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'''50edo''' divides the [[octave]] into 50 equal parts of precisely 24 [[cent|cents]] each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice; it is a somewhat sharp approximation of [[2/7-comma meantone]] (and is almost exactly 5/18-comma meantone). In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo where the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 11/9 and 18/11.


In the [[5-limit]], 50edo tempers out [[81/80]], making it a [[meantone]] system, and in that capacity has historically has drawn some notice. In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is the highest edo which maps [[9/8]] and [[10/9]] to the same interval in a [[consistent]] manner, with two stacked fifths falling almost precisely in the middle of the two.
It tempers out [[126/125]], [[225/224]] and [[3136/3125]] in the [[7-limit]], indicating it [[support]]s septimal meantone; [[245/242]], [[385/384]] and [[540/539]] in the [[11-limit]] and [[105/104]], [[144/143]] and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension [[meanpop]], it can be used to advantage for the [[coblack]] temperament (15 & 50), and provides the optimal patent val for 11- and 13-limit [[Meantone family #Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], {{monzo| 23 6 -14 }}, so that in 50edo seven chromatic semitones stack to a perfect fourth. By comparison, this gives a perfect fifth in 12edo, a doubly diminished fifth in 31edo, and a diminished fourth in 19edo.


50edo tempers out 126/125, 225/224 and 3136/3125 in the [[7-limit]], indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the [[11-limit]] and 105/104, 144/143 and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the [[Starling temperaments #Coblack temperament|coblack (15&50) temperament]], and provides the optimal patent val for 11 and 13 limit [[Meantone_family #Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], {{monzo|23 6 -14}};, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.
=== Odd harmonics ===
{{Harmonics in equal|50|columns=15}}


== Relations ==
=== Relations ===
The 50edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "Golden Tone System" ([[Das Goldene Tonsystem]]) of [[Thorvald Kornerup]] (and similarly as the next step from 31edo in [[Joseph Yasser]]'s "[http://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y A Theory of Evolving Tonality]").
The 50edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "[[Golden meantone|Golden Tone System]]" ([[Das Goldene Tonsystem]]) of [[Thorvald Kornerup]] (and similarly as the next step from 31edo in [[Joseph Yasser]]'s "[https://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y A Theory of Evolving Tonality]").


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! #
! #
! Cents
! Cents
! Ratios*
! Ratios<ref group="note">{{sg|13-limit}}</ref>
! Generator for*
! colspan="3" | [[Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and vvd2)
|-
|-
| 0
| 0
| 0
| 0
| 1/1
| 1/1
|  
| Perfect 1sn
| P1
| D
|-
|-
| 1
| 1
| 24
| 24
| 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168
| 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168
| [[Hemimean_clan#Sengagen|Sengagen]]
| Up 1sn
| ^1
| ^D
|-
|-
| 2
| 2
| 48
| 48
| 33/32, 36/35, 50/49, 55/54, 64/63
| 27/26, 33/32, 36/35, 50/49, 55/54, 64/63
|  
| Dim 2nd, Downaug 1sn
| d2, vA1
| Ebb, vD#
|-
|-
| 3
| 3
| 72
| 72
| 21/20, 25/24, 26/25, 27/26, 28/27
| 21/20, 25/24, 26/25, 28/27
| [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [[Starling temperaments #Coblack temperament|Coblack]] (5/oct)
| Aug 1sn, Updim 2nd
| A1, ^d2
| D#, ^Ebb
|-
|-
| 4
| 4
| 96
| 96
| 22/21
| 22/21
| [[Meantone_family#Injera|Injera]] (50d val, 2/oct)
| Downminor 2nd
| vm2
| vEb
|-
|-
| 5
| 5
| 120
| 120
| 16/15, 15/14, 14/13
| 16/15, 15/14, 14/13
|  
| Minor 2nd
| m2
| Eb
|-
|-
| 6
| 6
| 144
| 144
| 13/12, 12/11
| 13/12, 12/11
|  
| Upminor 2nd
| ^m2
| ^Eb
|-
|-
| 7
| 7
| 168
| 168
| 11/10
| 11/10
|  
| Downmajor 2nd
| vM2
| vE
|-
|-
| 8
| 8
| 192
| 192
| 9/8, 10/9
| 9/8, 10/9
|  
| Major 2nd
| M2
| E
|-
|-
| 9
| 9
| 216
| 216
| 25/22
| 25/22
| [http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&limit=2.3.5.11.13 Tremka], [[Subgroup_temperaments#x2.9.7.11-Machine|Machine]] (50b val)
| Upmajor 2nd
| ^M2
| ^E
|-
|-
| 10
| 10
| 240
| 240
| 8/7, 15/13
| 8/7, 15/13
|  
| Downaug 2nd, Dim 3rd
| vA2, d3
| vE#, Fb
|-
|-
| 11
| 11
| 264
| 264
| 7/6
| 7/6
| [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]
| Updim 3rd, Aug 2nd
| ^d3, A2
| ^Fb, E#
|-
|-
| 12
| 12
| 288
| 288
| 13/11
| 13/11
|  
| Downminor 3rd
| vm3
| vF
|-
|-
| 13
| 13
| 312
| 312
| 6/5
| 6/5
| [[Oolong]]
| Minor 3rd
| m3
| F
|-
|-
| 14
| 14
| 336
| 336
| 27/22, 39/32, 40/33, 49/40
| 27/22, 39/32, 40/33, 49/40
|  
| Upminor 3rd
| ^m3
| ^F
|-
|-
| 15
| 15
| 360
| 360
| 16/13, 11/9
| 16/13, 11/9
|  
| Downmajor 3rd
| vM3
| vF#
|-
|-
| 16
| 16
| 384
| 384
| 5/4
| 5/4
| [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)
| Major 3rd
| M3
| F#
|-
|-
| 17
| 17
| 408
| 408
| 14/11
| 14/11
| [[Ditonmic_family|Ditonic]]
| Upmajor 3rd
| ^M3
| ^F#
|-
|-
| 18
| 18
| 432
| 432
| 9/7
| 9/7
| [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)
| Downaug 3rd, Dim 4th
| vA3, d4
| vFx, Gb
|-
|-
| 19
| 19
| 456
| 456
| 13/10
| 13/10
| [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)
| Updim 4th, Aug 3rd
| A3, ^d4
| ^Gb, Fx
|-
|-
| 20
| 20
| 480
| 480
| 33/25, 55/42, 64/49
| 33/25, 55/42, 64/49
|  
| Down 4th
| v4
| vG
|-
|-
| 21
| 21
| 504
| 504
| 4/3
| 4/3
| [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
| Perfect 4th
| P4
| G
|-
|-
| 22
| 22
| 528
| 528
| 15/11
| 15/11
|  
| Up 4th
| ^4
| ^G
|-
|-
| 23
| 23
| 552
| 552
| 11/8, 18/13
| 11/8, 18/13
| [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|Emka]]
| Downaug 4th
| vA4
| vG#
|-
|-
| 24
| 24
| 576
| 576
| 7/5
| 7/5
|  
| Aug 4th
| A4
| G#
|-
|-
| 25
| 25
| 600
| 600
| 63/44, 88/63, 78/55, 55/39
| 63/44, 88/63, 78/55, 55/39
|  
| Upaug 4th, Downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
|-
| 26
| 26
| 624
| 624
| 10/7
| 10/7
|  
| Dim 5th
| d5
| Ab
|-
|-
| 27
| 27
| 648
| 648
| 16/11, 13/9
| 16/11, 13/9
|  
| Updim 5th
| ^d5
| ^Ab
|-
|-
| 28
| 28
| 672
| 672
| 22/15
| 22/15
|  
| Down 5th
| v5
| vA
|-
|-
| 29
| 29
| 696
| 696
| 3/2
| 3/2
|  
| Perfect 5th
| P5
| A
|-
|-
| 30
| 30
| 720
| 720
| 50/33, 84/55, 49/32
| 50/33, 84/55, 49/32
|  
| Up 5th
| ^5
| ^A
|-
|-
| 31
| 31
| 744
| 744
| 20/13
| 20/13
|  
| Downaug 5th, Dim 6th
| vA5, d6
| vA#, Bbb
|-
|-
| 32
| 32
| 768
| 768
| 14/9
| 14/9
|  
| Updim 6th, Aug 5th
| ^d6, A5
| ^Bbb, A#
|-
|-
| 33
| 33
| 792
| 792
| 11/7
| 11/7
|  
| Downminor 6th
| vm6
| vBb
|-
|-
| 34
| 34
| 816
| 816
| 8/5
| 8/5
|  
| Minor 6th
| m6
| Bb
|-
|-
| 35
| 35
| 840
| 840
| 13/8, 18/11
| 13/8, 18/11
|  
| Upminor 6th
| ^m6
| ^Bb
|-
|-
| 36
| 36
| 864
| 864
| 44/27, 64/39, 33/20, 80/49
| 44/27, 64/39, 33/20, 80/49
|  
| Downmajor 6th
| vM6
| vB
|-
|-
| 37
| 37
| 888
| 888
| 5/3
| 5/3
|  
| Major 6th
| M6
| B
|-
|-
| 38
| 38
| 912
| 912
| 22/13
| 22/13
|  
| Upmajor 6th
| ^M6
| ^B
|-
|-
| 39
| 39
| 936
| 936
| 12/7
| 12/7
|  
| Downaug 6th, Dim 7th
| vA6, d7
| vB#, Cb
|-
|-
| 40
| 40
| 960
| 960
| 7/4
| 7/4
|  
| Updim 7th, Aug 6th
| ^d7, A6
| ^Cb, B#
|-
|-
| 41
| 41
| 984
| 984
| 44/25
| 44/25
|  
| Downminor 7th
| vm7
| vC
|-
|-
| 42
| 42
| 1008
| 1008
| 16/9, 9/5
| 16/9, 9/5
|  
| Minor 7th
| m7
| C
|-
|-
| 43
| 43
| 1032
| 1032
| 20/11
| 20/11
|  
| Upminor 7th
| ^m7
| ^C
|-
|-
| 44
| 44
| 1056
| 1056
| 24/13, 11/6
| 24/13, 11/6
|  
| Downmajor 7th
| vM7
| vC#
|-
|-
| 45
| 45
| 1080
| 1080
| 15/8, 28/15, 13/7
| 15/8, 28/15, 13/7
|  
| Major 7th
| M7
| C#
|-
|-
| 46
| 46
| 1104
| 1104
| 21/11
| 21/11
|  
| Upmajor 7th
| ^M7
| ^C#
|-
|-
| 47
| 47
| 1128
| 1128
| 40/21, 48/25, 25/13, 52/27, 27/14
| 40/21, 48/25, 25/13, 27/14
|  
| Downaug 7th, Dim 8ve
| vA7, d8
| vCx, Db
|-
|-
| 48
| 48
| 1152
| 1152
| 64/33, 35/18, 49/25, 108/55, 63/32
| 52/27, 64/33, 35/18, 49/25, 108/55, 63/32
|  
| Updim 8ve, Aug 7th
| ^d8, A7
| ^Db, Cx
|-
|-
| 49
| 49
| 1176
| 1176
| 88/45, 96/49, 55/28, 128/65, 65/33, 77/39, 180/91, 196/99, 99/50, 240/121, 336/169
| 88/45, 96/49, 55/28, 128/65, 65/33, 77/39, 180/91, 196/99, 99/50, 240/121, 336/169
|  
| Down 8ve
| v8
| vD
|-
|-
| 50
| 50
| 1200
| 1200
| 2/1
| 2/1
|  
| Perfect 8ve
| P8
| D
|}
|}
<nowiki>*</nowiki> Using the 13-limit patent val, except as noted.


== Just approximation ==
== Notation ==
=== Selected just intervals ===
=== Ups and downs notation ===
{| class="wikitable center-all"
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
! colspan="2" |
{{sharpness-sharp3a}}
! prime 2
 
! prime 3
Using [[Helmholtz–Ellis]] accidentals, 50edo can also be notated using [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]:
! prime 5
{{Sharpness-sharp3}}
! prime 7
Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.
! prime 11
 
! prime 13
=== Sagittal notation ===
|-
This notation uses the same sagittal sequence as EDOs [[57edo#Sagittal notation|57]], [[64edo#Sagittal notation|64]], and [[71edo#Second-best fifth notation|71b]].
! rowspan="2" |Error
 
! absolute (¢)
==== Evo flavor ====
| 0.0
<imagemap>
| -6.0
File:50-EDO_Evo_Sagittal.svg
| -2.3
desc none
| -8.8
rect 80 0 300 50 [[Sagittal_notation]]
| +0.7
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| -0.5
rect 20 80 160 106 [[1053/1024]]
|-
default [[File:50-EDO_Evo_Sagittal.svg]]
! relative (%)
</imagemap>
| 0.0
 
| -24.8
==== Revo flavor ====
| -9.6
<imagemap>
| -36.8
File:50-EDO_Revo_Sagittal.svg
| +2.8
desc none
| -2.2
rect 80 0 300 50 [[Sagittal_notation]]
|}
rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:50-EDO_Revo_Sagittal.svg]]
</imagemap>
 
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.


==== 15-odd-limit mappings ====
== Approximation to JI ==
The following table shows how [[15-odd-limit intervals]] are represented in 50edo (ordered by absolute error). Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
[[File:50ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 29-limit intervals approximated in 50edo]]


{| class="wikitable center-all"
=== 15-odd-limit interval mappings ===
|+ Direct mapping (even if inconsistent)
{{Q-odd-limit intervals|50|15}}
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[16/13|16/13]], [[13/8|13/8]]'''
| '''0.528'''
|-
| [[15/14|15/14]], [[28/15|28/15]]
| 0.557
|-
| '''[[11/8|11/8]], [[16/11|16/11]]'''
| '''0.682'''
|-
| [[13/11|13/11]], [[22/13|22/13]]
| 1.210
|-
| [[13/10|13/10]], [[20/13|20/13]]
| 1.786
|-
| '''[[5/4|5/4]], [[8/5|8/5]]'''
| '''2.314'''
|-
| [[7/6|7/6]], [[12/7|12/7]]
| 2.871
|-
| [[11/10|11/10]], [[20/11|20/11]]
| 2.996
|-
| [[9/7|9/7]], [[14/9|14/9]]
| 3.084
|-
| [[6/5|6/5]], [[5/3|5/3]]
| 3.641
|-
| [[13/12|13/12]], [[24/13|24/13]]
| 5.427
|-
| '''[[4/3|4/3]], [[3/2|3/2]]'''
| '''5.955'''
|-
| [[7/5|7/5]], [[10/7|10/7]]
| 6.512
|-
| [[12/11|12/11]], [[11/6|11/6]]
| 6.637
|-
| [[15/13|15/13]], [[26/15|26/15]]
| 7.741
|-
| [[16/15|16/15]], [[15/8|15/8]]
| 8.269
|-
| [[14/13|14/13]], [[13/7|13/7]]
| 8.298
|-
| '''[[8/7|8/7]], [[7/4|7/4]]'''
| '''8.826'''
|-
| [[15/11|15/11]], [[22/15|22/15]]
| 8.951
|-
| [[14/11|14/11]], [[11/7|11/7]]
| 9.508
|-
| [[10/9|10/9]], [[9/5|9/5]]
| 9.596
|-
| [[18/13|18/13]], [[13/9|13/9]]
| 11.382
|-
| ''[[11/9|11/9]], [[18/11|18/11]]''
| ''11.408''
|-
| [[9/8|9/8]], [[16/9|16/9]]
| 11.910
|}


{| class="wikitable center-all"
== Regular temperament properties ==
|+Patent val mapping
=== Temperament measures ===
{| class="wikitable center-4 center-5 center-6"
|-
|-
! Interval, complement
! rowspan="2" | [[Subgroup]]
! Error (abs, [[cent|¢]])
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| '''[[16/13|16/13]], [[13/8|13/8]]'''
! [[TE error|Absolute]] (¢)
| '''0.528'''
! [[TE simple badness|Relative]] (%)
|-
|-
| [[15/14|15/14]], [[28/15|28/15]]
| 2.3
| 0.557
| {{monzo| -79 50 }}
| {{mapping| 50 79 }}
| +1.88
| 1.88
| 7.83
|-
|-
| '''[[11/8|11/8]], [[16/11|16/11]]'''
| 2.3.5
| '''0.682'''
| 81/80, {{monzo| -27 -2 13 }}
| {{mapping| 50 79 116 }}
| +1.58
| 1.59
| 6.62
|-
|-
| [[13/11|13/11]], [[22/13|22/13]]
| 2.3.5.7
| 1.210
| 81/80, 126/125, 84035/82944
| {{mapping| 50 79 116 140 }}
| +1.98
| 1.54
| 6.39
|-
|-
| [[13/10|13/10]], [[20/13|20/13]]
| 2.3.5.7.11
| 1.786
| 81/80, 126/125, 245/242, 385/384
|-
| {{mapping| 50 79 116 140 173 }}
| '''[[5/4|5/4]], [[8/5|8/5]]'''
| +1.54
| '''2.314'''
| 1.63
|-
| 6.76
| [[7/6|7/6]], [[12/7|12/7]]
| 2.871
|-
| [[11/10|11/10]], [[20/11|20/11]]
| 2.996
|-
| [[9/7|9/7]], [[14/9|14/9]]
| 3.084
|-
| [[6/5|6/5]], [[5/3|5/3]]
| 3.641
|-
| [[13/12|13/12]], [[24/13|24/13]]
| 5.427
|-
| '''[[4/3|4/3]], [[3/2|3/2]]'''
| '''5.955'''
|-
| [[7/5|7/5]], [[10/7|10/7]]
| 6.512
|-
| [[12/11|12/11]], [[11/6|11/6]]
| 6.637
|-
| [[15/13|15/13]], [[26/15|26/15]]
| 7.741
|-
| [[16/15|16/15]], [[15/8|15/8]]
| 8.269
|-
| [[14/13|14/13]], [[13/7|13/7]]
| 8.298
|-
| '''[[8/7|8/7]], [[7/4|7/4]]'''
| '''8.826'''
|-
| [[15/11|15/11]], [[22/15|22/15]]
| 8.951
|-
| [[14/11|14/11]], [[11/7|11/7]]
| 9.508
|-
| [[10/9|10/9]], [[9/5|9/5]]
| 9.596
|-
| [[18/13|18/13]], [[13/9|13/9]]
| 11.382
|-
| [[9/8|9/8]], [[16/9|16/9]]
| 11.910
|-
| ''[[11/9|11/9]], [[18/11|18/11]]''
| ''12.592''
|}
 
=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 50et.  
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 11-limit
! 13-limit
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5.7.11.13
| +1.88
| 81/80, 105/104, 126/125, 144/143, 245/242
| +1.58
| {{mapping| 50 79 116 140 173 185 }}
| +1.98
| +1.54
| +1.31
| +1.31
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 1.88
| 1.59
| 1.54
| 1.63
| 1.57
| 1.57
|-
! [[TE simple badness|relative]] (%)
| 7.83
| 6.62
| 6.39
| 6.76
| 6.54
| 6.54
|}
|}


== Commas ==
=== Commas ===
50 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val|50 79 116 140 173 185 204 212 226}}, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.
50et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 50 79 116 140 173 185 204 212 226 }}, comma values in cents rounded to 2 decimal places. This list is not all-inclusive, and is based on the interval table from Scala version 2.2.


{| class="wikitable center-all left-3 right-4"
{| class="commatable wikitable center-all left-3 right-4 left-5"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! Ratio
! [[Ratio]]<ref group="note">{{rd}}</ref>
! Monzo
! [[Monzo]]
! Cents
! [[Cent]]s
! Name 1
! Name
! Name 2
|-
| 3
| <abbr title="717897987691852588770249/604462909807314587353088">(20 digits)</abbr>
| {{monzo| -79 50 }}
| 297.75
| 50-comma
|-
|-
| 5
| 5
| 81/80
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| {{monzo| -4 4 -1 }}
| 21.51
| 21.51
| Syntonic comma
| Syntonic comma
| Didymus comma
|-
|-
| 5
| 5
|  
| <abbr title="1220703125/1207959552">(20 digits)</abbr>
| {{Monzo| -27 -2 13 }}
| {{monzo| -27 -2 13 }}
| 18.17
| 18.17
| Ditonma
| [[Ditonma]]
|
|-
|-
| 5
| 5
|  
| [[6115295232/6103515625|(20 digits)]]
| {{Monzo| 23 6 -14 }}
| {{monzo| 23 6 -14 }}
| 3.34
| 3.34
| Vishnu comma
| [[Vishnuzma]]
|
|-
|-
| 7
| 7
| 59049/57344
| [[59049/57344]]
| {{Monzo| -13 10 0 -1 }}
| {{monzo| -13 10 0 -1 }}
| 50.72
| 50.72
| Harrison's comma
| Harrison's comma
|
|-
|-
| 7
| 7
| 126/125
| [[16807/16384]]
| {{Monzo| 1 2 -3 1 }}
| {{monzo| -14 0 0 5}}
| 44.13
| Cloudy comma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Schismean comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Starling comma
| Starling comma
| Small septimal comma
|-
|-
| 7
| 7
| 225/224
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Septimal kleisma
| Marvel comma
| Marvel comma
|-
|-
| 7
| 7
| 3136/3125
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Hemimean
| Hemimean comma
| Middle second comma
|-
|-
| 7
| 7
|  
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
| {{Monzo| 11 -10 -10 10 }}
| {{monzo| 11 -10 -10 10 }}
| 5.57
| 5.57
| Linus
| [[Linus comma]]
|
|-
|-
| 7
| 7
|  
| [[703125/702464|(12 digits)]]
| {{Monzo| -11 2 7 -3 }}
| {{monzo| -11 2 7 -3 }}
| 1.63
| 1.63
| Meter
| [[Meter]]
|
|-
|-
| 7
| 7
|  
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo| -6 -8 2 5 }}
| {{monzo| -6 -8 2 5 }}
| 1.12
| 1.12
| Wizma
| [[Wizma]]
|
|-
|-
| 11
| 11
| 245/242
| [[245/242]]
| {{Monzo| -1 0 1 2 -2 }}
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| 21.33
| Cassacot
| Frostma
|
|-
|-
| 11
| 11
| 385/384
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Keenanisma
| Keenanisma
| Undecimal kleisma
|-
|-
| 11
| 11
| 540/539
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| 3.21
| Swets' comma
| Swetisma
| Swetisma
|-
|-
| 11
| 11
| 4000/3993
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3 }}
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| 3.03
| Wizardharry
| Wizardharry comma
| Undecimal schisma
|-
|-
| 11
| 11
| 9801/9800
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Kalisma
| Kalisma
| Gauss' comma
|-
|-
| 13
| 13
| 105/104
| [[105/104]]
| {{Monzo| -3 1 1 1 0 -1 }}
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| 16.57
| Animist comma
| Animist comma
| Small tridecimal comma
|-
|-
| 13
| 13
| 144/143
| [[144/143]]
| {{Monzo| 4 2 0 0 -1 -1 }}
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| 12.06
| Grossma
| Grossma
|
|-
|-
| 13
| 13
| 196/195
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| {{monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| 8.86
| Mynucuma
| Mynucuma
|
|-
|-
| 13
| 13
| 1188/1183
| [[1188/1183]]
| {{Monzo| 2 3 0 -1 1 -2 }}
| {{monzo| 2 3 0 -1 1 -2 }}
| 7.30
| 7.30
| Kestrel Comma
| Kestrel comma
|  
|-
| 13
| [[31213/31104]]
| {{monzo| -7 -5 0 4 0 1 }}
| 6.06
| Praveensma
|-
|-
| 13
| 13
| 364/363
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| {{monzo| 2 -1 0 1 -2 1 }}
| 4.76
| 4.76
| Gentle comma
| Minor minthma
|
|-
|-
| 13
| 13
| 2200/2197
| [[2200/2197]]
| {{Monzo| 3 0 2 0 1 -3 }}
| {{monzo| 3 0 2 0 1 -3 }}
| 2.36
| 2.36
| Petrma
| Petrma
| Parizek comma
|-
| 17
| [[170/169]]
| {{monzo| 1 0 1 0 0 -2 1 }}
| 10.21
| Major naiadma
|-
| 17
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| Minor naiadma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Semitonisma
|-
| 17
| [[375/374]]
| {{monzo| -1 1 3 0 -1 0 -1 }}
| 4.62
| Ursulisma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1 }}
| 11.35
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1}}
| 10.15
| Malcolmisma
|-
| 19
| [[210/209]]
| {{monzo| 1 1 1 1 -1 0 0 1}}
| 8.26
| Spleen comma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Go comma
|-
| 19
| [[495/494]]
| {{monzo| -1 2 1 0 1 -1 0 -1 }}
| 3.50
| Eulalisma
|-
| 23
| [[507/506]]
| 2.3.11.13.23 {{monzo| -1 1 -1 2 -1 }}
| 3.42
| Laodicisma
|-
|-
| 23
| 23
| 1288/1287
| [[529/528]]
| {{Monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 2.3.11.23 {{monzo| -4 -1 -1 2 }}
| 3.28
| Preziosisma
|-
| 23
| [[576/575]]
| 2.3.5.23 {{monzo| 6 2 -2 -1 }}
| 3.01
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| 1.34
| Triaphonisma
| Triaphonisma
|
|}
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 1\50
| 24.0
| 686/675
| [[Sengagen]]
|-
| 1
| 9\50
| 216.0
| 17/15
| [[Tremka]]
|-
| 1
| 11\50
| 264.0
| 7/6
| [[Septimin]]
|-
| 1
| 13\50
| 312.0
| 6/5
| [[Oolong]]
|-
| 1
| 17\50
| 408.0
| 325/256
| [[Coditone]]
|-
| 1
| 19\50
| 456.0
| 125/96
| [[Qak]]
|-
| 1
| 21\50
| 504.0
| 4/3
| [[Meantone]] / [[meanpop]]
|-
| 1
| 23\50
| 552.0
| 11/8
| [[Emka]]
|-
| 2
| 2\50
| 48.0
| 36/35
| [[Pombe]]
|-
| 2
| 3\50
| 72.0
| 25/24
| [[Vishnu]] / [[vishnean]]
|-
| 2
| 6\50
| 144.0
| 12/11
| [[Bisemidim]]
|-
| 2
| 9\50
| 216.0
| 17/15
| [[Wizard]] / [[lizard]] / [[gizzard]]
|-
| 2
| 12\50
| 288.0
| 13/11
| [[Vines]]
|-
| 2
| 21\50<br>(4\50)
| 504.0<br>(96.0)
| 4/3<br>(35/33)
| [[Bimeantone]]
|-
| 5
| 21\50<br>(1\50)
| 504.0<br>(24.0)
| 4/3<br>(49/48)
| [[Cloudtone]]
|-
| 5
| 23<br>(3\50)
| 552.0<br>(72.0)
| 11/8<br>(21/20)
| [[Coblack]]
|-
| 10
| 7\50<br>(3\50)
| 168.0<br>(72.0)
| 54/49<br>(25/24)
| [[Decavish]]
|-
| 10
| 21\50<br>(1\50)
| 504.0<br>(24.0)
| 4/3<br>(78/77)
| [[Decic]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Instruments ==
; Lumatone
See [[Lumatone mapping for 50edo]]
; Piano
A [[:Category:Piano|piano]] playing with a 50edo ensemble may wish to use the tuning [[116ed5]]. This tuning is almost exactly the same as 50edo, but with octaves [[octave stretch|stretched]] by 1 cent. Because pianos usually use stretched octaves, this tuning will sit better with the [[timbre]] of the piano, while still being close enough that it sounds perfectly in-tune with the other instruments tuned to 50edo.


== Music ==
== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
=== Modern renderings ===
* [http://soonlabel.com/xenharmonic/archives/1118 Fantasia Catalana by Claudi Meneghin]
; {{W|Johann Sebastian Bach}}
* [http://soonlabel.com/xenharmonic/archives/1929 Fugue on the Dragnet theme by Claudi Meneghin]
* [https://www.youtube.com/watch?v=RnYqc0NKMLM "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
* [https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album the late little xmas album by Cam Taylor]
* [https://www.youtube.com/watch?v=e6fMO-sue4Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) &ndash; rendered by Claudi Meneghin (2024)
* [https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor]
* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) &ndash; rendered by Claudi Meneghin (2024, organ sound rendering)
* [https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo Long improvisation 2 in 50EDO by Cam Taylor]
* [https://www.youtube.com/watch?v=qjb9DDM32Ic "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742-1749) &mdash; rendered by Claudi Meneghin (2025, harpsichord sound rendering)
* [https://soundcloud.com/camtaylor-1/chord-sequence-for-difference Chord sequence for Difference tones in 50EDO by Cam Taylor]
 
* [https://soundcloud.com/camtaylor-1/enharmonic-modulations-in Enharmonic Modulations in 50EDO by Cam Taylor]
; {{W|Nicolaus Bruhns}}
* [https://soundcloud.com/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq Harmonic Clusters on 50EDO Harpsichord by Cam Taylor]
* [https://www.youtube.com/watch?v=yrM50pvmD5c ''Prelude in E Minor "The Great"''] &ndash; rendered by Claudi Meneghin (2023)
* [https://soundcloud.com/camtaylor-1/fragment-in-fifty Fragment in Fifty] by Cam Taylor
 
; {{W|Gabriel Fauré}}
* [https://www.youtube.com/watch?v=7djfrUlw2ck  ''Pavane'', op. 50] (1887) &ndash; arranged for harpsichord and rendered by Claudi Meneghin (2020)
 
; {{W|Akira Kamiya}}
* [https://www.youtube.com/watch?v=5UnPAhRqmb4 ''funfunfun ta yo''] (2007) &ndash; rendered by MortisTheneRd (2024)
 
=== 21st century===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/zCsc5n6dr_I ''microtonal improv in 50edo''] (2024)
* [https://www.youtube.com/shorts/ynz5XvJOHiE ''Piano that may not be played that well - Deltarune (microtonal cover in 50edo)''] (2025)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=pH6E35hwUnM ''On My Way To Somewhere''] (2023)
 
; [[Claudi Meneghin]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon &ndash; 50 edo] {{dead link}}
* [https://www.youtube.com/watch?v=wcTVED9zFrU ''Blue Fugue for Organ''] (2018)
* [https://www.youtube.com/watch?v=Zh2jWoIXAf8 ''La Petite Poule Grise - Fugue''] (2019)
* [https://www.youtube.com/watch?v=28x3vqw9kDI ''Happy Birthday Canon'', 6-in-1 Canon in 50edo] (2019)
* [https://www.youtube.com/watch?v=szUpO3FAOes ''Fantasia Catalana''] (2020)
* [https://www.youtube.com/watch?v=38UMa3oWSIE ''Preludi Nocturn i Fuga sobre la Lluna la Pruna''] (2020)
* [https://www.youtube.com/watch?v=TRXy0FJOKIA ''Fugue on the Dragnet theme''] (2020)
* [https://www.youtube.com/watch?v=C4EkNEu4EeU ''Canon at the Semitone on The Mother's Malison Theme'', for Organ] (2022)
* [https://www.youtube.com/watch?v=FyDKSjS9Qtg ''Fugue on an Original Theme'', for Baroque Ensemble] (2023) ([https://www.youtube.com/watch?v=TXwlLV2TCsw for Organ])
* [https://www.youtube.com/watch?v=2nD_7Ot8-0A ''Catalan Fugue (La Santa Espina)''] (2023)
* [https://www.youtube.com/watch?v=TBxDmpM9Xa8 ''Canon in C='' for Baroque Wind Ensemble] (2023)
* [https://www.youtube.com/watch?v=sIr394fGEEg ''Fantasia Catalana'', for Baroque Ensemble] (2023)
 
; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album ''the late little xmas album''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone ''Harpsichord meantone improvisation 1 in 50EDO''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo ''Long improvisation 2 in 50EDO''] (2014)
* [https://soundcloud.com/camtaylor-1/chord-sequence-for-difference ''Chord sequence for Difference tones in 50EDO''] (2014)
* [https://soundcloud.com/camtaylor-1/enharmonic-modulations-in ''Enharmonic Modulations in 50EDO''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq ''Harmonic Clusters on 50EDO Harpsichord''] (2014)
* [https://soundcloud.com/camtaylor-1/fragment-in-fifty ''Fragment in Fifty''] (2014)


== Additional reading ==
== Additional reading ==
Line 682: Line 897:
* [http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html More information about Robert Smith's temperament]{{Dead link}}
* [http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html More information about Robert Smith's temperament]{{Dead link}}
* [https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]{{Dead link}}
* [https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]{{Dead link}}
* [http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]    
* [http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]    
 
== Notes ==
<references group="note" />


[[Category:50edo]]
[[Category:50edo]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:golden]]
[[Category:Golden meantone]]
[[Category:meantone]]
[[Category:Historical]]
[[Category:theory]]
[[Category:Listen]]
[[Category:Meantone]]
[[Category:Meanpop]]
Retrieved from "https://en.xen.wiki/w/50edo"