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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 [[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.  
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. It 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]].  
Using the 51c val {{val| 51 81 '''119''' 143 }}, the [[5/4]] is mapped to 1\3 (400 cents), [[support]]ing [[augmented (temperament)|augmented]]. In the 7-limit it tempers out [[245/243]] and supports [[hemiaug]] and [[rodan]]. Alternatively, the 51cd val {{val| 51 81 '''119''' '''144''' }} takes the same [[7/4]] from 17edo, and supports [[augene]]. The 51ce val {{val| 51 81 '''119''' 143 '''177''' 189 }} supports a variant of rodan called [[Gamelismic_clan#Aerodino|aerodino]].  


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.
51edo's step is the closest direct approximation to the [[Pythagorean comma]] by edosteps, though that comma itself is mapped to a different interval.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|51}}
{{Harmonics in equal|51|intervals=odd|prec=2|columns=14}}
{{Harmonics in equal|51|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 51edo (continued)}}


=== 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-1 right-2 center-6 center-7 center-8"
|-
|-
! Degrees
! rowspan="2" | #
! [[Cent]]s
! rowspan="2" | [[Cent]]s
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | Approximate ratios*
! rowspan="2" colspan="3" | [[Ups and downs notation]]
|-
! 2.3.7.11/5.13<br>subgroup
! Ratios of 5 and 11<br>tending flat (51 val)
! Ratios of 5 and 11<br>tending sharp (51ce val)
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 31: Line 42:
|-
|-
| 1
| 1
| 23.529
| 23.5
| [[64/63]], ''[[49/48]]''
| ''40/39''
| [[81/80]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 37: Line 51:
|-
|-
| 2
| 2
| 47.059
| 47.1
| ''[[28/27]]''
| [[33/32]], ''25/24'', ''81/80''
| [[36/35]], [[40/39]]
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 43: Line 60:
|-
|-
| 3
| 3
| 70.588
| 70.6
| [[27/26]]
| ''36/35''
| ''21/20'', ''33/32''
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 49: Line 69:
|-
|-
| 4
| 4
| 94.118
| 94.1
|
| [[21/20]]
| ''16/15'', ''25/24''
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 55: Line 78:
|-
|-
| 5
| 5
| 117.647
| 117.6
| [[14/13]]
| [[15/14]], [[16/15]]
|
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 61: Line 87:
|-
|-
| 6
| 6
| 141.176
| 141.2
| [[13/12]]
|
| [[12/11]], ''15/14''
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 67: Line 96:
|-
|-
| 7
| 7
| 164.706
| 164.7
| [[11/10]]
| ''10/9'', ''12/11''
|
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 73: Line 105:
|-
|-
| 8
| 8
| 188.235
| 188.2
|
|
| [[10/9]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 79: Line 114:
|-
|-
| 9
| 9
| 211.765
| 211.8
| [[9/8]]
|
|
| Major 2nd
| Major 2nd
| M2
| M2
Line 85: Line 123:
|-
|-
| 10
| 10
| 235.294
| 235.3
| [[8/7]]
| ''15/13''
|
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 91: Line 132:
|-
|-
| 11
| 11
| 258.824
| 258.8
| [[7/6]]
|
| [[15/13]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 97: Line 141:
|-
|-
| 12
| 12
| 282.353
| 282.4
| ''[[32/27]]''
|
| [[13/11]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 103: Line 150:
|-
|-
| 13
| 13
| 305.882
| 305.9
|
| ''13/11''
| [[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 109: Line 159:
|-
|-
| 14
| 14
| 329.412
| 329.4
| [[40/33]], [[63/52]]
| ''6/5'', ''11/9''
|
| Downmid 3rd
| Downmid 3rd
| v~3
| v~3
Line 115: Line 168:
|-
|-
| 15
| 15
| 352.941
| 352.9
| [[16/13]], [[39/32]]
|
| [[11/9]], [[27/22]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 121: Line 177:
|-
|-
| 16
| 16
| 376.471
| 376.5
| [[26/21]]
| [[5/4]], ''27/22''
|
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 127: Line 186:
|-
|-
| 17
| 17
| 400.000
| 400.0
|
|
| ''5/4'', ''14/11''
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 133: Line 195:
|-
|-
| 18
| 18
| 423.529
| 423.5
| ''[[81/64]]''
| [[14/11]]
|
| Major 3rd
| Major 3rd
| M3
| M3
Line 139: Line 204:
|-
|-
| 19
| 19
| 447.509
| 447.1
| ''[[9/7]]''
|
| [[13/10]]
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 145: Line 213:
|-
|-
| 20
| 20
| 470.588
| 470.6
| [[21/16]]
| ''13/10''
|
| Down 4th
| Down 4th
| v4
| v4
Line 151: Line 222:
|-
|-
| 21
| 21
| 494.118
| 494.1
| [[4/3]]
|
|
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 157: Line 231:
|-
|-
| 22
| 22
| 517.647
| 517.6
|
|
| [[27/20]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 163: Line 240:
|-
|-
| 23
| 23
| 541.176
| 541.2
| [[15/11]]
| [[11/8]], ''27/20''
|
| Downdim 5th
| Downdim 5th
| vd5
| vd5
Line 169: Line 249:
|-
|-
| 24
| 24
| 564.706
| 564.7
| [[18/13]]
|
| ''7/5'', ''11/8''
| Dim 5th
| Dim 5th
| d5
| d5
Line 175: Line 258:
|-
|-
| 25
| 25
| 588.235
| 588.2
| [[39/28]]
| [[7/5]]
|
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 181: Line 267:
|-
|-
| 26
| 26
| 611.765
| 611.8
| [[56/39]]
| [[10/7]]
|
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 187: Line 276:
|-
|-
| 27
| 27
| 635.294
| 635.3
| [[13/9]]
|
| ''10/7'', ''16/11''
| Aug 4th
| Aug 4th
| A4
| A4
Line 193: Line 285:
|-
|-
| 28
| 28
| 658.824
| 658.8
| [[22/15]]
| [[16/11]], ''40/27''
|
| Upaug 4th
| Upaug 4th
| ^A4
| ^A4
Line 199: Line 294:
|-
|-
| 29
| 29
| 682.353
| 682.4
|
|
| [[40/27]]
| Down 5th
| Down 5th
| v5
| v5
Line 205: Line 303:
|-
|-
| 30
| 30
| 705.882
| 705.9
| [[3/2]]
|
|
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 211: Line 312:
|-
|-
| 31
| 31
| 729.412
| 729.4
| [[32/21]]
| ''20/13''
|
| Up 5th
| Up 5th
| ^5
| ^5
Line 217: Line 321:
|-
|-
| 32
| 32
| 752.941
| 752.9
| ''[[14/9]]''
|
| [[20/13]]
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 223: Line 330:
|-
|-
| 33
| 33
| 776.471
| 776.5
| ''[[128/81]]''
| [[11/7]]
|
| Minor 6th
| Minor 6th
| m6
| m6
Line 229: Line 339:
|-
|-
| 34
| 34
| 800.000
| 800.0
|
|
| ''8/5'', ''11/7''
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 235: Line 348:
|-
|-
| 35
| 35
| 823.529
| 823.5
| [[21/13]]
| [[8/5]], ''44/27''
|
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 241: Line 357:
|-
|-
| 36
| 36
| 847.059
| 847.1
| [[13/8]], [[64/39]]
|
| [[18/11]], [[44/27]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 247: Line 366:
|-
|-
| 37
| 37
| 870.588
| 870.6
| [[33/20]], [[104/63]]
| ''5/3'', ''18/11''
|
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 253: Line 375:
|-
|-
| 38
| 38
| 894.118
| 894.1
|
| ''22/13''
| [[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 259: Line 384:
|-
|-
| 39
| 39
| 917.647
| 917.6
| ''[[27/16]]''
|
| [[22/13]]
| Major 6th
| Major 6th
| M6
| M6
Line 265: Line 393:
|-
|-
| 40
| 40
| 941.176
| 941.2
| [[12/7]]
|
| [[26/15]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 271: Line 402:
|-
|-
| 41
| 41
| 964.706
| 964.7
| [[7/4]]
| ''26/15''
|
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 277: Line 411:
|-
|-
| 42
| 42
| 988.235
| 988.2
| [[16/9]]
|
|
| Minor 7th
| Minor 7th
| m7
| m7
Line 283: Line 420:
|-
|-
| 43
| 43
| 1011.765
| 1011.8
|
|
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 289: Line 429:
|-
|-
| 44
| 44
| 1035.294
| 1035.3
| [[20/11]]
|
| ''9/5'', ''11/6''
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 295: Line 438:
|-
|-
| 45
| 45
| 1058.824
| 1058.8
| [[24/13]]
|
| [[11/6]], ''28/15''
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 301: Line 447:
|-
|-
| 46
| 46
| 1082.353
| 1082.4
| [[13/7]]
| [[15/8]], [[28/15]]
|
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 307: Line 456:
|-
|-
| 47
| 47
| 1105.882
| 1105.9
|
| [[40/21]]
| ''15/8'', ''48/25''
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 313: Line 465:
|-
|-
| 48
| 48
| 1129.412
| 1129.4
| [[52/27]]
| ''35/18''
| ''40/21'', ''64/33''
| Major 7th
| Major 7th
| M7
| M7
Line 319: Line 474:
|-
|-
| 49
| 49
| 1152.941
| 1152.9
| ''[[27/14]]''
| [[64/33]], ''48/25'', ''160/81''
| [[35/18]], [[39/20]]
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 325: Line 483:
|-
|-
| 50
| 50
| 1176.471
| 1176.5
| [[63/32]], ''[[96/49]]''
| ''39/20''
| [[160/81]]
| Down 8ve
| Down 8ve
| v8
| v8
Line 331: Line 492:
|-
|-
| 51
| 51
| 1200.000
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
|}
|}
<nowiki>*</nowiki> inconsistent intervals in italic.


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
In 58edo, 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.
[[Stein–Zimmermann–Gould notation]] for 51edo uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}


{{Sharpness-sharp6}}
If double arrows are not desirable, then arrows can be attached to quartertone accidentals:
{{Sharpness-sharp6-qt-szg}}


If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
=== Kite's ups and downs notation ===
51edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}


{{Sharpness-sharp6-qt}}
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}


=== Ivan Wyschnegradsky's notation ===
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
{{Sharpness-sharp6-iw}}
{{Sharpness-sharp6-iw}}


=== Sagittal notation ===
=== 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.
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 ====
==== Evo flavor ====
Line 389: Line 558:
default [[File:51-EDO_Evo-SZ_Sagittal.svg]]
default [[File:51-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
</imagemap>
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|51}}
{{Q-odd-limit intervals|51.1|apx=val|header=none|tag=none|title=15-odd-limit intervals in 51edo (51ce val mapping)}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 405: Line 579:
| 1029/1024, {{monzo| 17 -16 3 }}
| 1029/1024, {{monzo| 17 -16 3 }}
| {{Mapping| 51 81 143 }}
| {{Mapping| 51 81 143 }}
| -0.339
| −0.339
| 1.63
| 1.63
| 6.92
| 6.92
|-
| 2.3.7.13
| 343/338, 512/507, 2197/2187
| {{Mapping| 51 81 143 }}
| −0.695
| 1.54
| 6.54
|- style="border-top: double;"
|- style="border-top: double;"
| 2.3.5
| 2.3.5
| 128/125, {{monzo| -13 17 -6 }}
| 128/125, {{monzo| -13 17 -6 }}
| {{Mapping| 51 81 119 }} (51c)
| {{Mapping| 51 81 119 }} (51c)
| -2.789
| −2.789
| 2.41
| 2.41
| 10.3
| 10.3
|-
| 2.3.5.7
| 128/125, 245/243, 1029/1000
| {{Mapping| 51 81 119 143 }} (51c)
| −1.730
| 2.79
| 11.9
|- style="border-top: double;"
|- style="border-top: double;"
| 2.3.5
| 2.3.5
Line 422: Line 610:
| 2.77
| 2.77
| 11.8
| 11.8
|-
| 2.3.5.7
| 225/224, 250/243, 1029/1024
| {{Mapping| 51 81 118 143 }} (51)
| +0.803
| 2.43
| 10.3
|}
|}


Line 438: Line 633:
| 117.6
| 117.6
| 15/14
| 15/14
| [[Miracle]] (51e, out of tune) / oracle (51)
| [[Miracle]] (51e, out of tune)
|-
|-
| 1
| 1
Line 450: Line 645:
| 235.3
| 235.3
| 8/7
| 8/7
| [[Rodan]] (51cf…, out of tune) / aerodino (51ce)
| [[Rodan]] (51cf, out of tune) / aerodino (51ce)
|-
| 1
| 19\51
| 447.1
| 13/10
| [[Supersensi]] (51cde)
|-
| 1
| 22\51
| 517.6
| 27/20
| [[Gravity]] (51ce) / [[abergravity]] (51ce)
|-
|-
| 1
| 1
| 5\51
| 23\51
| 541.2
| 541.2
| 15/11
| 15/11
| [[Necromanteion]] (51ce)
| [[Necromanteion]] (51ce)<br>[[Oracle]] (51)<br>[[Cypress]] (51cde…)
|-
|-
| 3
| 3
Line 464: Line 671:
| [[Hemiaug]] (51ce)
| [[Hemiaug]] (51ce)
|-
|-
| 3
| rowspan="2" | 3
| 21\51<br>(4\51)
| rowspan="2" | 21\51<br>(4\51)
| 494.1<br>(94.1)
| rowspan="2" | 494.1<br>(94.1)
| 4/3<br>(16/15)
| 4/3<br>(16/15)
| [[Augmented]] (51c)
| [[Augmented (temperament)|Augmented]] (7-limit, 51cd)
|-
| style="text-align: center;" | 4/3<br>(21/20)
| style="text-align: left;" | [[Fog]] (51)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
<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


== 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 692:


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=sySLQUXnQ70 ''Preludio Sentimentale (microtonal improvisation in 28edo)''] (2023)
* [https://www.youtube.com/watch?v=sCE0MjUyRUk ''28edo blues''] (2023)
* [https://www.youtube.com/shorts/sTPJtuHUwkg ''51edo improv''] (2025-02-03)
* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025-05-02)
* [https://www.youtube.com/shorts/Fymg9vYO6iQ ''Northernlight - Deltarune (microtonal cover in 51edo)''] (2025)
* [https://www.youtube.com/shorts/SJW-JTHyeIA ''51edo prelude''] (2026)
* [https://www.youtube.com/watch?v=k3NOBYbiqpo ''51edo improv''] (2026-04-22)
; [[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
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