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Euler, perceiving that the relative pitches of all musical notes might be represented by 2“. 3”. 5P, formed different “genera musica” by allowingn andp to vary from 0 to fixed limits. His “genus diatonicum hodiernum” (op. cit . p, 135) limits into 3 and p to 2, and consists of 12 tones. These tones and 12 others are contained in his “genus cujus exponens est 2m . 37 . 52 ,” that is, which limitsn to 7 andp to 2. He has further (ib . p. 161) given a scheme in which each manual of an instrument should represent two sounds, the primary belonging to the first 12 tones, and the secondary to the additional 12. He says (ib . p. 162), “ Soni secundarii summo rigore ab iisdem clavibus edi nequeunt, quia vero tarn parum a primariis discrepant, ad eos exprimendos hse clavessine sensibili harmonics jactura tuto adhiberi possunt.Nam etiamsi ab bus comma seu diaschisma, quibus mtervallis soni secundarii a primariis differunt, distingui queat, tamen quia soni secundarii cum primariis nequein eademconsonantianeque in duavum consonantiarum successione misceri possunt , error etiam ab acutissimo auditu percipi non potent.” It will i appear in the sequel that these assertions, when tested by experiments on instruments with fixed tones, are all incorrect. The musical’scale has formed the subject of many recent investigations*; but I have been unable to find a complete account of the necessary conditions to be fulfilled by a perfect scale, the least number of fixed tones required, and the practical means of producing them uncurtailed without inconvenience to the performer, although instruments which produce alimited number of just tones have been practically used by Gen. Perronet Thompson, Mr. Poole, Prof. Helmholtz, Prof. Wheatstone, myself, and others. This is therefore the subject of the present paper.

II. On the conditions, extent, and realization of a perfect musical scale on instruments with fixed tones