 

 to my paper on Temperament in the current volume of the 

 Royal Society's c Proceedings/ or to the paper in the ' Transac- 

 tions ' of the Musical Association.) The values for the perfect 

 harmonic seventh are 



_ 14 __. 10 __. 3lm . 

 41-91 3^084" " ' 



whence the system of 41 would represent it best of the positive 

 systems enumerated, and the system of 31 best of the negative 

 systems ; the latter representation is rather close. 



Symmetrical Arrangement and Generalized Key-board. 



For an illustration of the application of the principle of sym- 

 metrical arrangement to positive systems, and for a woodcut of 

 the generalized key-board, I must refer to my papers in the Royal 

 Society's f Proceedings ' and the ' Transactions ' of the Musical 

 Association. 



Any interval formed in any regular system can be represented 

 by an expression of the form x + y8, where x is an integral num- 

 ber of E.T. semitones, and y the number of fifths whose depar- 

 ture is involved. The principle of symmetrical arrangement 

 which I adopt consists in representing notes by positions deter- 

 mined by x as an abscissa and y as an ordinate, where x and y 

 have integral values. The second term is always taken positive 

 for fifths up, and negative for fifths down, whatever be the sign 

 of the departure 8. The reason for this is the convenience of 

 having the same key-board for positive and negative systems. 

 The best mode of describing the key-board will probably be to 

 refer the middle points of the ends of the keys to three rectan- 

 gular axes, x being measured along the length of the k^y-board, 

 y along the depth [from the player), and z vertically upwards. 

 The note c on the left nearest the player may be taken for 

 origin. The numbers of units in y and z are the same. The 

 extremities of the keys all lie in a plane rising like the slope of 

 a desk at an angle tan -1 j. The values of x correspond to half- 

 inches, of y to quarter-inches, and of z to twelfths of an inch. 

 Thus, for the departure of 12 fifths, #=0, y — 3 inches, z—\ 

 inch; that is, the key /c is straight behind and a little above c, 

 3 inches back and 1 inch up. The octave is made 6 inches wide 

 (#=12). In ordinary keyboards this width is 6^ inches. The 

 middle lines of the columns of keys are \ an inch apart, and 

 the keys themselves § inch broad ; so that there is § inch alto- 

 gether between the two keys which rise on each side of any 



