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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 ==
Retrieved from "https://en.xen.wiki/w/50edo"