Systematic comma names explained: Difference between revisions

(9 intermediate revisions by 3 users not shown)

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 44:

Line 39:

== 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{{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 '''7~~/25~~ kleisma''' (7/25k), and the syntonic comma [[81/80]] is named '''1/5 comma''' (1/5C) or "5-comma" in some ~~early~~ sources. Because the ~~small size of~~ [[~~Mercator~~'s comma]] risks placing ~~two~~ commas ~~related~~ by ~~53 or more 3s~~ in the same size category, this categorization scheme is most rigorously defined only on the simplest representation of the ~~interval~~ 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 ~~occasionally get mixed up~~ 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.

Many comma pages with sagittal names were named using the spreadsheet

[[File:CommaNamer.xls]], which was made in 2004.

~~According to the~~ spreadsheet, these are the ~~interval~~ size ~~boundaries in cents~~ up to one decimal place:

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'')

* Less than 1.8 cents = schismina (or ''atom'')

* 1.8 to 4.5 = schisma (''or skisma, skhisma'')

* 1.8 to 4.5 = schisma (or ''skisma, skhisma'')

* 4.5 to 11.7 = kleisma (''or semicomma'')

* 4.5 to 11.7 = kleisma (or ''semicomma'')

* 11.7 to 35.2 = comma (''or diaschisma, diaskhisma, chroma'')

* 11.7 to 35.2 = comma (or ''diaschisma, diaskhisma, chroma'')

* 35.2 to 45.1 = ~~minor-~~diesis (''~~or small-~~diesis, 1/5-tone, chroma'')

* 35.2 to 45.1 = small diesis (or ''minor diesis, 1/5-tone, chroma'')

* 45.1 to 56.8 = diesis (''~~or medium-~~diesis, 1/4-tone, chroma, enharmonic-diesis, enharmonic'')

* 45.1 to 56.8 = medium diesis (or ''diesis, 1/4-tone, chroma, enharmonic-diesis, enharmonic'')

* 56.8 to 68.6 = ~~major-~~diesis (''~~or large-~~diesis, 1/3-tone'')

* 56.8 to 68.6 = large diesis (or ''major diesis, 1/3-tone'')

* 68.6 to 78.5 = ~~chromatic-~~semitone (''~~or small-~~semitone'')

* 68.6 to 78.5 = small semitone (or ''chromatic semitone'')

* 78.5 to 102.0 = ~~limma~~ (''~~or medium-semitone~~'')

* 78.5 to 102.0 = medium semitone (or ''limma'')

* 102.0 to 111.9 = ~~diatonic-~~semitone (''~~or large-~~semitone'')

* 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 = ~~minor~~-diesis-plus-apotome (''or neutral second'')

* 148.9 to 158.8 = small-diesis-plus-apotome (or ''neutral second'')

* 158.8 to 170.5 = diesis-plus-apotome

* 158.8 to 170.5 = medium-diesis-plus-apotome

* 170.5 to 182.3 = ~~major~~-diesis-plus-apotome

* 170.5 to 182.3 = large-diesis-plus-apotome

* 182.3 to 192.2 = ~~chromatic~~-semitone-plus-apotome

* 182.3 to 192.2 = small-semitone-plus-apotome

* 192.2 to 215.6 = ~~limma~~-plus-apotome

* 192.2 to 215.6 = medium-semitone-plus-apotome

* 215.6 to 225.6 = ~~diatonic~~-semitone-plus-apotome

* 215.6 to 225.6 = large-semitone-plus-apotome

* 225.6 to 229.2 = double-apotome

Intervals larger than 229.2{{cent}} are outside the scope of this system.

In this context, the term "chroma" implied 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).

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).

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

{{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)}}

{{todo|inline=1|~~expand~~|research|comment=explain how, exactly, ~~sagittal notation is used to name them~~ (the sagittal notation page doesn't explain it, nor do any of its internal or external links)}}

== 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|~~expand~~|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)}}

{{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. ==

{{todo|inline=1|~~expand~~|research|comment=please explain this type of comma name}}

{{todo|inline=1|complete section|research|comment=please explain this type of comma name}}

== Prima, secunda, etc. ==

{{todo|inline=1|~~expand~~|research|comment=please explain this type of comma name}}

{{todo|inline=1|complete section|research|comment=please explain this type of comma name}}

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

{{todo|inline=1|~~expand~~|research|comment=please explain this type of comma name; are these all the same type of name, or multiple different categories? Are they systematic?}}

{{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]]

@@ 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 44: / Line 39: @@
 == 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{{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 '''7/25 kleisma''' (7/25k), and the syntonic comma [[81/80]] is named '''1/5 comma''' (1/5C) or "5-comma" in some early sources. Because the small size of [[Mercator's comma]] risks placing two commas related by 53 or more 3s in the same size category, this categorization scheme is most rigorously defined only on the simplest representation of the interval 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 occasionally get mixed up 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.
 Many comma pages with sagittal names were named using the spreadsheet
 [[File:CommaNamer.xls]], which was made in 2004.
-According to the spreadsheet, these are the interval size boundaries in cents up to one decimal place:
+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'')
+* Less than 1.8 cents = schismina (or ''atom'')
-* 1.8 to 4.5 = schisma (''or skisma, skhisma'')
+* 1.8 to 4.5 = schisma (or ''skisma, skhisma'')
-* 4.5 to 11.7 = kleisma (''or semicomma'')
+* 4.5 to 11.7 = kleisma (or ''semicomma'')
-* 11.7 to 35.2 = comma (''or diaschisma, diaskhisma, chroma'')
+* 11.7 to 35.2 = comma (or ''diaschisma, diaskhisma, chroma'')
-* 35.2 to 45.1 = minor-diesis (''or small-diesis, 1/5-tone, chroma'')
+* 35.2 to 45.1 = small diesis (or ''minor diesis, 1/5-tone, chroma'')
-* 45.1 to 56.8 = diesis (''or medium-diesis, 1/4-tone, chroma, enharmonic-diesis, enharmonic'')
+* 45.1 to 56.8 = medium diesis (or ''diesis, 1/4-tone, chroma, enharmonic-diesis, enharmonic'')
-* 56.8 to 68.6 = major-diesis (''or large-diesis, 1/3-tone'')
+* 56.8 to 68.6 = large diesis (or ''major diesis, 1/3-tone'')
-* 68.6 to 78.5 = chromatic-semitone (''or small-semitone'')
+* 68.6 to 78.5 = small semitone (or ''chromatic semitone'')
-* 78.5 to 102.0 = limma (''or medium-semitone'')
+* 78.5 to 102.0 = medium semitone (or ''limma'')
-* 102.0 to 111.9 = diatonic-semitone (''or large-semitone'')
+* 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 = minor-diesis-plus-apotome (''or neutral second'')
+* 148.9 to 158.8 = small-diesis-plus-apotome (or ''neutral second'')
-* 158.8 to 170.5 = diesis-plus-apotome
+* 158.8 to 170.5 = medium-diesis-plus-apotome
-* 170.5 to 182.3 = major-diesis-plus-apotome
+* 170.5 to 182.3 = large-diesis-plus-apotome
-* 182.3 to 192.2 = chromatic-semitone-plus-apotome
+* 182.3 to 192.2 = small-semitone-plus-apotome
-* 192.2 to 215.6 = limma-plus-apotome
+* 192.2 to 215.6 = medium-semitone-plus-apotome
-* 215.6 to 225.6 = diatonic-semitone-plus-apotome
+* 215.6 to 225.6 = large-semitone-plus-apotome
 * 225.6 to 229.2 = double-apotome
 Intervals larger than 229.2{{cent}} are outside the scope of this system.
-In this context, the term "chroma" implied 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).
+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).
-The spreadsheet advised not to use the "plus-apotome" names unless the interval is being considered first and foremost as a comma and not a scale degree.
+{{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)}}
-{{todo|inline=1|expand|research|comment=explain how, exactly, sagittal notation is used to name them (the sagittal notation page doesn't explain it, nor do any of its internal or external links)}}
 == 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|expand|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)}}
+{{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. ==
-{{todo|inline=1|expand|research|comment=please explain this type of comma name}}
+{{todo|inline=1|complete section|research|comment=please explain this type of comma name}}
 == Prima, secunda, etc. ==
-{{todo|inline=1|expand|research|comment=please explain this type of comma name}}
+{{todo|inline=1|complete section|research|comment=please explain this type of comma name}}
 == 34-jubilismic, 23 semitone, 19-minor mediant, etc. ==
-{{todo|inline=1|expand|research|comment=please explain this type of comma name; are these all the same type of name, or multiple different categories? Are they systematic?}}
+{{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]]