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>. 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 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 {{OEIS|A135972}}. | |||
== List of Mersenne commas == | == List of Mersenne commas == | ||
{| class="wikitable" | {| class="wikitable center-1" | ||
|+Table of first Mersenne composite commas | |+Table of first Mersenne composite commas | ||
! Index | ! Index | ||
! Comma | ! Comma | ||
! Subgroup | ! Subgroup | ||
! S. | ! S. monzo | ||
! Comments | ! Comments | ||
|- | |- | ||
| Line 15: | Line 17: | ||
| [[16/15]] | | [[16/15]] | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| 4 -1 -1 }} | ||
| Classic diatonic semitone | | Classic diatonic semitone | ||
|- | |- | ||
| Line 21: | Line 23: | ||
| [[64/63]] | | [[64/63]] | ||
| 2.3.7 | | 2.3.7 | ||
| {{ | | {{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 }} | ||
| | | Charisma | ||
|- | |- | ||
| 9 | | 9 | ||
| [[512/511]] | | [[512/511]] | ||
| 2.7.73 | | 2.7.73 | ||
| {{ | | {{Monzo| 9 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| Line 39: | Line 41: | ||
| [[1024/1023]] | | [[1024/1023]] | ||
| 2.3.11.31 | | 2.3.11.31 | ||
| {{ | | {{Monzo| 10 -1 -1 -1 }} | ||
| | | Kibisma | ||
|- | |- | ||
| 11 | | 11 | ||
| [[2048/2047]] | | [[2048/2047]] | ||
| 2.23.89 | | 2.23.89 | ||
| {{ | | {{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 }} | ||
| | | Minisma | ||
|- | |- | ||
| | | 14 | ||
| [[16384/16383]] | | [[16384/16383]] | ||
| 2.3.43.127 | | 2.3.43.127 | ||
| {{ | | {{Monzo| 14 -1 -1 -1 }} | ||
| | | | ||
|- | |- | ||
| | | 15 | ||
| [[32768/32767]] | | [[32768/32767]] | ||
| 2.7.31.151 | | 2.7.31.151 | ||
| {{ | | {{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 }} | ||
| | | | ||
|- | |- | ||
| | | 18 | ||
| [[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]] | | [[1048576/1048575]] | ||
| 2.3.5.11.31.41 | | 2.3.5.11.31.41 | ||
| {{ | | {{Monzo| 20 -1 -2 -1 -1 -1 }} | ||
| | | Mebisma | ||
|- | |- | ||
| | | 21 | ||
| [[2097152/2097151]] | | [[2097152/2097151]] | ||
| 2.7.127.337 | | 2.7.127.337 | ||
| {{ | | {{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 }} | ||
| | | | ||
|- | |- | ||
| | | 23 | ||
| [[8388608/8388607]] | | [[8388608/8388607]] | ||
| 2.47.178481 | | 2.47.178481 | ||
| {{ | | {{Monzo| 23 -1 -1 }} | ||
| | | | ||
|- | |||
| 24 | |||
| [[16777216/16777215]] | |||
| 2.3.5.7.13.17.241 | |||
| {{Monzo| 24 -2 -1 -1 -1 -1 -1 }} | |||
|} | |} | ||
| Line 106: | Line 113: | ||
[[Category:Lists of commas]] | [[Category:Lists of commas]] | ||
[[Category:Octave-reduced subharmonics]] | [[Category:Octave-reduced subharmonics]] | ||
{{ | {{Todo|explain its xenharmonic value}} | ||