1864.] on Instruments with Fixed Tones. 103 



and therefore only contained the following perfect harmonies, and tivo 

 perfect scales, A major and a minor : — 



—XD F 



F A C Xdfa XDXF^A 

 tJEi?6?-Bb, c'^^g CFG ace AXC%F 



g^lb— G B — egh EXG^B. 



Prof. Helmholtz has tuned an harmonium with two boards of manuals, 

 somewhat in Euler's manner, as follows : — 



Helmholtz's Double Scheme. 

 Upper Board. 



Schism, equiv. [JC, d\}, XD, eb, F\}, F, ^b, A, b^7, Ci?] 



Tones tuned. . B^, \c%, Cx, d%, \F, t^tf, t/ft -FX, ^, G^X, t^, t^- 



Lower Board, 



Tones tuned.. C, D, X^%, F, F% f% G, X.^,fA, a% B 



Schism, equiv. [|D>t>, Jf^b, J^bb, Jeb, XF^, XF, J^b, -4bb, |«b, 5bb, J6b, t C^f ]. 



This scheme has nearly the same extent and the same defects as Euler's. 



The concertina, invented by Prof. Wheatstone, F.R.S., has 14 manuals 

 to the octave, which were originally tuned thus, as an extension of Euler's 

 12-tone scheme. 



tc#, B, Xd% F, teb, F,f^, G, Xg^, A, fab, B, fib. 



It possessed the perfect major and minor scales of C and E. The harsh- 

 ness of the chords -f^b B Fy D F^ A, for Bb XD F, D Fjf;, fA has, however, 

 led to the abandonment of this scheme, and to the introduction of a tem- 

 pered scale. I have taken advantage of the 14 manuals to contrive 4 dif- 

 ferent methods of tuning, so that 4 concertinas would play in all the common 

 major and minor scales. Two of these I have in use, and find them 

 effective and very useful for experimental purposes. The following gives 

 the arrangement of the manuals in each, together with the scales possessed 

 by each instrument, major in capitals, and minor in small letters. Where 

 commatic substitution makes the dominant chord too flat in major scales, 

 parentheses ( ) are used ; where it makes the subdominant chord too sharp, 

 brackets [ ] are used. Minor scales in brackets have only the subdominant 

 tone too sharp. 



The major chord GBD and the tone C being common to all four 

 instruments, determine their relative pitch. The method of tuning these 

 and all justly intoned or teleon * instruments is very simple. C being tuned 

 to any standard pitch, the fifths above and below it are tuned perfect. To 

 any convenient tone thus formed, as C itself, form the major thirds above, 



* A convenient name, formed from riXeov diaaTiifxa, a perfect interval. 



