51edo: Difference between revisions

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Line 1: Line 1:
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 17
{{ED intro}}
| Step size = 23.52941¢
| Fifth = 30\51 ≈ 706¢
| Major 2nd = 9\51 ≈ 212¢
| Minor 2nd = 3\51 ≈ 71¢
| Augmented 1sn = 6\51 ≈ 141¢
}}


'''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. 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.
 
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]].  


== Theory ==
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 ===
{{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 ===
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-1 right-2 center-6 center-7 center-8"
{| class="wikitable center-all right-2 left-3"
|-
! rowspan="2" | #
! rowspan="2" | [[Cent]]s
! colspan="3" | Approximate ratios*
! rowspan="2" colspan="3" | [[Ups and downs notation]]
|-
|-
! Degrees
! 2.3.7.11/5.13<br>subgroup
! [[Cents|Cents]]
! Ratios of 5 and 11<br>tending flat (51 val)
! colspan="3" | [[Ups and Downs Notation]]
! 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 30: Line 42:
|-
|-
| 1
| 1
| 23.529
| 23.5
| [[64/63]], ''[[49/48]]''
| ''40/39''
| [[81/80]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 36: 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 42: Line 60:
|-
|-
| 3
| 3
| 70.588
| 70.6
| [[27/26]]
| ''36/35''
| ''21/20'', ''33/32''
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 48: Line 69:
|-
|-
| 4
| 4
| 94.118
| 94.1
|
| [[21/20]]
| ''16/15'', ''25/24''
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 54: Line 78:
|-
|-
| 5
| 5
| 117.647
| 117.6
| [[14/13]]
| [[15/14]], [[16/15]]
|
| Downmid 2nd
| Downmid 2nd
| v~2
| v~2
Line 60: Line 87:
|-
|-
| 6
| 6
| 141.176
| 141.2
| [[13/12]]
|
| [[12/11]], ''15/14''
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 66: Line 96:
|-
|-
| 7
| 7
| 164.706
| 164.7
| [[11/10]]
| ''10/9'', ''12/11''
|
| Upmid 2nd
| Upmid 2nd
| ^~2
| ^~2
Line 72: Line 105:
|-
|-
| 8
| 8
| 188.235
| 188.2
|
|
| [[10/9]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 78: Line 114:
|-
|-
| 9
| 9
| 211.765
| 211.8
| [[9/8]]
|
|
| Major 2nd
| Major 2nd
| M2
| M2
Line 84: Line 123:
|-
|-
| 10
| 10
| 235.294
| 235.3
| [[8/7]]
| ''15/13''
|
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 90: Line 132:
|-
|-
| 11
| 11
| 258.824
| 258.8
| [[7/6]]
|
| [[15/13]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 96: Line 141:
|-
|-
| 12
| 12
| 282.353
| 282.4
| ''[[32/27]]''
|
| [[13/11]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 102: Line 150:
|-
|-
| 13
| 13
| 305.882
| 305.9
|
| ''13/11''
| [[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 108: 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 114: Line 168:
|-
|-
| 15
| 15
| 352.941
| 352.9
| [[16/13]], [[39/32]]
|
| [[11/9]], [[27/22]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 120: Line 177:
|-
|-
| 16
| 16
| 376.471
| 376.5
| [[26/21]]
| [[5/4]], ''27/22''
|
| Upmid 3rd
| Upmid 3rd
| ^~3
| ^~3
Line 126: Line 186:
|-
|-
| 17
| 17
| 400.000
| 400.0
|
|
| ''5/4'', ''14/11''
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 132: Line 195:
|-
|-
| 18
| 18
| 423.529
| 423.5
| ''[[81/64]]''
| [[14/11]]
|
| Major 3rd
| Major 3rd
| M3
| M3
Line 138: Line 204:
|-
|-
| 19
| 19
| 447.509
| 447.1
| ''[[9/7]]''
|
| [[13/10]]
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 144: Line 213:
|-
|-
| 20
| 20
| 470.588
| 470.6
| [[21/16]]
| ''13/10''
|
| Down 4th
| Down 4th
| v4
| v4
Line 150: Line 222:
|-
|-
| 21
| 21
| 494.118
| 494.1
| [[4/3]]
|
|
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 156: Line 231:
|-
|-
| 22
| 22
| 517.647
| 517.6
|
|
| [[27/20]]
| Up 4th
| Up 4th
| ^1
| ^4
| ^G
| ^G
|-
|-
| 23
| 23
| 541.176
| 541.2
| [[15/11]]
| [[11/8]], ''27/20''
|
| Downdim 5th
| Downdim 5th
| vd5
| vd5
Line 168: Line 249:
|-
|-
| 24
| 24
| 564.706
| 564.7
| [[18/13]]
|
| ''7/5'', ''11/8''
| Dim 5th
| Dim 5th
| d5
| d5
Line 174: Line 258:
|-
|-
| 25
| 25
| 588.235
| 588.2
| [[39/28]]
| [[7/5]]
|
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 180: Line 267:
|-
|-
| 26
| 26
| 611.765
| 611.8
| [[56/39]]
| [[10/7]]
|
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 186: Line 276:
|-
|-
| 27
| 27
| 635.294
| 635.3
| [[13/9]]
|
| ''10/7'', ''16/11''
| Aug 4th
| Aug 4th
| A4
| A4
Line 192: Line 285:
|-
|-
| 28
| 28
| 658.824
| 658.8
| [[22/15]]
| [[16/11]], ''40/27''
|
| Upaug 4th
| Upaug 4th
| ^A4
| ^A4
Line 198: Line 294:
|-
|-
| 29
| 29
| 682.353
| 682.4
|
|
| [[40/27]]
| Down 5th
| Down 5th
| v5
| v5
Line 204: Line 303:
|-
|-
| 30
| 30
| 705.882
| 705.9
| [[3/2]]
|
|
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 210: Line 312:
|-
|-
| 31
| 31
| 729.412
| 729.4
| [[32/21]]
| ''20/13''
|
| Up 5th
| Up 5th
| ^5
| ^5
Line 216: Line 321:
|-
|-
| 32
| 32
| 752.941
| 752.9
| ''[[14/9]]''
|
| [[20/13]]
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 222: Line 330:
|-
|-
| 33
| 33
| 776.471
| 776.5
| ''[[128/81]]''
| [[11/7]]
|
| Minor 6th
| Minor 6th
| m6
| m6
Line 228: Line 339:
|-
|-
| 34
| 34
| 800.000
| 800.0
|
|
| ''8/5'', ''11/7''
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 234: Line 348:
|-
|-
| 35
| 35
| 823.529
| 823.5
| [[21/13]]
| [[8/5]], ''44/27''
|
| Downmid 6th
| Downmid 6th
| v~6
| v~6
Line 240: Line 357:
|-
|-
| 36
| 36
| 847.059
| 847.1
| [[13/8]], [[64/39]]
|
| [[18/11]], [[44/27]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 246: Line 366:
|-
|-
| 37
| 37
| 870.588
| 870.6
| [[33/20]], [[104/63]]
| ''5/3'', ''18/11''
|
| Upmid 6th
| Upmid 6th
| ^~6
| ^~6
Line 252: Line 375:
|-
|-
| 38
| 38
| 894.118
| 894.1
|
| ''22/13''
| [[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 258: Line 384:
|-
|-
| 39
| 39
| 917.647
| 917.6
| ''[[27/16]]''
|
| [[22/13]]
| Major 6th
| Major 6th
| M6
| M6
Line 264: Line 393:
|-
|-
| 40
| 40
| 941.176
| 941.2
| [[12/7]]
|
| [[26/15]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 270: Line 402:
|-
|-
| 41
| 41
| 964.706
| 964.7
| [[7/4]]
| ''26/15''
|
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 276: Line 411:
|-
|-
| 42
| 42
| 988.235
| 988.2
| [[16/9]]
|
|
| Minor 7th
| Minor 7th
| m7
| m7
Line 282: Line 420:
|-
|-
| 43
| 43
| 1011.765
| 1011.8
|
|
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 288: Line 429:
|-
|-
| 44
| 44
| 1035.294
| 1035.3
| [[20/11]]
|
| ''9/5'', ''11/6''
| Downmid 7th
| Downmid 7th
| v~7
| v~7
Line 294: Line 438:
|-
|-
| 45
| 45
| 1058.824
| 1058.8
| [[24/13]]
|
| [[11/6]], ''28/15''
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 300: Line 447:
|-
|-
| 46
| 46
| 1082.353
| 1082.4
| [[13/7]]
| [[15/8]], [[28/15]]
|
| Upmid 7th
| Upmid 7th
| ^~7
| ^~7
Line 306: Line 456:
|-
|-
| 47
| 47
| 1105.882
| 1105.9
|
| [[40/21]]
| ''15/8'', ''48/25''
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 312: Line 465:
|-
|-
| 48
| 48
| 1129.412
| 1129.4
| [[52/27]]
| ''35/18''
| ''40/21'', ''64/33''
| Major 7th
| Major 7th
| M7
| M7
Line 318: 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 324: Line 483:
|-
|-
| 50
| 50
| 1176.471
| 1176.5
| [[63/32]], ''[[96/49]]''
| ''39/20''
| [[160/81]]
| Down 8ve
| Down 8ve
| v8
| v8
Line 330: 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 ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] for 51edo uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
If double arrows are not desirable, then arrows can be attached to quartertone accidentals:
{{Sharpness-sharp6-qt-szg}}
=== 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}}
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
=== 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>
== 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 ==
{| 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
|-
| 2.3.7.13
| 343/338, 512/507, 2197/2187
| {{Mapping| 51 81 143 }}
| −0.695
| 1.54
| 6.54
|- style="border-top: double;"
| 2.3.5
| 128/125, {{monzo| -13 17 -6 }}
| {{Mapping| 51 81 119 }} (51c)
| −2.789
| 2.41
| 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;"
| 2.3.5
| 250/243, 34171875/33554432
| {{Mapping| 51 81 118 }} (51)
| +0.581
| 2.77
| 11.8
|-
| 2.3.5.7
| 225/224, 250/243, 1029/1024
| {{Mapping| 51 81 118 143 }} (51)
| +0.803
| 2.43
| 10.3
|}
=== 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)
|-
| 1
| 7\51
| 164.7
| 11/10
| [[Porky]] (51)
|-
| 1
| 10\51
| 235.3
| 8/7
| [[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
| 23\51
| 541.2
| 15/11
| [[Necromanteion]] (51ce)<br>[[Oracle]] (51)<br>[[Cypress]] (51cde…)
|-
| 3
| 19\51<br>(2\51)
| 447.1<br>(47.1)
| 9/7<br>(36/35)
| [[Hemiaug]] (51ce)
|-
| rowspan="2" | 3
| rowspan="2" | 21\51<br>(4\51)
| rowspan="2" | 494.1<br>(94.1)
| 4/3<br>(16/15)
| [[Augmented (temperament)|Augmented]] (7-limit, 51cd)
|-
| style="text-align: center;" | 4/3<br>(21/20)
| style="text-align: left;" | [[Fog]] (51)
|}
<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 ==
* [[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) - [https://youtu.be/8GojBZSyqDw STARS (Thoughts and Prayers)] (51 EDO for 51 States)
; [[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]]
* ''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 -->

Latest revision as of 03:35, 28 May 2026

← 50edo 51edo 52edo →
Prime factorization 3 × 17
Step size 23.5294 ¢ 
Fifth 30\51 (705.882 ¢) (→ 10\17)
Semitones (A1:m2) 6:3 (141.2 ¢ : 70.59 ¢)
Consistency limit 3
Distinct consistency limit 3

51 equal divisions of the octave (abbreviated 51edo or 51ed2), also called 51-tone equal temperament (51tet) or 51 equal temperament (51et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 51 equal parts of about 23.5 ¢ each. Each step represents a frequency ratio of 21/51, or the 51st root of 2.

Theory

Since 51 = 3 × 17, 51edo shares its 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 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.

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

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

Approximation of odd harmonics in 51edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Error Absolute (¢) +3.93 -9.84 -4.12 +7.85 -10.14 +6.53 -5.92 -10.84 +8.37 -0.19 +7.02 +3.84 -11.75 +5.72
Relative (%) +16.7 -41.8 -17.5 +33.4 -43.1 +27.8 -25.1 -46.1 +35.6 -0.8 +29.8 +16.3 -49.9 +24.3
Steps
(reduced) 		81
(30) 		118
(16) 		143
(41) 		162
(9) 		176
(23) 		189
(36) 		199
(46) 		208
(4) 		217
(13) 		224
(20) 		231
(27) 		237
(33) 		242
(38) 		248
(44)
Approximation of odd harmonics in 51edo (continued)
Harmonic 31 33 35 37 39 41 43 45 47 49 51 53 55 57
Error Absolute (¢) +7.91 -6.21 +9.57 +7.48 +10.46 -5.53 +6.13 -1.99 -6.68 -8.24 -6.91 -2.92 +3.54 -11.23
Relative (%) +33.6 -26.4 +40.7 +31.8 +44.4 -23.5 +26.0 -8.5 -28.4 -35.0 -29.4 -12.4 +15.1 -47.7
Steps
(reduced) 		253
(49) 		257
(2) 		262
(7) 		266
(11) 		270
(15) 		273
(18) 		277
(22) 		280
(25) 		283
(28) 		286
(31) 		289
(34) 		292
(37) 		295
(40) 		297
(42)

Subsets and supersets

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

# Cents Approximate ratios* Ups and downs notation
2.3.7.11/5.13
subgroup 		Ratios of 5 and 11
tending flat (51 val) 		Ratios of 5 and 11
tending sharp (51ce val)
0 0.0 1/1 Perfect 1sn P1 D
1 23.5 64/63, 49/48 40/39 81/80 Up 1sn ^1 ^D
2 47.1 28/27 33/32, 25/24, 81/80 36/35, 40/39 Downminor 2nd vm2 vEb
3 70.6 27/26 36/35 21/20, 33/32 Minor 2nd m2 Eb
4 94.1 21/20 16/15, 25/24 Upminor 2nd ^m2 ^Eb
5 117.6 14/13 15/14, 16/15 Downmid 2nd v~2 ^^Eb
6 141.2 13/12 12/11, 15/14 Mid 2nd ~2 vvvE, ^^^Eb
7 164.7 11/10 10/9, 12/11 Upmid 2nd ^~2 vvE
8 188.2 10/9 Downmajor 2nd vM2 vE
9 211.8 9/8 Major 2nd M2 E
10 235.3 8/7 15/13 Upmajor 2nd ^M2 ^E
11 258.8 7/6 15/13 Downminor 3rd vm3 vF
12 282.4 32/27 13/11 Minor 3rd m3 F
13 305.9 13/11 6/5 Upminor 3rd ^m3 ^F
14 329.4 40/33, 63/52 6/5, 11/9 Downmid 3rd v~3 ^^F
15 352.9 16/13, 39/32 11/9, 27/22 Mid 3rd ~3 ^^^F, vvvF#
16 376.5 26/21 5/4, 27/22 Upmid 3rd ^~3 vvF#
17 400.0 5/4, 14/11 Downmajor 3rd vM3 vF#
18 423.5 81/64 14/11 Major 3rd M3 F#
19 447.1 9/7 13/10 Upmajor 3rd ^M3 ^F#
20 470.6 21/16 13/10 Down 4th v4 vG
21 494.1 4/3 Perfect 4th P4 G
22 517.6 27/20 Up 4th ^4 ^G
23 541.2 15/11 11/8, 27/20 Downdim 5th vd5 vAb
24 564.7 18/13 7/5, 11/8 Dim 5th d5 Ab
25 588.2 39/28 7/5 Updim 5th ^d5 ^Ab
26 611.8 56/39 10/7 Downaug 4th vA4 vG#
27 635.3 13/9 10/7, 16/11 Aug 4th A4 G#
28 658.8 22/15 16/11, 40/27 Upaug 4th ^A4 ^G#
29 682.4 40/27 Down 5th v5 vA
30 705.9 3/2 Perfect 5th P5 A
31 729.4 32/21 20/13 Up 5th ^5 ^A
32 752.9 14/9 20/13 Downminor 6th vm6 vBb
33 776.5 128/81 11/7 Minor 6th m6 Bb
34 800.0 8/5, 11/7 Upminor 6th ^m6 ^Bb
35 823.5 21/13 8/5, 44/27 Downmid 6th v~6 ^^Bb
36 847.1 13/8, 64/39 18/11, 44/27 Mid 6th ~6 vvvB, ^^^Bb
37 870.6 33/20, 104/63 5/3, 18/11 Upmid 6th ^~6 vvB
38 894.1 22/13 5/3 Downmajor 6th vM6 vB
39 917.6 27/16 22/13 Major 6th M6 B
40 941.2 12/7 26/15 Upmajor 6th ^M6 ^B
41 964.7 7/4 26/15 Downminor 7th vm7 vC
42 988.2 16/9 Minor 7th m7 C
43 1011.8 9/5 Upminor 7th ^m7 ^C
44 1035.3 20/11 9/5, 11/6 Downmid 7th v~7 ^^C
45 1058.8 24/13 11/6, 28/15 Mid 7th ~7 ^^^C, vvvC#
46 1082.4 13/7 15/8, 28/15 Upmid 7th ^~7 vvC#
47 1105.9 40/21 15/8, 48/25 Downmajor 7th vM7 vC#
48 1129.4 52/27 35/18 40/21, 64/33 Major 7th M7 C#
49 1152.9 27/14 64/33, 48/25, 160/81 35/18, 39/20 Upmajor 7th ^M7 ^C#
50 1176.5 63/32, 96/49 39/20 160/81 Down 8ve v8 vD
51 1200.0 2/1 Perfect 8ve P8 D

* inconsistent intervals in italic.

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation for 51edo uses sharps and flats combined with quartertone accidentals and arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

If double arrows are not desirable, then arrows can be attached to quartertone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol

Kite's ups and downs notation

51edo can also be notated with Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Sagittal notation

In the following diagrams, 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.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8027/26

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8027/26

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8027/26

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 51edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 51edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/10, 20/11 0.298 1.3
13/9, 18/13 1.324 5.6
15/14, 28/15 1.796 7.6
13/12, 24/13 2.604 11.1
3/2, 4/3 3.927 16.7
7/4, 8/7 4.120 17.5
15/11, 22/15 4.226 18.0
11/9, 18/11 5.533 23.5
7/5, 10/7 5.723 24.3
9/5, 10/9 5.832 24.8
15/8, 16/15 5.916 25.1
11/7, 14/11 6.021 25.6
13/8, 16/13 6.531 27.8
13/11, 22/13 6.857 29.1
13/10, 20/13 7.155 30.4
9/8, 16/9 7.855 33.4
7/6, 12/7 8.047 34.2
11/6, 12/11 9.461 40.2
5/3, 6/5 9.759 41.5
5/4, 8/5 9.843 41.8
11/8, 16/11 10.141 43.1
13/7, 14/13 10.651 45.3
15/13, 26/15 11.082 47.1
9/7, 14/9 11.555 49.1
15-odd-limit intervals in 51edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/10, 20/11 0.298 1.3
13/9, 18/13 1.324 5.6
15/14, 28/15 1.796 7.6
13/12, 24/13 2.604 11.1
3/2, 4/3 3.927 16.7
7/4, 8/7 4.120 17.5
15/11, 22/15 4.226 18.0
7/5, 10/7 5.723 24.3
15/8, 16/15 5.916 25.1
11/7, 14/11 6.021 25.6
13/8, 16/13 6.531 27.8
9/8, 16/9 7.855 33.4
7/6, 12/7 8.047 34.2
5/4, 8/5 9.843 41.8
11/8, 16/11 10.141 43.1
13/7, 14/13 10.651 45.3
9/7, 14/9 11.975 50.9
15/13, 26/15 12.447 52.9
5/3, 6/5 13.770 58.5
11/6, 12/11 14.069 59.8
13/10, 20/13 16.374 69.6
13/11, 22/13 16.673 70.9
9/5, 10/9 17.698 75.2
11/9, 18/11 17.996 76.5
15-odd-limit intervals in 51edo (51ce val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/10, 20/11 0.298 1.3
13/9, 18/13 1.324 5.6
13/12, 24/13 2.604 11.1
3/2, 4/3 3.927 16.7
7/4, 8/7 4.120 17.5
15/11, 22/15 4.226 18.0
11/9, 18/11 5.533 23.5
9/5, 10/9 5.832 24.8
13/8, 16/13 6.531 27.8
13/11, 22/13 6.857 29.1
13/10, 20/13 7.155 30.4
9/8, 16/9 7.855 33.4
7/6, 12/7 8.047 34.2
11/6, 12/11 9.461 40.2
5/3, 6/5 9.759 41.5
13/7, 14/13 10.651 45.3
15/13, 26/15 11.082 47.1
9/7, 14/9 11.975 50.9
11/8, 16/11 13.388 56.9
5/4, 8/5 13.686 58.2
11/7, 14/11 17.508 74.4
15/8, 16/15 17.614 74.9
7/5, 10/7 17.806 75.7
15/14, 28/15 21.734 92.4

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.7 1029/1024, [17 -16 3⟩ [⟨51 81 143]] −0.339 1.63 6.92
2.3.7.13 343/338, 512/507, 2197/2187 [⟨51 81 143]] −0.695 1.54 6.54
2.3.5 128/125, [-13 17 -6⟩ [⟨51 81 119]] (51c) −2.789 2.41 10.3
2.3.5.7 128/125, 245/243, 1029/1000 [⟨51 81 119 143]] (51c) −1.730 2.79 11.9
2.3.5 250/243, 34171875/33554432 [⟨51 81 118]] (51) +0.581 2.77 11.8
2.3.5.7 225/224, 250/243, 1029/1024 [⟨51 81 118 143]] (51) +0.803 2.43 10.3

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 5\51 117.6 15/14 Miracle (51e, out of tune)
1 7\51 164.7 11/10 Porky (51)
1 10\51 235.3 8/7 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 23\51 541.2 15/11 Necromanteion (51ce)
Oracle (51)
Cypress (51cde…)
3 19\51
(2\51) 		447.1
(47.1) 		9/7
(36/35) 		Hemiaug (51ce)
3 21\51
(4\51) 		494.1
(94.1) 		4/3
(16/15) 		Augmented (7-limit, 51cd)
4/3
(21/20) 		Fog (51)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Instruments

Lumatone
See Lumatone mapping for 51edo.

Music

Bryan Deister
Frédéric Gagné
James Mulvale (FASTFAST)
Ray Perlner
  • Fugue (2023) – for organ in 51edo Porcupine[7] ssssssL "Pandian"
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