Systematic comma names explained: Difference between revisions

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== 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.

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

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.)

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

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.

~~=== 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 rather than just ''n''-7.

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.)

=== 87-fold starling comma, etc. ===

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== 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|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)}}

@@ Line 21: / Line 21: @@
 == 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.
-For example, the ''31-5-comma'' is the difference between a stack of 31 5/4's (5/4 is the octave reduces harmonic 5), and 10 octaves, which is tempered out in 31edo.
+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.)
-If the harmonic in question is the reduced third harmonic (3/2), then it is left out of the comma name. For example the 31-comma is the difference between a stack of 3/2's and a stack of octaves in 31edo.
+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.
-=== 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 rather than just ''n''-7.
-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.)
 === 87-fold starling comma, etc. ===
@@ Line 80: / Line 75: @@
 == 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|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)}}