1868. ] On Musical Scales. 341 
plement of the two latter to the Octave (1,000—585 = 415 and 
1,000 —322 = 678), and those formed by the addition to them of 
one or more octaves (as 1,000 + 585 = 1,585, and 2,000 + 322 = 
2,322), are perfectly harmonious, and the only intervals in music 
which are so: for the Minor Third (263), though not unpleasing, 
or in any way discordant, leaves the ear in some degree unsatis- 
fied, giving that melancholy expression to the Minor key which 
conveys the impression of something wanting to perfect happiness. 
Assigning, then, to each note in the scale Do, Re, Mi, Fa, Sol, La, 
Si, do, constituting the Octave, a numerical value expressing its 
interval or distance from the fundamental note Do, we have the two 
extremes Do, do designated respectively by 0 and 1,000, and three 
of the intermediate ones (Mi =322; Fa=415; and Sol = 585) 
will claim admission as perfect harmonics with the fundamental 
note or its octave. On the same ground also might 678 claim a 
place in the scale; but as it differs only by 93, or less than the 
tenth part of the octave from Sol already fixed, its admission would 
go to break up the octave into too small subdivisions; so that 
although an excellent candidate for admission into a chromatic scale 
of twelve semitones, as an intermediate between Sol and La it 
cannot be received into the scale of the natural notes. Regarding 
it, then, as an open question how to break the intervals between 0 
and 822 on the one hand, and between 585 and 1,000 on the other, 
by introducing three more natural notes Re, La, and Si; it is evi- 
dent that they ought to be determined (since they cannot harmonize 
with Do or do) by the condition of forming, each, if possible, per- 
fectly harmonious combinations with one or more of the other three 
already fixed. This condition is satisfied by assigning to Re the 
number 170, which thus forms a [Vth (585 —170 = 415) with Sol; 
by 737 assigned to La, which thus affords a [Vth (787 —322=415) 
with Mi, and a IlIrd (737 —415= 822) with Fa; and by 907 
assigned to Sz, giving a Vth (907-822 = 585) with Mz, and a 
IIIvd (907 — 585 = 522) with Sol. 
Thus, then, the natural scale ig filled up. But here already 
occurs an alternative, and a choice of difficulties. As respects the 
positions of La and Si there can be no doubt. La besides standing 
in perfect harmony with Mi and Fa gives a minor Third (1,000 — 
737 =263) with the upper octave Do, an interval though somewhat 
less than harmonious (as. already observed), yet indispensable in 
music, and without which a large region of human emotion would 
lack its musical expression; and La being thus fixed, S¢ can no 
otherwise be determined. As regards Re, however, it may equally 
well be determined by making it form a Vth with La (giving 737 
—585=152 for its value) as a IVth with Sol, which, as we have 
seen, gives 170. This latter, however, makes the interval between 
