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Line 4: Line 4:
| Fifth = 29\50 = 696¢
| Fifth = 29\50 = 696¢
| Major 2nd = 8\50 = 192¢
| Major 2nd = 8\50 = 192¢
| Minor 2nd = 5\50 = 120¢
| Semitones = 3:5 (72¢:120¢)
| Augmented 1sn = 3\50 = 72¢
| Consistency = 9
| Monotonicity = 19
}}
}}


Line 11: Line 12:


== Theory ==
== Theory ==
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.
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 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 [[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.
50edo 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 [[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.
{{harmonics in equal|50}}
{{harmonics in equal|50}}
== 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 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]").
== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
Line 26: Line 28:
! Ratios*
! Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! Generator for*
|-
|-
| 0
| 0
Line 34: Line 35:
| P1
| P1
| D
| D
|
|-
|-
| 1
| 1
Line 42: Line 42:
| ^1
| ^1
| ^D
| ^D
| [[Hemimean_clan#Sengagen|Sengagen]]
|-
|-
| 2
| 2
Line 50: Line 49:
| d2, vA1
| d2, vA1
| Ebb, vD#
| Ebb, vD#
|
|-
|-
| 3
| 3
Line 58: Line 56:
| A1, ^d2
| A1, ^d2
| D#, ^Ebb
| D#, ^Ebb
| [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [[Starling temperaments #Coblack temperament|Coblack]] (5/oct)
|-
|-
| 4
| 4
Line 66: Line 63:
| vm2
| vm2
| vEb
| vEb
| [[Meantone_family#Injera|Injera]] (50d val, 2/oct)
|-
|-
| 5
| 5
Line 74: Line 70:
| m2
| m2
| Eb
| Eb
|
|-
|-
| 6
| 6
Line 82: Line 77:
| ^m2
| ^m2
| ^Eb
| ^Eb
|
|-
|-
| 7
| 7
Line 90: Line 84:
| vM2
| vM2
| vE
| vE
|
|-
|-
| 8
| 8
Line 98: Line 91:
| M2
| M2
| E
| E
|
|-
|-
| 9
| 9
Line 106: Line 98:
| ^M2
| ^M2
| ^E
| ^E
| [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)
|-
|-
| 10
| 10
Line 114: Line 105:
| vA2, d3
| vA2, d3
| vE#, Fb
| vE#, Fb
|
|-
|-
| 11
| 11
Line 122: Line 112:
| ^d3, A2
| ^d3, A2
| ^Fb, E#
| ^Fb, E#
| [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]
|-
|-
| 12
| 12
Line 130: Line 119:
| vm3
| vm3
| vF
| vF
|
|-
|-
| 13
| 13
Line 138: Line 126:
| m3
| m3
| F
| F
| [[Oolong]]
|-
|-
| 14
| 14
Line 146: Line 133:
| ^m3
| ^m3
| ^F
| ^F
|
|-
|-
| 15
| 15
Line 154: Line 140:
| vM3
| vM3
| vF#
| vF#
|
|-
|-
| 16
| 16
Line 162: Line 147:
| M3
| M3
| F#
| F#
| [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)
|-
|-
| 17
| 17
Line 170: Line 154:
| ^M3
| ^M3
| ^F#
| ^F#
| [[Ditonmic_family|Ditonic]]
|-
|-
| 18
| 18
Line 178: Line 161:
| vA3, d4
| vA3, d4
| vFx, Gb
| vFx, Gb
| [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)
|-
|-
| 19
| 19
Line 186: Line 168:
| A3, ^d4
| A3, ^d4
| ^Gb, Fx
| ^Gb, Fx
| [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)
|-
|-
| 20
| 20
Line 194: Line 175:
| v4
| v4
| vG
| vG
|
|-
|-
| 21
| 21
Line 202: Line 182:
| P4
| P4
| G
| G
| [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
|-
|-
| 22
| 22
Line 210: Line 189:
| ^4
| ^4
| ^G
| ^G
|
|-
|-
| 23
| 23
Line 218: Line 196:
| vA4
| vA4
| vG#
| vG#
| [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|Emka]]
|-
|-
| 24
| 24
Line 226: Line 203:
| A4
| A4
| G#
| G#
|
|-
|-
| 25
| 25
Line 234: Line 210:
| ^A4, vd5
| ^A4, vd5
| ^G#, vAb
| ^G#, vAb
|
|-
|-
| 26
| 26
Line 242: Line 217:
| d5
| d5
| Ab
| Ab
|
|-
|-
| 27
| 27
Line 250: Line 224:
| ^d5
| ^d5
| ^Ab
| ^Ab
|
|-
|-
| 28
| 28
Line 258: Line 231:
| v5
| v5
| vA
| vA
|
|-
|-
| 29
| 29
Line 266: Line 238:
| P5
| P5
| A
| A
|
|-
|-
| 30
| 30
Line 274: Line 245:
| ^5
| ^5
| ^A
| ^A
|
|-
|-
| 31
| 31
Line 282: Line 252:
| vA5, d6
| vA5, d6
| vA#, Bbb
| vA#, Bbb
|
|-
|-
| 32
| 32
Line 290: Line 259:
| ^d6, A5
| ^d6, A5
| ^Bbb, A#
| ^Bbb, A#
|
|-
|-
| 33
| 33
Line 298: Line 266:
| vm6
| vm6
| vBb
| vBb
|
|-
|-
| 34
| 34
Line 306: Line 273:
| m6
| m6
| Bb
| Bb
|
|-
|-
| 35
| 35
Line 314: Line 280:
| ^m6
| ^m6
| ^Bb
| ^Bb
|
|-
|-
| 36
| 36
Line 322: Line 287:
| vM6
| vM6
| vB
| vB
|
|-
|-
| 37
| 37
Line 330: Line 294:
| M6
| M6
| B
| B
|
|-
|-
| 38
| 38
Line 338: Line 301:
| ^M6
| ^M6
| ^B
| ^B
|
|-
|-
| 39
| 39
Line 346: Line 308:
| vA6, d7
| vA6, d7
| vB#, Cb
| vB#, Cb
|
|-
|-
| 40
| 40
Line 354: Line 315:
| ^d7, A6
| ^d7, A6
| ^Cb, B#
| ^Cb, B#
|
|-
|-
| 41
| 41
Line 362: Line 322:
| vm7
| vm7
| vC
| vC
|
|-
|-
| 42
| 42
Line 370: Line 329:
| m7
| m7
| C
| C
|
|-
|-
| 43
| 43
Line 378: Line 336:
| ^m7
| ^m7
| ^C
| ^C
|
|-
|-
| 44
| 44
Line 386: Line 343:
| vM7
| vM7
| vC#
| vC#
|
|-
|-
| 45
| 45
Line 394: Line 350:
| M7
| M7
| C#
| C#
|
|-
|-
| 46
| 46
Line 402: Line 357:
| ^M7
| ^M7
| ^C#
| ^C#
|
|-
|-
| 47
| 47
Line 410: Line 364:
| vA7, d8
| vA7, d8
| vCx, Db
| vCx, Db
|
|-
|-
| 48
| 48
Line 418: Line 371:
| ^d8, A7
| ^d8, A7
| ^Db, Cx
| ^Db, Cx
|
|-
|-
| 49
| 49
Line 426: Line 378:
| v8
| v8
| vD
| vD
|
|-
|-
| 50
| 50
Line 434: Line 385:
| P8
| P8
| D
| D
|
|}
|}
<nowiki>*</nowiki> Using the 13-limit patent val, except as noted.
<nowiki>*</nowiki> using the patent val


== Just approximation ==
== Just approximation ==
{{Primes in edo|50|columns=9}}
{{Primes in edo|50|columns=9}}


Line 451: Line 400:
! Error (abs, [[cent|¢]])
! Error (abs, [[cent|¢]])
|-
|-
| '''[[16/13|16/13]], [[13/8|13/8]]'''
| '''[[16/13]], [[13/8]]'''
| '''0.528'''
| '''0.528'''
|-
|-
| [[15/14|15/14]], [[28/15|28/15]]
| [[15/14]], [[28/15]]
| 0.557
| 0.557
|-
|-
| '''[[11/8|11/8]], [[16/11|16/11]]'''
| '''[[11/8]], [[16/11]]'''
| '''0.682'''
| '''0.682'''
|-
|-
| [[13/11|13/11]], [[22/13|22/13]]
| [[13/11]], [[22/13]]
| 1.210
| 1.210
|-
|-
| [[13/10|13/10]], [[20/13|20/13]]
| [[13/10]], [[20/13]]
| 1.786
| 1.786
|-
|-
| '''[[5/4|5/4]], [[8/5|8/5]]'''
| '''[[5/4]], [[8/5]]'''
| '''2.314'''
| '''2.314'''
|-
|-
| [[7/6|7/6]], [[12/7|12/7]]
| [[7/6]], [[12/7]]
| 2.871
| 2.871
|-
|-
| [[11/10|11/10]], [[20/11|20/11]]
| [[11/10]], [[20/11]]
| 2.996
| 2.996
|-
|-
| [[9/7|9/7]], [[14/9|14/9]]
| [[9/7]], [[14/9]]
| 3.084
| 3.084
|-
|-
| [[6/5|6/5]], [[5/3|5/3]]
| [[6/5]], [[5/3]]
| 3.641
| 3.641
|-
|-
| [[13/12|13/12]], [[24/13|24/13]]
| [[13/12]], [[24/13]]
| 5.427
| 5.427
|-
|-
| '''[[4/3|4/3]], [[3/2|3/2]]'''
| '''[[4/3]], [[3/2]]'''
| '''5.955'''
| '''5.955'''
|-
|-
| [[7/5|7/5]], [[10/7|10/7]]
| [[7/5]], [[10/7]]
| 6.512
| 6.512
|-
|-
| [[12/11|12/11]], [[11/6|11/6]]
| [[12/11]], [[11/6]]
| 6.637
| 6.637
|-
|-
| [[15/13|15/13]], [[26/15|26/15]]
| [[15/13]], [[26/15]]
| 7.741
| 7.741
|-
|-
| [[16/15|16/15]], [[15/8|15/8]]
| [[16/15]], [[15/8]]
| 8.269
| 8.269
|-
|-
| [[14/13|14/13]], [[13/7|13/7]]
| [[14/13]], [[13/7]]
| 8.298
| 8.298
|-
|-
| '''[[8/7|8/7]], [[7/4|7/4]]'''
| '''[[8/7]], [[7/4]]'''
| '''8.826'''
| '''8.826'''
|-
|-
| [[15/11|15/11]], [[22/15|22/15]]
| [[15/11]], [[22/15]]
| 8.951
| 8.951
|-
|-
| [[14/11|14/11]], [[11/7|11/7]]
| [[14/11]], [[11/7]]
| 9.508
| 9.508
|-
|-
| [[10/9|10/9]], [[9/5|9/5]]
| [[10/9]], [[9/5]]
| 9.596
| 9.596
|-
|-
| [[18/13|18/13]], [[13/9|13/9]]
| [[18/13]], [[13/9]]
| 11.382
| 11.382
|-
|-
| ''[[11/9|11/9]], [[18/11|18/11]]''
| ''[[11/9]], [[18/11]]''
| ''11.408''
| ''11.408''
|-
|-
| [[9/8|9/8]], [[16/9|16/9]]
| [[9/8]], [[16/9]]
| 11.910
| 11.910
|}
|}


{| class="wikitable center-all"
== Regular temperament properties ==
|+Patent val mapping
=== Temperament measures ===
|-
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 50et.
! 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
|-
| [[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"
{| class="wikitable center-all"
! colspan="2" |
! colspan="2" |
Line 636: Line 508:
|}
|}


== 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.
50edo [[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.


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
Line 840: Line 712:
|}
|}
<references/>
<references/>
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ Table of rank-2 temperaments by generator
|-
! Periods<br> per Octave
! Generator
! Cents
! Associated<br>Ratio
! Temperament
|-
| 1
| 1\50
| 24.00
| 686/675
| [[Sengagen]]
|-
| 1
| 9\50
| 216.00
| 17/15
| [[Tremka]]
|-
| 1
| 11\50
| 264.00
| 7/6
| [[Septimin]]
|-
| 1
| 13\50
| 312.00
| 6/5
| [[Oolong]]
|-
| 1
| 17\50
| 408.00
| 15625/12288
| [[Ditonic]] / [[coditone]]
|-
| 1
| 19\50
| 456.00
| 125/96
| [[Qak]]
|-
| 1
| 21\50
| 504.00
| 4/3
| [[Meantone]] / [[meanpop]]
|-
| 1
| 23\50
| 552.00
| 11/8
| [[Emka]]
|-
| 2
| 2\50
| 48.00
| 36/35
| [[Pombe]]
|-
| 2
| 3\50
| 72.00
| 25/24
| [[Vishnu]] / [[vishnean]]
|-
| 2
| 4\50
| 96.00
| 35/33
| [[Bimeantone]]
|-
| 2
| 6\50
| 144.00
| 12/11
| [[Bisemidim]]
|-
| 2
| 9\50
| 216.00
| 17/15
| [[Wizard]] / [[lizard]] / [[gizzard]]
|-
| 2
| 12\50
| 288.00
| 13/11
| [[Vines]]
|-
| 5
| 21\50 <br>(1\50)
| 504.00 <br>(24.00)
| 4/3 <br>&nbsp;
| [[Cloudtone]]
|-
| 5
| 3\50
| 72.00
| 21/20, 25/24
| [[Coblack]]
|-
| 10
| 21\50 <br>(1\50)
| 504.00 <br>(24.00)
| 4/3 <br>&nbsp;
| [[Decic]]
|-
| 10
| 2\50 <br>(3\50)
| 48.00 <br>(72.00)
| 36/35 <br>(25/24)
| [[Decavish]]
|}


== Music ==
== Music ==

Revision as of 12:33, 5 February 2022

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

50edo divides the octave into 50 equal parts of precisely 24 cents each.

Theory

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 "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 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 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 (15&50) temperament, 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 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.


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)

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* Ups and Downs Notation
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 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, 27/26, 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, 52/27, 27/14 Downaug 7th, Dim 8ve vA7, d8 vCx, Db
48 1152 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

* using the patent val

Just approximation

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15-odd-limit mappings

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.

Direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢)
16/13, 13/8 0.528
15/14, 28/15 0.557
11/8, 16/11 0.682
13/11, 22/13 1.210
13/10, 20/13 1.786
5/4, 8/5 2.314
7/6, 12/7 2.871
11/10, 20/11 2.996
9/7, 14/9 3.084
6/5, 5/3 3.641
13/12, 24/13 5.427
4/3, 3/2 5.955
7/5, 10/7 6.512
12/11, 11/6 6.637
15/13, 26/15 7.741
16/15, 15/8 8.269
14/13, 13/7 8.298
8/7, 7/4 8.826
15/11, 22/15 8.951
14/11, 11/7 9.508
10/9, 9/5 9.596
18/13, 13/9 11.382
11/9, 18/11 11.408
9/8, 16/9 11.910

Regular temperament properties

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 50et.

3-limit 5-limit 7-limit 11-limit 13-limit
Octave stretch (¢) +1.88 +1.58 +1.98 +1.54 +1.31
Error absolute (¢) 1.88 1.59 1.54 1.63 1.57
relative (%) 7.83 6.62 6.39 6.76 6.54

Commas

50edo tempers out the following commas. (Note: 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[1] Monzo Cents Name(s)
5 81/80 [-4 4 -1⟩ 21.51 Syntonic comma, Didymus comma
5 (20 digits) [-27 -2 13⟩ 18.17 Ditonma
5 (20 digits) [23 6 -14⟩ 3.34 Vishnuzma, Vishnu comma
7 59049/57344 [-13 10 0 -1⟩ 50.72 Harrison's comma
7 126/125 [1 2 -3 1⟩ 13.79 Starling comma, Small septimal comma
7 225/224 [-5 2 2 -1⟩ 7.71 Septimal kleisma, Marvel comma
7 3136/3125 [6 0 -5 2⟩ 6.08 Hemimean, Middle second comma
7 (24 digits) [11 -10 -10 10⟩ 5.57 Linus
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 Cassacot
11 385/384 [-7 -1 1 1 1⟩ 4.50 Keenanisma, undecimal kleisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Swets' comma, Swetisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.03 Wizardharry, undecimal schisma
11 9801/9800 [-3 4 -2 -2 2⟩ 0.18 Kalisma, Gauss' comma
13 105/104 [-3 1 1 1 0 -1⟩ 16.57 Animist comma, small tridecimal 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
13 364/363 [2 -1 0 1 -2 1⟩ 4.76 Gentle comma
13 2200/2197 [3 0 2 0 1 -3⟩ 2.36 Petrma, Parizek comma
17 170/169 [1 0 1 0 0 -2 1⟩ 10.21
17 221/220 [-2 0 -1 0 -1 1 1⟩ 7.85
17 289/288 [-5 -2 0 0 0 0 2⟩ 6.00 minor seconds comma
17 375/374 [-1 1 3 0 -1 0 -1⟩ 4.62
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
19 210/209 [1 1 1 1 -1 0 0 1⟩ 8.26
19 324/323 [2 4 0 0 0 0 -1 -1⟩ 5.35
19 361/360 [-3 -2 -1 0 0 0 0 2⟩ 4.80
19 495/494 [-1 2 1 0 1 -1 0 -1⟩ 3.50
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 Octave 		Generator Cents Associated
Ratio 		Temperament
1 1\50 24.00 686/675 Sengagen
1 9\50 216.00 17/15 Tremka
1 11\50 264.00 7/6 Septimin
1 13\50 312.00 6/5 Oolong
1 17\50 408.00 15625/12288 Ditonic / coditone
1 19\50 456.00 125/96 Qak
1 21\50 504.00 4/3 Meantone / meanpop
1 23\50 552.00 11/8 Emka
2 2\50 48.00 36/35 Pombe
2 3\50 72.00 25/24 Vishnu / vishnean
2 4\50 96.00 35/33 Bimeantone
2 6\50 144.00 12/11 Bisemidim
2 9\50 216.00 17/15 Wizard / lizard / gizzard
2 12\50 288.00 13/11 Vines
5 21\50
(1\50) 		504.00
(24.00) 		4/3
  		Cloudtone
5 3\50 72.00 21/20, 25/24 Coblack
10 21\50
(1\50) 		504.00
(24.00) 		4/3
  		Decic
10 2\50
(3\50) 		48.00
(72.00) 		36/35
(25/24) 		Decavish

Music

Additional reading

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