81/80: Difference between revisions
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The '''syntonic''' or '''Didymus''' or '''meantone comma''' (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 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 tempering out 81/80 is desirable. This gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]], hence the name meantone comma. | The '''syntonic''' or '''Didymus''' or '''meantone comma''' (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 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 tempering out 81/80 is desirable. This gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]], hence the name meantone comma. | ||
81/80 is the smallest [[superparticular | 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<sup>2</sup>/(n<sup>2</sup>-1) = n/(n-1) ÷ (n+1)/n. 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. Tempering out a comma does not just depend on an edo's size; [[105edo]] tempers it out, while [[15edo|3edo]] does not. | ||
Youtube video of "[http://www.youtube.com/watch?v=IpWiEWFRGAY Five senses of 81/80]", demonstratory video by Jacob Barton. | Youtube video of "[http://www.youtube.com/watch?v=IpWiEWFRGAY Five senses of 81/80]", demonstratory video by Jacob Barton. | ||
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According to [http://untwelve.org/interviews/golden.html this interview], Monroe Golden's ''Incongruity'' uses just-intonation chord progressions that exploit this comma. | According to [http://untwelve.org/interviews/golden.html this interview], Monroe Golden's ''Incongruity'' uses just-intonation chord progressions that exploit this comma. | ||
=Relations to other Superparticular Ratios= | == 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. | 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. | ||
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==External Links== | == External Links == | ||
* [ | |||
* [[Wikipedia: Syntonic comma]] | |||
[[Category:5-limit]] | [[Category:5-limit]] | ||
Revision as of 16:46, 27 November 2020
| Interval information |
Didymus comma,
meantone comma
gu unison
reduced
The syntonic or Didymus or meantone comma (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 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 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. 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. Tempering out a comma does not just depend on an edo's size; 105edo tempers it out, while 3edo does not.
Youtube video of "Five senses of 81/80", demonstratory video by Jacob Barton.
According to this interview, Monroe Golden's Incongruity uses just-intonation chord progressions that exploit this comma.
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.
| 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 | - |