Diesis

The musical interval by which an octave exceeds three major thirds,

$${2\over ({\textstyle{5\over 4}})^3} = {2^7\over 5^3} = {128\over 125} = 1.024.$$

Taking Continued Fraction Convergents of $\log(5/4)/\log(2)$ gives the increasing accurate approximations $m/n$ of $m$ octaves and $n$ major thirds: 1/3, 9/28, 19/59, 47/146, 207/643, 1289/4004, ... (Sloane's A046103 and A046104). Other near equalities of $m$ octaves and $n$ major thirds having

$$R\equiv {2^m\over ({\textstyle{5\over 4}})^n} = {2^{m+2n}\over 5^n}$$

with $\vert R-1\vert<0.02$ are given in the following table.

$m$	$n$	Ratio	$m$	$n$	Ratio
9	28	0.9903520314	104	323	1.012011267
10	31	1.01412048	113	351	1.002247414
18	56	0.9807971462	122	379	0.9925777621
19	59	1.004336278	123	382	1.016399628
28	87	0.9946464728	131	407	0.983001403
29	90	1.018517988	132	410	1.006593437
37	115	0.9850501549	141	438	0.9968818549
38	118	1.008691359	150	466	0.9872639701
47	146	0.9989595361	151	469	1.010958305
56	174	0.9893216059	160	497	1.001204611
57	177	1.013065324	169	525	0.9915450208
66	205	1.003291302	170	528	1.015342101
75	233	0.9936115791	178	553	0.9819786256
76	236	1.017458257	179	556	1.005546113
84	261	0.9840252458	188	584	0.9958446353
85	264	1.007641852	189	587	1.019744907
94	292	0.9979201548	197	612	0.9862367575
103	320	0.9882922525	198	615	1.00990644

See also Comma of Didymus, Comma of Pythagoras, Schisma

References

Sloane, N. J. A. A046103 and A046104 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.