Comma of Pythagoras

The musical interval by which twelve fifths exceed seven octaves,

$${({\textstyle{3\over 2}})^{12}\over 2^7} = {3^{12}\over 2^{19}} = {531441\over 524288} = 1.013643265.$$

Successive Continued Fraction Convergents to $\log 2/\log(3/2)$ give increasingly close approximations $m/n$ of $m$ fifths by $n$ octaves as 1, 2, 5/3, 12/7, 41/24, 53/31, 306/179, 665/389, ... (Sloane's A005664 and A046102; Jeans 1968, p. 188), shown in bold in the table below. All near-equalities of $m$ fifths and $n$ octaves having

$$R\equiv {({\textstyle{3\over 2}})^m\over 2^n} = {3^m\over 2^{m+n}}$$

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

$m$	$n$	Ratio	$m$	$n$	Ratio
12	7	1.013643265	265	155	1.010495356
41	24	0.9886025477	294	172	0.9855324037
53	31	1.002090314	306	179	0.9989782832
65	38	1.015762098	318	186	1.012607608
94	55	0.9906690375	347	203	0.9875924759
106	62	1.004184997	359	210	1.001066462
118	69	1.017885359	371	217	1.014724276
147	86	0.9927398469	400	234	0.9896568543
159	93	1.006284059	412	241	1.003159005
188	110	0.9814251419	424	248	1.016845369
200	117	0.994814985	453	265	0.9917255479
212	124	1.008387509	465	272	1.005255922
241	141	0.9834766286	477	279	1.018970895
253	148	0.9968944607	494	289	0.9804224033

See also Comma of Didymus, Diesis, Schisma

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 257, 1995.

Guy, R. K. ``Small Differences Between Powers of 2 and 3.'' §F23 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 261, 1994.

Sloane, N. J. A. Sequences A005664/M1428 and A046102 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.