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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, <31 1 -14|
== 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 26411/25920 26411/26244 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
| | Ratios it approximates
! Approximate ratios
! colspan="3" | [[Ups and downs notation]]
|-
|-
| | 0
| 0
| | 0
| 0.0
| | 1/1
| 1/1
| P1
| perfect 1sn
| D
|-
|-
| | 1
| 1
| | 21.818
| 21.8
| | 128/125
| 65/64, 78/77, 99/98, ''128/125''
| ^1
| up 1sn
| ^D
|-
|-
| | 2
| 2
| | 43.636
| 43.6
| |  
| 36/35, ''64/63''
| ^^1
| dup 1sn
| ^^D
|-
|-
| | 3
| 3
| | 65.455
| 65.5
| |  
| 28/27
| vvm2
| dudminor 2nd
| vvEb
|-
|-
| | 4
| 4
| | 87.273
| 87.3
| | 25/24
| 21/20, ''18/17'', ''25/24''
| vm2
| downminor 2nd
| vEb
|-
|-
| | 5
| 5
| | 109.091
| 109.1
| | 16/15
| 16/15, 17/16
| m2
| minor 2nd
| Eb
|-
|-
| | 6
| 6
| | 130.909
| 130.9
| |
| 13/12, 14/13
| ^m2
| upminor 2nd
| ^Eb
|-
|-
| | 7
| 7
| | 152.727
| 152.7
| |
| 12/11, ''11/10''
| ~2
| mid 2nd
| vvE
|-
|-
| | 8
| 8
| | 174.545
| 174.5
| |
|
| vM2
| downmajor 2nd
| vE
|-
|-
| | 9
| 9
| | 196.364
| 196.4
| | 9/8, 10/9
| 9/8, ''10/9''
| M2
| major 2nd
| E
|-
|-
| | 10
| 10
| | 218.182
| 218.2
| 17/15
| ^M2
| upmajor 2nd
| ^E
|-
|-
| | 11
| 11
| | 240.000
| 240.0
| 8/7
| ^^M2
| dupmajor 2nd
| ^^E
|-
|-
| | 12
| 12
| | 261.818
| 261.8
| 7/6
| vvm3
| dudminor 3rd
| vvF
|-
|-
| | 13
| 13
| | 283.636
| 283.6
| 13/11
| vm3
| downminor 3rd
| vF
|-
|-
| | 14
| 14
| | 305.455
| 305.5
| 6/5
| m3
| minor 3rd
| F
|-
|-
| | 15
| 15
| | 327.273
| 327.3
|
| ^m3
| upminor 3rd
| ^F
|-
|-
| | 16
| 16
| | 349.091
| 349.1
| 11/9, 27/22
| ~3
| mid 3rd
| ^^F
|-
|-
| | 17
| 17
| | 370.909
| 370.9
| 26/21, ''16/13''
| vM3
| downmajor 3rd
| vF#
|-
|-
| | 18
| 18
| | 392.727
| 392.7
| 5/4
| M3
| major 3rd
| F#
|-
|-
| | 19
| 19
| | 414.545
| 414.5
| 14/11
| ^M3
| upmajor 3rd
| ^F#
|-
|-
| | 20
| 20
| | 436.364
| 436.4
| 9/7
| ^^M3
| dupmajor 3rd
| ^^F#
|-
|-
| | 21
| 21
| | 458.182
| 458.2
| ''21/16''
| vv4
| dud 4th
| vvG
|-
|-
| | 22
| 22
| | 480.000
| 480.0
|
| v4
| down 4th
| vG
|-
|-
| | 23
| 23
| | 501.818
| 501.8
| 4/3, ''27/20''
| P4
| perfect 4th
| G
|-
|-
| | 24
| 24
| | 523.636
| 523.6
|
| ^4
| up 4th
| ^G
|-
|-
| | 25
| 25
| | 545.455
| 545.5
| 11/8, 15/11
| ~4
| mid 4th
| ^^G
|-
|-
| | 26
| 26
| | 567.273
| 567.3
| [[7/5]], [[18/13]]
| vA4
| downaug 4th
| vG#
|-
|-
| | 27
| 27
| | 589.091
| 589.1
| 24/17
| A4, vd5
| aug 4th, downdim 5th
| G#, vAb
|-
|-
| | 28
| 28
| | 610.909
| 610.9
| 17/12
| ^A4, d5
| upaug 4th, dim 5th
| ^G#, Ab
|-
|-
| | 29
| 29
| | 632.727
| 632.7
| [[10/7]], [[13/9]]
| ^d5
| updim 5th
| ^Ab
|-
|-
| | 30
| 30
| | 654.545
| 654.5
| 16/11, 22/15
| ~5
| mid 5th
| vvA
|-
|-
| | 31
| 31
| | 676.364
| 676.4
|
| v5
| down 5th
| vA
|-
|-
| | 32
| 32
| | 698.182
| 698.2
| 3/2, ''40/27''
| P5
| perfect 5th
| A
|-
|-
| | 33
| 33
| | 720.000
| 720.0
|
| ^5
| up 5th
| ^A
|-
|-
| | 34
| 34
| | 741.818
| 741.8
| ''32/21''
| ^^5
| dup 5th
| ^^A
|-
|-
| | 35
| 35
| | 763.636
| 763.6
| 14/9
| vvm6
| dudminor 6th
| vvBb
|-
|-
| | 36
| 36
| | 785.455
| 785.5
| 11/7
| vm6
| downminor 6th
| vBb
|-
|-
| | 37
| 37
| | 807.273
| 807.3
| 8/5
| m6
| minor 6th
| Bb
|-
|-
| | 38
| 38
| | 829.091
| 829.1
| 21/13, ''13/8''
| ^m6
| upminor 6th
| ^Bb
|-
|-
| | 39
| 39
| | 850.909
| 850.9
| 18/11, 44/27
| ~6
| mid 6th
| vvB
|-
|-
| | 40
| 40
| | 872.727
| 872.7
|
| vM6
| downmajor 6th
| vB
|-
|-
| | 41
| 41
| | 894.545
| 894.5
| 5/3
| M6
| major 6th
| B
|-
|-
| | 42
| 42
| | 916.364
| 916.4
| 22/13
| ^M6
| upmajor 6th
| ^B
|-
|-
| | 43
| 43
| | 938.182
| 938.2
| 12/7
| ^^M6
| dupmajor 6th
| ^^B
|-
|-
| | 44
| 44
| | 960.000
| 960.0
| 7/4
| vvm7
| dudminor 7th
| vvC
|-
|-
| | 45
| 45
| | 981.818
| 981.8
| 30/17
| vm7
| downminor 7th
| vC
|-
|-
| | 46
| 46
| | 1003.636
| 1003.6
| 16/9, ''9/5''
| m7
| minor 7th
| C
|-
|-
| | 47
| 47
| | 1025.455
| 1025.5
|
| ^m7
| upminor 7th
| ^C
|-
|-
| | 48
| 48
| | 1047.273
| 1047.3
| 11/6, ''20/11''
| ~7
| mid 7th
| ^^C
|-
|-
| | 49
| 49
| | 1069.091
| 1069.1
| 13/7, 24/13
| vM7
| downmajor 7th
| vC#
|-
|-
| | 50
| 50
| | 1090.909
| 1090.9
| 15/8, ''32/17''
| M7
| major 7th
| C#
|-
|-
| | 51
| 51
| | 1112.727
| 1112.7
| 40/21, ''17/9'', ''48/25''
| ^M7
| upmajor 7th
| ^C#
|-
|-
| | 52
| 52
| | 1134.545
| 1134.5
| 56/27
| ^^M7
| dupmajor 7th
| ^^C#
|-
|-
| | 53
| 53
| | 1156.364
| 1156.4
| 35/18, ''63/32''
| vv8
| dud 8ve
| vvD
|-
|-
| | 54
| 54
| | 1178.182
| 1178.2
| 128/65, 77/39, 196/99, ''125/64''
| v8
| down 8ve
| vD
|-
|-
| | 55
| 55
| | 1200.000
| 1200.0
| 2/1
| P8
| 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;" | [[16/15|16/15]], [[15/8|15/8]]
! Periods<br>per 8ve
| style="text-align:center;" | 2.640
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| 1
| style="text-align:center;" | 2.963
| 6\55
| 130.9
| 14/13
| [[Twothirdtonic]] (55f)
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
|1
| style="text-align:center;" | 3.773
|8\55
|174.5
|[[10/9]]~[[11/10]]
|[[Tetracot]] (55c)
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 1
| style="text-align:center;" | 3.968
| 16\55
| 349.1
| 11/9
| [[Mohaha]]
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 1
| style="text-align:center;" | 5.053
| 23\55
| 501.8
| 4/3
| [[Meantone]] (55d)
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 1
| style="text-align:center;" | 5.863
| 26\55
| 567.3
| 7/5
| [[Liese]] (55)
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 1
| style="text-align:center;" | 6.414
| 27\55
| 589.1
| 45/32
| [[Untriton]] (55d) / [[aufo]] (55)
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| 5
| style="text-align:center;" | 7.546
| 17\55<br>(5\55)
| 370.9<br>(109.1)
| 99/80<br>(16/15)
| [[Quintosec]]
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| 11
| style="text-align:center;" | 7.741
| 23\55<br>(3\55)
|-
| 501.8<br>(65.5)
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| 4/3<br>(36/35)
| style="text-align:center;" | 8.504
| [[Hendecatonic]] (55)
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 8.826
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 10.187
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 10.381
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 11.466
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 12.277
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 13.960
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 14.155
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 15.239
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 16.245
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 17.928
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 19.207
|}
|}
<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]]

Latest revision as of 10:46, 19 August 2025

← 54edo 55edo 56edo →
Prime factorization 5 × 11
Step size 21.8182 ¢ 
Fifth 32\55 (698.182 ¢)
Semitones (A1:m2) 4:5 (87.27 ¢ : 109.1 ¢)
Consistency limit 5
Distinct consistency limit 5

55 equal divisions of the octave (abbreviated 55edo or 55ed2), also called 55-tone equal temperament (55tet) or 55 equal temperament (55et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 55 equal parts of about 21.8 ¢ each. Each step represents a frequency ratio of 21/55, or the 55th root of 2.

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). Telemann suggested it as a theoretical basis for analyzing the intervals of meantone. Leopold and 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.[1] It can also be used for Mohajira and Liese temperaments. It also supports an extremely sharp tuning of Huygens/undecimal meantone using the 55de val, meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.

Odd harmonics

Approximation of odd harmonics in 55edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -3.77 +6.41 -8.83 -7.55 -5.86 +10.38 +2.64 +4.14 +7.94 +9.22 +4.45
Relative (%) -17.3 +29.4 -40.5 -34.6 -26.9 +47.6 +12.1 +19.0 +36.4 +42.3 +20.4
Steps
(reduced) 		87
(32) 		128
(18) 		154
(44) 		174
(9) 		190
(25) 		204
(39) 		215
(50) 		225
(5) 		234
(14) 		242
(22) 		249
(29)

Subsets and supersets

Since 55 factors into 5 × 11, 55edo contains 5edo and 11edo as its subsets.

Intervals

# Cents Approximate ratios Ups and downs notation
0 0.0 1/1 P1 perfect 1sn D
1 21.8 65/64, 78/77, 99/98, 128/125 ^1 up 1sn ^D
2 43.6 36/35, 64/63 ^^1 dup 1sn ^^D
3 65.5 28/27 vvm2 dudminor 2nd vvEb
4 87.3 21/20, 18/17, 25/24 vm2 downminor 2nd vEb
5 109.1 16/15, 17/16 m2 minor 2nd Eb
6 130.9 13/12, 14/13 ^m2 upminor 2nd ^Eb
7 152.7 12/11, 11/10 ~2 mid 2nd vvE
8 174.5 vM2 downmajor 2nd vE
9 196.4 9/8, 10/9 M2 major 2nd E
10 218.2 17/15 ^M2 upmajor 2nd ^E
11 240.0 8/7 ^^M2 dupmajor 2nd ^^E
12 261.8 7/6 vvm3 dudminor 3rd vvF
13 283.6 13/11 vm3 downminor 3rd vF
14 305.5 6/5 m3 minor 3rd F
15 327.3 ^m3 upminor 3rd ^F
16 349.1 11/9, 27/22 ~3 mid 3rd ^^F
17 370.9 26/21, 16/13 vM3 downmajor 3rd vF#
18 392.7 5/4 M3 major 3rd F#
19 414.5 14/11 ^M3 upmajor 3rd ^F#
20 436.4 9/7 ^^M3 dupmajor 3rd ^^F#
21 458.2 21/16 vv4 dud 4th vvG
22 480.0 v4 down 4th vG
23 501.8 4/3, 27/20 P4 perfect 4th G
24 523.6 ^4 up 4th ^G
25 545.5 11/8, 15/11 ~4 mid 4th ^^G
26 567.3 7/5, 18/13 vA4 downaug 4th vG#
27 589.1 24/17 A4, vd5 aug 4th, downdim 5th G#, vAb
28 610.9 17/12 ^A4, d5 upaug 4th, dim 5th ^G#, Ab
29 632.7 10/7, 13/9 ^d5 updim 5th ^Ab
30 654.5 16/11, 22/15 ~5 mid 5th vvA
31 676.4 v5 down 5th vA
32 698.2 3/2, 40/27 P5 perfect 5th A
33 720.0 ^5 up 5th ^A
34 741.8 32/21 ^^5 dup 5th ^^A
35 763.6 14/9 vvm6 dudminor 6th vvBb
36 785.5 11/7 vm6 downminor 6th vBb
37 807.3 8/5 m6 minor 6th Bb
38 829.1 21/13, 13/8 ^m6 upminor 6th ^Bb
39 850.9 18/11, 44/27 ~6 mid 6th vvB
40 872.7 vM6 downmajor 6th vB
41 894.5 5/3 M6 major 6th B
42 916.4 22/13 ^M6 upmajor 6th ^B
43 938.2 12/7 ^^M6 dupmajor 6th ^^B
44 960.0 7/4 vvm7 dudminor 7th vvC
45 981.8 30/17 vm7 downminor 7th vC
46 1003.6 16/9, 9/5 m7 minor 7th C
47 1025.5 ^m7 upminor 7th ^C
48 1047.3 11/6, 20/11 ~7 mid 7th ^^C
49 1069.1 13/7, 24/13 vM7 downmajor 7th vC#
50 1090.9 15/8, 32/17 M7 major 7th C#
51 1112.7 40/21, 17/9, 48/25 ^M7 upmajor 7th ^C#
52 1134.5 56/27 ^^M7 dupmajor 7th ^^C#
53 1156.4 35/18, 63/32 vv8 dud 8ve vvD
54 1178.2 128/65, 77/39, 196/99, 125/64 v8 down 8ve vD
55 1200.0 2/1 P8 perfect 8ve D

* 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.

Step offset 0 1 2 3 4 5 6 7 8 9
Sharp symbol               
Flat symbol
                

Alternative symbols for ups and downs notation uses sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:

Step offset 0 1 2 3 4 5 6 7 8 9
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 
 
 
 		 

Sagittal notation

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation896/89133/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation896/89133/32

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation896/89133/32

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.

Diagram of 31-tone subset of 55edo using plain Western notation, by Joe Monzo.

Approximation to JI

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Selected 19-limit intervals approximated in 55edo

Selected just intervals by error

The following tables show how 15-odd-limit intervals are represented in 55edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 55edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 5.9
11/9, 18/11 1.683 7.7
11/6, 12/11 2.090 9.6
13/7, 14/13 2.611 12.0
15/8, 16/15 2.640 12.1
11/7, 14/11 2.963 13.6
3/2, 4/3 3.773 17.3
13/9, 18/13 3.890 17.8
13/10, 20/13 3.968 18.2
7/6, 12/7 5.053 23.2
13/11, 22/13 5.573 25.5
11/8, 16/11 5.863 26.9
5/4, 8/5 6.414 29.4
7/5, 10/7 6.579 30.2
9/8, 16/9 7.546 34.6
13/12, 24/13 7.664 35.1
15/13, 26/15 7.741 35.5
9/5, 10/9 7.858 36.0
15/11, 22/15 8.504 39.0
7/4, 8/7 8.826 40.5
11/10, 20/11 9.541 43.7
5/3, 6/5 10.187 46.7
15/14, 28/15 10.352 47.4
13/8, 16/13 10.381 47.6
15-odd-limit intervals in 55edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 5.9
11/9, 18/11 1.683 7.7
11/6, 12/11 2.090 9.6
15/8, 16/15 2.640 12.1
11/7, 14/11 2.963 13.6
3/2, 4/3 3.773 17.3
13/10, 20/13 3.968 18.2
7/6, 12/7 5.053 23.2
11/8, 16/11 5.863 26.9
5/4, 8/5 6.414 29.4
9/8, 16/9 7.546 34.6
15/13, 26/15 7.741 35.5
15/11, 22/15 8.504 39.0
7/4, 8/7 8.826 40.5
5/3, 6/5 10.187 46.7
13/8, 16/13 10.381 47.6
15/14, 28/15 11.466 52.6
11/10, 20/11 12.277 56.3
9/5, 10/9 13.960 64.0
13/12, 24/13 14.155 64.9
7/5, 10/7 15.239 69.8
13/11, 22/13 16.245 74.5
13/9, 18/13 17.928 82.2
13/7, 14/13 19.207 88.0
15-odd-limit intervals by 55d val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/9, 18/11 1.683 7.7
11/6, 12/11 2.090 9.6
13/7, 14/13 2.611 12.0
15/8, 16/15 2.640 12.1
3/2, 4/3 3.773 17.3
13/10, 20/13 3.968 18.2
11/8, 16/11 5.863 26.9
5/4, 8/5 6.414 29.4
7/5, 10/7 6.579 30.2
9/8, 16/9 7.546 34.6
15/13, 26/15 7.741 35.5
15/11, 22/15 8.504 39.0
5/3, 6/5 10.187 46.7
15/14, 28/15 10.352 47.4
13/8, 16/13 10.381 47.6
11/10, 20/11 12.277 56.3
7/4, 8/7 12.992 59.5
9/5, 10/9 13.960 64.0
13/12, 24/13 14.155 64.9
13/11, 22/13 16.245 74.5
7/6, 12/7 16.765 76.8
13/9, 18/13 17.928 82.2
11/7, 14/11 18.856 86.4
9/7, 14/9 20.539 94.1

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-87 55⟩ [⟨55 87]] +1.31 1.19 7.21
2.3.5 81/80, [31 1 -14⟩ [⟨55 87 128]] −0.13 2.10 9.63

Uniform maps

13-limit uniform maps between 54.8 and 55.2
Min. size Max. size Wart notation Map
54.7778 54.9113 55cf ⟨55 87 127 154 190 203]
54.9113 54.9935 55f ⟨55 87 128 154 190 203]
54.9935 55.0340 55 ⟨55 87 128 154 190 204]
55.0340 55.0668 55d ⟨55 87 128 155 190 204]
55.0668 55.2064 55de ⟨55 87 128 155 191 204]

Commas

Todo: cleanup

5-limit commas: 81/80, [47 -15 -10⟩, [31 1 -14⟩, [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, 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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 6\55 130.9 14/13 Twothirdtonic (55f)
1 8\55 174.5 10/9~11/10 Tetracot (55c)
1 16\55 349.1 11/9 Mohaha
1 23\55 501.8 4/3 Meantone (55d)
1 26\55 567.3 7/5 Liese (55)
1 27\55 589.1 45/32 Untriton (55d) / aufo (55)
5 17\55
(5\55) 		370.9
(109.1) 		99/80
(16/15) 		Quintosec
11 23\55
(3\55) 		501.8
(65.5) 		4/3
(36/35) 		Hendecatonic (55)

* Octave-reduced form, reduced to the first half-octave, and 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

Music

Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns
Georg Frideric Handel
Scott Joplin
Wolfgang Amadeus Mozart
Keiichi Okabe

21st century

Bryan Deister
James Kukula
Budjarn Lambeth
Claudi Meneghin
Herman Miller

External links

References

  1. ↑ 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) doi.org/10.2307/831219, https://www.jstor.org/stable/831219
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