26edo: Difference between revisions

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{{~~interwiki~~

{{Interwiki

| en = 26edo

| de = 26-EDO

~~| en = 26edo~~

| es =

| ja =

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# In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which provides an interesting structure but unsatisfying intonation due mainly to the poorly tuned thirds. Extending meantone harmony to the 7-limit is quite intuitive; for example, augmented becomes supermajor, and diminished becomes subminor. Simple mappings for harmonics up to 13 are also achieved.

# As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, [[38edo]]) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of [[14edo]].

# 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas [[65536/65219]] and ~~{{monzo| -3 0 0 6 -4 }}~~. The 65536/65219 comma, the orgonisma, leads to the [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with mos scales of size 7, 11 and 15. The ~~{{monzo| -3 0 0 6 -4 }}~~ comma leads to a half-octave period and an approximate [[49/44]] generator of 4\26, leading to mos of size 8 and 14.

# 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas [[65536/65219]] and [[117649/117128]]. The 65536/65219 comma, the orgonisma, leads to the [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with mos scales of size 7, 11 and 15. The 117649/117128 comma leads to a half-octave period and an approximate [[49/44]] generator of 4\26, leading to mos of size 8 and 14.

# We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).

# It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fourths give a 17:14 and four fifths give a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and [[33edo]] even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.

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=== Subsets and supersets ===

26edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing the 2.9.5.21.11.13.17.19-subgroup with 26edo. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. [[104edo]] is a notable dual-5s system. [[130edo]], [[364edo]], [[494edo]], and [[624edo]] do well in approximating JI, though they are more complex.

26edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing the 2.9.5.21.11.13.17.19-subgroup with 26edo.

26edo tempers out [[Fynn's comma]], which sets ~7/4 to 21\26. This is shared by several notable superset edos. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. [[104edo]] is a notable dual-5's system. [[130edo]], [[364edo]], [[494edo]], and [[624edo]] do well in approximating JI, though they are more complex.

== Intervals ==

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| 8

| 369.23

| [[5/4]], [[11/9]], [[16/13]]

| [[5/4]], [[11/9]], [[16/13]], [[26/21]]

| M3

| F#

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| 10

| 461.54

| [[21/16]], [[13/10]]

| [[21/16]], [[13/10]], [[64/49]]

| d4

| Gb

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| 16

| 738.46

| [[32/21]], [[20/13]]

| [[32/21]], [[20/13]], [[49/32]]

| A5

| A#

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| 18

| 830.77

| [[13/8]], [[8/5]]

| [[8/5]], [[13/8]], [[21/13]]

| m6

| Bb

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== Approximation to irrational intervals ==

26edo approximates both [[acoustic phi]] (the [[golden ratio]]) and [[pi]] quite accurately. Not until 1076edo do we find a better edo in terms of relative error on these intervals.

26edo approximates both [[acoustic phi]] (the [[golden ratio]]) and [[pi]] quite accurately. Not until 1076edo do we find a better edo in terms of relative error on these intervals{{Clarify}}.

{| class="wikitable center-all"

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=== Modern renderings ===

=== 26 equal divisions of the octave (26edo proper) ===

==== Modern renderings ====

; {{W|Johann Sebastian Bach}}

* [https://www.youtube.com/watch?v=LUNOFjiyZ0Y ''Contrapunctus 4'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

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* [https://www.youtube.com/watch?v=-EVO5ntuoSM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

=== 21st century ===

==== 21st century ====

; [[Abnormality]]

* [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)

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* [https://www.youtube.com/shorts/1aCl6tuVS0c ''Happy Together - The Turtles (microtonal cover in 26edo)''] (2026)

* ''My Violet - 26edo'' (2026)

** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>'']

** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>] (Lumatone view)

** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>'']

** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>] (Lumatone view)

** [https://www.youtube.com/watch?v=XplpKE_Tc38 <nowiki>[full version[</nowiki>] (with animation by WIRED0006)

* [https://www.youtube.com/shorts/wHGLOaeAkt8 ''26edo groove''] (2026)

; [[User:Eboone|Ebooone]]

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; [[YoVariable]]

* [https://www.youtube.com/watch?v=01w70PbbT3o ''Jingle Bells (26edo microtonal Lumatone cover + Mystery Song)''] (2025)

=== Unequal Derivatives of 26edo ===

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/mzUGcki6H0Y ''<nowiki>Daisy Bell - Harry Dacre (microtonal cover in unequal 26ish tone [displaced from 26edo in dozens])</nowiki>''] (2026) — from Bryan Deister's video comments, "displacement in cents roughly: 0, -8, -3, -12, 14, 5, 18, 3, -12, 13, -15, 10" (these repeat every 12 notes, NOT every 13 note semi-octave, thus causing each octave to be different)

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-{{interwiki
+{{Interwiki
+| en = 26edo
 | de = 26-EDO
-| en = 26edo
 | es =
 | ja =
@@ Line 21: / Line 21: @@
 # In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which provides an interesting structure but unsatisfying intonation due mainly to the poorly tuned thirds. Extending meantone harmony to the 7-limit is quite intuitive; for example, augmented becomes supermajor, and diminished becomes subminor. Simple mappings for harmonics up to 13 are also achieved.
 # As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, [[38edo]]) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&amp;26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of [[14edo]].
-# 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas [[65536/65219]] and {{monzo| -3 0 0 6 -4 }}. The 65536/65219 comma, the orgonisma, leads to the [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with mos scales of size 7, 11 and 15. The {{monzo| -3 0 0 6 -4 }} comma leads to a half-octave period and an approximate [[49/44]] generator of 4\26, leading to mos of size 8 and 14.
+# 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas [[65536/65219]] and [[117649/117128]]. The 65536/65219 comma, the orgonisma, leads to the [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with mos scales of size 7, 11 and 15. The 117649/117128 comma leads to a half-octave period and an approximate [[49/44]] generator of 4\26, leading to mos of size 8 and 14.
 # We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).
 # It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fourths give a 17:14 and four fifths give a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and [[33edo]] even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.
@@ Line 33: / Line 33: @@
 === Subsets and supersets ===
-edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing the 2.9.5.21.11.13.17.19-subgroup with 26edo. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. [[104edo]] is a notable dual-5s system. [[130edo]], [[364edo]], [[494edo]], and [[624edo]] do well in approximating JI, though they are more complex.
+edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing the 2.9.5.21.11.13.17.19-subgroup with 26edo.
+edo tempers out [[Fynn's comma]], which sets ~7/4 to 21\26. This is shared by several notable superset edos. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. [[104edo]] is a notable dual-5's system. [[130edo]], [[364edo]], [[494edo]], and [[624edo]] do well in approximating JI, though they are more complex.
 == Intervals ==
@@ Line 129: / Line 131: @@
 | 8
 | 369.23
-| [[5/4]], [[11/9]], [[16/13]]
+| [[5/4]], [[11/9]], [[16/13]], [[26/21]]
 | M3
 | F#
@@ Line 149: / Line 151: @@
 | 10
 | 461.54
-| [[21/16]], [[13/10]]
+| [[21/16]], [[13/10]], [[64/49]]
 | d4
 | Gb
@@ Line 209: / Line 211: @@
 | 16
 | 738.46
-| [[32/21]], [[20/13]]
+| [[32/21]], [[20/13]], [[49/32]]
 | A5
 | A#
@@ Line 229: / Line 231: @@
 | 18
 | 830.77
-| [[13/8]], [[8/5]]
+| [[8/5]], [[13/8]], [[21/13]]
 | m6
 | Bb
@@ Line 437: / Line 439: @@
 == Approximation to irrational intervals ==
-edo approximates both [[acoustic phi]] (the [[golden ratio]]) and [[pi]] quite accurately. Not until 1076edo do we find a better edo in terms of relative error on these intervals.
+edo approximates both [[acoustic phi]] (the [[golden ratio]]) and [[pi]] quite accurately. Not until 1076edo do we find a better edo in terms of relative error on these intervals{{Clarify}}.
 {| class="wikitable center-all"
@@ Line 802: / Line 804: @@
 {{Catrel|26edo tracks}}
-=== Modern renderings ===
+=== 26 equal divisions of the octave (26edo proper) ===
+==== Modern renderings ====
 ; {{W|Johann Sebastian Bach}}
 * [https://www.youtube.com/watch?v=LUNOFjiyZ0Y ''Contrapunctus 4'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
@@ Line 814: / Line 817: @@
 * [https://www.youtube.com/watch?v=-EVO5ntuoSM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
-=== 21st century ===
+==== 21st century ====
 ; [[Abnormality]]
 * [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)
@@ Line 850: / Line 853: @@
 * [https://www.youtube.com/shorts/1aCl6tuVS0c ''Happy Together - The Turtles (microtonal cover in 26edo)''] (2026)
 * ''My Violet - 26edo'' (2026)
-** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>'']
+** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>] (Lumatone view)
-** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>'']
+** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>] (Lumatone view)
+** [https://www.youtube.com/watch?v=XplpKE_Tc38 <nowiki>[full version[</nowiki>] (with animation by WIRED0006)
+* [https://www.youtube.com/shorts/wHGLOaeAkt8 ''26edo groove''] (2026)
 ; [[User:Eboone|Ebooone]]
@@ Line 925: / Line 930: @@
 ; [[YoVariable]]
 * [https://www.youtube.com/watch?v=01w70PbbT3o ''Jingle Bells (26edo microtonal Lumatone cover + Mystery Song)''] (2025)
+=== Unequal Derivatives of 26edo ===
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/mzUGcki6H0Y ''<nowiki>Daisy Bell - Harry Dacre (microtonal cover in unequal 26ish tone [displaced from 26edo in dozens])</nowiki>''] (2026) &mdash; from Bryan Deister's video comments, "displacement in cents roughly: 0, -8, -3, -12, 14, 5, 18, 3, -12, 13, -15, 10" (these repeat every 12 notes, NOT every 13 note semi-octave, thus causing each octave to be different)
 [[Category:Listen]]