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'''81/80'''
'''81/80'''
|-4 4 -1>


21.506290 cents
{{Monzo| -4 4 -1 }}


The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit|5-limit]]. Like [[16/15|16/15]], [[625/624|625/624]], [[2401/2400|2401/2400]] and [[4096/4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10/9|10/9]] and [[9/8|9/8]], the product of which is the just major third, [[5/4|5/4]]. That the numerator is a fourth power entails that the larger 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|105edo]] tempers it out, while [[15edo|3edo]] does not.
21.506290 [[cents]]


Tempering out 81/80 gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone_family|meantone temperament]].
The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|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 comma with a square numerator is the ratio between two larger successive superparticular intervals; it 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 larger 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.
 
Tempering out 81/80 gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].


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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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.


Names in brackets refer to 7-limit [[Meantone_family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus_rank_three_family|Didymus family]] that temper out the respective ratios as commas.
Names in brackets refer to 7-limit [[Meantone family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus rank three family|Didymus family]] that temper out the respective ratios as commas.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Limit
! Limit
! | r1 * r2
! r1 * r2
! | r2 / r1
! r2 / r1
|-
|-
| | 5
| 5
| | -
| -
| | 9/8 * 9/10
| 9/8 * 9/10
|-
|-
| | 7
| 7
| | 126/125 * 225/224 (septimal meantone)
| 126/125 * 225/224 (septimal meantone)
| | 21/20 * 27/28 (sharptone), 36/35 * 63/64 (dominant)
| 21/20 * 27/28 (sharptone), 36/35 * 63/64 (dominant)
|-
|-
| | 11
| 11
| | 99/98 * 441/440 (euterpe), 121/120 * 243/242 (urania)
| 99/98 * 441/440 (euterpe), 121/120 * 243/242 (urania)
| | 33/32 * 54/55 (thalia), 45/44 * 99/100 (calliope)
| 33/32 * 54/55 (thalia), 45/44 * 99/100 (calliope)
|-
|-
| | 13
| 13
| | 91/90 * 729/728, 105/104 * 351/350
| 91/90 * 729/728, 105/104 * 351/350
| | 27/26 * 39/40, 65/64 * 324/325, 66/65 * 351/352, 78/77 * 2079/2080
| 27/26 * 39/40, 65/64 * 324/325, 66/65 * 351/352, 78/77 * 2079/2080
|-
|-
| | 17
| 17
| | 85/84 * 1701/1700
| 85/84 * 1701/1700
| | 51/50 * 135/136
| 51/50 * 135/136
|-
|-
| | 19
| 19
| | 96/95 * 513/512, 153/152 * 171/170
| 96/95 * 513/512, 153/152 * 171/170
| | 57/56 * 189/190, 76/75 * 1215/1216, 77/76 * 1539/1540
| 57/56 * 189/190, 76/75 * 1215/1216, 77/76 * 1539/1540
|-
|-
| | 23
| 23
| | 161/160 * 162/161
| 161/160 * 162/161
| | 69/68 * 459/460
| 69/68 * 459/460
|-
|-
| | 29
| 29
| | 117/116 * 261/260
| 117/116 * 261/260
| | -
| -
|-
|-
| | 31
| 31
| | 93/92 * 621/620
| 93/92 * 621/620
| | 63/62 * 279/280
| 63/62 * 279/280
|-
|-
| | 37
| 37
| | 111/110 * 297/296
| 111/110 * 297/296
| | 75/74 * 999/1000
| 75/74 * 999/1000
|-
|-
| | 41
| 41
| | 82/81 * 6561/6560
| 82/81 * 6561/6560
| | 41/40 * 81/82
| 41/40 * 81/82
|-
|-
| | 43
| 43
| | 86/85 * 1377/1376, 87/86 * 1161/1160, 129/128 * 216/215
| 86/85 * 1377/1376, 87/86 * 1161/1160, 129/128 * 216/215
| | -
| -
|-
|-
| | 47
| 47
| | 141/140 * 189/188
| 141/140 * 189/188
| | -
| -
|-
|-
| | 53
| 53
| | -
| -
| | 54/53 * 159/160
| 54/53 * 159/160
|-
|-
| | 59
| 59
| | -
| -
| | -
| -
|-
|-
| | 61
| 61
| | -
| -
| | 61/60 * 243/244
| 61/60 * 243/244
|-
|-
| | 67
| 67
| | 135/134 * 201/200
| 135/134 * 201/200
| | -
| -
|-
|-
| | 71
| 71
| | -
| -
| | 71/70 * 567/568, 72/71 * 639/640
| 71/70 * 567/568, 72/71 * 639/640
|-
|-
| | 73
| 73
| | -
| -
| | 73/72 * 729/730
| 73/72 * 729/730
|-
|-
| | 79
| 79
| | -
| -
| | 79/78 * 3159/3160, 80/79 * 6399/6400
| 79/78 * 3159/3160, 80/79 * 6399/6400
|-
|-
| | 83
| 83
| | 83/82 * 3321/3320, 84/83 * 2241/2240
| 83/82 * 3321/3320, 84/83 * 2241/2240
| | -
| -
|-
|-
| | 89
| 89
| | 89/88 * 891/890, 90/89 * 801/800
| 89/88 * 891/890, 90/89 * 801/800
| | -
| -
|-
|-
| | 97
| 97
| | 97/96 * 486/485
| 97/96 * 486/485
| | -
| -
|-
|-
| | 101
| 101
| | 101/100 * 405/404
| 101/100 * 405/404
| | -
| -
|-
|-
| | 103
| 103
| | -
| -
| | -
| -
|-
|-
| | 107
| 107
| | 108/107 * 321/320
| 108/107 * 321/320
| | -
| -
|}
|}


==External Links==
==External Links==
[http://en.wikipedia.org/wiki/Syntonic_comma http://en.wikipedia.org/wiki/Syntonic_comma]     [[Category:5-limit]]
* [https://en.wikipedia.org/wiki/Syntonic_comma Syntonic comma - Wikipedia]
 
[[Category:5-limit]]
[[Category:comma]]
[[Category:comma]]
[[Category:definition]]
[[Category:definition]]
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[[Category:superparticular]]
[[Category:superparticular]]
[[Category:syntonic]]
[[Category:syntonic]]
<!-- interwiki -->
[[de:81/80]]

Revision as of 13:16, 1 October 2018

81/80

[-4 4 -1⟩

21.506290 cents

The syntonic or Didymus comma (frequency ratio 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 comma with a square numerator is the ratio between two larger successive superparticular intervals; it 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 larger 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.

Tempering out 81/80 gives a tuning for the whole tone which is intermediate between 10/9 and 9/8, and leads to meantone temperament.

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 -

External Links

Retrieved from "https://en.xen.wiki/index.php?title=81/80&oldid=34292"