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Mersenne commas are a series of commas 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.


Since Mersenne prime commas effectively set their own prime limit, they are of no interest to EDO theory. Therefore, this time Mersenne composite numbers enter the stage - sequence [https://oeis.org/A135972 A135972] in OEIS.  
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.


==Theory==
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}}.
{| class="wikitable"
 
== List of Mersenne commas ==
{| class="wikitable center-1"
|+Table of first Mersenne composite commas
|+Table of first Mersenne composite commas
!Index
! Index
!Comma
! Comma
!Prime
! Subgroup
Subgroup
! S. monzo
!Monzo
! Comments
(zeroes skipped)
|-
!Comments
| 4
| [[16/15]]
| 2.3.5
| {{Monzo| 4 -1 -1 }}
| Classic diatonic semitone
|-
| 6
| [[64/63]]
| 2.3.7
| {{Monzo| 6 -2 -1 }}
| Septimal comma (Archytas' comma)
|-
| 8
| [[256/255]]
| 2.3.5.17
| {{Monzo| 8 -1 -1 -1 }}
| Charisma
|-
| 9
| [[512/511]]
| 2.7.73
| {{Monzo| 9 -1 -1 }}
|
|-
| 10
| [[1024/1023]]
| 2.3.11.31
| {{Monzo| 10 -1 -1 -1 }}
| Kibisma
|-
| 11
| [[2048/2047]]
| 2.23.89
| {{Monzo| 11 -1 -1 }}
|
|-
|-
|4
| 12
|[[16/15]]
| [[4096/4095]]
|2.3.5
| 2.3.5.7.13
|[4 -1 -1⟩
| {{Monzo| 12 -2 -1 -1 -1 }}
|Classic diatonic semitone.
| Minisma
|-
|-
|6
| 14
|[[64/63]]
| [[16384/16383]]
|2.3.7
| 2.3.43.127
|[6 -2 -1⟩
| {{Monzo| 14 -1 -1 -1 }}
|Septimal comma.
|  
|-
|-
|8
| 15
|[[256/255]]
| [[32768/32767]]
|2.3.5.17
| 2.7.31.151
|[8 -1 -1 -1⟩
| {{Monzo| 15 -1 -1 -1 }}
|Septendecimal kleisma.
|  
|-
|-
|9
| 16
|[[512/511]]
| [[65536/65535]]
|2.7.73
| 2.3.5.17.257
|[9 -1 -1⟩
| {{Monzo| 16 -1 -1 -1 -1 }}
|
|
|-
|-
|10
| 18
|[[1024/1023]]
| [[262144/262143]]
|2.3.11.31
| 2.3.7.19.73
|[10 -1 -1 -1⟩
| {{monzo| 18 -3 -1 -1 -1 }}
|
|
|-
| 20
| [[1048576/1048575]]
| 2.3.5.11.31.41
| {{Monzo| 20 -1 -2 -1 -1 -1 }}
| Mebisma
|-
| 21
| [[2097152/2097151]]
| 2.7.127.337
| {{Monzo| 21 -2 -1 -1 }}
|
|-
| 22
| [[4194304/4194303]]
| 2.3.23.89.683
| {{Monzo| 22 -1 -1 -1 -1 }}
|  
|-
|-
|11
| 23
|[[2048/2047]]
| [[8388608/8388607]]
|2.23.89
| 2.47.178481
|[11 -1 -1⟩
| {{Monzo| 23 -1 -1 }}
|
|  
|-
|-
|12
| 24
|[[4096/4095]]
| [[16777216/16777215]]
|2.3.5.7.13
| 2.3.5.7.13.17.241
|[12 -2 -1 -1 -1⟩
| {{Monzo| 24 -2 -1 -1 -1 -1 -1 }}
|Schismina.
|}
|}


[[Category:Ratio]]
[[Category:Lists of commas]]
[[Category:Octave-reduced subharmonics]]
[[Category:Octave-reduced subharmonics]]
[[Category:superparticular ratios]]
{{Todo|explain its xenharmonic value}}
[[Category:Rational intervals]]
Retrieved from "https://en.xen.wiki/w/Mersenne_comma"