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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.


Since Mersenne prime commas effectively set their own prime limit, they are of no interest to EDO theory{{clarify}}. 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.
 
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. Monzo
! S. monzo
! Comments
! Comments
|-
|-
Line 15: Line 17:
| [[16/15]]
| [[16/15]]
| 2.3.5
| 2.3.5
| {{monzo| 4 -1 -1 }}
| {{Monzo| 4 -1 -1 }}
| Classic diatonic semitone
| Classic diatonic semitone
|-
|-
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| [[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
| Charisma
|-
|-
| 9
| 9
| [[512/511]]
| [[512/511]]
| 2.7.73
| 2.7.73
| {{monzo| 9 -1 -1 }}
| {{Monzo| 9 -1 -1 }}
|  
|  
|-
|-
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| [[1024/1023]]
| [[1024/1023]]
| 2.3.11.31
| 2.3.11.31
| {{monzo| 10 -1 -1 -1 }}
| {{Monzo| 10 -1 -1 -1 }}
|  
| Kibisma
|-
|-
| 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
| Minisma
|-
| 14
| [[16384/16383]]
| 2.3.43.127
| {{Monzo| 14 -1 -1 -1 }}
|
|-
| 15
| [[32768/32767]]
| 2.7.31.151
| {{Monzo| 15 -1 -1 -1 }}
|
|-
| 16
| [[65536/65535]]
| 2.3.5.17.257
| {{Monzo| 16 -1 -1 -1 -1 }}
|
|-
| 18
| [[262144/262143]]
| 2.3.7.19.73
| {{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 }}
|
|-
| 23
| [[8388608/8388607]]
| 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 }}
|}
|}


[[Category:Ratio]]
[[Category:Ratio]]
[[Category:Lists of commas]]
[[Category:Octave-reduced subharmonics]]
{{Todo|explain its xenharmonic value}}
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