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.

Interval information

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	[math]\displaystyle{ \text{P1}_{5} }[/math]
Special properties	square superparticular, reduced
Tenney norm (log2 nd)	12.6618
Weil norm (log2 max(n, d))	12.6797
Wilson norm (sopfr(nd))	25
Comma size	small
S-expressions	S9, S6/S8
[sound info]
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Syntonic 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.

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 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	-

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]]