# 81/80

(Redirected from Didymus comma)
 Ratio 81/80 Factorization 2-4 × 34 × 5-1 Monzo [-4 4 -1⟩ Size in cents 21.50629¢ Names syntonic comma,Didymus comma,meantone comma,Ptolemaic comma Color name g1, Gu comma, gu unison FJS name $\text{P1}_{5}$ Special properties square superparticular,reduced Tenney height (log2 nd) 12.6618 Weil height (log2 max(n, d)) 12.6797 Wilson height (sopfr (nd)) 25 Harmonic entropy(Shannon, $\sqrt{nd}$) ~3.94063 bits Comma size small S-expressions S9,S6 / S8 https://en.xen.wiki/w/File:Audacity_pluck_81_80.wav[sound info] open this interval in xen-calc
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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.

Monroe Golden's Incongruity uses just-intonation chord progressions that exploit this comma[1].

## 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:

## 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 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 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.

## 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 -