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Mr. A. J. Ellis on a Perfect Musical Scale {Jan. 21, 



I!\>XG^, ^9tDF, FAC, CEG, GBB, BF^\A, t^C#t^, and form 

 the five related chords of each as before. The result will be Jive keys, as 

 those of ^b, Ft C, Gj B, such that the primary major scales of each will 

 have either two major chords, or one major chord in common with the 

 original primary major scale. I call these five keys the jjostdominant, sub- 

 dominant, tonicj dominant, and superdominant keys, and the whole group 

 of 21 major and 21 minor chords, with the 30 tones which they contain, 

 I term the system of the first tone of the tonic chord of the original pri- 

 mary major scale, which tone may be called the tonic of the system. 



A piece of music is written in a certain system, determined by the com- 

 pass or quality of tone of the instruments or voices which have to perform 

 it, and rarely exceeds that system*. It is only in the system that the true 

 relation of the tones of a piece of music, the course and intention of the 

 modulation, and the return to the original key or scale can be appreciated. 

 I have not yet found these relations fully expressed in any theoretical work 

 on music ; but their full expression was necessary to the solution of the 

 problem here proposed. 



It will be found practically that only 11 systems are used in music. 

 These are, in dominative order, the systems of $Db, J>, J5'l>, ^b, F, C, G, 

 B, fA, fE, fB, which contain the 11 keys of the same name, together 

 with the 4 keys of $Cb, $G^b, and fF^, fCtf. In Table V., columns III. 

 to VIII., the whole of the major and minor chords of these 15 keys are 

 exhibited in dominative order §. This Table, therefore, furnishes the tones 

 which must be contained in a perfect musical scale of fixed tones, or the 

 conditions of the problem. 



On examination it will be found that these six columns contain 72 dif- 

 ferent notes. Hence the extent of a perfect scale is fixed at 72 tones to 

 the octave. It is therefore six times as extensive as the equally tempered 

 scale. Some means of reducing this unwieldy extent is required. The 

 most obvious is that proposed by Euler, in the passage already quoted, 

 namely, the use of certain tones for others which differ from them by a 

 comma or diaschisma. Such substitution within the same chord creates 

 intolerable dissonance. But in melody and in successions of chords it might 

 seem feasible. I have had a concertina tuned, so that the three chords of 



* The use of the equally tempered scale has much dimmislied the feeling for 

 the relations of the system^ by confounding tones originally distinct, and has 

 thus led to the confusion of the corresponding notes. Thus such a note as 

 will have to be read as Xg% g% t^jj; Jab; d? or fci^, according to the re- 

 quirements of the system, for all six tones are represented by one on the equally 

 tempered scale. 



§ The Table of Key-relationships (TonaHenvenvandtschaficn) in Gottfried 

 Weber's Theorie der TonsetzUmst (3rd ed. 1830, vol. ii. p. 86), may be formed 

 from Table V., by suppressing the signs f, X, supposing all the notes to repre- 

 sent tempered tones, contracting the names of the chords to their first notes, and 

 extending- the Table indefinitely in all directions. 



