81/80

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 unison, gM, guma
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]
Open this interval in xen-calc

English Wikipedia has an article on:

Syntonic comma

The syntonic comma, also known as the Didymus' comma, the meantone comma or the Ptolemaic comma, with a frequency ratio 81/80, is the difference between many 3-limit and 5-limit ratios in just intonation. Adding or subtracting this comma to/from any complex 3-limit ratio (such as 32/27 or 81/64) creates a 5-limit ratio with a much lower odd-limit (such as 6/5 or 5/4). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the pythagorean major triad, 64:81:96, is quite dissonant, but flattening the 81/64 major third by 81/80 leads to the much more consonant 4:5:6 chord, with a 5/4 major third in place of 81/64. However, adding/subtracting this comma to/from the perfect fourth, fifth, or octave creates a wolf interval of odd limit 27 or higher, such as the 40/27 wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to make intervals and chords pure in one key will create wolf intervals in others; thus, for those who wish to avoid such wolves in composition, tempering out 81/80 is desirable. This leads to meantone temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones. This also equates 10/9 with 9/8, giving a tuning for the whole tone which is intermediate between them; hence the name "meantone".

81/80 is the smallest superparticular interval which belongs to the 5-limit, and in fact 81/80 is a square superparticular, being the difference between 10/9 and 9/8, the product of which is the just major third, 5/4.

Comma pumps

The familiar vi–ii–V–I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.

A passage (listen) from Ben Johnston's 9th string quartet, near the end of movement 1, makes a sudden and prominent use of the 81/80 comma, which demonstrates how a simple progression with held common tones can quickly lead to severe interference beating, rupturing the diatonic collection routinely associated with the 5-limit and exposing "C major" as anything but simple.

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

Adam Neely's harmonization of the licc pumps upward by 81/80 every measure. After 9 iterations, D modulates nearly to E.

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 648/625 exceeds 128/125.
• 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.

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 .

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, thus tempering out the barium comma ([-225 224 -56⟩).

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 rank-3 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.)

Relations between 81/80 and other superparticular ratios

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
• 1ed81/80 – its equal multiplication
• Pythagorean comma
• 64/63 – the septimal comma or Archytas' comma
• Small comma
• List of superparticular intervals