Mersenne comma: Difference between revisions
Jump to navigation
Jump to search
m →List of Mersenne commas: corrected names |
explain relevance of octave-reduced subharmonic quality |
||
| Line 1: | Line 1: | ||
A '''Mersenne comma''' is a [[comma]] of the form <math>\frac{2^n}{2^n-1}</math>. | A '''Mersenne comma''' is a [[comma]] of the form <math>\frac{2^n}{2^n-1}</math>. As such, they are also by definition [[octave-reduced]] [[subharmonic]]s. | ||
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 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. | ||
Revision as of 21:00, 29 September 2024
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 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⟩ | Kibisma |
| 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⟩ | 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⟩ |