81/80: Difference between revisions

| en = 81/80

| de = 81/80

| Color name = g1, Gu comma, <br/> gu unison

| Comma = yes

| Sound = audacity pluck 81 80.wav

{{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.  

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.

[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.

== 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.

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 "-".

 

 

== 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.)

 

{| class="wikitable"

! ''r''₁ ⋅ ''r''₂

! ''r''₂ / ''r''₁

|-

|-

|-

| -

| 79/78 ⋅ 3159/3160, 80/79 ⋅ 6399/6400

| 89/88 ⋅ 891/890, 90/89 ⋅ 801/800

| 97/96 ⋅ 486/485

| 101/100 ⋅ 405/404

| 108/107 ⋅ 321/320

 

* [[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]]