81/80
Ratio | 81/80 |
Factorization | 2^{-4} × 3^{4} × 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]\text{P1}_{5}[/math] |
Special properties | square superparticular, reduced |
Tenney height (log_{2} nd) | 12.6618 |
Weil height (log_{2} max(n, d)) | 12.6797 |
Wilson height (sopfr (nd)) | 25 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~3.94063 bits |
Comma size | small |
S-expressions | S9, S6 / S8 |
[sound info] | |
open this interval in xen-calc |
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 n^{2}/(n^{2}-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 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.
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 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]]