81/80
| Interval information |
Didymus comma,
meantone comma
gu unison
reduced
The syntonic or Didymus or meantone comma (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 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 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. 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.
Tempering out a comma does not just depend on an edo's size; 105edo tempers 81/80 out, while 3edo does not.
YouTube video of "Five senses of 81/80" [dead link], demonstratory video by Jacob Barton.
According to this interview, Monroe Golden's Incongruity uses just-intonation chord progressions that exploit this comma.
If one should be so bold as 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 appoggitura, an acciaccatura, or a quick passing tone. Furthermore, it is also very easy to exploit in comma pump modulations, as among the known examples of this kind of thing are familiar chord progressions.
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 three temperaments from the Didymus family that temper out the respective ratios as commas.
| 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 | - |