81/80
| 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- 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.
If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out (which in higher-accuracy contexts causes significant damage to the 5-limit), one can easily use it in melodies as either an appoggitura, 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, 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.
Monroe Golden's Incongruity uses just-intonation chord progressions that exploit this comma[1].
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.
Temperaments
If one wants to extend meantone beyond 5-limit, there is a number of ways to do so discussed in the meantone family, usually by decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, a unique opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into n parts leave the part closer to just than usual, because we can allow — and indeed want — more (flatwards) tempering on the fifth, so may be recommended for this reason. However, as 9/8 is typically flat in meantone, we might mention that an opportunity not based on splitting the fifth comes from interpreting the tritone (~9/8)3 as 7/5, leading to septimal meantone, a very elegant extension to the 7-limit.
Splitting the meantone fifth into two (243/242)
By tempering 243/242 we equate the distance from 9/8 to 10/9 (= S9) with the distance between 11/10 to 12/11 (= S11), leading to mohaha which is in some sense thus a trivial tuning of rastmic (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone rastmic temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full 11-limit by finding 7/4 as the semi-diminished seventh, leading to mohajira, which inflates 64/63 to equate it with a small quarter-tone, which is characteristic. (Mohajira can also be thought of as equating a slightly sharpened (5/4)2 with 11/7, which is also natural as meantone tempering usually has 5/4 slightly sharp.) There is also the consideration that tempering 121/120 leads to similarly high damage in the 11-limit as tempering 81/80 in the 5-limit, because both erase key distinctions of their respective JI subgroups.
Splitting the meantone fifth into three (1029/1024)
By tempering 1029/1024 we equate the distance from 7/6 to 8/7 (= S7) with the distance from 8/7 to 9/8 (= S8), so that (8/7)3 is equated with 3/2, because of being able to be rewritten as 9/8 * 8/7 * 7/6 (this observation can be generalized to define the family of ultraparticular commas). This is an unusually natural extension, because of a surprising coincidence: (36/35)/(64/63) = 81/80, or using the shorthand notation, S6/S8 = S9. This means that as S6/S8 is already tempered in meantone, it is natural to want 49/48 = S7 (which is bigger than S8 and smaller than S6) to be equated, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)3 = 1728/1715 = S6/S7, the orwellisma.
This strategy leads to the 7-limit version of mothra, which is also sometimes called cynder, though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6 = S7 = S8 with S9 tempered, we can try S8 = S10 by tempering 176/175 = S8/S10 = (11/7)/(5/4)2 , taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 11/8 at
31edo as splitting the fifth into two, three and nine
31edo is unique as combining all aforementioned tempering strategies into one elegant 11-limit meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate 5/4 and 7/4 and an even more accurate 35/32, so that it is very strong in the 2.5.7 subgroup. A tempering strategy not mentioned is splitting a flattened 3/2 into nine sharpened 25/24's, resulting in the 5-limit version of valentine so that 31edo is uniquely meantone + valentine. Valentine is a natural 11-limit temperament that tempers 121/120 so for this reason might be natural to combine with meantone. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has 31edo as a patent val tuning.
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. (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
- Syntonisma, the difference by which a stack of seven 81/80s falls short of 12/11
- Small comma
- List of superparticular intervals
Notes
[[1]]