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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = 81/80
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-08 14:19:30 UTC</tt>.<br>
| de = 81/80
: The original revision id was <tt>244871623</tt>.<br>
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{{Infobox Interval
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma
<h4>Original Wikitext content:</h4>
| Color name = g1, Gu comma, <br/> gu unison
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The syntonic or Didymus 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 denominator.
| Comma = yes
| Sound = audacity pluck 81 80.wav
}}
{{Wikipedia|Syntonic comma}}


[[http://en.wikipedia.org/wiki/Syntonic_comma]]</pre></div>
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.
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;81_80&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The syntonic or Didymus 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 denominator.&lt;br /&gt;
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.
&lt;br /&gt;
 
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Syntonic_comma&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
== 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>.
 
[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 ==
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 [[25/24]] exceeds [[250/243]].
* The amount by which [[135/128]] exceeds [[25/24]].
* 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 ==
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| /| }}.
 
== 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.)
 
{| class="wikitable"
|-
! Limit
! ''r''<sub>1</sub> ⋅ ''r''<sub>2</sub>
! ''r''<sub>2</sub> / ''r''<sub>1</sub>
|-
| 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 ==
* [[160/81]] – its [[octave complement]]
* [[40/27]] – its [[fifth complement]]
* [[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]]
* [[List of superparticular intervals]]
 
== Notes ==
 
[[Category:Meantone]]
[[Category:Commas named for their regular temperament properties]]
[[Category:Commas named after polymaths]]
[[Category:Commas named for the intervals they stack]]
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