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



perament, it is most convenient to use ordinary logarithms to five places, 

 because the actual pitches, and the length of the monochord (which is the 

 reciprocal of the relative pitch), can be thus most easily found. In Table I. 

 the principal intervals are given as fractions, logarithms, and degrees. If 

 we call 0-00568 one degree, then 53 degrees =0-301 04 =log 2—0-00001, 

 and 31 degrees= 0*1 7608 =log f— 0-00001. If we moreover represent 

 the addition and subtraction of 0-00035 (or one-sixteenth of a degree) by 

 an acute or grave accent respectively, then 17' degrees=0'09691=log f, 

 and r degree = 0-00533=log -1^—0-00007. Two numbers of degrees which 

 differ by a single accent of the same kind, as 1 7', 1 7" represent notes whose 

 real interval is a schisma (thus e has 17' degrees; and dx, =%e, has 

 17" degrees), having a difference of logarithm = 0-00049 or 0' degrees 

 + 0-00014. By observing this, degrees may be very conveniently used 

 for all calculation of intervals between tones of pitches represented by 

 2"" . 3" . 5^. Table IV. contains a hst of tones which differ from each 

 other by a schisma, and will be useful hereafter. 



The conditions of a perfect musical scale are not discovered by taking 

 all the tones which can be expressed by one of Euler's '* exponents," nor 

 by forming all the tones which are consonant with a certain tone, and then 

 all the tones consonant with these, as Drobiscli has done. Such processes 

 produce many useless, and omit many necessary tones. Since modern 

 music depends on the relations of harmonies, and not on scales, it is 

 necessary to find what consonant chords of three tones are most closely 

 connected *. 



Three tones whose pitches are as 4 : 5 : 6, or 10 : 12 : 15 form a major 

 or minor consonant chord respectively. The same names are used when 

 any one or more of the pitches is multiplied or divided by a power of 2, 

 notwithstanding the dissonant effect in some cases. Thus, C:E: G= 

 4 : 5 : 6 is a major, and c : teb : ^=10 : 12 : 15 is a minor chord, and the 

 same names are applied to e : : c^ = 5 : 2 x 6 : 2^ x 4, and G : feb : c^= 

 15 : 2 X 12 : 2^ X 10, although these chords are really dissonant (Helmholtz, 

 ib. p. 333-4). I shall consequently use a group of capitals, as CFG, to 

 represent a major chord, and a group of small letters, as c f^g, to represent 

 a minor chord, irrespective of the octaves. The three notes in this order, 

 being the first, third and fifth of the major or minor scale commencing 

 with the first, are called the first, third and fifth of the chords respectively. 

 Both chords contain a fifth, a major and a minor third. If the interval of 

 the ^/th is contained by the same tones in a major and minor chord, as 



* There are consonant chords of four tones, such as gb (P g/^, and these are 

 insisted on by Poole (he. cit) ; but, though they are quite consonant and agree- 

 able, and much pleasanter than the dissonant chords by which they are replaced, 

 such as ^7 6 cPf^, they do not form a part of modern music, for reasons clearly laid 

 down by Hehnholtz (o}). cit. p. 295). Dissonant chords must always arise from the 

 union of tones belonging to two consonant chords, or from the inversions of con- 

 sonant chords ; and therefore their tones are determined with those of the others. 



