Mersenne comma: Difference between revisions
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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>. | ||
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. | |||
== List of Mersenne commas == | Mersenne composite commas, on the other hand, can have other uses, and the table below includes such notable examples of these as the [[64/63|Septimal comma]]. Mersenne composite numbers can be found in The On-Line Encyclopedia of Integer Sequences (OEIS) at [[oeis:A135972|sequence A135972]]. | ||
==List of Mersenne commas== | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Table of first Mersenne composite commas | |+Table of first Mersenne composite commas | ||
! Index | ! Index | ||
! Comma | !Comma | ||
! Subgroup | !Subgroup | ||
! S. Monzo | !S. Monzo | ||
! Comments | !Comments | ||
|- | |- | ||
| 4 | | 4 | ||
| [[16/15]] | |[[16/15]] | ||
| 2.3.5 | |2.3.5 | ||
| {{monzo| 4 -1 -1 }} | |{{monzo| 4 -1 -1 }} | ||
| Classic diatonic semitone | |Classic diatonic semitone | ||
|- | |- | ||
| 6 | |6 | ||
| [[64/63]] | |[[64/63]] | ||
| 2.3.7 | |2.3.7 | ||
| {{monzo| 6 -2 -1 }} | |{{monzo| 6 -2 -1 }} | ||
| Septimal comma | | Septimal comma (Archytas comma) | ||
|- | |- | ||
| 8 | |8 | ||
| [[256/255]] | |[[256/255]] | ||
| 2.3.5.17 | |2.3.5.17 | ||
| {{monzo| 8 -1 -1 -1 }} | |{{monzo| 8 -1 -1 -1 }} | ||
| Septendecimal kleisma | | Septendecimal kleisma | ||
|- | |- | ||
| 9 | |9 | ||
| [[512/511]] | |[[512/511]] | ||
| 2.7.73 | |2.7.73 | ||
| {{monzo| 9 -1 -1 }} | |{{monzo| 9 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| 10 | |10 | ||
| [[1024/1023]] | |[[1024/1023]] | ||
| 2.3.11.31 | | 2.3.11.31 | ||
| {{monzo| 10 -1 -1 -1 }} | |{{monzo| 10 -1 -1 -1 }} | ||
| Kilobyte comma | |Kilobyte comma | ||
|- | |- | ||
| 11 | |11 | ||
| [[2048/2047]] | |[[2048/2047]] | ||
| 2.23.89 | |2.23.89 | ||
| {{monzo| 11 -1 -1 }} | |{{monzo| 11 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| 12 | |12 | ||
| [[4096/4095]] | |[[4096/4095]] | ||
| 2.3.5.7.13 | | 2.3.5.7.13 | ||
| {{monzo| 12 -2 -1 -1 -1 }} | |{{monzo| 12 -2 -1 -1 -1 }} | ||
| Schismina | |Schismina | ||
|- | |- | ||
| | |14 | ||
| [[16384/16383]] | |[[16384/16383]] | ||
| 2.3.43.127 | |2.3.43.127 | ||
| {{monzo| 14 -1 -1 -1 }} | |{{monzo| 14 -1 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| | |15 | ||
| [[32768/32767]] | |[[32768/32767]] | ||
| 2.7.31.151 | | 2.7.31.151 | ||
| {{monzo| 15 -1 -1 -1 }} | |{{monzo| 15 -1 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| | |16 | ||
| [[65536/65535]] | |[[65536/65535]] | ||
| 2.3.5.17.257 | |2.3.5.17.257 | ||
| {{monzo| 16 -1 -1 -1 -1 }} | |{{monzo| 16 -1 -1 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| | |19 | ||
| [[262144/262143]] | |[[262144/262143]] | ||
| 2.3.7.19.73 | |2.3.7.19.73 | ||
| {{monzo| 18 -3 -1 -1 -1 }} | |{{monzo| 18 -3 -1 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| | |20 | ||
| [[1048576/1048575]] | |[[[[Megabyte comma|1048576/1048575]]]] | ||
| 2.3.5.11.31.41 | |2.3.5.11.31.41 | ||
| {{monzo| 20 -1 -2 -1 -1 -1 }} | |{{monzo| 20 -1 -2 -1 -1 -1 }} | ||
| | |Megabyte comma | ||
|- | |- | ||
| | |21 | ||
| [[2097152/2097151]] | |[[2097152/2097151]] | ||
| 2.7.127.337 | |2.7.127.337 | ||
| {{monzo| 21 -2 -1 -1 }} | |{{monzo| 21 -2 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| | |22 | ||
| [[4194304/4194303]] | |[[4194304/4194303]] | ||
| 2.3.23.89.683 | |2.3.23.89.683 | ||
| {{monzo| 22 -1 -1 -1 -1 }} | |{{monzo| 22 -1 -1 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| | |23 | ||
| [[8388608/8388607]] | |[[8388608/8388607]] | ||
| 2.47.178481 | |2.47.178481 | ||
| {{monzo| 23 -1 -1 }} | |{{monzo| 23 -1 -1 }} | ||
| | | | ||
|} | |} | ||