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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 is also almost consistent to the no-21s [[25-odd-limit]], only barely missing consistent mapping 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.
50edo is also quite strong in the realm of tertian harmony for a meantone system, as the errors on [[7/6]], [[6/5]], [[5/4]], and [[9/7]] are all balanced to be roughly half as flat as the fifth, meaning that this set of thirds taken as a whole is minimally out-of-tune given the damage induced by meantone. Though it fails to approximate [[11/9]] well by virtue of not having a perfect hemififth, it inherits the excellent [[16/13]] from [[10edo]] and additionally has a 1.2{{c}} flat [[13/11]], providing even more qualities of roughly just thirds alongside their more complex [[fifth complement]]s.


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|intervals=odd|columns=11}}
{{Harmonics in equal|50|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 50edo (continued)}}


== Relations ==
=== As a tuning of other temperaments ===
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]").
50et 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.
 
=== Relations ===
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|50edo|limit=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]],<br> [[66/65]], [[78/77]], [[91/90]], [[99/98]],<br> [[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]],<br> ''[[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]],<br> ''[[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]],<br> [[65/33]], [[77/39]], [[180/91]], [[196/99]],<br> [[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.
<references group="note" />
 
== Notation ==
=== Stein–Zimmermann–Gould notation ===
50edo can be notated with [[Stein–Zimmermann–Gould notation]]:
{{Sharpness-sharp3-szg}}
 
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.
 
=== Kite's ups and downs notation ===
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{Ups and downs sharpness}}
 
=== 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]].
 
==== Evo flavor ====
<imagemap>
File:50-EDO_Evo_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 [[1053/1024]]
default [[File:50-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:50-EDO_Revo_Sagittal.svg
desc none
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>


== Just approximation ==
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.
=== Selected just intervals ===
{{Primes in edo|50}}


==== 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]]
| 2.3.5.7.11.13
| 11.910
| 81/80, 105/104, 126/125, 144/143, 245/242
|-
| {{mapping| 50 79 116 140 173 185 }}
| ''[[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 (¢)
| +1.88
| +1.58
| +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="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-5"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
! Name(s)
! Name
|-
| 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, Didymus comma
| Syntonic comma
|-
|-
| 5
| 5
| <abbr title="1220703125/1207959552">(20 digits)</abbr>
| <abbr title="1220703125/1207959552">(20 digits)</abbr>
| {{Monzo| -27 -2 13 }}
| {{monzo| -27 -2 13 }}
| 18.17
| 18.17
| [[Ditonma]]
| [[Ditonma]]
Line 499: Line 507:
| 5
| 5
| [[6115295232/6103515625|(20 digits)]]
| [[6115295232/6103515625|(20 digits)]]
| {{Monzo| 23 6 -14 }}
| {{monzo| 23 6 -14 }}
| 3.34
| 3.34
| [[Vishnuzma]], 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
| [[16807/16384]]
| {{monzo| -14 0 0 5}}
| 44.13
| Cloudy comma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Schismean comma
|-
|-
| 7
| 7
| [[126/125]]
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Starling comma, Small septimal comma
| Starling 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, Middle second comma
| Hemimean comma
|-
|-
| 7
| 7
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
| <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)]]
| [[703125/702464|(12 digits)]]
| {{Monzo| -11 2 7 -3 }}
| {{monzo| -11 2 7 -3 }}
| 1.63
| 1.63
| [[Meter]]
| [[Meter]]
Line 541: Line 561:
| 7
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo| -6 -8 2 5 }}
| {{monzo| -6 -8 2 5 }}
| 1.12
| 1.12
| [[Wizma]]
| [[Wizma]]
Line 547: Line 567:
| 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, undecimal kleisma
| Keenanisma
|-
|-
| 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, undecimal schisma
| Wizardharry comma
|-
|-
| 11
| 11
| [[9801/9800]]
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Kalisma, Gauss' comma
| Kalisma
|-
|-
| 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, small tridecimal comma
| Animist 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
Line 589: Line 609:
| 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
Line 595: Line 615:
| 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, Parizek comma
| Petrma
|-
| 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
| [[529/528]]
| 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
| 23
| [[1288/1287]]
| [[1288/1287]]
| {{Monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| 1.34
| Triaphonisma
| Triaphonisma
|}
|}
<references/>
<references group="note" />
 
=== 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\50<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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Octave stretch or compression ==
50edo's [[prime]]s 3, 5, 7, 17, 19, and 23 are all tuned flat and its 11 and 13 have close to no error, so 50edo can benefit from slight [[octave stretching]]. Some slightly stretched-octave tunings of 50edo include (least to most stretch): [[equal tuning|166ed10]], [[ed5|116ed5]], [[zpi|238zpi]] and [[ed12|179ed12]].
 
== 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|David Belasco}}
* [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/shorts/WcExL9W2Gyc ''The Prettiest Little Song Of All''] (1908) - microtonal cover in 50edo by [[Bryan Deister]] (2025)
* [https://soundcloud.com/camtaylor-1/fragment-in-fifty Fragment in Fifty] by Cam Taylor
 
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=yrM50pvmD5c ''Prelude in E Minor "The Great"''] &ndash; rendered by Claudi Meneghin (2023)
 
; {{W|John Bull (composer)|John Bull}}
* [https://www.youtube.com/watch?v=6RewllRJ5rU ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2026)
 
; {{w|Frédéric Chopin}}
* [https://www.youtube.com/shorts/7Bisk0I2H4o ''Prelude Op. 28, No. 7 in A major''] (1839), arranged for fortepiano, tuned into 50-edo – rendered by [[Claudi Meneghin]] (2025)
 
; {{W|Louis Couperin}}
* [https://www.youtube.com/shorts/NSzakO66Roc ''«La Piémontoise»''] (1658?) &ndash; rendered by Claudi Meneghin (2026)
 
; {{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|Toby Fox}}
* [https://www.youtube.com/shorts/ynz5XvJOHiE ''Piano that may not be played that well''] via ''{{W|Deltarune}} Chapters 3 + 4'' (2025) (microtonal cover in 50edo) by [[Bryan Deister]] (2025)
 
; {{W|Iyowa}}
* [https://www.youtube.com/shorts/L6jF5_HEGkM ''Heat Abnormal''] (2024) - microtonal cover in 50edo by [[Bryan Deister]] (2025)
 
; {{W|Akira Kamiya}}
* [https://www.youtube.com/watch?v=5UnPAhRqmb4 ''funfunfun ta yo''] (2007) &ndash; rendered by MortisTheneRd (2024)
 
; {{W|Laufey_(singer)|Laufey}}
* [https://www.youtube.com/shorts/J34qt45jZW4 ''Snow White''] (2025) - microtonal cover in 50edo by [[Bryan Deister]] (2025)
 
; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=YK_kFs4PL2g&list=WL&index=347 ''Gigue KV 574 («Leipziger Gigue»)''] (1789) – rendered by Claudi Meneghin (2026)
 
; {{W|Akiko Shikata}}
* [https://www.youtube.com/shorts/eXGcC52YMh8 ''Mother''] via ''{{W|Umineko_When_They_Cry|Umineko no Naku Koro ni}}'' (2007) - microtonal cover in 50edo by [[Bryan Deister]] (2026)
 
=== 21st century===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/zCsc5n6dr_I ''microtonal improv in 50edo''] (2024)
* [https://www.youtube.com/shorts/dAyMY-14yZo ''50edo improv''] (2025-10-13)
* [https://www.youtube.com/shorts/DIiLORDPfUI ''50edo improv''] (2026-05-25)
* [https://www.youtube.com/watch?v=LqzPpl01WXc ''Fantasy in 50edo''] (2026)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=pH6E35hwUnM ''On My Way To Somewhere''] (2023)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Zh2jWoIXAf8 ''La Petite Poule Grise - Fugue''] (2014, uploaded 2019)
* [https://www.youtube.com/shorts/IhKVro5YEcA ''Canon on «Twinkle Twinkle Little Star» in 50-edo, for Organ''] (≤2014, restored and re-hosted 2025)
* [https://www.youtube.com/watch?v=TRXy0FJOKIA ''Fugue on the Dragnet theme''] (2014)
* [https://www.youtube.com/watch?v=wcTVED9zFrU ''Blue Fugue for Organ''] (2018)
* [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=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)
* ''Chord Progression: The Octave Divided into Five Parts in 50 edo'' (intended for demonstrating chord progressions, as the title indicates, but actually works as a short composition)
** [https://www.youtube.com/shorts/7_kROtWc4Sw <nowiki>organ rendition</nowiki>] (2023)
** [https://www.youtube.com/shorts/qsuM1sA2-A0 <nowiki>harpsichord rendition</nowiki>] (2024)
* [https://www.youtube.com/shorts/2x5atFuN6WA ''Baroque Blues - 4-Part Fugue for Baroque Consort''] (2026)
* ''Fugue on the French Lullaby «La Petite Poule Grise»'' (2026)
** [https://www.youtube.com/watch?v=RrsZ-bzf1mE Baroque ensemble rendition]
** [https://www.youtube.com/watch?v=wgsWzn_vfIM organ rendition]
 
; [[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 635: Line 943:
* [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]    


[[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]]

Latest revision as of 04:55, 3 June 2026

← 49edo 50edo 51edo →
Prime factorization 2 × 52
Step size 24 ¢ 
Fifth 29\50 (696 ¢)
Semitones (A1:m2) 3:5 (72 ¢ : 120 ¢)
Consistency limit 9
Distinct consistency limit 7

50 equal divisions of the octave (abbreviated 50edo or 50ed2), also called 50-tone equal temperament (50tet) or 50 equal temperament (50et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 50 equal parts of exactly 24 ¢ each. Each step represents a frequency ratio of 21/50, or the 50th root of 2.

Theory

As an equal temperament, 50et 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 "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 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 is also almost consistent to the no-21s 25-odd-limit, only barely missing consistent mapping of 11/9 and 18/11.

50edo is also quite strong in the realm of tertian harmony for a meantone system, as the errors on 7/6, 6/5, 5/4, and 9/7 are all balanced to be roughly half as flat as the fifth, meaning that this set of thirds taken as a whole is minimally out-of-tune given the damage induced by meantone. Though it fails to approximate 11/9 well by virtue of not having a perfect hemififth, it inherits the excellent 16/13 from 10edo and additionally has a 1.2 ¢ flat 13/11, providing even more qualities of roughly just thirds alongside their more complex fifth complements.

Odd harmonics

Approximation of odd harmonics in 50edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -6.0 -2.3 -8.8 -11.9 +0.7 -0.5 -8.3 -9.0 -9.5 +9.2 -4.3
Relative (%) -24.8 -9.6 -36.8 -49.6 +2.8 -2.2 -34.5 -37.3 -39.6 +38.4 -17.8
Steps
(reduced) 		79
(29) 		116
(16) 		140
(40) 		158
(8) 		173
(23) 		185
(35) 		195
(45) 		204
(4) 		212
(12) 		220
(20) 		226
(26)
Approximation of odd harmonics in 50edo (continued)
Harmonic 25 27 29 31 33 35 37 39 41 43 45 47
Error Absolute (¢) -4.6 +6.1 +2.4 +7.0 -5.3 -11.1 -11.3 -6.5 +2.9 -7.5 +9.8 +6.5
Relative (%) -19.3 +25.6 +10.1 +29.0 -22.0 -46.4 -47.3 -27.0 +12.2 -31.3 +40.7 +27.1
Steps
(reduced) 		232
(32) 		238
(38) 		243
(43) 		248
(48) 		252
(2) 		256
(6) 		260
(10) 		264
(14) 		268
(18) 		271
(21) 		275
(25) 		278
(28)

As a tuning of other temperaments

50et 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 coblack temperament (15 & 50), and provides the optimal patent val for 11- and 13-limit bimeantone. It is also the unique equal temperament tempering out both 81/80 and the vishnuzma, [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.

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 "A Theory of Evolving Tonality").

Intervals

# Cents Ratios[note 1] Ups and downs notation

(EUs: v3A1 and vvd2)

0 0 1/1 Perfect 1sn P1 D
1 24 45/44, 49/48, 56/55, 65/64,
66/65, 78/77, 91/90, 99/98,
100/99, 121/120, 169/168 		Up 1sn ^1 ^D
2 48 27/26, 33/32, 36/35,
50/49, 55/54, 64/63 		Dim 2nd, Downaug 1sn d2, vA1 Ebb, vD#
3 72 21/20, 25/24, 26/25, 28/27 Aug 1sn, Updim 2nd A1, ^d2 D#, ^Ebb
4 96 22/21 Downminor 2nd vm2 vEb
5 120 16/15, 15/14, 14/13 Minor 2nd m2 Eb
6 144 13/12, 12/11 Upminor 2nd ^m2 ^Eb
7 168 11/10 Downmajor 2nd vM2 vE
8 192 9/8, 10/9 Major 2nd M2 E
9 216 25/22 Upmajor 2nd ^M2 ^E
10 240 8/7, 15/13 Downaug 2nd, Dim 3rd vA2, d3 vE#, Fb
11 264 7/6 Updim 3rd, Aug 2nd ^d3, A2 ^Fb, E#
12 288 13/11 Downminor 3rd vm3 vF
13 312 6/5 Minor 3rd m3 F
14 336 27/22, 39/32, 40/33, 49/40 Upminor 3rd ^m3 ^F
15 360 16/13, 11/9 Downmajor 3rd vM3 vF#
16 384 5/4 Major 3rd M3 F#
17 408 14/11 Upmajor 3rd ^M3 ^F#
18 432 9/7 Downaug 3rd, Dim 4th vA3, d4 vFx, Gb
19 456 13/10 Updim 4th, Aug 3rd A3, ^d4 ^Gb, Fx
20 480 33/25, 55/42, 64/49 Down 4th v4 vG
21 504 4/3 Perfect 4th P4 G
22 528 15/11 Up 4th ^4 ^G
23 552 11/8, 18/13 Downaug 4th vA4 vG#
24 576 7/5 Aug 4th A4 G#
25 600 63/44, 88/63, 78/55, 55/39 Upaug 4th, Downdim 5th ^A4, vd5 ^G#, vAb
26 624 10/7 Dim 5th d5 Ab
27 648 16/11, 13/9 Updim 5th ^d5 ^Ab
28 672 22/15 Down 5th v5 vA
29 696 3/2 Perfect 5th P5 A
30 720 50/33, 84/55, 49/32 Up 5th ^5 ^A
31 744 20/13 Downaug 5th, Dim 6th vA5, d6 vA#, Bbb
32 768 14/9 Updim 6th, Aug 5th ^d6, A5 ^Bbb, A#
33 792 11/7 Downminor 6th vm6 vBb
34 816 8/5 Minor 6th m6 Bb
35 840 13/8, 18/11 Upminor 6th ^m6 ^Bb
36 864 44/27, 64/39, 33/20, 80/49 Downmajor 6th vM6 vB
37 888 5/3 Major 6th M6 B
38 912 22/13 Upmajor 6th ^M6 ^B
39 936 12/7 Downaug 6th, Dim 7th vA6, d7 vB#, Cb
40 960 7/4 Updim 7th, Aug 6th ^d7, A6 ^Cb, B#
41 984 44/25 Downminor 7th vm7 vC
42 1008 16/9, 9/5 Minor 7th m7 C
43 1032 20/11 Upminor 7th ^m7 ^C
44 1056 24/13, 11/6 Downmajor 7th vM7 vC#
45 1080 15/8, 28/15, 13/7 Major 7th M7 C#
46 1104 21/11 Upmajor 7th ^M7 ^C#
47 1128 40/21, 48/25, 25/13, 27/14 Downaug 7th, Dim 8ve vA7, d8 vCx, Db
48 1152 52/27, 64/33, 35/18,
49/25, 108/55, 63/32 		Updim 8ve, Aug 7th ^d8, A7 ^Db, Cx
49 1176 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 1200 2/1 Perfect 8ve P8 D
  1. ↑ Based on treating 50edo as a 13-limit temperament; other approaches are also possible.

Notation

Stein–Zimmermann–Gould notation

50edo can be notated with Stein–Zimmermann–Gould notation:

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

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.

Kite's ups and downs notation

Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

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

Sagittal notation

This notation uses the same sagittal sequence as edos 57, 64, and 71b.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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.

Approximation to JI

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Selected 29-limit intervals approximated in 50edo

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 50edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 50edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.528 2.2
15/14, 28/15 0.557 2.3
11/8, 16/11 0.682 2.8
13/11, 22/13 1.210 5.0
13/10, 20/13 1.786 7.4
5/4, 8/5 2.314 9.6
7/6, 12/7 2.871 12.0
11/10, 20/11 2.996 12.5
9/7, 14/9 3.084 12.9
5/3, 6/5 3.641 15.2
13/12, 24/13 5.427 22.6
3/2, 4/3 5.955 24.8
7/5, 10/7 6.512 27.1
11/6, 12/11 6.637 27.7
15/13, 26/15 7.741 32.3
15/8, 16/15 8.269 34.5
13/7, 14/13 8.298 34.6
7/4, 8/7 8.826 36.8
15/11, 22/15 8.951 37.3
11/7, 14/11 9.508 39.6
9/5, 10/9 9.596 40.0
13/9, 18/13 11.382 47.4
11/9, 18/11 11.408 47.5
9/8, 16/9 11.910 49.6
15-odd-limit intervals in 50edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.528 2.2
15/14, 28/15 0.557 2.3
11/8, 16/11 0.682 2.8
13/11, 22/13 1.210 5.0
13/10, 20/13 1.786 7.4
5/4, 8/5 2.314 9.6
7/6, 12/7 2.871 12.0
11/10, 20/11 2.996 12.5
9/7, 14/9 3.084 12.9
5/3, 6/5 3.641 15.2
13/12, 24/13 5.427 22.6
3/2, 4/3 5.955 24.8
7/5, 10/7 6.512 27.1
11/6, 12/11 6.637 27.7
15/13, 26/15 7.741 32.3
15/8, 16/15 8.269 34.5
13/7, 14/13 8.298 34.6
7/4, 8/7 8.826 36.8
15/11, 22/15 8.951 37.3
11/7, 14/11 9.508 39.6
9/5, 10/9 9.596 40.0
13/9, 18/13 11.382 47.4
9/8, 16/9 11.910 49.6
11/9, 18/11 12.592 52.5

Regular temperament properties

Temperament measures

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-79 50⟩ [⟨50 79]] +1.88 1.88 7.83
2.3.5 81/80, [-27 -2 13⟩ [⟨50 79 116]] +1.58 1.59 6.62
2.3.5.7 81/80, 126/125, 84035/82944 [⟨50 79 116 140]] +1.98 1.54 6.39
2.3.5.7.11 81/80, 126/125, 245/242, 385/384 [⟨50 79 116 140 173]] +1.54 1.63 6.76
2.3.5.7.11.13 81/80, 105/104, 126/125, 144/143, 245/242 [⟨50 79 116 140 173 185]] +1.31 1.57 6.54

Commas

50et tempers out the following commas. This assumes the 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.

Prime
limit 		Ratio[note 1] Monzo Cents Name
3 (20 digits) [-79 50⟩ 297.75 50-comma
5 81/80 [-4 4 -1⟩ 21.51 Syntonic comma
5 (20 digits) [-27 -2 13⟩ 18.17 Ditonma
5 (20 digits) [23 6 -14⟩ 3.34 Vishnuzma
7 59049/57344 [-13 10 0 -1⟩ 50.72 Harrison's comma
7 16807/16384 [-14 0 0 5⟩ 44.13 Cloudy comma
7 3645/3584 [-9 6 1 -1⟩ 29.22 Schismean comma
7 126/125 [1 2 -3 1⟩ 13.79 Starling comma
7 225/224 [-5 2 2 -1⟩ 7.71 Marvel comma
7 3136/3125 [6 0 -5 2⟩ 6.08 Hemimean comma
7 (24 digits) [11 -10 -10 10⟩ 5.57 Linus comma
7 (12 digits) [-11 2 7 -3⟩ 1.63 Meter
7 (12 digits) [-6 -8 2 5⟩ 1.12 Wizma
11 245/242 [-1 0 1 2 -2⟩ 21.33 Frostma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Keenanisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Swetisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.03 Wizardharry comma
11 9801/9800 [-3 4 -2 -2 2⟩ 0.18 Kalisma
13 105/104 [-3 1 1 1 0 -1⟩ 16.57 Animist comma
13 144/143 [4 2 0 0 -1 -1⟩ 12.06 Grossma
13 196/195 [2 -1 -1 2 0 -1⟩ 8.86 Mynucuma
13 1188/1183 [2 3 0 -1 1 -2⟩ 7.30 Kestrel comma
13 31213/31104 [-7 -5 0 4 0 1⟩ 6.06 Praveensma
13 364/363 [2 -1 0 1 -2 1⟩ 4.76 Minor minthma
13 2200/2197 [3 0 2 0 1 -3⟩ 2.36 Petrma
17 170/169 [1 0 1 0 0 -2 1⟩ 10.21 Major naiadma
17 221/220 [-2 0 -1 0 -1 1 1⟩ 7.85 Minor naiadma
17 289/288 [-5 -2 0 0 0 0 2⟩ 6.00 Semitonisma
17 375/374 [-1 1 3 0 -1 0 -1⟩ 4.62 Ursulisma
19 153/152 [-3 2 0 0 0 0 1 -1⟩ 11.35 Ganassisma
19 171/170 [-1 2 -1 0 0 0 -1 1⟩ 10.15 Malcolmisma
19 210/209 [1 1 1 1 -1 0 0 1⟩ 8.26 Spleen comma
19 324/323 [2 4 0 0 0 0 -1 -1⟩ 5.35 Photisma
19 361/360 [-3 -2 -1 0 0 0 0 2⟩ 4.80 Go comma
19 495/494 [-1 2 1 0 1 -1 0 -1⟩ 3.50 Eulalisma
23 507/506 2.3.11.13.23 [-1 1 -1 2 -1⟩ 3.42 Laodicisma
23 529/528 2.3.11.23 [-4 -1 -1 2⟩ 3.28 Preziosisma
23 576/575 2.3.5.23 [6 2 -2 -1⟩ 3.01 Worcester comma
23 1288/1287 [3 -2 0 1 -1 -1 0 0 1⟩ 1.34 Triaphonisma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(4\50) 		504.0
(96.0) 		4/3
(35/33) 		Bimeantone
5 21\50
(1\50) 		504.0
(24.0) 		4/3
(49/48) 		Cloudtone
5 23\50
(3\50) 		552.0
(72.0) 		11/8
(21/20) 		Coblack
10 7\50
(3\50) 		168.0
(72.0) 		54/49
(25/24) 		Decavish
10 21\50
(1\50) 		504.0
(24.0) 		4/3
(78/77) 		Decic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

50edo's primes 3, 5, 7, 17, 19, and 23 are all tuned flat and its 11 and 13 have close to no error, so 50edo can benefit from slight octave stretching. Some slightly stretched-octave tunings of 50edo include (least to most stretch): 166ed10, 116ed5, 238zpi and 179ed12.

Instruments

Lumatone

See Lumatone mapping for 50edo.

Piano

A 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 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

Modern renderings

Johann Sebastian Bach
David Belasco
Nicolaus Bruhns
John Bull
Frédéric Chopin
Louis Couperin
Gabriel Fauré
  • Pavane, op. 50 (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2020)
Toby Fox
Iyowa
Akira Kamiya
Laufey
Wolfgang Amadeus Mozart
Akiko Shikata

21st century

Bryan Deister
Francium
Claudi Meneghin
Cam Taylor

Additional reading

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