Systematic comma names explained

Comma names are one of the most frustrating bits of xenharmonic theory to deal with. This is largely the result of smart and well-meaning theorists developing new standards to replace previous, flawed ones, only to end up with an XKCD 927 situation.

This page aims to document some of the ways that commas are named, to help the reader make a bit more sense of some of the comma names out there.

Trienstonic, hendecatonic, etc.

These are commas that generate a fractional-octave temperament, or a family of them. For example hendecatonic means it generates an 11th-octave temperament, or a family of those.

Trientone, hexadecatone, etc.

These are commas that are a fraction of a whole tone (~200 cents or ~9/8). For example, a trientone is about one third of a whole tone. A hexadecatone is about one sixteenth of a whole tone.

31-comma, 21-23-comma, etc.

These types of comma names are from sagittal notation. They show the closing error of a specific interval in a specific EDO.

For example, the 31-5-comma is the difference between a stack of 31 5/4s (5/4 is the octave reduces harmonic #5), and 10 octaves, which is tempered out in 31edo.

If the harmonic in question is the reduced third harmonic (3/2), then it is left out of the comma name. For example 31-comma is the difference between a stack of 3/2s and a stack of octaves in 31edo.

11-3/5 comma, 45-7/5 comma, etc.

These are the same type of name as above, but they involve stacks of intervals that are not octave reduced harmonics. For example 7/4 is an octave reduced harmonic, so a comma tempering a stack of those would be an "n-7 comma", but 7/5 is not a reduced harmonic, so a comma tempering a stack of those would be an "n-7/5 comma".

An interval with a bigger denominator than numerator, like 3/5, indicates a negative interval. 3/5 for example is about -884 cents. A comma can still temper a stack of these. Just imagine it like a stack of 5/3s but going down instead of up. (In an EDO, intervals that go down still wrap back around every octave, so this is possible.)

Todo: review double check this is correct

5/7-kleisma, 35/11-kleisma, etc.

These types of comma names come from an attempt to systematically name commas in around the year 2004. The commas were named according to the spreadsheet File:CommaNamer.xls.

The naming rules are a little difficult to work out, but "kleisma" definitely refers to the size of the interval - between 4.5 and 11.7 cents. The number in the name appears to have something to do with the prime factorization of the comma, but it is a bit difficult to reverse-engineer.

The full range of size classifications (rounded to 1 decimal place) was:

• Less than 1.8 cents = schismina or atom
• 1.8 to 4.5 = schisma (or skisma, skhisma)
• 4.5 to 11.7 = kleisma (or semicomma)
• 11.7 to 35.2 = comma (or dischisma, diaskhisma, chroma)
• 35.2 to 45.1 = minor-diesis (or small-diesis, 1/5-tone, chroma)
• 45.1 to 56.8 = diesis (or medium-diesis, 1/4-tone, chroma, enharmonic-diesis, enharmonic)
• 56.8 to 68.6 = major-diesis (or large-diesis, 1/3-tone)
• 68.6 to 78.5 = chromatic-semitone (or small-semitone)
• 78.5 to 102.0 = limma (or medium-semitone)
• 102.0 to 111.9 = diatonic-semitone (or large-semitone)
• 111.9 to 115.5 = apotome
• 115.5 to 118.2 = schisma-plus-apotome
• 118.2 to 125.4 = kleisma-plus-apotome
• 125.4 to 148.9 = comma-plus-apotome
• 148.9 to 158.8 = minor-diesis-plus-apotome (or neutral second)
• 158.8 to 170.5 = diesis-plus-apotome
• 170.5 to 182.3 = major-diesis-plus-apotome
• 182.3 to 192.2 = chromatic-semitone-plus-apotome
• 192.2 to 215.6 = limma-plus-apotome
• 215.6 to 225.6 = diatonic-semitone-plus-apotome
• 225.6 to 229.2 = double-apotome
• Over 229.2 = outside the scope of this system

The term "chroma" implies an absolute 5-exponent of 1 within this system. (But in wider xenharmonic usage, chroma is pretty vaguely defined and that does not necessarily apply).

The "plus-apotome" names are advised not to be used unless the interval is being considered first and foremost as a comma and not a scale degree.

Todo: expand reverse engineer the system and explain how it works

35-cycle, 21-cycle, etc.

Todo: expand please explain this type of comma name

19th partial chroma, 29th partial chroma, etc.

Todo: expand please explain this type of comma name

Prima, secunda, etc.

Todo: expand please explain this type of comma name

34-jubilismic, 23 semitone, 19-minor mediant, etc

Todo: expand please explain this type of comma name; are these all the same type of name, or multiple different categories

87-fold, as in 87-fold starling comma

Todo: expand please explain this type of comma name