Systematic comma names explained: Difference between revisions
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== Closing error == | == Closing error == | ||
=== 31-comma, 21-23-comma, etc. === | === 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]]. | 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. | 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. === | === 87-fold starling comma, etc. === | ||
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== Sagittal == | == Sagittal == | ||
=== 5-comma, 5/7-kleisma, 35/11-kleisma, etc. === | === 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 [[2.3-equivalent_class_and_Pythagorean-commatic_interval_naming_system|resulting ratio]] may be broken into smaller factors if it is too complex{{clarify}} 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''' (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.{{clarify}} | These types of comma names were developed for [[sagittal notation]]. After removing all factors of 2 and 3 from the comma, the [[2.3-equivalent_class_and_Pythagorean-commatic_interval_naming_system|resulting ratio]] may be broken into smaller factors if it is too complex{{clarify}} 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.{{clarify}} | ||
These sagittal names can be confused on occasion with the closing-error type of name described earlier. For example, [[81/80|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. | These sagittal names can be confused on occasion with the closing-error type of name described earlier. For example, [[81/80|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. | ||
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In this context, the term "chroma" implied an absolute 5-exponent of 1 within this system.{{clarify}} (But in wider xenharmonic usage, [[chroma]] is pretty vaguely defined and that does not necessarily apply). | In this context, the term "chroma" implied an absolute 5-exponent of 1 within this system.{{clarify}} (But in wider xenharmonic usage, [[chroma]] is pretty vaguely defined and that does not necessarily apply). | ||
{{todo|inline=1| | {{todo|inline=1|clarify|research|comment=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 == | == Johnston == | ||
=== 19th partial chroma, 29th partial chroma, etc. === | === 19th-partial chroma, 29th-partial chroma, etc. === | ||
These are commas named according to [[Ben Johnston's notation]]. | 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|inline=1| | {{todo|inline=1|complete section|research|comment=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. == | == 35-cycle, 21-cycle, etc. == | ||
{{todo|inline=1| | {{todo|inline=1|complete section|research|comment=please explain this type of comma name}} | ||
== Prima, secunda, etc. == | == Prima, secunda, etc. == | ||
{{todo|inline=1| | {{todo|inline=1|complete section|research|comment=please explain this type of comma name}} | ||
== 34-jubilismic, 23 semitone, 19-minor mediant, etc. == | == 34-jubilismic, 23 semitone, 19-minor mediant, etc. == | ||
{{todo|inline=1| | {{todo|inline=1|complete section|research|comment=please explain this type of comma name; are these all the same type of name, or multiple different categories? Are they systematic?}} | ||
[[Category:Comma]][[Category:Terms]] | |||
== See also == | |||
* [[Comma-prefix names]] | |||
* [[Temperament naming]] | |||
* [[:Category:Commas by name]] | |||
* [[Glossary]] | |||
* [[Acronyms]] | |||
[[Category:Comma]][[Category:Terms]][[Category:Interval naming]] | |||