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→Music: Add Bryan Deister's ''Fantasy in 50edo'' (2026); convert 5 microtonal covers to Modern Renderings format |
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== 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 | 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]]. | ||
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. | |||
=== 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)}} | |||
=== | === As a tuning of other temperaments === | ||
{{ | 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 === | === Relations === | ||
| Line 21: | Line 22: | ||
! # | ! # | ||
! Cents | ! Cents | ||
! Ratios<ref group="note">{{sg|13-limit}}</ref> | ! Ratios<ref group="note">{{sg|50edo|limit=13-limit}}</ref> | ||
! colspan="3" | [[Ups and downs notation]] | ! 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 | | Perfect 1sn | ||
| P1 | | P1 | ||
| Line 33: | Line 35: | ||
| 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]]'' | ||
| Up 1sn | | Up 1sn | ||
| ^1 | | ^1 | ||
| Line 40: | Line 42: | ||
| 2 | | 2 | ||
| 48 | | 48 | ||
| 27/26, 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 | | Dim 2nd, Downaug 1sn | ||
| d2, vA1 | | d2, vA1 | ||
| Line 47: | Line 49: | ||
| 3 | | 3 | ||
| 72 | | 72 | ||
| 21/20, 25/24, 26/25, 28/27 | | ''[[21/20]]'', [[25/24]], [[26/25]], [[28/27]] | ||
| Aug 1sn, Updim 2nd | | Aug 1sn, Updim 2nd | ||
| A1, ^d2 | | A1, ^d2 | ||
| Line 54: | Line 56: | ||
| 4 | | 4 | ||
| 96 | | 96 | ||
| 22/21 | | ''[[22/21]]'' | ||
| Downminor 2nd | | Downminor 2nd | ||
| vm2 | | vm2 | ||
| Line 61: | Line 63: | ||
| 5 | | 5 | ||
| 120 | | 120 | ||
| 16/15, 15/14, 14/13 | | [[16/15]], [[15/14]], [[14/13]] | ||
| Minor 2nd | | Minor 2nd | ||
| m2 | | m2 | ||
| Line 68: | Line 70: | ||
| 6 | | 6 | ||
| 144 | | 144 | ||
| 13/12, 12/11 | | [[13/12]], [[12/11]] | ||
| Upminor 2nd | | Upminor 2nd | ||
| ^m2 | | ^m2 | ||
| Line 75: | Line 77: | ||
| 7 | | 7 | ||
| 168 | | 168 | ||
| 11/10 | | [[11/10]] | ||
| Downmajor 2nd | | Downmajor 2nd | ||
| vM2 | | vM2 | ||
| Line 82: | Line 84: | ||
| 8 | | 8 | ||
| 192 | | 192 | ||
| 9/8, 10/9 | | [[9/8]], [[10/9]] | ||
| Major 2nd | | Major 2nd | ||
| M2 | | M2 | ||
| Line 89: | Line 91: | ||
| 9 | | 9 | ||
| 216 | | 216 | ||
| 25/22 | | [[25/22]] | ||
| Upmajor 2nd | | Upmajor 2nd | ||
| ^M2 | | ^M2 | ||
| Line 96: | Line 98: | ||
| 10 | | 10 | ||
| 240 | | 240 | ||
| 8/7, 15/13 | | [[8/7]], [[15/13]] | ||
| Downaug 2nd, Dim 3rd | | Downaug 2nd, Dim 3rd | ||
| vA2, d3 | | vA2, d3 | ||
| Line 103: | Line 105: | ||
| 11 | | 11 | ||
| 264 | | 264 | ||
| 7/6 | | [[7/6]] | ||
| Updim 3rd, Aug 2nd | | Updim 3rd, Aug 2nd | ||
| ^d3, A2 | | ^d3, A2 | ||
| Line 110: | Line 112: | ||
| 12 | | 12 | ||
| 288 | | 288 | ||
| 13/11 | | [[13/11]] | ||
| Downminor 3rd | | Downminor 3rd | ||
| vm3 | | vm3 | ||
| Line 117: | Line 119: | ||
| 13 | | 13 | ||
| 312 | | 312 | ||
| 6/5 | | [[6/5]] | ||
| Minor 3rd | | Minor 3rd | ||
| m3 | | m3 | ||
| Line 124: | Line 126: | ||
| 14 | | 14 | ||
| 336 | | 336 | ||
| 27/22, 39/32, 40/33, 49/40 | | ''[[27/22]]'', [[39/32]], [[40/33]], ''[[49/40]]'' | ||
| Upminor 3rd | | Upminor 3rd | ||
| ^m3 | | ^m3 | ||
| Line 131: | Line 133: | ||
| 15 | | 15 | ||
| 360 | | 360 | ||
| 16/13, 11/9 | | [[16/13]], ''[[11/9]]'' | ||
| Downmajor 3rd | | Downmajor 3rd | ||
| vM3 | | vM3 | ||
| Line 138: | Line 140: | ||
| 16 | | 16 | ||
| 384 | | 384 | ||
| 5/4 | | [[5/4]] | ||
| Major 3rd | | Major 3rd | ||
| M3 | | M3 | ||
| Line 145: | Line 147: | ||
| 17 | | 17 | ||
| 408 | | 408 | ||
| 14/11 | | [[14/11]] | ||
| Upmajor 3rd | | Upmajor 3rd | ||
| ^M3 | | ^M3 | ||
| Line 152: | Line 154: | ||
| 18 | | 18 | ||
| 432 | | 432 | ||
| 9/7 | | [[9/7]] | ||
| Downaug 3rd, Dim 4th | | Downaug 3rd, Dim 4th | ||
| vA3, d4 | | vA3, d4 | ||
| Line 159: | Line 161: | ||
| 19 | | 19 | ||
| 456 | | 456 | ||
| 13/10 | | [[13/10]] | ||
| Updim 4th, Aug 3rd | | Updim 4th, Aug 3rd | ||
| A3, ^d4 | | A3, ^d4 | ||
| Line 166: | Line 168: | ||
| 20 | | 20 | ||
| 480 | | 480 | ||
| 33/25, 55/42, 64/49 | | [[33/25]], ''[[55/42]]'', ''[[64/49]]'' | ||
| Down 4th | | Down 4th | ||
| v4 | | v4 | ||
| Line 173: | Line 175: | ||
| 21 | | 21 | ||
| 504 | | 504 | ||
| 4/3 | | [[4/3]] | ||
| Perfect 4th | | Perfect 4th | ||
| P4 | | P4 | ||
| Line 180: | Line 182: | ||
| 22 | | 22 | ||
| 528 | | 528 | ||
| 15/11 | | [[15/11]] | ||
| Up 4th | | Up 4th | ||
| ^4 | | ^4 | ||
| Line 187: | Line 189: | ||
| 23 | | 23 | ||
| 552 | | 552 | ||
| 11/8, 18/13 | | [[11/8]], [[18/13]] | ||
| Downaug 4th | | Downaug 4th | ||
| vA4 | | vA4 | ||
| Line 194: | Line 196: | ||
| 24 | | 24 | ||
| 576 | | 576 | ||
| 7/5 | | [[7/5]] | ||
| Aug 4th | | Aug 4th | ||
| A4 | | A4 | ||
| Line 201: | Line 203: | ||
| 25 | | 25 | ||
| 600 | | 600 | ||
| 63/44, 88/63, 78/55, 55/39 | | ''[[63/44]]'', ''[[88/63]]'', [[78/55]], [[55/39]] | ||
| Upaug 4th, Downdim 5th | | Upaug 4th, Downdim 5th | ||
| ^A4, vd5 | | ^A4, vd5 | ||
| Line 208: | Line 210: | ||
| 26 | | 26 | ||
| 624 | | 624 | ||
| 10/7 | | [[10/7]] | ||
| Dim 5th | | Dim 5th | ||
| d5 | | d5 | ||
| Line 215: | Line 217: | ||
| 27 | | 27 | ||
| 648 | | 648 | ||
| 16/11, 13/9 | | [[16/11]], [[13/9]] | ||
| Updim 5th | | Updim 5th | ||
| ^d5 | | ^d5 | ||
| Line 222: | Line 224: | ||
| 28 | | 28 | ||
| 672 | | 672 | ||
| 22/15 | | [[22/15]] | ||
| Down 5th | | Down 5th | ||
| v5 | | v5 | ||
| Line 229: | Line 231: | ||
| 29 | | 29 | ||
| 696 | | 696 | ||
| 3/2 | | [[3/2]] | ||
| Perfect 5th | | Perfect 5th | ||
| P5 | | P5 | ||
| Line 236: | Line 238: | ||
| 30 | | 30 | ||
| 720 | | 720 | ||
| 50/33, 84/55, 49/32 | | [[50/33]], ''[[84/55]]'', ''[[49/32]]'' | ||
| Up 5th | | Up 5th | ||
| ^5 | | ^5 | ||
| Line 243: | Line 245: | ||
| 31 | | 31 | ||
| 744 | | 744 | ||
| 20/13 | | [[20/13]] | ||
| Downaug 5th, Dim 6th | | Downaug 5th, Dim 6th | ||
| vA5, d6 | | vA5, d6 | ||
| Line 250: | Line 252: | ||
| 32 | | 32 | ||
| 768 | | 768 | ||
| 14/9 | | [[14/9]] | ||
| Updim 6th, Aug 5th | | Updim 6th, Aug 5th | ||
| ^d6, A5 | | ^d6, A5 | ||
| Line 257: | Line 259: | ||
| 33 | | 33 | ||
| 792 | | 792 | ||
| 11/7 | | [[11/7]] | ||
| Downminor 6th | | Downminor 6th | ||
| vm6 | | vm6 | ||
| Line 264: | Line 266: | ||
| 34 | | 34 | ||
| 816 | | 816 | ||
| 8/5 | | [[8/5]] | ||
| Minor 6th | | Minor 6th | ||
| m6 | | m6 | ||
| Line 271: | Line 273: | ||
| 35 | | 35 | ||
| 840 | | 840 | ||
| 13/8, 18/11 | | [[13/8]], ''[[18/11]]'' | ||
| Upminor 6th | | Upminor 6th | ||
| ^m6 | | ^m6 | ||
| Line 278: | Line 280: | ||
| 36 | | 36 | ||
| 864 | | 864 | ||
| 44/27, 64/39, 33/20, 80/49 | | ''[[44/27]]'', [[64/39]], [[33/20]], ''[[80/49]]'' | ||
| Downmajor 6th | | Downmajor 6th | ||
| vM6 | | vM6 | ||
| Line 285: | Line 287: | ||
| 37 | | 37 | ||
| 888 | | 888 | ||
| 5/3 | | [[5/3]] | ||
| Major 6th | | Major 6th | ||
| M6 | | M6 | ||
| Line 292: | Line 294: | ||
| 38 | | 38 | ||
| 912 | | 912 | ||
| 22/13 | | [[22/13]] | ||
| Upmajor 6th | | Upmajor 6th | ||
| ^M6 | | ^M6 | ||
| Line 299: | Line 301: | ||
| 39 | | 39 | ||
| 936 | | 936 | ||
| 12/7 | | [[12/7]] | ||
| Downaug 6th, Dim 7th | | Downaug 6th, Dim 7th | ||
| vA6, d7 | | vA6, d7 | ||
| Line 306: | Line 308: | ||
| 40 | | 40 | ||
| 960 | | 960 | ||
| 7/4 | | [[7/4]] | ||
| Updim 7th, Aug 6th | | Updim 7th, Aug 6th | ||
| ^d7, A6 | | ^d7, A6 | ||
| Line 313: | Line 315: | ||
| 41 | | 41 | ||
| 984 | | 984 | ||
| 44/25 | | [[44/25]] | ||
| Downminor 7th | | Downminor 7th | ||
| vm7 | | vm7 | ||
| Line 320: | Line 322: | ||
| 42 | | 42 | ||
| 1008 | | 1008 | ||
| 16/9, 9/5 | | [[16/9]], [[9/5]] | ||
| Minor 7th | | Minor 7th | ||
| m7 | | m7 | ||
| Line 327: | Line 329: | ||
| 43 | | 43 | ||
| 1032 | | 1032 | ||
| 20/11 | | [[20/11]] | ||
| Upminor 7th | | Upminor 7th | ||
| ^m7 | | ^m7 | ||
| Line 334: | Line 336: | ||
| 44 | | 44 | ||
| 1056 | | 1056 | ||
| 24/13, 11/6 | | [[24/13]], [[11/6]] | ||
| Downmajor 7th | | Downmajor 7th | ||
| vM7 | | vM7 | ||
| Line 341: | Line 343: | ||
| 45 | | 45 | ||
| 1080 | | 1080 | ||
| 15/8, 28/15, 13/7 | | [[15/8]], [[28/15]], [[13/7]] | ||
| Major 7th | | Major 7th | ||
| M7 | | M7 | ||
| Line 348: | Line 350: | ||
| 46 | | 46 | ||
| 1104 | | 1104 | ||
| 21/11 | | ''[[21/11]]'' | ||
| Upmajor 7th | | Upmajor 7th | ||
| ^M7 | | ^M7 | ||
| Line 355: | Line 357: | ||
| 47 | | 47 | ||
| 1128 | | 1128 | ||
| 40/21, 48/25, 25/13, 27/14 | | ''[[40/21]]'', [[48/25]], [[25/13]], [[27/14]] | ||
| Downaug 7th, Dim 8ve | | Downaug 7th, Dim 8ve | ||
| vA7, d8 | | vA7, d8 | ||
| Line 362: | Line 364: | ||
| 48 | | 48 | ||
| 1152 | | 1152 | ||
| 52/27, 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 | | Updim 8ve, Aug 7th | ||
| ^d8, A7 | | ^d8, A7 | ||
| Line 369: | Line 371: | ||
| 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 | | Down 8ve | ||
| v8 | | v8 | ||
| Line 376: | Line 378: | ||
| 50 | | 50 | ||
| 1200 | | 1200 | ||
| 2/1 | | [[2/1]] | ||
| Perfect 8ve | | Perfect 8ve | ||
| P8 | | P8 | ||
| D | | D | ||
|} | |} | ||
<references group="note" /> | |||
== Notation == | == 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. | 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 === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as | 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 ==== | ==== Evo flavor ==== | ||
| Line 414: | Line 419: | ||
</imagemap> | </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 | 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. | ||
== Approximation to JI == | == Approximation to JI == | ||
| Line 421: | Line 426: | ||
=== 15-odd-limit interval mappings === | === 15-odd-limit interval mappings === | ||
{{Q-odd-limit intervals|50|15}} | {{Q-odd-limit intervals|50|15}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 730: | Line 721: | ||
| Triaphonisma | | Triaphonisma | ||
|} | |} | ||
<references group="note" /> | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
| Line 834: | Line 824: | ||
|- | |- | ||
| 5 | | 5 | ||
| 23<br>(3\50) | | 23\50<br>(3\50) | ||
| 552.0<br>(72.0) | | 552.0<br>(72.0) | ||
| 11/8<br>(21/20) | | 11/8<br>(21/20) | ||
| Line 851: | Line 841: | ||
| [[Decic]] | | [[Decic]] | ||
|} | |} | ||
<nowiki/>* [[Normal | <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 == | == Instruments == | ||
; Lumatone | ; Lumatone | ||
See [[Lumatone mapping for 50edo]] | See [[Lumatone mapping for 50edo]]. | ||
; Piano | ; Piano | ||
| Line 868: | Line 860: | ||
* [https://www.youtube.com/watch?v=RnYqc0NKMLM "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024) | * [https://www.youtube.com/watch?v=RnYqc0NKMLM "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024) | ||
* [https://www.youtube.com/watch?v=e6fMO-sue4Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024) | * [https://www.youtube.com/watch?v=e6fMO-sue4Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024) | ||
* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024) | * [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024, organ sound rendering) | ||
* [https://www.youtube.com/watch?v=qjb9DDM32Ic "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742-1749) — rendered by Claudi Meneghin (2025, harpsichord sound rendering) | |||
; {{W|David Belasco}} | |||
* [https://www.youtube.com/shorts/WcExL9W2Gyc ''The Prettiest Little Song Of All''] (1908) - microtonal cover in 50edo by [[Bryan Deister]] (2025) | |||
; {{W|Nicolaus Bruhns}} | ; {{W|Nicolaus Bruhns}} | ||
* [https://www.youtube.com/watch?v=yrM50pvmD5c ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023) | * [https://www.youtube.com/watch?v=yrM50pvmD5c ''Prelude in E Minor "The Great"''] – 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?) – rendered by Claudi Meneghin (2026) | |||
; {{W|Gabriel Fauré}} | ; {{W|Gabriel Fauré}} | ||
* [https://www.youtube.com/watch?v=7djfrUlw2ck ''Pavane'', op. 50] (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2020) | * [https://www.youtube.com/watch?v=7djfrUlw2ck ''Pavane'', op. 50] (1887) – 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}} | ; {{W|Akira Kamiya}} | ||
* [https://www.youtube.com/watch?v=5UnPAhRqmb4 ''funfunfun ta yo''] (2007) – rendered by MortisTheneRd (2024) | * [https://www.youtube.com/watch?v=5UnPAhRqmb4 ''funfunfun ta yo''] (2007) – 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=== | === 21st century=== | ||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/shorts/zCsc5n6dr_I ''microtonal improv in 50edo''] (2024) | * [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]] | ; [[Francium]] | ||
| Line 887: | Line 910: | ||
; [[Claudi Meneghin]] | ; [[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=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=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=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=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=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=FyDKSjS9Qtg ''Fugue on an Original Theme'', for Baroque Ensemble] (2023) ([https://www.youtube.com/watch?v=TXwlLV2TCsw for Organ]) | ||
| Line 899: | Line 922: | ||
* [https://www.youtube.com/watch?v=TBxDmpM9Xa8 ''Canon in C='' for Baroque Wind Ensemble] (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) | * [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]] | ; [[Cam Taylor]] | ||
| Line 914: | Line 944: | ||
* [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]] | ||
Latest revision as of 04:55, 3 June 2026
| ← 49edo | 50edo | 51edo → |
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
| 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) | |
| 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 |
- ↑ 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
Revo flavor
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
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.
| 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 |
| 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 |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.
Rank-2 temperaments
| 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
- "Ricercar a 3" from The Musical Offering, BWV 1079 (1747) – rendered by Claudi Meneghin (2024)
- "Contrapunctus 4" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)
- "Contrapunctus 11" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024, organ sound rendering)
- "Contrapunctus 11" from The Art of Fugue, BWV 1080 (1742-1749) — rendered by Claudi Meneghin (2025, harpsichord sound rendering)
- The Prettiest Little Song Of All (1908) - microtonal cover in 50edo by Bryan Deister (2025)
- Prelude in E Minor "The Great" – rendered by Claudi Meneghin (2023)
- Fantasia «Ut Re Mi Fa Sol La» (late 1500s/early 1600s, from Fitzwilliam Virginal Book Vol.1 No.51) – rendered by Claudi Meneghin (2026)
- Prelude Op. 28, No. 7 in A major (1839), arranged for fortepiano, tuned into 50-edo – rendered by Claudi Meneghin (2025)
- «La Piémontoise» (1658?) – rendered by Claudi Meneghin (2026)
- Pavane, op. 50 (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2020)
- Piano that may not be played that well via Deltarune Chapters 3 + 4 (2025) (microtonal cover in 50edo) by Bryan Deister (2025)
- Heat Abnormal (2024) - microtonal cover in 50edo by Bryan Deister (2025)
- funfunfun ta yo (2007) – rendered by MortisTheneRd (2024)
- Snow White (2025) - microtonal cover in 50edo by Bryan Deister (2025)
- Gigue KV 574 («Leipziger Gigue») (1789) – rendered by Claudi Meneghin (2026)
- Mother via Umineko no Naku Koro ni (2007) - microtonal cover in 50edo by Bryan Deister (2026)
21st century
- microtonal improv in 50edo (2024)
- 50edo improv (2025-10-13)
- 50edo improv (2026-05-25)
- Fantasy in 50edo (2026)
- On My Way To Somewhere (2023)
- La Petite Poule Grise - Fugue (2014, uploaded 2019)
- Canon on «Twinkle Twinkle Little Star» in 50-edo, for Organ (≤2014, restored and re-hosted 2025)
- Fugue on the Dragnet theme (2014)
- Blue Fugue for Organ (2018)
- Happy Birthday Canon, 6-in-1 Canon in 50edo (2019)
- Fantasia Catalana (2020)
- Preludi Nocturn i Fuga sobre la Lluna la Pruna (2020)
- Canon at the Semitone on The Mother's Malison Theme, for Organ (2022)
- Fugue on an Original Theme, for Baroque Ensemble (2023) (for Organ)
- Catalan Fugue (La Santa Espina) (2023)
- Canon in C= for Baroque Wind Ensemble (2023)
- 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)
- organ rendition (2023)
- harpsichord rendition (2024)
- Baroque Blues - 4-Part Fugue for Baroque Consort (2026)
- Fugue on the French Lullaby «La Petite Poule Grise» (2026)
- the late little xmas album (2014)
- Harpsichord meantone improvisation 1 in 50EDO (2014)
- Long improvisation 2 in 50EDO (2014)
- Chord sequence for Difference tones in 50EDO (2014)
- Enharmonic Modulations in 50EDO (2014)
- Harmonic Clusters on 50EDO Harpsichord (2014)
- Fragment in Fifty (2014)