1864] 



on Instruments with Fixed Tones. 



101 



capitals red*. By this arrangement the fingering of every key would be 

 the same. The performer would disregard the signature except as naming 

 the pedal, and play as if the signature were natural. Table V. would 

 inform him whether the accidentals belonged to the key, its dominant, or 

 any other key ; and if they indicated another key, he would change the 

 pedal. It would be convenient to mark where a new pedal had to be used ; 

 but no change would be required in the established notation §. 



Mr. Poole's organ, which suggested the above arrangement, has 11 stops, 

 from 5b to 5j?, and only 12 manuals, which appear to be associated with 

 the following tones on each stop : 



BJach.. \2 (t2S) (4S) JSft gTb 



WJiHe.. 1 2 3 4 5 6 7 



The two manuals whose notes are put in parentheses are inadequately de- 

 scribed. Mr. Poole's scale does not include the synonymous minor chords, 

 which he plays by commatic substitution. 



Another method of realizing such a scale is by additional manuals and 

 additional boards of manuals. Thus three boards of manuals, each with 

 23 manuals, containing the tones in Table V. cols. III. to VIII., lines 4 



* On examining Table II. it will be foimd that 10 difterent tones lie on each 

 pair of manuals, so that there are only 70 different tones. The two missing tones 

 are, necessarily, tt/it (the acute fom-th of the key of ^C^), and Wlb (the gTave 

 seventh of the key of J Cb) ; and to this extent the scheme is defective. It would 

 probably be more convenient to the instrimient-maker to use all the 70 tones in this 

 arrangement than to take the inferior number 45 due to schismatic substitution. 

 A full-sized harmonium at present employs from 48 to 60 vibrators to the 

 octave, so that the mechanical difficulties to be overcome in introducing 70 are 

 comparatively slight. By omitting the two very imusual keys of % d\> and f C% 

 the 8 tones denoted by ft^/b, Ji^b, \X f, (b\> and fi^g:,, ^X, t^fi, tt^ in Table II. 

 would be saved, and the number of vibrators required would be reduced to 62, 

 nearly the same as that actually in use. As each new key introduces 4 addi- 

 tional tones, and the key of C has 14 tones, the number of vibrators required for 

 any extent of scale is readily calculated. Thus for the 11 keys from 5 flats to 5 

 shai-ps, or XW, A), BO, -Sb, -F, C, G, D, A, E, B, which is Mr. Poole's range, 

 and is sufficiently extensive for almost all pm-poses, only 4 X 10 + 14= 54 vibrators 

 to the octave would be requii-ed, distributed over 11 stops (exclusive of the tem- 

 pered notes) ; and such a number of vibrators and stops is in common use. 



§ If in Table V. we reject the marks f, t, consider 16 ^ = 27 C, 64^=81 C, 

 2187 2048 



128 B = 243 C, ii=^^7r^» b = xT-^> leaving- the value of the other letters un- 

 ' 2048' 218/^ ° 



changed, the Table will represent the Pythagorean relations expressed by the 

 usual notation (which is quite unsuited to the equally tempered scale). The 

 chords thus formed were too dissonant for the Greek or Arabic ear to endure, 

 although Drobisch and Naumann Qoc. cit. ad finem) desire this system to be ac- 

 knowledged as " the sole, really sufficient acoustical foundation for the theory of 

 music " {ctls einzige, wahrliaft genUyencle aJcustische Orundlage der theoretisch-mu- 

 kcdischen Lehre), 



I 2 



