81/80

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:

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.

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.

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 ⁠ ⁠.

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 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	-

Notes

↑ UnTwelve's interview to Monroe Golden

[1] UnTwelve's interview to Monroe Golden

[1]

81/80

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