Comma: Difference between revisions

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In just intonation, a '''comma''' is a small [[interval]] that occurs between two intervals which are close in [[pitch]]. Commas are always greater than [[1/1]] (0 [[cent]]s) and not a power such as a square or cube of any other just interval, and generally in a low [[prime limit]].

~~Some commas~~ are considered [[dissonant]] due to their small but noticeable [[interval size measure|size]] which induces an audible [[beat]]. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]] and [[~~temperament~~]].

Commas are often considered [[dissonant]] due to their small but noticeable [[interval size measure|size]] which induces an audible [[beat]]. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features to be desired or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]], [[temperament|temperaments]], and [[fudging]].

In [[regular temperament theory]], a comma ~~can~~ be ''[[tempered out]]'', ~~which is to say reduced to a [[unison]], by a temperament, thereby~~ equating ~~the~~ intervals ~~separated by that comma~~. For example, the [[syntonic comma]] (81/80), which occurs between [[10/9]] and [[9/8]], as well as [[81/64]] and [[5/4]], is tempered out by [[meantone]] temperament.

In [[regular temperament theory]], a comma is something to be ''[[tempered out]]'', equating two mathematically distinct intervals. For example, the [[syntonic comma]] (81/80), which occurs between [[10/9]] and [[9/8]] as well as between [[81/64]] and [[5/4]], is tempered out by [[meantone]] temperament and thus conflates these two pairs of intervals.

[[File:EufaJIcommasFinal1.svg|thumb|A visualization of intervals commonly used as commas, built from stacked intervals, to scale.]]

Commas are usually written as [[frequency ratio]]s, but they can also be written as ~~vectors~~, sometimes called [[monzo]]s or '''unison vectors'''~~. The following lists mostly give both forms, as well as the size in cents~~. The [[color name]] refers to both the comma and the temperament created when it is tempered out, except for 3-limit commas, which create [[edo]]s.

Commas are usually written as [[frequency ratio]]s, but they can also be written as products of primes, sometimes called [[monzo]]s or '''unison vectors'''. The [[color name]] refers to both the comma and the temperament created when it is tempered out, except for 3-limit commas, which create [[edo]]s, commonly called n-commas, such as the [[29-comma]], or the [[41-comma]].

== As an interval region ==

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== Lists of commas by size ==

Commas can theoretically have any size, but in practice most are smaller than a [[12edo]] semitone (100{{cent}}). The following categories, while arbitrary, are used on the Xenharmonic Wiki to classify commas by size.

Commas can theoretically have any size, but in practice most are much smaller than a [[12edo]] semitone (100{{cent}}). The following categories, while arbitrary, are used on the Xenharmonic Wiki to classify commas by size.

* [[Unnoticeable comma]]: under 3.5 cents in size~~. The 3.5-cent boundary roughly correlates to~~ the melodic [[just-noticeable difference]] ~~of pitch perception~~.

* [[Unnoticeable comma]]: under 3.5 cents in size; below the melodic [[just-noticeable difference]] and thus imperceptible. Tempering these commas out produces [[very high accuracy temperaments]] or [[Microtemperament|microtemperaments]].

* [[Small comma]]: between 3.5 and 30 cents.

* [[Small comma]]: between 3.5 and 30 cents. Tempering these commas out produces moderate accuracy temperaments.

* [[Medium comma]]: between 30 and 100 cents.

* [[Medium comma]]: between 30 and 100 cents. Tempering these commas out produces low accuracy temperaments.

* [[Large comma]]: over 100 cents in size. ~~The 100-cent limit matches the 12edo semitone~~.

* [[Large comma]]: over 100 cents in size; much wider than what is typically considered to be a comma. Tempering these commas out produces [[very low accuracy temperaments]] or [[Exotemperament|exotemperaments]].

== See also ==

@@ Line 9: / Line 9: @@
 In just intonation, a '''comma''' is a small [[interval]] that occurs between two intervals which are close in [[pitch]]. Commas are always greater than [[1/1]] (0 [[cent]]s) and not a power such as a square or cube of any other just interval, and generally in a low [[prime limit]].
-Some commas are considered [[dissonant]] due to their small but noticeable [[interval size measure|size]] which induces an audible [[beat]]. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]] and [[temperament]].
+Commas are often considered [[dissonant]] due to their small but noticeable [[interval size measure|size]] which induces an audible [[beat]]. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features to be desired or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]], [[temperament|temperaments]], and [[fudging]].
-In [[regular temperament theory]], a comma can be ''[[tempered out]]'', which is to say reduced to a [[unison]], by a temperament, thereby equating the intervals separated by that comma. For example, the [[syntonic comma]] (81/80), which occurs between [[10/9]] and [[9/8]], as well as [[81/64]] and [[5/4]], is tempered out by [[meantone]] temperament.
+In [[regular temperament theory]], a comma is something to be ''[[tempered out]]'', equating two mathematically distinct intervals. For example, the [[syntonic comma]] (81/80), which occurs between [[10/9]] and [[9/8]] as well as between [[81/64]] and [[5/4]], is tempered out by [[meantone]] temperament and thus conflates these two pairs of intervals.
+[[File:EufaJIcommasFinal1.svg|thumb|A visualization of intervals commonly used as commas, built from stacked intervals, to scale.]]
-Commas are usually written as [[frequency ratio]]s, but they can also be written as vectors, sometimes called [[monzo]]s or '''unison vectors'''. The following lists mostly give both forms, as well as the size in cents. The [[color name]] refers to both the comma and the temperament created when it is tempered out, except for 3-limit commas, which create [[edo]]s.
+Commas are usually written as [[frequency ratio]]s, but they can also be written as products of primes, sometimes called [[monzo]]s or '''unison vectors'''. The [[color name]] refers to both the comma and the temperament created when it is tempered out, except for 3-limit commas, which create [[edo]]s, commonly called n-commas, such as the [[29-comma]], or the [[41-comma]].
 == As an interval region ==
@@ Line 21: / Line 21: @@
 == Lists of commas by size ==
-Commas can theoretically have any size, but in practice most are smaller than a [[12edo]] semitone (100{{cent}}). The following categories, while arbitrary, are used on the Xenharmonic Wiki to classify commas by size.
+Commas can theoretically have any size, but in practice most are much smaller than a [[12edo]] semitone (100{{cent}}). The following categories, while arbitrary, are used on the Xenharmonic Wiki to classify commas by size.
-* [[Unnoticeable comma]]: under 3.5 cents in size. The 3.5-cent boundary roughly correlates to the melodic [[just-noticeable difference]] of pitch perception.
+* [[Unnoticeable comma]]: under 3.5 cents in size; below the melodic [[just-noticeable difference]] and thus imperceptible. Tempering these commas out produces [[very high accuracy temperaments]] or [[Microtemperament|microtemperaments]].
-* [[Small comma]]: between 3.5 and 30 cents.
+* [[Small comma]]: between 3.5 and 30 cents. Tempering these commas out produces moderate accuracy temperaments.
-* [[Medium comma]]: between 30 and 100 cents.
+* [[Medium comma]]: between 30 and 100 cents. Tempering these commas out produces low accuracy temperaments.
-* [[Large comma]]: over 100 cents in size. The 100-cent limit matches the 12edo semitone.
+* [[Large comma]]: over 100 cents in size; much wider than what is typically considered to be a comma. Tempering these commas out produces [[very low accuracy temperaments]] or [[Exotemperament|exotemperaments]].
 == See also ==