81/80: Difference between revisions

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{{Infobox Interval

| Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma

| Color name = g1, ~~Gu comma~~, ~~gu unison~~

| Color name = g1, gu unison, gM, guma

| Comma = yes

| Sound = audacity pluck 81 80.wav

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The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is ~~helpful for comparing~~ [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] ~~with an~~ [[~~odd limit~~]] ~~of 27~~ or ~~higher~~ creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from ~~any~~ 3~~-limit ratio of odd limit~~ 3 ~~or less (the 4th~~, ~~5th~~ or ~~8ve),~~ creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to ~~5-limit just intonation~~ will create ~~such wolves,~~ thus, for those who ~~have no interest or desire~~ to ~~utilize~~ such wolves in composition, [[tempering out]] 81/80 is desirable. This ~~gives a tuning for the~~ [[~~tone|whole tone~~]] which ~~is intermediate between~~ 10/9 ~~and~~ 9/8, ~~and leads to~~ [[~~meantone~~|~~meantone temperament~~]], hence the name meantone ~~comma~~.

The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the pythagorean major triad, [[64:81:96]], is quite dissonant, but flattening the 81/64 major third by 81/80 leads to the much more consonant [[4:5:6]] chord, with a 5/4 major third in place of 81/64. However, adding/subtracting this comma to/from the [[4/3|perfect fourth]], [[3/2|fifth]], or [[2/1|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to make intervals and chords pure in one key will create wolf intervals in others; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones. This also equates [[10/9]] with [[9/8]], giving a tuning for the [[tone|whole tone]] which is intermediate between them; hence the name "meantone".

81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit~~]]. Like [[16/15]], [[625/624~~]], ~~[[2401/2400]]~~ and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because {{nowrap|''n''2/(''n''2 − 1) {{=}} ''n''/(''n'' − 1) ÷ (''n'' + 1)/''n''}} (which is to say 81/80 is a [[square superparticular]]~~). 81/80 is in fact~~ the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. ~~That~~ the ~~numerator~~ is ~~a fourth power entails that~~ the ~~wider~~ of ~~these two intervals itself has~~ a ~~square numerator; 9~~/~~8 is~~ the ~~interval between~~ the ~~successive superparticulars 4/3~~ and ~~3/2~~.

81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].

== Comma pumps ==

The familiar vi–ii–V–I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.

A passage ([https://youtu.be/DO7yTiM-YJk?si=e4wVU4IlbITCAaNG&t=325 listen]) from [[Ben Johnston]]'s 9th string quartet, near the end of movement 1, makes a sudden and prominent use of the 81/80 comma, which demonstrates how a simple progression with held common tones can quickly lead to severe interference [[Beat|beating]], rupturing the diatonic collection routinely associated with the [[5-limit]] and exposing "C major" as anything but simple.

[[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.

[https://x.com/its_adamneely/status/1249700624003989508 Adam Neely's harmonization] of ''the licc'' pumps upward by 81/80 every measure. After 9 iterations, D modulates nearly to E.

== Temperaments ==

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* The amount by which [[25/24]] exceeds [[250/243]].

* The amount by which [[135/128]] exceeds [[25/24]].

* The amount by which [[648/625]] exceeds [[128/125]].

* The amount by which [[128/125]] exceeds [[2048/2025]].

* The amount by which [[27/25]] exceeds [[16/15]].

* The amount by which [[16/15]] exceeds [[256/243]].

~~== Approximation ==~~

If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggiatura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave.

== Notation ==

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=== Sagittal notation ===

In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \! }} and is called the '''5 comma''', or '''5C''' for short, because the simplest interval it notates is 5/1 (equiv. 5/4), as for example in C–E{{nbhsp}}{{sagittal | \! }}. The upward version is called '''1/5C''' or '''5C up''' and is represented by {{sagittal| /| }}.

== Approximation ==

If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggiatura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave, thus tempering out the [[barium comma]] ({{monzo| -225 224 -56 }}).

== Relations to other superparticular ratios ==

Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations ''r''1 ⋅ ''r''2 or ''r''2 / ''r''1 of 81/80, where ''r''1 and ''r''2 are other superparticular ratios.

Names in brackets refer to 7-limit [[meantone family|meantone extensions]], or 11-limit rank-3 temperaments from the [[~~Didymus~~ rank ~~three family|Didymus~~ family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)

Names in brackets refer to 7-limit [[meantone family|meantone extensions]], or 11-limit rank-3 temperaments from the [[didymus rank-3 family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)

{| class="wikitable"

{| class="wikitable mw-collapsible mw-collapsed"

|+ Relations between 81/80 and other superparticular ratios

|-

! Limit

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* [[40/27]] – its [[fifth complement]]

* [[1ed81/80]] – its equal multiplication

* [[Syntonisma]], the difference by which a stack of seven 81/80s falls short of [[12/11]]

* [[Mercator's comma]]

* [[Pythagorean comma]]

* [[64/63]] – the septimal comma or Archytas' comma

* [[Small comma]]

* [[List of superparticular intervals]]

@@ Line 5: / Line 5: @@
 {{Infobox Interval
 | Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma
-| Color name = g1, Gu comma, <br/> gu unison
+| Color name = g1, gu unison,<br/>gM, guma
 | Comma = yes
 | Sound = audacity pluck 81 80.wav
@@ Line 11: / Line 11: @@
 {{Wikipedia|Syntonic comma}}
-The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is helpful for comparing [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who have no interest or desire to utilize such wolves in composition, [[tempering out]] 81/80 is desirable. This gives a tuning for the [[tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[meantone|meantone temperament]], hence the name meantone comma.
+The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the pythagorean major triad, [[64:81:96]], is quite dissonant, but flattening the 81/64 major third by 81/80 leads to the much more consonant [[4:5:6]] chord, with a 5/4 major third in place of 81/64. However, adding/subtracting this comma to/from the [[4/3|perfect fourth]], [[3/2|fifth]], or [[2/1|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to make intervals and chords pure in one key will create wolf intervals in others; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones. This also equates [[10/9]] with [[9/8]], giving a tuning for the [[tone|whole tone]] which is intermediate between them; hence the name "meantone".
-/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because {{nowrap|''n''<sup>2</sup>/(''n''<sup>2</sup> − 1) {{=}} ''n''/(''n'' − 1) ÷ (''n'' + 1)/''n''}} (which is to say 81/80 is a [[square superparticular]]). 81/80 is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.
+/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].
+== Comma pumps ==
+The familiar vi–ii–V–I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.
+A passage ([https://youtu.be/DO7yTiM-YJk?si=e4wVU4IlbITCAaNG&t=325 listen]) from [[Ben Johnston]]'s 9th string quartet, near the end of movement 1, makes a sudden and prominent use of the 81/80 comma, which demonstrates how a simple progression with held common tones can quickly lead to severe interference [[Beat|beating]], rupturing the diatonic collection routinely associated with the [[5-limit]] and exposing "C major" as anything but simple.
 [[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.
+[https://x.com/its_adamneely/status/1249700624003989508 Adam Neely's harmonization] of ''the licc'' pumps upward by 81/80 every measure. After 9 iterations, D modulates nearly to E.
 == Temperaments ==
@@ Line 25: / Line 32: @@
 * The amount by which [[25/24]] exceeds [[250/243]].
 * The amount by which [[135/128]] exceeds [[25/24]].
+* The amount by which [[648/625]] exceeds [[128/125]].
 * The amount by which [[128/125]] exceeds [[2048/2025]].
 * The amount by which [[27/25]] exceeds [[16/15]].
 * The amount by which [[16/15]] exceeds [[256/243]].
-== Approximation ==
-If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggiatura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions.  Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave.
 == Notation ==
@@ Line 40: / Line 45: @@
 === Sagittal notation ===
 In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \! }} and is called the '''5 comma''', or '''5C''' for short, because the simplest interval it notates is 5/1 (equiv. 5/4), as for example in C–E{{nbhsp}}{{sagittal | \! }}. The upward version is called '''1/5C''' or '''5C up''' and is represented by {{sagittal| /| }}.
+== Approximation ==
+If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggiatura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions.  Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave, thus tempering out the [[barium comma]] ({{monzo| -225 224 -56 }}).
 == Relations to other superparticular ratios ==
 Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations ''r''<sub>1</sub> ⋅ ''r''<sub>2</sub> or ''r''<sub>2</sub> / ''r''<sub>1</sub> of 81/80, where ''r''<sub>1</sub> and ''r''<sub>2</sub> are other superparticular ratios.
-Names in brackets refer to 7-limit [[meantone family|meantone extensions]], or 11-limit rank-3 temperaments from the [[Didymus rank three family|Didymus family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)
+Names in brackets refer to 7-limit [[meantone family|meantone extensions]], or 11-limit rank-3 temperaments from the [[didymus rank-3 family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)
-{| class="wikitable"
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ Relations&nbsp;between&nbsp;81/80&nbsp;and&nbsp;other&nbsp;superparticular&nbsp;ratios
 |-
 ! Limit
@@ Line 161: / Line 170: @@
 * [[40/27]] – its [[fifth complement]]
 * [[1ed81/80]] – its equal multiplication
-* [[Syntonisma]], the difference by which a stack of seven 81/80s falls short of [[12/11]]
-* [[Mercator's comma]]
 * [[Pythagorean comma]]
+* [[64/63]] – the septimal comma or Archytas' comma
 * [[Small comma]]
 * [[List of superparticular intervals]]