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{{Interwiki
| en = 81/80
| de = 81/80
}}
{{Infobox Interval
{{Infobox Interval
| Name = syntonic comma, Didymus comma, meantone comma, Ptolemaic comma
| 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
| Comma = yes
| Sound = audacity pluck 81 80.wav
| Sound = audacity pluck 81 80.wav
Line 7: Line 11:
{{Wikipedia|Syntonic comma}}
{{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".  


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 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.
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>.
[[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 ==
== Temperaments ==
See [[Meantone_family#Extensions]] for a discussion on possible extensions.
See [[Meantone family #Extensions]] for a discussion on possible extensions.


== Relations to other 5-limit intervals ==
== Relations to other 5-limit intervals ==
[[81/80]] is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see [[#Relations to other superparticular ratios]]. A few important ones are that 81/80 is:
81/80 is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see [[#Relations to other superparticular ratios]]. A few important ones are that 81/80 is:
* The amount by which [[2187/2048]] exceeds [[135/128]].
* The amount by which [[2187/2048]] exceeds [[135/128]].
* The amount by which [[25/24]] exceeds [[250/243]].
* The amount by which [[25/24]] exceeds [[250/243]].
* The amount by which [[135/128]] exceeds [[25/24]].
* 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 [[128/125]] exceeds [[2048/2025]].
* The amount by which [[27/25]] exceeds [[16/15]].
* The amount by which [[27/25]] exceeds [[16/15]].
* The amount by which [[16/15]] exceeds [[256/243]].
* The amount by which [[16/15]] exceeds [[256/243]].
== Notation ==
This interval is significant in the [[Functional Just System]] and [[Helmholtz-Ellis notation]] as the classical (5-limit) formal comma which translates a Pythagorean interval to a nearby classical interval.
=== Ben Johnston's notation ===
In [[Ben Johnston's notation]], this interval is denoted with "+" and its reciprocal with "-".
=== 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 ==
== 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|appoggitura}}, 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.
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 }}).
 
== 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 ratio it notates is 4:5, as for example in C:E{{sagittal | \! }}.


== Relations to other superparticular ratios ==
== 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 r1 * r2 or r2 / r1 of 81/80, where r1 and r2 are 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 three 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
! Limit
! r1 * r2
! ''r''<sub>1</sub> ⋅ ''r''<sub>2</sub>
! r2 / r1
! ''r''<sub>2</sub> / ''r''<sub>1</sub>
|-
|-
| 5
| 5
| -
| -
| 9/8 * 9/10
| 9/8 ⋅ 9/10
|-
|-
| 7
| 7
| 126/125 * 225/224 (septimal meantone)
| 126/125 ⋅ 225/224 (septimal meantone)
| 21/20 * 27/28 (sharptone), 36/35 * 63/64 (dominant)
| 21/20 ⋅ 27/28 (sharptone), 36/35 ⋅ 63/64 (dominant)
|-
|-
| 11
| 11
| 99/98 * 441/440 (euterpe), 121/120 * 243/242 (urania)
| 99/98 ⋅ 441/440 (euterpe), 121/120 ⋅ 243/242 (urania)
| 33/32 * 54/55 (thalia), 45/44 * 99/100 (calliope)
| 33/32 ⋅ 54/55 (thalia), 45/44 ⋅ 99/100 (calliope)
|-
|-
| 13
| 13
| 91/90 * 729/728, 105/104 * 351/350
| 91/90 ⋅ 729/728, 105/104 ⋅ 351/350
| 27/26 * 39/40, 65/64 * 324/325, 66/65 * 351/352, 78/77 * 2079/2080
| 27/26 ⋅ 39/40, 65/64 ⋅ 324/325, 66/65 ⋅ 351/352, 78/77 ⋅ 2079/2080
|-
|-
| 17
| 17
| 85/84 * 1701/1700
| 85/84 ⋅ 1701/1700
| 51/50 * 135/136
| 51/50 ⋅ 135/136
|-
|-
| 19
| 19
| 96/95 * 513/512, 153/152 * 171/170
| 96/95 ⋅ 513/512, 153/152 ⋅ 171/170
| 57/56 * 189/190, 76/75 * 1215/1216, 77/76 * 1539/1540
| 57/56 ⋅ 189/190, 76/75 ⋅ 1215/1216, 77/76 ⋅ 1539/1540
|-
|-
| 23
| 23
| 161/160 * 162/161
| 161/160 ⋅ 162/161
| 69/68 * 459/460
| 69/68 ⋅ 459/460
|-
|-
| 29
| 29
| 117/116 * 261/260
| 117/116 ⋅ 261/260
| -
| -
|-
|-
| 31
| 31
| 93/92 * 621/620
| 93/92 ⋅ 621/620
| 63/62 * 279/280
| 63/62 ⋅ 279/280
|-
|-
| 37
| 37
| 111/110 * 297/296
| 111/110 ⋅ 297/296
| 75/74 * 999/1000
| 75/74 ⋅ 999/1000
|-
|-
| 41
| 41
| 82/81 * 6561/6560
| 82/81 ⋅ 6561/6560
| 41/40 * 81/82
| 41/40 ⋅ 81/82
|-
|-
| 43
| 43
| 86/85 * 1377/1376, 87/86 * 1161/1160, 129/128 * 216/215
| 86/85 ⋅ 1377/1376, 87/86 ⋅ 1161/1160, 129/128 ⋅ 216/215
| -
| -
|-
|-
| 47
| 47
| 141/140 * 189/188
| 141/140 ⋅ 189/188
| -
| -
|-
|-
| 53
| 53
| -
| -
| 54/53 * 159/160
| 54/53 ⋅ 159/160
|-
|-
| 59
| 59
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| 61
| 61
| -
| -
| 61/60 * 243/244
| 61/60 ⋅ 243/244
|-
|-
| 67
| 67
| 135/134 * 201/200
| 135/134 ⋅ 201/200
| -
| -
|-
|-
| 71
| 71
| -
| -
| 71/70 * 567/568, 72/71 * 639/640
| 71/70 ⋅ 567/568, 72/71 ⋅ 639/640
|-
|-
| 73
| 73
| -
| -
| 73/72 * 729/730
| 73/72 ⋅ 729/730
|-
|-
| 79
| 79
| -
| -
| 79/78 * 3159/3160, 80/79 * 6399/6400
| 79/78 ⋅ 3159/3160, 80/79 ⋅ 6399/6400
|-
|-
| 83
| 83
| 83/82 * 3321/3320, 84/83 * 2241/2240
| 83/82 ⋅ 3321/3320, 84/83 ⋅ 2241/2240
| -
| -
|-
|-
| 89
| 89
| 89/88 * 891/890, 90/89 * 801/800
| 89/88 ⋅ 891/890, 90/89 ⋅ 801/800
| -
| -
|-
|-
| 97
| 97
| 97/96 * 486/485
| 97/96 ⋅ 486/485
| -
| -
|-
|-
| 101
| 101
| 101/100 * 405/404
| 101/100 ⋅ 405/404
| -
| -
|-
|-
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|-
|-
| 107
| 107
| 108/107 * 321/320
| 108/107 ⋅ 321/320
| -
| -
|}
|}
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* [[160/81]] – its [[octave complement]]
* [[160/81]] – its [[octave complement]]
* [[40/27]] – its [[fifth complement]]
* [[40/27]] – its [[fifth complement]]
* [[Syntonisma]], the difference by which a stack of seven 81/80s falls short of [[12/11]]
* [[1ed81/80]] – its equal multiplication
* [[Pythagorean comma]]
* [[64/63]] – the septimal comma or Archytas' comma
* [[Small comma]]
* [[Small comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
Line 157: Line 178:


[[Category:Meantone]]
[[Category:Meantone]]
 
[[Category:Commas named for their regular temperament properties]]
<!-- interwiki -->
[[Category:Commas named after polymaths]]
[[https://de.xen.wiki/w/81/80]]
[[Category:Commas named for the intervals they stack]]

Latest revision as of 03:55, 29 April 2026

Interval information
Ratio 81/80
Factorization 2-4 × 34 × 5-1
Monzo [-4 4 -1⟩
Size in cents 21.50629¢
Names syntonic comma,
Didymus' comma,
meantone comma,
Ptolemaic comma
Color name g1, gu unison,
gM, guma
FJS name [math]\displaystyle{ \text{P1}_{5} }[/math]
Special properties square superparticular,
reduced
Tenney norm (log2 nd) 12.6618
Weil norm (log2 max(n, d)) 12.6797
Wilson norm (sopfr(nd)) 25
Comma size small
S-expressions S9,
S6/S8

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

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 perfect fourth, fifth, or 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 whole tone which is intermediate between them; hence the name "meantone".

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 (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 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[1].

Adam Neely's harmonization of the licc pumps upward by 81/80 every measure. After 9 iterations, D modulates nearly to E.

Temperaments

See Meantone family #Extensions for a discussion on possible extensions.

Relations to other 5-limit intervals

81/80 is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see #Relations to other superparticular ratios. A few important ones are that 81/80 is:

Notation

This interval is significant in the Functional Just System and Helmholtz-Ellis notation as the classical (5-limit) formal comma which translates a Pythagorean interval to a nearby classical interval.

Ben Johnston's notation

In Ben Johnston's notation, this interval is denoted with "+" and its reciprocal with "-".

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the 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 ⁠. The upward version is called 1/5C or 5C up and is represented by .

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 appoggiatura, an acciaccatura, or a quick passing tone. It is also very easy to exploit in comma pump modulations, as among the 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 ([-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 r1 ⋅ r2 or r2 / r1 of 81/80, where r1 and r2 are other superparticular ratios.

Names in brackets refer to 7-limit 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 exotemperaments.)

Relations between 81/80 and other superparticular ratios
Limit r1 ⋅ r2 r2 / r1
5 - 9/8 ⋅ 9/10
7 126/125 ⋅ 225/224 (septimal meantone) 21/20 ⋅ 27/28 (sharptone), 36/35 ⋅ 63/64 (dominant)
11 99/98 ⋅ 441/440 (euterpe), 121/120 ⋅ 243/242 (urania) 33/32 ⋅ 54/55 (thalia), 45/44 ⋅ 99/100 (calliope)
13 91/90 ⋅ 729/728, 105/104 ⋅ 351/350 27/26 ⋅ 39/40, 65/64 ⋅ 324/325, 66/65 ⋅ 351/352, 78/77 ⋅ 2079/2080
17 85/84 ⋅ 1701/1700 51/50 ⋅ 135/136
19 96/95 ⋅ 513/512, 153/152 ⋅ 171/170 57/56 ⋅ 189/190, 76/75 ⋅ 1215/1216, 77/76 ⋅ 1539/1540
23 161/160 ⋅ 162/161 69/68 ⋅ 459/460
29 117/116 ⋅ 261/260 -
31 93/92 ⋅ 621/620 63/62 ⋅ 279/280
37 111/110 ⋅ 297/296 75/74 ⋅ 999/1000
41 82/81 ⋅ 6561/6560 41/40 ⋅ 81/82
43 86/85 ⋅ 1377/1376, 87/86 ⋅ 1161/1160, 129/128 ⋅ 216/215 -
47 141/140 ⋅ 189/188 -
53 - 54/53 ⋅ 159/160
59 - -
61 - 61/60 ⋅ 243/244
67 135/134 ⋅ 201/200 -
71 - 71/70 ⋅ 567/568, 72/71 ⋅ 639/640
73 - 73/72 ⋅ 729/730
79 - 79/78 ⋅ 3159/3160, 80/79 ⋅ 6399/6400
83 83/82 ⋅ 3321/3320, 84/83 ⋅ 2241/2240 -
89 89/88 ⋅ 891/890, 90/89 ⋅ 801/800 -
97 97/96 ⋅ 486/485 -
101 101/100 ⋅ 405/404 -
103 - -
107 108/107 ⋅ 321/320 -

See also

Notes

  1. ↑ UnTwelve's interview to Monroe Golden
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