65edo: Difference between revisions

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== Notation ==

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

=== Stein–Zimmermann–Gould notation ===

65edo 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]] uses sharps and flats combined with quartertone accidentals and arrows:

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

=== Kite's ups and downs notation ===

65edo 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:

~~[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have arrows borrowed from extended [[Helmholtz–Ellis notation]]:~~

~~{{Sharpness-sharp6}}~~

~~If double arrows are not desirable, 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 ===

This notation uses the same sagittal sequence as ~~EDOs~~ [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]].

This notation uses the same sagittal sequence as edos [[72edo #Sagittal notation|72]] and [[79edo #Sagittal notation|79]].

==== Evo flavor ====

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default [[File:65-EDO_Evo-SZ_Sagittal.svg]]

</imagemap>

== Approximation to JI ==

=== ~~Zeta peak index~~ ===

=== 15-odd-limit interval mappings ===

{{~~ZPI~~

| ~~zpi = 334~~

{{Q-odd-limit intervals|65.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 65d val mapping}}

| ~~steps =~~ 65.~~0158450885860~~

~~| step size = 18.4570391781413~~

~~| tempered height = 7.813349~~

~~| pure height = 7.642373~~

~~| integral =~~ 1~~.269821~~

| ~~gap~~ = ~~16.514861~~

| ~~octave~~ = ~~1199.70754657919~~

| ~~consistent~~ = 6

| ~~distinct~~ = 6

}}

== Regular temperament properties ==

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| [[Countdown]]

|}

<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

== Octave stretch or compression ==

65edo tunes [[primes]] 2, 3, 5 and 11 very well, but its 7 and 13 have two about equally-bad mappings. [[Stretched and compressed tuning|Stretching or shrinking the octave]] of 65edo for improvements in its approximations of [[JI]] therefore depends on which mapping is used: the sharp tending 65d val wants octave shrinking, whereas the flat tending 65f val wants octave stretching; both can be achieved at the cost of relatively little damage to other primes.

Compressed tunings of 65edo that well approximate JI include [[zpi|334zpi]], [[ed5|151ed5]] and [[equal tuning|225ed11]].

Stretched tunings of 65edo that well approximate JI include [[WE|13-lim WE-tuned 65f]] (18.473cET) and [[TE|13-lim TE-tuned 65f]] (18.474cET).

== Scales ==

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== Music ==

; [[Bryan Deister]

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025).

* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025)

* [https://www.youtube.com/shorts/UJZw9NQuGnY ''Zanarkand - Nobuo Uematsu (microtonal cover in 65edo)''] (2026)

* [https://www.youtube.com/shorts/zxgVvwXnIGQ ''Waltz in 65edo''] (2026)

* [https://www.youtube.com/shorts/OtbEDFhjNkc ''65edo prelude''] (2026)

* [https://www.youtube.com/shorts/c0eWd7UvNQU ''Black Hole Sun - Soundgarden (microtonal cover in 65edo)''] (2026)

[[Category:Listen]]

@@ Line 422: / Line 422: @@
 == 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]] uses sharps and flats combined with quartertone accidentals and arrows:
+{{Sharpness-sharp6-szg}}
+If double arrows are not desirable, 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}}
-[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have arrows borrowed from extended [[Helmholtz–Ellis notation]]:
-{{Sharpness-sharp6}}
-If double arrows are not desirable, 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}}
-{{sharpness-sharp6-iw}}
 === Sagittal notation ===
-This notation uses the same sagittal sequence as EDOs [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]].
+This notation uses the same sagittal sequence as edos [[72edo #Sagittal notation|72]] and [[79edo #Sagittal notation|79]].
 ==== Evo flavor ====
@@ Line 478: / Line 478: @@
 default [[File:65-EDO_Evo-SZ_Sagittal.svg]]
 </imagemap>
 == Approximation to JI ==
-=== Zeta peak index ===
+=== 15-odd-limit interval mappings ===
-{{ZPI
+{{Q-odd-limit intervals|65}}
-| zpi = 334
+{{Q-odd-limit intervals|65.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 65d val mapping}}
-| steps = 65.0158450885860
-| step size = 18.4570391781413
-| tempered height = 7.813349
-| pure height = 7.642373
-| integral = 1.269821
-| gap = 16.514861
-| octave = 1199.70754657919
-| consistent = 6
-| distinct = 6
-}}
 == Regular temperament properties ==
@@ Line 604: / Line 593: @@
 | [[Countdown]]
 |}
-<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
+== Octave stretch or compression ==
+edo tunes [[primes]] 2, 3, 5 and 11 very well, but its 7 and 13 have two about equally-bad mappings. [[Stretched and compressed tuning|Stretching or shrinking the octave]] of 65edo for improvements in its approximations of [[JI]] therefore depends on which mapping is used: the sharp tending 65d val wants octave shrinking, whereas the flat tending 65f val wants octave stretching; both can be achieved at the cost of relatively little damage to other primes.
+Compressed tunings of 65edo that well approximate JI include [[zpi|334zpi]], [[ed5|151ed5]] and [[equal tuning|225ed11]].
+Stretched tunings of 65edo that well approximate JI include [[WE|13-lim WE-tuned 65f]] (18.473cET) and [[TE|13-lim TE-tuned 65f]] (18.474cET).
 == Scales ==
@@ Line 616: / Line 612: @@
 == Music ==
-; [[Bryan Deister]
+; [[Bryan Deister]]
-* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025).
+* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025)
+* [https://www.youtube.com/shorts/UJZw9NQuGnY ''Zanarkand - Nobuo Uematsu (microtonal cover in 65edo)''] (2026)
+* [https://www.youtube.com/shorts/zxgVvwXnIGQ ''Waltz in 65edo''] (2026)
+* [https://www.youtube.com/shorts/OtbEDFhjNkc ''65edo prelude''] (2026)
+* [https://www.youtube.com/shorts/c0eWd7UvNQU ''Black Hole Sun - Soundgarden (microtonal cover in 65edo)''] (2026)
 [[Category:Listen]]