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'''55EDO''' divides the octave into 55 parts of 21.8182 cents. It can be used for a meantone tuning, and is close to [[1-6_Syntonic_Comma_Meantone|1/6 comma meantone]] (and is almost exactly 10/57 comma meantone.) [http://en.wikipedia.org/wiki/Georg_Philipp_Telemann Telemann] suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by [http://en.wikipedia.org/wiki/Leopold_Mozart Leopold] and [http://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart Wolfgang Mozart]. It can also be used for [[Meantone_family|mohajira and liese]] temperaments.
'''55edo''' divides the octave into 55 parts of 21.818{{cent}}. It can be used for a meantone tuning, and is close to [[1-6_Syntonic_Comma_Meantone|1/6 comma meantone]] (and is almost exactly 10/57 comma meantone.) [http://en.wikipedia.org/wiki/Georg_Philipp_Telemann Telemann] suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by [http://en.wikipedia.org/wiki/Leopold_Mozart Leopold] and [http://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart Wolfgang Mozart]. It can also be used for [[Meantone_family|mohajira and liese]] temperaments.


5-limit commas: [[81/80]], &lt;31 1 -14|, <27 5 -15|
== Theory ==
{{Harmonics in equal|55}}
 
5-limit commas: [[81/80]], {{monzo|31 1 -14}}, {{monzo|27 5 -15}}


7-limit commas: 31104/30625, [[6144/6125]], 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, 686/675, 10976/10935, [[Cloudy comma|16807/16384]], 84035/82944
7-limit commas: 31104/30625, [[6144/6125]], 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, 686/675, 10976/10935, [[Cloudy comma|16807/16384]], 84035/82944
Line 8: Line 11:


13-limit commas: 59535/57122, 29400/28561, 29568/28561, 29645/28561, 24576/24167, 99225/96668, 24500/24167, 50421/48334, 45927/43940, 2268/2197, 2240/2197, 57624/54925, 61875/61516, 57024/54925, 11264/10985, 72765/70304, 13475/13182, 22869/21970, 6776/6591, 20736/20449, 20480/20449, 84035/81796, 91125/91091, 65536/65065, 15309/14872, 1890/1859, 5600/5577, 9604/9295, 59049/57967, 58320/57967, 4374/4225, 864/845, 512/507, 11025/10816, 6125/6084, 21952/21125, 16807/16224, 84035/82134, 66825/66248, 90112/88725, 56133/54080, 693/676, 1540/1521, 26411/25350, 58806/57967, 58080/57967, 88209/84500, 4356/4225, 7744/7605, 88935/86528, 33275/33124, 27951/27040, 9317/9126, 58564/57967, 43923/42250, 17496/17303, 87808/86515, 55296/55055, 25515/25168, 1575/1573, 64827/62920, 4802/4719, 98415/98098, 59049/57200, 729/715, [[144/143]], 18375/18304, 18522/17875, 10976/10725, 84035/82368, 59049/56875, 11664/11375, 2304/2275, 4096/4095, 1701/1664, [[105/104]], 42336/40625, 25088/24375, 21609/20800, 2401/2340, 9604/9477, 72171/71344, 2673/2600, [[66/65]], [[352/351]], 13475/13312, 33957/32500, 15092/14625, 81675/81536, 58806/56875, 11616/11375, 61952/61425, 68607/66560, 847/832, 4235/4212, 35937/35672, 1331/1300, 5324/5265, 58564/56875, 85293/85184, 13377/13310, 85293/84700, 15288/15125, 31213/30976, 67392/67375, 28431/28160, 34944/34375, 4459/4400, 4459/4455, 28431/28000, 351/350, 79872/78125, 66339/65536, 51597/50000, 637/625, 10192/10125, 31213/30720, [[31213/31104]], 30888/30625, 1287/1280, 81081/78125, 16016/15625, 49049/48000, 49049/48600, 14157/14000, 33033/32768, 77077/75000, 51909/51200, 17303/17280, 75712/75625, 8281/8250, 41067/40960, 31941/31250, 9464/9375, 57967/57600, 91091/90000, 61347/61250, 79092/78125
13-limit commas: 59535/57122, 29400/28561, 29568/28561, 29645/28561, 24576/24167, 99225/96668, 24500/24167, 50421/48334, 45927/43940, 2268/2197, 2240/2197, 57624/54925, 61875/61516, 57024/54925, 11264/10985, 72765/70304, 13475/13182, 22869/21970, 6776/6591, 20736/20449, 20480/20449, 84035/81796, 91125/91091, 65536/65065, 15309/14872, 1890/1859, 5600/5577, 9604/9295, 59049/57967, 58320/57967, 4374/4225, 864/845, 512/507, 11025/10816, 6125/6084, 21952/21125, 16807/16224, 84035/82134, 66825/66248, 90112/88725, 56133/54080, 693/676, 1540/1521, 26411/25350, 58806/57967, 58080/57967, 88209/84500, 4356/4225, 7744/7605, 88935/86528, 33275/33124, 27951/27040, 9317/9126, 58564/57967, 43923/42250, 17496/17303, 87808/86515, 55296/55055, 25515/25168, 1575/1573, 64827/62920, 4802/4719, 98415/98098, 59049/57200, 729/715, [[144/143]], 18375/18304, 18522/17875, 10976/10725, 84035/82368, 59049/56875, 11664/11375, 2304/2275, 4096/4095, 1701/1664, [[105/104]], 42336/40625, 25088/24375, 21609/20800, 2401/2340, 9604/9477, 72171/71344, 2673/2600, [[66/65]], [[352/351]], 13475/13312, 33957/32500, 15092/14625, 81675/81536, 58806/56875, 11616/11375, 61952/61425, 68607/66560, 847/832, 4235/4212, 35937/35672, 1331/1300, 5324/5265, 58564/56875, 85293/85184, 13377/13310, 85293/84700, 15288/15125, 31213/30976, 67392/67375, 28431/28160, 34944/34375, 4459/4400, 4459/4455, 28431/28000, 351/350, 79872/78125, 66339/65536, 51597/50000, 637/625, 10192/10125, 31213/30720, [[31213/31104]], 30888/30625, 1287/1280, 81081/78125, 16016/15625, 49049/48000, 49049/48600, 14157/14000, 33033/32768, 77077/75000, 51909/51200, 17303/17280, 75712/75625, 8281/8250, 41067/40960, 31941/31250, 9464/9375, 57967/57600, 91091/90000, 61347/61250, 79092/78125
==Prime intervals==
{{Primes in edo|55}}


==Intervals==
==Intervals==
 
{| class="wikitable center-1 right-2 left-3"
{| class="wikitable"
|-
|-
| | Degrees of 55-EDO
! [[Degree|#]]
| | Cents value
! [[Cent]]s
| | Ratios it approximates
! Approximate ratios
|-
|-
| 0
| 0
|0.000
| 0.000
| | 1/1
| 1/1
|-
|-
| | 1
| 1
| | 21.818
| 21.818
| | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80''
| 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80''
|-
|-
| | 2
| 2
| | 43.636
| 43.636
| |36/35
| 36/35
|-
|-
| | 3
| 3
| | 65.4545
| 65.4545
| |28/27, ''25/24''
| 28/27, ''25/24''
|-
|-
| | 4
| 4
| | 87.273
| 87.273
| | 25/24, 21/20
| 25/24, 21/20
|-
|-
| | 5
| 5
| | 109.091
| 109.091
| | 16/15
| 16/15
|-
|-
| | 6
| 6
| | 130.909
| 130.909
| |14/13, ''13/12''
| 14/13, ''13/12''
|-
|-
| | 7
| 7
| | 152.727
| 152.727
| |13/12, 12/11
| 13/12, 12/11
|-
|-
| | 8
| 8
| | 174.5455
| 174.5455
| |11/10, ''10/9''
| 11/10, ''10/9''
|-
|-
| | 9
| 9
| | 196.364
| 196.364
| | 9/8, 10/9
| 9/8, 10/9
|-
|-
| | 10
| 10
| | 218.182
| 218.182
|17/15
| 17/15
|-
|-
| | 11
| 11
| | 240
| 240
|8/7, 15/13
| 8/7, 15/13
|-
|-
| | 12
| 12
| | 261.818
| 261.818
|7/6
| 7/6
|-
|-
| | 13
| 13
| | 283.636
| 283.636
|13/11
| 13/11
|-
|-
| | 14
| 14
| | 305.4545
| 305.4545
|6/5-
| 6/5-
|-
|-
| | 15
| 15
| | 327.273
| 327.273
|6/5+
| 6/5+
|-
|-
| | 16
| 16
| | 349.091
| 349.091
|11/9, 27/22
| 11/9, 27/22
|-
|-
| | 17
| 17
| | 370.909
| 370.909
|16/13
| 16/13
|-
|-
| | 18
| 18
| | 392.727
| 392.727
|5/4
| 5/4
|-
|-
| | 19
| 19
| | 414.5455
| 414.5455
|14/11
| 14/11
|-
|-
| | 20
| 20
| | 436.364
| 436.364
|9/7
| 9/7
|-
|-
| | 21
| 21
| | 458.182
| 458.182
|13/10
| 13/10
|-
|-
| | 22
| 22
| | 480
| 480
|21/16
| 21/16
|-
|-
| | 23
| 23
| | 501.818
| 501.818
|4/3, 27/20
| 4/3, 27/20
|-
|-
| | 24
| 24
| | 523.636
| 523.636
|''27/20''
| ''27/20''
|-
|-
| | 25
| 25
| | 545.4545
| 545.4545
|11/8
| 11/8
|-
|-
| | 26
| 26
| | 567.273
| 567.273
|18/13, 25/18
| 18/13, 25/18
|-
|-
| | 27
| 27
| | 589.091
| 589.091
|7/5
| 7/5
|-
|-
| | 28
| 28
| | 610.909
| 610.909
|10/7
| 10/7
|-
|-
| | 29
| 29
| | 632.727
| 632.727
|13/9, 36/25
| 13/9, 36/25
|-
|-
| | 30
| 30
| | 654.5455
| 654.5455
|16/11
| 16/11
|-
|-
| | 31
| 31
| | 676.364
| 676.364
|''40/27''
| ''40/27''
|-
|-
| | 32
| 32
| | 698.182
| 698.182
|3/2, 40/27
| 3/2, 40/27
|-
|-
| | 33
| 33
| | 720
| 720
|32/21
| 32/21
|-
|-
| | 34
| 34
| | 741.818
| 741.818
|20/13
| 20/13
|-
|-
| | 35
| 35
| | 763.636
| 763.636
|14/9
| 14/9
|-
|-
| | 36
| 36
| | 785.4545
| 785.4545
|11/7
| 11/7
|-
|-
| | 37
| 37
| | 807.273
| 807.273
|8/5
| 8/5
|-
|-
| | 38
| 38
| | 829.091
| 829.091
|13/8
| 13/8
|-
|-
| | 39
| 39
| | 850.909
| 850.909
|18/11, 44/27
| 18/11, 44/27
|-
|-
| | 40
| 40
| | 872.727
| 872.727
|5/3-
| 5/3-
|-
|-
| | 41
| 41
| | 894.5455
| 894.5455
|5/3+
| 5/3+
|-
|-
| | 42
| 42
| | 916.364
| 916.364
|22/13
| 22/13
|-
|-
| | 43
| 43
| | 938.182
| 938.182
|12/7
| 12/7
|-
|-
| | 44
| 44
| | 960
| 960
|7/4, 26/15
| 7/4, 26/15
|-
|-
| | 45
| 45
| | 981.818
| 981.818
|30/17
| 30/17
|-
|-
| | 46
| 46
| | 1003.636
| 1003.636
|16/9, 9/5
| 16/9, 9/5
|-
|-
| | 47
| 47
| | 1025.4545
| 1025.4545
|''9/5'', 20/11
| ''9/5'', 20/11
|-
|-
| | 48
| 48
| | 1047.273
| 1047.273
|11/6, 24/13
| 11/6, 24/13
|-
|-
| | 49
| 49
| | 1069.091
| 1069.091
|''24/13'', 13/7
| ''24/13'', 13/7
|-
|-
| | 50
| 50
| | 1090.909
| 1090.909
|15/8
| 15/8
|-
|-
| | 51
| 51
| | 1112.727
| 1112.727
|40/21, 48/25
| 40/21, 48/25
|-
|-
| | 52
| 52
| | 1134.5455
| 1134.5455
|56/27, ''48/25''
| 56/27, ''48/25''
|-
|-
| | 53
| 53
| | 1156.364
| 1156.364
|35/18
| 35/18
|-
|-
| | 54
| 54
| | 1178.182
| 1178.182
|125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81''
| 125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81''
|-
|-
| | 55
| 55
| | 1200
| 1200
|2/1
| 2/1
|}
|}


==Selected just intervals by error==
== Selected just intervals by error ==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 55edo (ordered by absolute error).
The following table shows how [[15-odd-limit]] just intervals are represented in 55edo (ordered by absolute error).
 
{{15-odd-limit|55}}
{| class="wikitable"
|-
! | Interval, complement
! | Error (abs., in [[cent|cents]])
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 1.280
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 1.683
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 2.090
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 2.611
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 2.640
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 2.963
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 3.773
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 3.890
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 3.968
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 5.053
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 5.573
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 5.863
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 6.414
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 6.579
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 7.546
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 7.664
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 7.741
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 7.858
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 8.504
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 8.826
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 9.541
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 10.187
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 10.352
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 10.381
|}


[http://www.seraph.it/dep/int/AdagioKV540.mp3 Mozart - Adagio in B minor KV 540] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])
== Music ==
* [http://www.seraph.it/dep/int/AdagioKV540.mp3 Mozart - Adagio in B minor KV 540] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])


[http://tonalsoft.com/monzo/55edo/55edo.aspx "Mozart's tuning: 55edo"] (containing another listening example) in the [[tonalsoft_encyclopedia|tonalsoft encyclopedia]]
== External links ==
* [http://tonalsoft.com/monzo/55edo/55edo.aspx "Mozart's tuning: 55edo"] (containing another listening example) in the [[Tonalsoft Encyclopedia]]


[[Category:55edo]]
[[Category:55edo]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Meantone]]
[[Category:Meantone]]
Retrieved from "https://en.xen.wiki/w/55edo"