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{{Infobox ET}}
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{{ED intro}}


'''51 EDO''' divides the [[octave]] into 51 equal parts of 23.5 [[cent]]s each, which is about the size of the [[Pythagorean comma]] (even though this comma itself is mapped to a different interval).
== Theory ==
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.
 
Alternatively, using the 51c val {{val| 51 81 '''119''' 143 }}, the [[5/4]] is mapped to 1\3 (400 cents), [[support]]ing [[augmented]]. In the 7-limit it tempers out [[245/243]] and supports [[hemiaug]] and [[rodan]]. The 51cd val {{val| 51 81 '''119''' '''144''' }} takes the same [[7/4]] from 17edo, and supports [[augene]].
 
51edo's step is the closest direct approximation to the [[Pythagorean comma]] by edo steps, though that comma itself is mapped to a different interval.


== Theory ==
=== Odd harmonics ===
{{Harmonics in equal|51}}
 
=== Subsets and supersets ===
51edo contains [[3edo]] and [[17edo]] as subsets.


{{primes in edo|51}}
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.
51 EDO 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 [[Porcupine rank three family #Sonic|sonic]], the rank three temperament tempering out 250/243, 55/54 and 100/99, and also for the rank four temperament tempering out 55/54. It provides an alternative tuning to [[22edo]] for [[Porcupine family #Porcupine|porcupine temperament]], with a nice fifth but a rather flat major third, and the optimal patent val for 7 and 11-limit [[Porcupine family #Porky|porky temperament]], which is sonic plus 225/224. 51 contains an [[6L 1s|Archeotonic]] 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.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! #
! [[Cents|Cents]]
! [[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 23: Line 33:
|-
|-
| 1
| 1
| 23.529
| 23.5
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 29: Line 39:
|-
|-
| 2
| 2
| 47.059
| 47.1
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 35: Line 45:
|-
|-
| 3
| 3
| 70.588
| 70.6
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 41: Line 51:
|-
|-
| 4
| 4
| 94.118
| 94.1
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 47: Line 57:
|-
|-
| 5
| 5
| 117.647
| 117.6
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 53: Line 63:
|-
|-
| 6
| 6
| 141.176
| 141.2
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 59: Line 69:
|-
|-
| 7
| 7
| 164.706
| 164.7
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 65: Line 75:
|-
|-
| 8
| 8
| 188.235
| 188.2
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 71: Line 81:
|-
|-
| 9
| 9
| 211.765
| 211.8
| Major 2nd
| Major 2nd
| M2
| M2
Line 77: Line 87:
|-
|-
| 10
| 10
| 235.294
| 235.3
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 83: Line 93:
|-
|-
| 11
| 11
| 258.824
| 258.8
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 89: Line 99:
|-
|-
| 12
| 12
| 282.353
| 282.4
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 95: Line 105:
|-
|-
| 13
| 13
| 305.882
| 305.9
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 101: Line 111:
|-
|-
| 14
| 14
| 329.412
| 329.4
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 107: Line 117:
|-
|-
| 15
| 15
| 352.941
| 352.9
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 113: Line 123:
|-
|-
| 16
| 16
| 376.471
| 376.5
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 119: Line 129:
|-
|-
| 17
| 17
| 400.000
| 400.0
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 125: Line 135:
|-
|-
| 18
| 18
| 423.529
| 423.5
| Major 3rd
| Major 3rd
| M3
| M3
Line 131: Line 141:
|-
|-
| 19
| 19
| 447.509
| 447.1
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 137: Line 147:
|-
|-
| 20
| 20
| 470.588
| 470.6
| Down 4th
| Down 4th
| v4
| v4
Line 143: Line 153:
|-
|-
| 21
| 21
| 494.118
| 494.1
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 149: Line 159:
|-
|-
| 22
| 22
| 517.647
| 517.6
| Up 4th
| Up 4th
| ^4
| ^4
Line 155: Line 165:
|-
|-
| 23
| 23
| 541.176
| 541.2
| Downdim 5th
| Downdim 5th
| vd5
| vd5
Line 161: Line 171:
|-
|-
| 24
| 24
| 564.706
| 564.7
| Dim 5th
| Dim 5th
| d5
| d5
Line 167: Line 177:
|-
|-
| 25
| 25
| 588.235
| 588.2
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 173: Line 183:
|-
|-
| 26
| 26
| 611.765
| 611.8
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 179: Line 189:
|-
|-
| 27
| 27
| 635.294
| 635.3
| Aug 4th
| Aug 4th
| A4
| A4
Line 185: Line 195:
|-
|-
| 28
| 28
| 658.824
| 658.8
| Upaug 4th
| Upaug 4th
| ^A4
| ^A4
Line 191: Line 201:
|-
|-
| 29
| 29
| 682.353
| 682.4
| Down 5th
| Down 5th
| v5
| v5
Line 197: Line 207:
|-
|-
| 30
| 30
| 705.882
| 705.9
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 203: Line 213:
|-
|-
| 31
| 31
| 729.412
| 729.4
| Up 5th
| Up 5th
| ^5
| ^5
Line 209: Line 219:
|-
|-
| 32
| 32
| 752.941
| 752.9
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 215: Line 225:
|-
|-
| 33
| 33
| 776.471
| 776.5
| Minor 6th
| Minor 6th
| m6
| m6
Line 221: Line 231:
|-
|-
| 34
| 34
| 800.000
| 800.0
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 227: Line 237:
|-
|-
| 35
| 35
| 823.529
| 823.5
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 233: Line 243:
|-
|-
| 36
| 36
| 847.059
| 847.1
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 239: Line 249:
|-
|-
| 37
| 37
| 870.588
| 870.6
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 245: Line 255:
|-
|-
| 38
| 38
| 894.118
| 894.1
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 251: Line 261:
|-
|-
| 39
| 39
| 917.647
| 917.6
| Major 6th
| Major 6th
| M6
| M6
Line 257: Line 267:
|-
|-
| 40
| 40
| 941.176
| 941.2
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 263: Line 273:
|-
|-
| 41
| 41
| 964.706
| 964.7
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 269: Line 279:
|-
|-
| 42
| 42
| 988.235
| 988.2
| Minor 7th
| Minor 7th
| m7
| m7
Line 275: Line 285:
|-
|-
| 43
| 43
| 1011.765
| 1011.8
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 281: Line 291:
|-
|-
| 44
| 44
| 1035.294
| 1035.3
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 287: Line 297:
|-
|-
| 45
| 45
| 1058.824
| 1058.8
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 293: Line 303:
|-
|-
| 46
| 46
| 1082.353
| 1082.4
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 299: Line 309:
|-
|-
| 47
| 47
| 1105.882
| 1105.9
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 305: Line 315:
|-
|-
| 48
| 48
| 1129.412
| 1129.4
| Major 7th
| Major 7th
| M7
| M7
Line 311: Line 321:
|-
|-
| 49
| 49
| 1152.941
| 1152.9
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 317: Line 327:
|-
|-
| 50
| 50
| 1176.471
| 1176.5
| Down 8ve
| Down 8ve
| v8
| v8
Line 323: Line 333:
|-
|-
| 51
| 51
| 1200.000
| 1200.0
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
|}
|}
== Notation ==
=== Ups and downs notation ===
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}}
If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
{{Sharpness-sharp6-iw}}
=== Sagittal notation ===
In the following diagrams, 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.
==== Evo flavor ====
<imagemap>
File:51-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 519 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[27/26]]
</imagemap>
==== Revo flavor ====
<imagemap>
File:51-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[27/26]]
default [[File:51-EDO_Revo_Sagittal.svg]]
</imagemap>
==== Evo-SZ flavor ====
<imagemap>
File:51-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[27/26]]
default [[File:51-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.7
| 1029/1024, {{monzo| 17 -16 3 }}
| {{Mapping| 51 81 143 }}
| −0.339
| 1.63
| 6.92
|- style="border-top: double;"
| 2.3.5
| 128/125, {{monzo| -13 17 -6 }}
| {{Mapping| 51 81 119 }} (51c)
| −2.789
| 2.41
| 10.3
|- style="border-top: double;"
| 2.3.5
| 250/243, 34171875/33554432
| {{Mapping| 51 81 118 }} (51)
| +0.581
| 2.77
| 11.8
|}
=== 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
| 5\51
| 117.6
| 15/14
| [[Miracle]] (51e, out of tune) / oracle (51)
|-
| 1
| 7\51
| 164.7
| 11/10
| [[Porky]] (51)
|-
| 1
| 10\51
| 235.3
| 8/7
| [[Rodan]] (51cf…, out of tune) / aerodino (51ce)
|-
| 1
| 5\51
| 541.2
| 15/11
| [[Necromanteion]] (51ce)
|-
| 3
| 19\51<br>(2\51)
| 447.1<br>(47.1)
| 9/7<br>(36/35)
| [[Hemiaug]] (51ce)
|-
| 3
| 21\51<br>(4\51)
| 494.1<br>(94.1)
| 4/3<br>(16/15)
| [[Augmented]] (51c)
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
== Scales ==
* [[Porky]][7] (Palace{{idio}}): 7 7 7 9 7 7 7
* 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{{idio}} subset of UFO scale): 21 9 4 9 8
== Instruments ==
; Lumatone
: See [[Lumatone mapping for 51edo]].


== Music ==
== Music ==
'''James Mulvale (FAST-fast)'''
; [[Bryan Deister]]
* [https://youtu.be/8GojBZSyqDw STARS (Thoughts and Prayers)] (51 EDO for 51 States)
* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025)
'''[[Ray Perlner]]'''
 
* [https://www.youtube.com/watch?v=peidZ1jEafQ Fugue for Organ in 51EDO Porcupine[7] ssssssL “Pandian”]
; [[Frédéric Gagné]]
* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – 7:4 [[semiquartal]] 4|4 mode
 
; [[James Mulvale]] (FASTFAST)
* [https://youtu.be/8GojBZSyqDw ''STARS (Thoughts and Prayers)''] (2020)
 
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=peidZ1jEafQ ''Fugue''] (2023) – for organ in 51edo Porcupine[7] ssssssL "Pandian"


[[Category:51edo]]
[[Category:Listen]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
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