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'''55edo''' divides the octave into 55 parts of 21.818 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.
{{interwiki
| de = 55-EDO
| en = 55edo
| es = 55 EDO
| ja =
}}
{{Infobox ET}}
{{ED intro}}


5-limit commas: 81/80, &lt;31 1 -14|, <-165 220 55|
== Theory ==
55edo can be used for a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[Meantone_family#Mohajira|Mohajira]] and [[Meantone_family#Liese|Liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.


7-limit commas: 31104/30625, 6144/6125, 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, 686/675, 10976/10935, 16807/16384, 84035/82944
=== Odd harmonics ===
{{Harmonics in equal|55}}


11-limit commas: 59049/58564, 74088/73205, 46656/46585, 21609/21296, 12005/11979, 19683/19360, 243/242, 3087/3025, 5488/5445, 19683/19250, 1944/1925, 45927/45056, 2835/2816, 35721/34375, 7056/6875, 12544/12375, 7203/7040, 2401/2376, 24057/24010, 72171/70000, 891/875, 176/175, 2079/2048, 385/384, 3234/3125, 17248/16875, 26411/25600, 26411/2592, 26411/262404, 88209/87808, 30976/30625, 3267/3200, 121/120, 81312/78125, 41503/40000, 41503/40500, 35937/35000, 2662/2625, 42592/42525, 83853/81920, 9317/9216, 65219/62500, 43923/43904, 14641/14400, 14641/14580
=== Subsets and supersets ===
Since 55 factors into {{factorization|55}}, 55edo contains [[5edo]] and [[11edo]] as its subsets.


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
== Intervals ==
 
{| class="wikitable center-1 right-2 left-3"
==Intervals==
 
{| class="wikitable"
|-
|-
| | Degrees of 55-EDO
! [[Degree|&#35;]]
| | Cents value
! [[Cent]]s
|pions
! Approximate ratios
|7mus
! colspan="3" | [[Ups and downs notation]]
| | Ratios it approximates
|-
|-
| colspan="4"| 0
| 0
| | 1/1
| 0.0
| 1/1
| P1
| perfect 1sn
| D
|-
|-
| | 1
| 1
| | 21.818
| 21.8
|23.127
| 65/64, 78/77, 99/98, ''128/125''
|27.927 (1B.ED6<sub>16</sub>)
| ^1
| | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80''
| up 1sn
| ^D
|-
|-
| | 2
| 2
| | 43.636
| 43.6
|46.2545
| 36/35, ''64/63''
|55.8545 (37.DAC<sub>16</sub>)
| ^^1
| |36/35
| dup 1sn
| ^^D
|-
|-
| | 3
| 3
| | 65.455
| 65.5
|69.382
| 28/27
|83.782 (53.C82<sub>16</sub>)
| vvm2
| |28/27, ''25/24''
| dudminor 2nd
| vvEb
|-
|-
| | 4
| 4
| | 87.273
| 87.3
|92.509
| 21/20, ''18/17'', ''25/24''
|111.709 (6F.B58<sub>16</sub>)
| vm2
| | 25/24, 21/20
| downminor 2nd
| vEb
|-
|-
| | 5
| 5
| | 109.091
| 109.1
|115.636
| 16/15, 17/16
|139.636 (8B.A2E8<sub>16</sub>)
| m2
| | 16/15
| minor 2nd
| Eb
|-
|-
| | 6
| 6
| | 130.909
| 130.9
|138.764
| 13/12, 14/13
|167.564 (A7.905<sub>16</sub>)
| ^m2
| |14/13, ''13/12''
| upminor 2nd
| ^Eb
|-
|-
| | 7
| 7
| | 152.727
| 152.7
|161.891
| 12/11, ''11/10''
|195.491 (C3.7DA<sub>16</sub>)
| ~2
| |13/12, 12/11
| mid 2nd
| vvE
|-
|-
| | 8
| 8
| | 174.5455
| 174.5
|185.018
|  
|223.418 (DF.6B1<sub>16</sub>)
| vM2
| |11/10, ''10/9''
| downmajor 2nd
| vE
|-
|-
| | 9
| 9
| | 196.364
| 196.4
|208.1455
| 9/8, ''10/9''
|251.3455 (FB.587<sub>16</sub>)
| M2
| | 9/8, 10/9
| major 2nd
| E
|-
|-
| | 10
| 10
| | 218.182
| 218.2
|231.273
| 17/15
|279.273 (117.467<sub>16</sub>)
| ^M2
|17/15
| upmajor 2nd
| ^E
|-
|-
| | 11
| 11
| | 240
| 240.0
|254.4
| 8/7
|307.2 (133.333<sub>16</sub>)
| ^^M2
|8/7, 15/13
| dupmajor 2nd
| ^^E
|-
|-
| | 12
| 12
| | 261.818
| 261.8
|277.527
| 7/6
|335.127 (14F.209<sub>16</sub>)
| vvm3
|7/6
| dudminor 3rd
| vvF
|-
|-
| | 13
| 13
| | 283.636
| 283.6
|300.6545
| 13/11
|363.0545 (16B.0DF<sub>16</sub>)
| vm3
|13/11
| downminor 3rd
| vF
|-
|-
| | 14
| 14
| | 305.4545
| 305.5
|323.782
| 6/5
|390.982 (186.FB48<sub>16</sub>)
| m3
|6/5-
| minor 3rd
| F
|-
|-
| | 15
| 15
| | 327.273
| 327.3
|346.909
|
|418.909 (1A2.E8C<sub>16</sub>)
| ^m3
|6/5+
| upminor 3rd
| ^F
|-
|-
| | 16
| 16
| | 349.091
| 349.1
|370.036
| 11/9, 27/22
|446.836 (1BE.D62<sub>16</sub>)
| ~3
|11/9, 27/22
| mid 3rd
| ^^F
|-
|-
| | 17
| 17
| | 370.909
| 370.9
|393.164
| 26/21, ''16/13''
|474.763 (1DA.C56<sub>16</sub>)
| vM3
|16/13
| downmajor 3rd
| vF#
|-
|-
| | 18
| 18
| | 392.727
| 392.7
|416.291
| 5/4
|502.691 (1F6.A0E<sub>16</sub>)
| M3
|5/4
| major 3rd
| F#
|-
|-
| | 19
| 19
| | 414.5455
| 414.5
|439.418
| 14/11
|530.618 (212.9ED<sub>16</sub>)
| ^M3
|14/11
| upmajor 3rd
| ^F#
|-
|-
| | 20
| 20
| | 436.364
| 436.4
|462.5455
| 9/7
|558.5455 (22E.8BA<sub>16</sub>)
| ^^M3
|9/7
| dupmajor 3rd
| ^^F#
|-
|-
| | 21
| 21
| | 458.182
| 458.2
|485.673
| ''21/16''
|586.473 (24A.79<sub>16</sub>)
| vv4
|13/10
| dud 4th
| vvG
|-
|-
| | 22
| 22
| | 480
| 480.0
|508.8
|  
|614.4 (266.666<sub>16</sub>)
| v4
|21/16
| down 4th
| vG
|-
|-
| | 23
| 23
| | 501.818
| 501.8
|531.927
| 4/3, ''27/20''
|642.327 (282.5508<sub>16</sub>)
| P4
|4/3, 27/20
| perfect 4th
| G
|-
|-
| | 24
| 24
| | 523.636
| 523.6
|555.0545
|
|670.2545 (29E.413<sub>16</sub>)
| ^4
|''27/20''
| up 4th
| ^G
|-
|-
| | 25
| 25
| | 545.4545
| 545.5
|578.182
| 11/8, 15/11
|698.182 (2BA.2E9<sub>16</sub>)
| ~4
|11/8
| mid 4th
| ^^G
|-
|-
| | 26
| 26
| | 567.273
| 567.3
|601.309
| [[7/5]], [[18/13]]
|726.109 (2D6.1BF<sub>16</sub>)
| vA4
|18/13, 25/18
| downaug 4th
| vG#
|-
|-
| | 27
| 27
| | 589.091
| 589.1
|624.436
| 24/17
|754.036 (2F2.095<sub>16</sub>)
| A4, vd5
|7/5
| aug 4th, downdim 5th
| G#, vAb
|-
|-
| | 28
| 28
| | 610.909
| 610.9
|647.564
| 17/12
|781.944 (30D.F7B<sub>16</sub>)
| ^A4, d5
|10/7
| upaug 4th, dim 5th
| ^G#, Ab
|-
|-
| | 29
| 29
| | 632.727
| 632.7
|670.691
| [[10/7]], [[13/9]]
|809.891 (329.E41<sub>16</sub>)
| ^d5
|13/9, 36/25
| updim 5th
| ^Ab
|-
|-
| | 30
| 30
| | 654.5455
| 654.5
|693.818
| 16/11, 22/15
|837.818 (345.D17<sub>16</sub>)
| ~5
|16/11
| mid 5th
| vvA
|-
|-
| | 31
| 31
| | 676.364
| 676.4
|716.9455
|
|865.7455 (361.BEC<sub>16</sub>)
| v5
|''40/27''
| down 5th
| vA
|-
|-
| | 32
| 32
| | 698.182
| 698.2
|740.073
| 3/2, ''40/27''
|893.673 (37D.AAF8<sub>16</sub>)
| P5
|3/2, 40/27
| perfect 5th
| A
|-
|-
| | 33
| 33
| | 720
| 720.0
|763.2
|  
|921.6 (399.99A<sub>16</sub>)
| ^5
|32/21
| up 5th
| ^A
|-
|-
| | 34
| 34
| | 741.818
| 741.8
|786.327
| ''32/21''
|949.527 (3B5.87<sub>16</sub>)
| ^^5
|20/13
| dup 5th
| ^^A
|-
|-
| | 35
| 35
| | 763.636
| 763.6
|809.4545
| 14/9
|977.4545 (3D1.746<sub>16</sub>)
| vvm6
|14/9
| dudminor 6th
| vvBb
|-
|-
| | 36
| 36
| | 785.4545
| 785.5
|832.582
| 11/7
|1005.382 (FED.612<sub>16</sub>)
| vm6
|11/7
| downminor 6th
| vBb
|-
|-
| | 37
| 37
| | 807.273
| 807.3
|855.709
| 8/5
|1033.309 (409.5F2<sub>16</sub>)
| m6
|8/5
| minor 6th
| Bb
|-
|-
| | 38
| 38
| | 829.091
| 829.1
|878.836
| 21/13, ''13/8''
|1061.327 (425.3A9<sub>16</sub>)
| ^m6
|13/8
| upminor 6th
| ^Bb
|-
|-
| | 39
| 39
| | 850.909
| 850.9
|901.964
| 18/11, 44/27
|1089.164 (441.19E<sub>16</sub>)
| ~6
|18/11, 44/27
| mid 6th
| vvB
|-
|-
| | 40
| 40
| | 872.727
| 872.7
|925.091
|
|1117.091 (45D.174<sub>16</sub>)
| vM6
|5/3-
| downmajor 6th
| vB
|-
|-
| | 41
| 41
| | 894.5455
| 894.5
|948.218
| 5/3
|1145.018 (479.04A8<sub>16</sub>)
| M6
|5/3+
| major 6th
| B
|-
|-
| | 42
| 42
| | 916.364
| 916.4
|971.3455
| 22/13
|1172.9455 (494.F21<sub>16</sub>)
| ^M6
|22/13
| upmajor 6th
| ^B
|-
|-
| | 43
| 43
| | 938.182
| 938.2
|994.473
| 12/7
|1200.873 (4B0.DF7<sub>16</sub>)
| ^^M6
|12/7
| dupmajor 6th
| ^^B
|-
|-
| | 44
| 44
| | 960
| 960.0
|1017.6
| 7/4
|1228.8 (4CC.CCD<sub>16</sub>)
| vvm7
|7/4, 26/15
| dudminor 7th
| vvC
|-
|-
| | 45
| 45
| | 981.818
| 981.8
|1040.727
| 30/17
|1256.727 (4E8.B99<sub>16</sub>)
| vm7
|30/17
| downminor 7th
| vC
|-
|-
| | 46
| 46
| | 1003.636
| 1003.6
|1063.8545
| 16/9, ''9/5''
|1284.6545 (504.A79<sub>16</sub>)
| m7
|16/9, 9/5
| minor 7th
| C
|-
|-
| | 47
| 47
| | 1025.4545
| 1025.5
|1058.982
|  
|1312.582 (520.94F<sub>16</sub>)
| ^m7
|''9/5'', 20/11
| upminor 7th
| ^C
|-
|-
| | 48
| 48
| | 1047.273
| 1047.3
|1110.109
| 11/6, ''20/11''
|1340.509 (53C.816<sub>16</sub>)
| ~7
|11/6, 24/13
| mid 7th
| ^^C
|-
|-
| | 49
| 49
| | 1069.091
| 1069.1
|1133.236
| 13/7, 24/13
|1368.436 (558.6FB<sub>16</sub>)
| vM7
|''24/13'', 13/7
| downmajor 7th
| vC#
|-
|-
| | 50
| 50
| | 1090.909
| 1090.9
|1156.364
| 15/8, ''32/17''
|1396.364 (574.5D18<sub>16</sub>)
| M7
|15/8
| major 7th
| C#
|-
|-
| | 51
| 51
| | 1112.727
| 1112.7
|1179.491
| 40/21, ''17/9'', ''48/25''
|1424.291 (590.4A7<sub>16</sub>)
| ^M7
|40/21, 48/25
| upmajor 7th
| ^C#
|-
|-
| | 52
| 52
| | 1134.5455
| 1134.5
|1202.618
| 56/27
|1452.218 (FAC.37D<sub>16</sub>)
| ^^M7
|56/27, ''48/25''
| dupmajor 7th
| ^^C#
|-
|-
| | 53
| 53
| | 1156.364
| 1156.4
|1225.7455
| 35/18, ''63/32''
|1480.1455 (FC8.253<sub>16</sub>)
| vv8
|35/18
| dud 8ve
| vvD
|-
|-
| | 54
| 54
| | 1178.182
| 1178.2
|1248.873
| 128/65, 77/39, 196/99, ''125/64''
|1508.073 (FE4.12A<sub>16</sub>)
| v8
|125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81''
| down 8ve
| vD
|-
|-
| | 55
| 55
| | 1200
| 1200.0
|1272
| 2/1
|1536 (600<sub>16</sub>)
| P8
|2/1
| perfect 8ve
| D
|}
|}
<nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic''
== Notation ==
=== Ups and downs notation ===
55edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}
[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}
=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:55-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 615 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Evo_Sagittal.svg]]
</imagemap>
==== Revo flavor ====
<imagemap>
File:55-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Revo_Sagittal.svg]]
</imagemap>


==Selected just intervals by error==
==== Evo-SZ flavor ====
The following table shows how [[Just-24|some prominent just intervals]] are represented in 55edo (ordered by absolute error).
<imagemap>
File:55-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>


{| class="wikitable"
=== 31-tone subset ===
The 31-out-of-55edo subset can be notated entirely with the standard notation of 7 each of naturals/sharps/flats, and 5 each of doublesharps/doubleflats, as a 31-tone chain-of-5ths from Gbb to Ax.
 
[[File:Monzo55Notation.jpeg|400px|frameless|alt=Diagram of 31-tone subset of 55edo using plain Western notation, by Joe Monzo.|Diagram of 31-tone subset of 55edo using plain Western notation, by [[Joe Monzo]].]]
 
== Approximation to JI ==
[[File:55ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 55edo]]
 
=== Selected just intervals by error ===
{{Q-odd-limit intervals|55}}
{{Q-odd-limit intervals|55.05|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55d val mapping}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
! | Interval, complement
! rowspan="2" | [[Subgroup]]
! | Error (abs., in [[cent|cents]])
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
! [[TE error|Absolute]] (¢)
| style="text-align:center;" | 1.280
! [[TE simple badness|Relative]] (%)
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| 2.3
| style="text-align:center;" | 1.683
| {{monzo| -87 55 }}
| {{mapping| 55 87 }}
| +1.31
| 1.19
| 7.21
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| 2.3.5
| style="text-align:center;" | 2.090
| 81/80, {{monzo| 31 1 -14 }}
| {{mapping| 55 87 128 }}
| −0.13
| 2.10
| 9.63
|}
 
=== Uniform maps ===
{{Uniform map|edo=55}}
 
=== Commas ===
{{Todo|cleanup|inline=true}}
 
'''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{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
 
'''11-limit commas''': 59049/58564, 74088/73205, 46656/46585, 21609/21296, 12005/11979, 19683/19360, [[243/242]], 3087/3025, 5488/5445, 19683/19250, 1944/1925, 45927/45056, 2835/2816, 35721/34375, 7056/6875, 12544/12375, 7203/7040, 2401/2376, 24057/24010, 72171/70000, 891/875, [[176/175]], 2079/2048, [[385/384]], 3234/3125, 17248/16875, 26411/25600, 26411/2592, 26411/262404, 88209/87808, 30976/30625, 3267/3200, [[121/120]], 81312/78125, 41503/40000, 41503/40500, 35937/35000, 2662/2625, 42592/42525, 83853/81920, 9317/9216, 65219/62500, 43923/43904, 14641/14400, [[14641/14580]]
 
'''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
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
! Periods<br>per 8ve
| style="text-align:center;" |2.611
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| 1
| style="text-align:center;" | 2.640
| 6\55
| 130.9
| 14/13
| [[Twothirdtonic]] (55f)
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
|1
| style="text-align:center;" | 2.963
|8\55
|174.5
|[[10/9]]~[[11/10]]
|[[Tetracot]] (55c)
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| 1
| style="text-align:center;" | 3.773
| 16\55
| 349.1
| 11/9
| [[Mohaha]]
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| 1
| style="text-align:center;" |3.890
| 23\55
| 501.8
| 4/3
| [[Meantone]] (55d)
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 1
| style="text-align:center;" | 3.968
| 26\55
| 567.3
| 7/5
| [[Liese]] (55)
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 1
| style="text-align:center;" | 5.053
| 27\55
| 589.1
| 45/32
| [[Untriton]] (55d) / [[aufo]] (55)
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| 5
| style="text-align:center;" |5.573
| 17\55<br>(5\55)
| 370.9<br>(109.1)
| 99/80<br>(16/15)
| [[Quintosec]]
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 11
| style="text-align:center;" | 5.863
| 23\55<br>(3\55)
|-
| 501.8<br>(65.5)
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 4/3<br>(36/35)
| style="text-align:center;" | 6.414
| [[Hendecatonic]] (55)
|-
| 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
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Scales ==
; Subsets of twothirdtonic[37]
* Undecimal otonal-like pentatonic: 17 8 7 12 11
; Subsets of hendecatonic[33]
* Septimal pentatonic-like: 10 13 9 13 10
* Septimal minor blues-like: 13 10 4 5 13 10
* Septimal heptatonic blues-like: 13 10 4 5 8 5 10
; Others
* Sakura-like scale containing [[phi]]: 9 6 18 5 17
* Quasi-[[equiheptatonic]] scale: 8 8 7 9 7 9 7
== Instruments ==
* [[Lumatone mapping for 55edo]]
== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=xoCNOIsjfeU "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by [[Claudi Meneghin]] (2024)
* [https://www.youtube.com/watch?v=OkRVNo19guo "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025)
* [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by [[Claudi Meneghin]] (2023)
* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
; {{W|Georg Frideric Handel}}
* [https://www.youtube.com/watch?v=rDvKPuzsno8 ''Fugue'' from "Suite in E minor", HWV 429] (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025)
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] (2024)
; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', KV 401] (1782) – rendered by Francium (2023)
* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', KV 540] (1788) – rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])
* [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 ''Mozart's Gigue KV 574, Arranged for Fortepiano (55-edo)''] – rendered by [[Claudi Meneghin]] (2025)
; {{W|Keiichi Okabe}}
* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) – rendered by MortisTheneRd (2024)
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025)
* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025)
; [[James Kukula]]
* ''[https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg 55edo Melted Syntonic]'' (2025)
; [[Budjarn Lambeth]]
* ''[https://www.youtube.com/watch?v=9c5MtrZFNhA Improvisation One in 55edo]'' (2025)
* ''[https://www.youtube.com/watch?v=ggFGUn1Ya2A Improvisation Two in 55edo]'' (2025)
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020)
* [https://www.youtube.com/watch?v=rAbbvyotIr4 ''Canon at the Diatonic Semitone on an Ancient Lombard Theme''] (2021)
* [https://www.youtube.com/watch?v=hCUIx1RzvEk ''Chacony "Lament & Deception"'' for Two Violins and Cello] (2021), [https://www.youtube.com/watch?v=abJP4euMlsg for Baroque Wind Ensemble] (2023)
* [https://www.youtube.com/watch?v=9zfWeO0eJdA Fantasy "Almost a Fugue" on a Theme by Giuliani, for String Quartet] (2021)
* [https://www.youtube.com/watch?v=jOiub14Cskw ''Double Fugue on "Old McDonald" + "Shave & a Haircut"''] (2024)
; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere Road Trip to Nowhere]'' (2021)
* ''[https://soundcloud.com/morphosyntax-1/migration Migration]'' (2025)


[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])
== External links ==
* ''[http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone]'' (containing another listening example) on [[Tonalsoft Encyclopedia]]


[http://tonalsoft.com/monzo/55edo/55edo.aspx "Mozart's tuning: 55edo"] (containing another listening example) in the [[tonalsoft_encyclopedia|tonalsoft encyclopedia]]
== References ==
<references />


[[Category:55edo]]
[[Category:Meantone]]
[[Category:edo]]
[[Category:Historical]]
[[Category:intervals]]
[[Category:Listen]]
[[Category:meantone]]
[[Category:theory]]
Retrieved from "https://en.xen.wiki/w/55edo"