344 On Musical Seales. [July, 
The remaining 15 equations, when for C, D, HE, &., we substi- 
stute their assigned numerical values, divide themselves into two 
classes; those which give explicit values to the unknown quantities 
a, 8, y, &c., and those which give differential relations between 
them, thus :— 
(III.)\—Expuiicir VALUES. 
a=59; a=93; B=229; B=263; y=492; 5=644; 5=678; «= 848. 
DIFFERENTIAL RELATIONS. 
e-— y= 822. 
(1V.)—Exp.icir VALUES. 
y=492; «= 830. 
DIFFERENTIAL RELATIONS. 
y—a=415; 5-a=585; 5—B=415; e—B=585; 
of which the system (III.) contains those whose fulfilment secures 
perfect Thirds, and (1V.) those which secure perfect Fifths. And 
on them both we have to remark—Ist. That not more than five 
distinct Thirds, in addition to the three which the natural notes 
afford, can be secured by any choice of a, B, y, &c., by employing 
the equations (III.) 2ndly. That the same is true of Fifths as 
secured by the equations (IV.). And 38rdly. That these Thirds and 
Fifths are mutually exclusive, with the exception that the adoption 
of 492 as the value of y, secures at once both a Fifth and a Third. 
To show this, it suffices to compare the values assigned by each set 
of equations separately, which are— 
(V.), THOSE DEDUCED FRomM (III.), viz. :— 
a=59 or 93 
5=644 or 678 
¥y = 492 7 = 492 y = 526 
and or and or and 
e = 448 e= 814 e= 848 
Anp (VI.), THOSE DERIVED FROM (IV.), BEING Srx Distinct Systems oF 
VALUES OBTAINED : 
Ist, by excluding y=492; viz. a=75, B=245, y=—490, 5=660, e=830 
2nd, ” e— COU) ss Ciel 7y=492, 5=662, «=832 
3rd, =f; y—a=415; a t= oO. B20: 7y=492, 5=660, «=830 
4th, 3-885 he. G= 77 B= 24s, y= 402, SE Gab. cee 
5th, 3 5—B=415; , a=77, B=245, y=492, 5=662, «=830 
Gth, i e-B=585; - a=77, B=247, y=492, 8=662, «= 830 
These, however, are not the only Fifths obtainable; for among 
the value of a, B, y, &c., given in (Y.), there are six pairs which 
satisfy one or other of the differential relations in (IV.), wz. (« = 59, 
5 = 644), and (2 = 93, 5 = 678), either of which pairs satisfies 
8 —a«a=585. And again, (@ = 229, 6= 644), and (8 = 263, 
