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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 comma, <br/> gu unison
| Comma = yes
| Comma = yes
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{{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 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.  


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]]. 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.


If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out (which in higher-accuracy contexts causes significant damage to the [[5-limit]]), one can easily use it in melodies as either an [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|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, 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.
== Use in recorded music ==
[https://youtu.be/DO7yTiM-YJk?si=e4wVU4IlbITCAaNG&t=325 This passage] 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 ==
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 [[128/125]] exceeds [[2048/2025]].
* The amount by which [[128/125]] exceeds [[2048/2025]].
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* The amount by which [[16/15]] exceeds [[256/243]].
* The amount by which [[16/15]] exceeds [[256/243]].


== Temperaments ==
== Approximation ==
If one wants to extend meantone beyond 5-limit, there is a number of ways to do so discussed in the [[meantone family]], usually by decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, a unique opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow — and indeed want — more (flatwards) tempering on the fifth, so may be recommended for this reason. However, as [[9/8]] is typically flat in meantone, we might mention that an opportunity not based on splitting the fifth comes from interpreting the tritone (~9/8)<sup>3</sup> as [[7/5]], leading to [[septimal meantone]], a very elegant extension to the [[7-limit]].
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.


=== Splitting the meantone fifth into two (243/242) ===
== Notation ==
By tempering [[243/242]] we equate the distance from 9/8 to 10/9 (= [[81/80|S9]]) with the distance between 11/10 to 12/11 (= [[121/120|S11]]), leading to [[mohaha]] which is in some sense thus a trivial tuning of [[rastmic]] (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone [[rastmic]] temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full [[11-limit]] by finding [[7/4]] as the semi-diminished seventh, leading to [[mohajira]], which inflates [[64/63]] to equate it with a small quarter-tone, which is characteristic. (Mohajira can also be thought of as equating a slightly sharpened [[25/16|(5/4)<sup>2</sup>]] with [[11/7]], which is also natural as meantone tempering usually has [[5/4]] slightly sharp.) There is also the consideration that tempering [[121/120]] leads to similarly high damage in the 11-limit as tempering [[81/80]] in the 5-limit, because both erase key distinctions of their respective JI subgroups.
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.  


=== Splitting the meantone fifth into three (1029/1024) ===
=== Ben Johnston's notation ===
By tempering [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as 9/8 * 8/7 * 7/6 (this observation can be generalized to define the family of [[ultraparticular]] commas). This is an unusually natural extension, because of a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. This means that as [[81/80|S6/S8]] is already tempered in meantone, it is natural to want [[49/48|49/48 = S7]] (which is bigger than S8 and smaller than S6) to be equated, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] = S6/S7, the orwellisma.
In [[Ben Johnston's notation]], this interval is denoted with "+" and its reciprocal with "-".  


This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6 = S7 = S8 with S9 tempered, we can try S8 = S10 by tempering [[176/175|176/175 = S8/S10 = (11/7)/(5/4)<sup>2</sup>]] , taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])<sup>2</sup> = [[36/25]] = ([[3/2]])/([[25/24]]).
=== 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| /| }}.
=== 31edo as splitting the fifth into two, three and nine ===
[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]], so that it is very strong in the 2.5.7 subgroup. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is uniquely meantone + valentine. Valentine is a natural [[11-limit]] temperament that tempers [[121/120]] so for this reason might be natural to combine with meantone. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has [[31edo]] as a [[patent val]] tuning.


== 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 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.)


{| class="wikitable"
{| class="wikitable"
|-
|-
! 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
* [[Syntonoschisma]], the difference by which a stack of seven 81/80s falls short of [[12/11]]
* [[Mercator's comma]]
* [[Pythagorean comma]]
* [[Small comma]]
* [[Small comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
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[[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]]
Retrieved from "https://en.xen.wiki/w/81/80"