65edo: Difference between revisions

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== Theory ==

65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects ~~and that~~ instead elegantly connects primes 7, 13, 37 and 41 to nestoria. Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].

65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects (which instead elegantly connects primes 7, 13, 37, and 41 to nestoria). Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].

=== Prime harmonics ===

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=== Subsets and supersets ===

65edo contains [[13edo]] as ~~a subset~~. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[Andrew Heathwaite]]'s composition [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].

65edo contains [[5edo]] and [[13edo]] as subsets. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[Andrew Heathwaite]]'s composition [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].

[[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13.

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{| class="wikitable center-all right-2 left-3"

|-

! #

! [[Cent]]s

! Approximate Ratios *

! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref>

! colspan="2" | [[Ups and ~~Downs Notation~~]]

! colspan="2" | [[Ups and downs notation]]

|-

| 0

Line 419:

| D

|}

<~~nowiki>*<~~/~~nowiki~~> ~~based on treating~~ 65edo as a 2.3.5.11.13/7.17.19.23.29.31.~~47 subgroup temperament~~.

== Notation ==

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

[[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:

=== 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]].

==== Evo flavor ====

File:65-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[81/80]]

rect 120 80 220 106 [[64/63]]

rect 220 80 340 106 [[33/32]]

default [[File:65-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:65-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 650 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[81/80]]

rect 120 80 220 106 [[64/63]]

rect 220 80 340 106 [[33/32]]

default [[File:65-EDO_Revo_Sagittal.svg]]

</imagemap>

==== Evo-SZ flavor ====

File:65-EDO_Evo-SZ_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[81/80]]

rect 120 80 220 106 [[64/63]]

rect 220 80 340 106 [[33/32]]

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

</imagemap>

== Approximation to JI ==

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

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

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 442:

Line 505:

| 32805/32768, 78732/78125

| {{mapping| 65 103 151 }}

| -0.110

| −0.110

| 0.358

| 1.94

Line 449:

Line 512:

| 243/242, 4000/3993, 5632/5625

| {{mapping| 65 103 151 225 }}

| -0.266

| −0.266

| 0.410

| 2.22

Line 456:

Line 519:

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! ~~Temperaments~~

! Temperament

|-

| 1

Line 503:

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| 498.46

| 4/3

| [[Helmholtz]] / [[nestoria]] / [[photia]]

| [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]

|-

| 1

Line 521:

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| 498.46 (18.46)

| 4/3 (81/80)

| [[~~Pental (temperament)|Pental~~]]

| [[Quintile]]

|-

| 5

Line 529:

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| [[Countdown]]

|}

<nowiki>*~~</nowiki>~~ [[Normal ~~lists~~|~~octave~~-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if it is ~~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 ==

* Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[würschmidt]]): 5 3 5 5 3 5 12 5 5 3 5 12 5

* [[Photia7]]

* [[Photia12]]

* [[Skateboard7]]

== Instruments ==

[[Lumatone mapping for 65edo]]

== Music ==

; [[Bryan Deister]]

* [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]]

~~[[Category:Subgroup temperaments]]~~

[[Category:Schismic]]

[[Category:Sensipent]]

[[Category:Subgroup temperaments]]

[[Category:Würschmidt]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|65}}
+{{ED intro}}
 == Theory ==
 et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide.
-edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects and that instead elegantly connects primes 7, 13, 37 and 41 to nestoria. Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].
+edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects (which instead elegantly connects primes 7, 13, 37, and 41 to nestoria). Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]].
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-edo contains [[13edo]] as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[Andrew Heathwaite]]'s composition [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].
+edo contains [[5edo]] and [[13edo]] as subsets. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[Andrew Heathwaite]]'s composition [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].
 [[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13.
@@ Line 18: / Line 18: @@
 {| class="wikitable center-all right-2 left-3"
 |-
-! #
+! &#35;
 ! [[Cent]]s
-! Approximate Ratios *
+! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref>
-! colspan="2" | [[Ups and Downs Notation]]
+! colspan="2" | [[Ups and downs notation]]
 |-
 | 0
@@ Line 419: / Line 419: @@
 | D
 |}
-<nowiki>*</nowiki> based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament.
+<references group="note" />
+== Notation ==
+=== Stein–Zimmermann–Gould notation ===
+[[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}}
+=== 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 ===
+This notation uses the same sagittal sequence as edos [[72edo #Sagittal notation|72]] and [[79edo #Sagittal notation|79]].
+==== Evo flavor ====
+<imagemap>
+File:65-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[81/80]]
+rect 120 80 220 106 [[64/63]]
+rect 220 80 340 106 [[33/32]]
+default [[File:65-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:65-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 650 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[81/80]]
+rect 120 80 220 106 [[64/63]]
+rect 220 80 340 106 [[33/32]]
+default [[File:65-EDO_Revo_Sagittal.svg]]
+</imagemap>
+==== Evo-SZ flavor ====
+<imagemap>
+File:65-EDO_Evo-SZ_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[81/80]]
+rect 120 80 220 106 [[64/63]]
+rect 220 80 340 106 [[33/32]]
+default [[File:65-EDO_Evo-SZ_Sagittal.svg]]
+</imagemap>
+== Approximation to JI ==
+=== 15-odd-limit interval mappings ===
+{{Q-odd-limit intervals|65}}
+{{Q-odd-limit intervals|65.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 65d val mapping}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 442: / Line 505: @@
 | 32805/32768, 78732/78125
 | {{mapping| 65 103 151 }}
-| -0.110
+| −0.110
 | 0.358
 | 1.94
@@ Line 449: / Line 512: @@
 | 243/242, 4000/3993, 5632/5625
 | {{mapping| 65 103 151 225 }}
-| -0.266
+| −0.266
 | 0.410
 | 2.22
@@ Line 456: / Line 519: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br>ratio*
-! Temperaments
+! Temperament
 |-
 | 1
@@ Line 503: / Line 567: @@
 | 498.46
 | 4/3
-| [[Helmholtz]] / [[nestoria]] / [[photia]]
+| [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]
 |-
 | 1
@@ Line 521: / Line 585: @@
 | 498.46<br>(18.46)
 | 4/3<br>(81/80)
-| [[Pental (temperament)|Pental]]
+| [[Quintile]]
 |-
 | 5
@@ Line 529: / Line 593: @@
 | [[Countdown]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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 ==
+* Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[würschmidt]]): 5 3 5 5 3 5 12 5 5 3 5 12 5
 * [[Photia7]]
 * [[Photia12]]
 * [[Skateboard7]]
+== Instruments ==
+[[Lumatone mapping for 65edo]]
+== Music ==
+; [[Bryan Deister]]
+* [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]]
-[[Category:Subgroup temperaments]]
 [[Category:Schismic]]
 [[Category:Sensipent]]
+[[Category:Subgroup temperaments]]
 [[Category:Würschmidt]]