Systematic comma names explained

This page aims to document some of the methods of systematically naming commas, to help the reader make a bit more sense of some of the comma names out there.

This page does not cover common names for commas, it only covers names that were generated using some systematic process.

Pseudo-systematic names

Trienstonic, hendecatonic, etc.

Often, these are commas that generate a fractional-octave temperament, but this type of name is not actually systematic. Usually these types of commas are named after the temperament, and not the other way around. To learn about some of these temperament names, visit Temperament naming.

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.

This method of naming is only semi-systematic, as there is still a level of subjectivity and vagueness involved, but it's still worth mentioning because it is used often.

Color notation

Lala-negu, Triyo, etc.

Main article: Color notation/Temperament names

Color notation uses syllables derived from English color names and number names to express the prime factors, including repetitions and accounting for location (over or under, i.e. in the numerator or the denominator), that compose the ratio of a JI interval. Therefore, a comma's ratio can be derived from its color name. Octave equivalence is assumed, so prime 2 is not indicated in the color name, while prime 3 is treated as a way to disambiguate simple and complex commas, instead of counting repetitions in the factorization.

For example, 250/243 is Triyo, since it has three "over" 5s in its numerator, the rest being 2s and 3s only; for comparison, 273375/262144 is Latriyo and 64000/59049 is Satriyo, both having three over 5s as well, but having more complex structures in primes 2 and 3.

Closing error

31-comma, 21-23-comma, 11-3/5 comma, etc.

These types of comma names show the closing error of a specific interval in a specific edo. In general, an n-m-comma, where n is a positive integer and m is a frequency ratio, is the difference between a stack of n instances of m and a number of octaves. m can be an integer, which means it is a harmonic. If the harmonic in question is the third harmonic (3/1), then it is left out of the comma name.

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 5/3's but going down instead of up. (In an edo, intervals that go down still wrap back around every octave, so this is possible.)

For example, the 31-5-comma is the difference between a stack of thirty-one 5/4's (5/4 is the octave-reduced harmonic 5) and 10 octaves, which is tempered out in 31edo. Meanwhile, the 31-comma is the difference between a stack of thirty-one 3/2's and eighteen octaves. As another example, the 11-3/5-comma is the difference between a stack of eleven 3/5's and minus eight octaves.

These kinds of names can sometimes be mistaken for sagittal names (discussed later on this page) and vice versa, so be wary of that.

87-fold starling comma, etc.

This is another type of closing error name. It is for more complex commas that are created by other commas. It is easiest to understand with a couple examples:

87-fold starling comma means the difference between a stack of octaves, and a stack of 87 starling commas (126/125's). This results in an 87th-octave temperament.

12-fold wesley comma means the difference between a stack of octaves (in this case 1 octave), and a stack of 12 wesley commas (78125/73728's).

Sagittal

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

These types of comma names were developed for sagittal notation. After removing all factors of 2 and 3 from the comma, the resulting ratio may be broken into smaller factors if it is too complex⁠ ⁠^{[clarification needed ]} and is used as the first part of the comma's name. This ratio is followed by the comma's size category, distinguishing 10 categories below the apotome. For example, the septimal kleisma 225/224 is named 25/7 kleisma (25/7k or 7/25k), and the syntonic comma 81/80 is named 1/5 comma (1/5C) or "5-comma" in some sources. Because complementation by the pythagorean comma (and adjustments by mercator's comma) risks placing commas and their inversions differing by factors of 2 and 3 in the same size category, this categorization scheme is most rigorously defined only on the simplest representation of the comma in its size category.⁠ ⁠^{[clarification needed ]}

These sagittal names can be confused on occasion with the closing-error type of name described earlier. For example, 5-comma (81/80) is a sagittal name, but the most common meaning of 31-comma uses a closing-error type name (even though "31-comma" is a valid sagittal name for a different interval). These clashes are unfortunate, but not fatal, as a look at the comma's page should reveal which system makes the most sense for interpreting its name.

Many comma pages with sagittal names were named using the spreadsheet File:CommaNamer.xls, which was made in 2004. From this spreadsheet, these are the cent values of the size categories up to one decimal place:

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 diaschisma, diaskhisma, chroma)
35.2 to 45.1 = small diesis (or minor diesis, 1/5-tone, chroma)
45.1 to 56.8 = medium diesis (or diesis, 1/4-tone, chroma, enharmonic-diesis, enharmonic)
56.8 to 68.6 = large diesis (or major diesis, 1/3-tone)
68.6 to 78.5 = small semitone (or chromatic semitone)
78.5 to 102.0 = medium semitone (or limma)
102.0 to 111.9 = large semitone (or diatonic semitone)
111.9 to 115.5 = apotome

For intervals larger than the apotome, "plus-apotome" names are provided, although they are far less popular:

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 = small-diesis-plus-apotome (or neutral second)
158.8 to 170.5 = medium-diesis-plus-apotome
170.5 to 182.3 = large-diesis-plus-apotome
182.3 to 192.2 = small-semitone-plus-apotome
192.2 to 215.6 = medium-semitone-plus-apotome
215.6 to 225.6 = large-semitone-plus-apotome
225.6 to 229.2 = double-apotome

Intervals larger than 229.2 ¢ are outside the scope of this system.

In this context, the term "chroma" implied an absolute 5-exponent of 1 within this system.⁠ ⁠^{[clarification needed ]} (But in wider xenharmonic usage, chroma is pretty vaguely defined and that does not necessarily apply).

Todo: clarify, research
explain how, exactly, the representative commas are chosen (the sagittal notation page doesn't explain it, and nor do any of its internal or external links)

Johnston

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

These are commas named according to Ben Johnston's notation. In general, the p-th-partial chroma is the formal comma that translates a basic interval to an interval of the corresponding harmonic, or "partial". For example, the 19th-partial chroma is the difference between 6/5 and 19/16, so that using it on a 6/5 minor third converts it to 19/16.

Todo: complete section, research
explain how, exactly, Ben Johnston's notation is used to name them (the Ben Johnston notation page doesn't explain it, nor do any of its internal or external links)

35-cycle, 21-cycle, etc.

Todo: complete section, research
please explain this type of comma name

Prima, secunda, etc.