51edo: Difference between revisions

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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 [[aerodino]].

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 ===

{{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 ===

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== Notation ==

=== ~~Ups~~ and downs notation ===

=== Stein–Zimmermann–Gould 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.

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

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

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.~~

~~{{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:

=== 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 ====

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| 235.3

| 8/7

| [[Rodan]] (~~51cf…~~, out of tune) / aerodino (51ce)

| [[Rodan]] (51cf, out of tune) / aerodino (51ce)

|-

| 1

Line 658:

Line 657:

| 517.6

| 27/20

| [[Gravity]] (51ce)

| [[Gravity]] (51ce) / [[abergravity]] (51ce)

|-

| 1

Line 681:

Line 680:

| style="text-align: left;" | [[Fog]] (51)

|}

<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|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 ==

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== 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]]

@@ Line 7: / Line 7: @@
 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 [[aerodino]].
+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]].
-edo's step is the closest direct approximation to the [[Pythagorean comma]] by edo steps, though that comma itself is mapped to a different interval.
+edo'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}}
+{{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 ===
@@ Line 502: / Line 503: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
+[[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 ===
+edo 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}}
-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.
+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 ====
@@ Line 646: / Line 645: @@
 | 235.3
 | 8/7
-| [[Rodan]] (51cf…, out of tune) / aerodino (51ce)
+| [[Rodan]] (51cf, out of tune) / aerodino (51ce)
 |-
 | 1
@@ Line 658: / Line 657: @@
 | 517.6
 | 27/20
-| [[Gravity]] (51ce)
+| [[Gravity]] (51ce) / [[abergravity]] (51ce)
 |-
 | 1
@@ Line 681: / Line 680: @@
 | style="text-align: left;" | [[Fog]] (51)
 |}
-<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|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 ==
@@ Line 694: / Line 693: @@
 == 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]]