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



to 8, / to 11, and 10 to 14 respectively would be nearly complete. The 

 manuals might be similar to those on General T. Perronet Thompson's 

 Enharmonic Organ, which has 3 boards, with 20, 23 and 22 manuals re- 

 spectively, and contains the chords in Table V. cols. III., lines 6 to 11 ; 

 IV. 6 to 12 ; v., VI., VII., 5 to 12 ; VIII. and IX., 6 to 12 (four chords 

 belonging to col. IX., lines 6 to 9, are not in the Table, but can be readily 

 supplied, as well as the additional lines 0, — 1, named below). 



Euler's "genus cujus exponens est 2"'. 3"^. 5^," as developed in his 

 Tentamen, p. 161, must be considered as adapted for an instrument with 

 two boards of ordinary manuals, such as some harmoniums are now con- 

 structed. His "soni primarii" would occupy the lower, and his "soni 

 secuudarii " the upper board. If to these we add their schismatic equiva- 

 lents, inclosed in brackets, and distinguish white and black manuals by 

 capital and small letters as in Table II., Euler's scheme will appear as 

 follows, wliere the notation interprets his arithmetical expressions of pitch 

 ("soni"), and not his notes ("signa sonora"), which are too vague. 



Euler's Double Scheme. § 

 TJppei' Board. 



Schism. Equival.. . [tC, ID, eb, F'\), XF, (p, XG, cb, JBi^,l)?, Cb| 



'^SomSecundarii" B% c% Cx, cl% W, F% Fx, U, U^, t^. 



Lower Board. 



^'SoniPnmarii" . C, Xc% B, Xd%F, F, f%G, Xff^A 



Schism. Equival.. . [Ji^b, JJcZb, FW, X^, XF'), X^)^, Xg), -^^b, tCf 



Although it is evident from his notation that Euler regarded schismatic 

 equivalents as identities, he has not especially alluded to them. The above 

 scheme would contain Table V. col. V., lines to 14, and the major third 

 '\F^ fJ^ in 15 (with the schismatic error of %B'7 ^XD F for ^> J7) F), 

 col. VI. 1 to 15 ; VII. P to 24 ; VIII. 10 to 24 ; IX. 18 to 24 ; III. 

 — 1 to 5 ; IV. to 6. It would be therefore nearly complete in major 

 scales, but would have only Xd, a, e, b, f%, c%, ff^ minor, and their compa- 

 ratively useless schismatic equivalents. It would have no single complete 

 key, and would therefore require many commatic substitutions in modula- 

 tion, and the use of the Pythagorean major third in the major chords of 

 the comparatively common minor scales of J/, %€, %g. If only the " soni 

 pi imarii" of the lower board are used the substitutions become very harsh, 

 as for example A^DF, DF^A for B? tD F, DF^-fA. 



Euler's "soni primarii" maybe compared with Rameau's scale*, which 

 was as follows, 



C, Xc% XD, Vh, E, F, Xn, G, X9% A, t5b, B, 



* Traite de I'Harmoiiie, 1721. The values of the tones are determined from Lis 

 arithmetical expression of the intervals. 



