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<b>51-EDO</b> divides the [[Octave|octave]] into 51 equal parts of 23.529 [[cent|cent]]s each, which is about the size of the [http://en.wikipedia.org/wiki/Pythagorean_comma Pythagorean comma] (even though this comma itself is mapped to a different interval). It tempers out [[250/243|250/243]] in the [[5-limit|5-limit]], [[225/224|225/224]] and [[2401/2400|2401/2400]] in the [[7-limit|7-limit]], and [[55/54|55/54]] and [[100/99|100/99]] in the [[11-limit|11-limit]]. It is the [[Optimal_patent_val|optimal patent val]] for [[Porcupine_rank_three_family|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|22edo]] for [[Porcupine_family|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.
{{Infobox ET}}
[[Category:51edo]]
{{ED intro}}
[[Category:edo]]
 
[[Category:theory]]
== 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&nbsp;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]].
 
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.
 
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 ==
{| class="wikitable center-1 right-2 center-6 center-7 center-8"
|-
! rowspan="2" | #
! rowspan="2" | [[Cent]]s
! 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
| [[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
|}
<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 ==
; [[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:Listen]]
Retrieved from "https://en.xen.wiki/w/51edo"