Mersenne comma
A Mersenne comma is a comma of the form [math]\displaystyle{ \frac{2^n}{2^n-1} }[/math].
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 The On-Line Encyclopedia of Integer Sequences (OEIS) at sequence A135972.
List of Mersenne 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⟩ | Septendecimal kleisma |
| 9 | 512/511 | 2.7.73 | [9 -1 -1⟩ | |
| 10 | 1024/1023 | 2.3.11.31 | [10 -1 -1 -1⟩ | Kilobyte comma |
| 11 | 2048/2047 | 2.23.89 | [11 -1 -1⟩ | |
| 12 | 4096/4095 | 2.3.5.7.13 | [12 -2 -1 -1 -1⟩ | Schismina |
| 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⟩ | |
| 19 | 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⟩ | Megabyte comma |
| 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⟩ |