51edo: Difference between revisions

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== Theory ==

51 = 3 × 17, ~~and~~ 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 ~~very good~~ approximations of [[11/10]] and [[21/16]].

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.

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=== 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 ==

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== 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.

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

In 51edo, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.

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== 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 ==

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== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025)

; [[Frédéric Gagné]]

* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – 7:4 [[semiquartal]] 4|4 mode

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|51}}
+{{ED intro}}
 == Theory ==
-= 3 × 17, and 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 very good approximations of [[11/10]] and [[21/16]].
+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&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.
@@ Line 15: / Line 15: @@
 === Subsets and supersets ===
-edo contains [[3edo]] and [[17edo]] as subsets. A step of 51edo is exactly 12 [[skisma]]s.
+edo 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 ==
@@ Line 339: / Line 341: @@
 == 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.
+edo 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}}
@@ Line 473: / Line 481: @@
 == 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 ==
@@ Line 482: / Line 490: @@
 == Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025)
 ; [[Frédéric Gagné]]
 * ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – 7:4 [[semiquartal]] 4|4 mode