81/80
| Interval information |
Didymus' comma,
meantone comma,
Ptolemaic comma
gM, guma
reduced
S6/S8
[sound info]
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:
- 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 648/625 exceeds 128/125.
- 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.
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.)
| 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
- Pythagorean comma
- 64/63 – the septimal comma or Archytas' comma
- Small comma
- List of superparticular intervals