Mersenne comma: Difference between revisions

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.

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"

!Comma

!Subgroup

!S. Monzo

!Comments

|[[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 }}

| Septendecimal kleisma

|9

|[[512/511]]

|2.7.73

|{{monzo| 9 -1 -1 }}

|

|10

|[[1024/1023]]

|{{monzo| 10 -1 -1 -1 }}

|Kibisma

|11

|[[2048/2047]]

|2.23.89

|{{monzo| 11 -1 -1 }}

|

|12

|[[4096/4095]]

|{{monzo| 12 -2 -1 -1 -1 }}

|Schismina

|14

|[[16384/16383]]

|2.3.43.127

|{{monzo| 14 -1 -1 -1 }}

|

|15

|[[32768/32767]]

|{{monzo| 15 -1 -1 -1 }}

|16

|[[65536/65535]]

|2.3.5.17.257

|{{monzo| 16 -1 -1 -1 -1 }}

|

|19

|[[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 }}

{{todo|complete table|review|correct maths|comment=check and complete the Index column}}