81/80

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Interval information
Ratio 81/80
Monzo |-4 4 -1>
Size in cents 21.506290
Name(s) syntonic comma,
Didymus comma
Color name g1, gu comma,
gu unison

The syntonic or Didymus comma (frequency ratio 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 comma with a square numerator is the ratio between two larger successive superparticular intervals; it 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 larger 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 it out, while 3edo does not.

Tempering out 81/80 gives a tuning for the whole tone which is intermediate between 10/9 and 9/8, and leads to meantone temperament.

Youtube video of "Five senses of 81/80", demonstratory video by Jacob Barton.

According to this interview, Monroe Golden's Incongruity uses just-intonation chord progressions that exploit this comma.

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 -

External Links