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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''' (thus systematically the '''syntonisma''' and '''ptolemaisma''', not to be confused with the [[ptolemisma]]) 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''<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. | ||
Revision as of 20:48, 14 September 2025
| Interval information |
Didymus' comma,
meantone comma,
Ptolemaic comma
gu unison
reduced
S6/S8
[sound info]
The syntonic comma, also known as the Didymus' comma, the meantone comma or the Ptolemaic comma (thus systematically the syntonisma and ptolemaisma, not to be confused with the ptolemisma) 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 whole tone which is intermediate between 10/9 and 9/8, and leads to 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 n2/(n2 − 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.
Use in recorded music
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 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:
- 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 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.
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 .
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.)
| 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
- 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