Mersenne comma

A Mersenne comma is a comma of the form [math]\displaystyle{ \frac{2^n}{2^n-1} }[/math]. As such, they are also by definition octave-reduced subharmonics.

Mersenne prime commas equate a specific prime harmonic with the octave, so they are generally not of interest to edo theory, with the possible exception of certain equal divisions of a compressed octave.

Mersenne composite commas, on the other hand, can have other uses, and the table below includes such notable examples of these as the septimal comma. Mersenne composite numbers can be found in OEIS: A135972.

List of Mersenne commas

Table of first Mersenne composite commas

Index	Comma	Subgroup	S. monzo	Comments
4	16/15	2.3.5	[4 -1 -1⟩	Classic diatonic semitone
6	64/63	2.3.7	[6 -2 -1⟩	Septimal comma (Archytas' comma)
8	256/255	2.3.5.17	[8 -1 -1 -1⟩	Charisma
9	512/511	2.7.73	[9 -1 -1⟩
10	1024/1023	2.3.11.31	[10 -1 -1 -1⟩	Kibisma
11	2048/2047	2.23.89	[11 -1 -1⟩
12	4096/4095	2.3.5.7.13	[12 -2 -1 -1 -1⟩	Minisma
14	16384/16383	2.3.43.127	[14 -1 -1 -1⟩
15	32768/32767	2.7.31.151	[15 -1 -1 -1⟩
16	65536/65535	2.3.5.17.257	[16 -1 -1 -1 -1⟩
18	262144/262143	2.3.7.19.73	[18 -3 -1 -1 -1⟩
20	1048576/1048575	2.3.5.11.31.41	[20 -1 -2 -1 -1 -1⟩	Mebisma
21	2097152/2097151	2.7.127.337	[21 -2 -1 -1⟩
22	4194304/4194303	2.3.23.89.683	[22 -1 -1 -1 -1⟩
23	8388608/8388607	2.47.178481	[23 -1 -1⟩
24	16777216/16777215	2.3.5.7.13.17.241	[24 -2 -1 -1 -1 -1 -1⟩