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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|51}}
{{ED intro}}


== Theory ==
== Theory ==
51 = 3 × 17, and 51edo shares its [[3/2|fifth]] with [[17edo]].  
Since {{nowrap|51 {{=}} 3 × 17}}, 51edo shares its [[3/2|fifth]] with [[17edo]]. Compared to other multiples of 17edo, notably [[34edo]] and [[68edo]], 51edo's harmonic inventory seems lacking, getting few harmonics very well considering its step size. However, it does possess excellent approximations of [[11/10]] and [[21/16]], only about 0.3 cents off in each case.


Using the [[patent val]], 51et [[tempering out|tempers out]] [[250/243]] in the [[5-limit]], [[225/224]] and [[2401/2400]] in the [[7-limit]], and [[55/54]] and [[100/99]] in the [[11-limit]]. It is the [[optimal patent val]] for [[sonic]], the rank-3 temperament tempering out 55/54, 100/99, and 250/243, and also for the rank-4 temperament tempering out 55/54. It provides an alternative tuning to [[22edo]] for [[porcupine]], with a nice fifth but a rather flat major third, and the optimal patent val for the 7- and 11-limit [[porky]] temperament, which is sonic plus 225/224. 51 contains an archeotonic [[6L 1s]] scale based on repetitions of 8\51, creating a scale with a whole-tone-like drive towards the tonic through the 17edo semitone at the top.  
Using the [[patent val]], 51et [[tempering out|tempers out]] [[250/243]] in the [[5-limit]], [[225/224]] and [[2401/2400]] in the [[7-limit]], and [[55/54]] and [[100/99]] in the [[11-limit]]. It is the [[optimal patent val]] for [[sonic]], the rank-3 temperament tempering out 55/54, 100/99, and 250/243, and also for the rank-4 temperament tempering out 55/54. It provides an alternative tuning to [[22edo]] for [[porcupine]], with a nice fifth but a rather flat major third, and the optimal patent val for the 7- and 11-limit [[porky]] temperament, which is sonic plus 225/224. 51 contains an archeotonic [[6L 1s]] scale based on repetitions of 8\51, creating a scale with a whole-tone-like drive towards the tonic through the 17edo semitone at the top.  
Line 15: Line 15:


=== Subsets and supersets ===
=== Subsets and supersets ===
51edo contains [[3edo]] and [[17edo]] as subsets. A step of 51edo is exactly 12 [[skisma]]s.
51edo contains [[3edo]] and [[17edo]] as subsets.
 
One of the very powerful (but very complex) supersets of 51edo is [[612edo]], which divides each step of 51edo into 12 equal parts, for which the name "skisma" has been proposed.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! #
! [[Cent]]s
! [[Cent]]s
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 31: Line 33:
|-
|-
| 1
| 1
| 23.529
| 23.5
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 37: Line 39:
|-
|-
| 2
| 2
| 47.059
| 47.1
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 43: Line 45:
|-
|-
| 3
| 3
| 70.588
| 70.6
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 49: Line 51:
|-
|-
| 4
| 4
| 94.118
| 94.1
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 55: Line 57:
|-
|-
| 5
| 5
| 117.647
| 117.6
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 61: Line 63:
|-
|-
| 6
| 6
| 141.176
| 141.2
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 67: Line 69:
|-
|-
| 7
| 7
| 164.706
| 164.7
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 73: Line 75:
|-
|-
| 8
| 8
| 188.235
| 188.2
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 79: Line 81:
|-
|-
| 9
| 9
| 211.765
| 211.8
| Major 2nd
| Major 2nd
| M2
| M2
Line 85: Line 87:
|-
|-
| 10
| 10
| 235.294
| 235.3
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 91: Line 93:
|-
|-
| 11
| 11
| 258.824
| 258.8
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 97: Line 99:
|-
|-
| 12
| 12
| 282.353
| 282.4
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 103: Line 105:
|-
|-
| 13
| 13
| 305.882
| 305.9
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 109: Line 111:
|-
|-
| 14
| 14
| 329.412
| 329.4
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 115: Line 117:
|-
|-
| 15
| 15
| 352.941
| 352.9
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 121: Line 123:
|-
|-
| 16
| 16
| 376.471
| 376.5
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 127: Line 129:
|-
|-
| 17
| 17
| 400.000
| 400.0
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 133: Line 135:
|-
|-
| 18
| 18
| 423.529
| 423.5
| Major 3rd
| Major 3rd
| M3
| M3
Line 139: Line 141:
|-
|-
| 19
| 19
| 447.509
| 447.1
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 145: Line 147:
|-
|-
| 20
| 20
| 470.588
| 470.6
| Down 4th
| Down 4th
| v4
| v4
Line 151: Line 153:
|-
|-
| 21
| 21
| 494.118
| 494.1
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 157: Line 159:
|-
|-
| 22
| 22
| 517.647
| 517.6
| Up 4th
| Up 4th
| ^4
| ^4
Line 163: Line 165:
|-
|-
| 23
| 23
| 541.176
| 541.2
| Downdim 5th
| Downdim 5th
| vd5
| vd5
Line 169: Line 171:
|-
|-
| 24
| 24
| 564.706
| 564.7
| Dim 5th
| Dim 5th
| d5
| d5
Line 175: Line 177:
|-
|-
| 25
| 25
| 588.235
| 588.2
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 181: Line 183:
|-
|-
| 26
| 26
| 611.765
| 611.8
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 187: Line 189:
|-
|-
| 27
| 27
| 635.294
| 635.3
| Aug 4th
| Aug 4th
| A4
| A4
Line 193: Line 195:
|-
|-
| 28
| 28
| 658.824
| 658.8
| Upaug 4th
| Upaug 4th
| ^A4
| ^A4
Line 199: Line 201:
|-
|-
| 29
| 29
| 682.353
| 682.4
| Down 5th
| Down 5th
| v5
| v5
Line 205: Line 207:
|-
|-
| 30
| 30
| 705.882
| 705.9
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 211: Line 213:
|-
|-
| 31
| 31
| 729.412
| 729.4
| Up 5th
| Up 5th
| ^5
| ^5
Line 217: Line 219:
|-
|-
| 32
| 32
| 752.941
| 752.9
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 223: Line 225:
|-
|-
| 33
| 33
| 776.471
| 776.5
| Minor 6th
| Minor 6th
| m6
| m6
Line 229: Line 231:
|-
|-
| 34
| 34
| 800.000
| 800.0
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 235: Line 237:
|-
|-
| 35
| 35
| 823.529
| 823.5
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 241: Line 243:
|-
|-
| 36
| 36
| 847.059
| 847.1
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 247: Line 249:
|-
|-
| 37
| 37
| 870.588
| 870.6
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 253: Line 255:
|-
|-
| 38
| 38
| 894.118
| 894.1
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 259: Line 261:
|-
|-
| 39
| 39
| 917.647
| 917.6
| Major 6th
| Major 6th
| M6
| M6
Line 265: Line 267:
|-
|-
| 40
| 40
| 941.176
| 941.2
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 271: Line 273:
|-
|-
| 41
| 41
| 964.706
| 964.7
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 277: Line 279:
|-
|-
| 42
| 42
| 988.235
| 988.2
| Minor 7th
| Minor 7th
| m7
| m7
Line 283: Line 285:
|-
|-
| 43
| 43
| 1011.765
| 1011.8
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 289: Line 291:
|-
|-
| 44
| 44
| 1035.294
| 1035.3
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 295: Line 297:
|-
|-
| 45
| 45
| 1058.824
| 1058.8
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 301: Line 303:
|-
|-
| 46
| 46
| 1082.353
| 1082.4
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 307: Line 309:
|-
|-
| 47
| 47
| 1105.882
| 1105.9
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 313: Line 315:
|-
|-
| 48
| 48
| 1129.412
| 1129.4
| Major 7th
| Major 7th
| M7
| M7
Line 319: Line 321:
|-
|-
| 49
| 49
| 1152.941
| 1152.9
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 325: Line 327:
|-
|-
| 50
| 50
| 1176.471
| 1176.5
| Down 8ve
| Down 8ve
| v8
| v8
Line 331: Line 333:
|-
|-
| 51
| 51
| 1200.000
| 1200.0
| Perfect 8ve
| Perfect 8ve
| P8
| P8
Line 339: Line 341:
== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Ups and downs notation ===
In 51edo, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used to fill in the gaps.
51edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
 
In 51edo, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.


{{Sharpness-sharp6}}
{{Sharpness-sharp6}}
Line 473: Line 481:


== Scales ==
== Scales ==
* [[Porky]][7], Palace: 7 7 7 9 7 7 7
* [[Porky]][7] (Palace{{idio}}): 7 7 7 9 7 7 7
* UFO scale ([[inflected MOS]] of [[Batch 89 temperaments#Teefs|Teefs]][19]): 2 2 4 1 2 2 2 4 2 5 2 4 4 2 2 1 4 2 2
* UFO scale{{idio}} ([[inflected MOS]] of [[Batch 89 temperaments#Teefs|Teefs]][19]{{idio}}): 2 2 4 1 2 2 2 4 2 5 2 4 4 2 2 1 4 2 2
* Cosmic scale (subset of UFO scale): 21 9 4 9 8
* Cosmic scale{{idio}} subset of UFO scale): 21 9 4 9 8


== Instruments ==
== Instruments ==
Line 482: Line 490:


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025)
; [[Frédéric Gagné]]
; [[Frédéric Gagné]]
* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – 7:4 [[semiquartal]] 4|4 mode
* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – 7:4 [[semiquartal]] 4|4 mode
Retrieved from "https://en.xen.wiki/w/51edo"