26edo: Difference between revisions

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== Theory ==

26edo tempers out [[81/80]] in the [[5-limit]], making it a [[meantone]] tuning ~~with a very flat fifth~~ (0.~~088957¢~~ flat of the [[4/9-comma meantone]] fifth).

26edo has a [[3/2|perfect fifth]] of about 692 cents and [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a very flat [[meantone]] tuning (0.088957{{c}} flat of the [[4/9-comma meantone]] fifth) with a very soft [[5L 2s|diatonic scale]].

In the [[7-limit]], it tempers out 50/49, 525/512, and 875/864, and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.

In the [[7-limit]], it tempers out [[50/49]], [[525/512]], and [[875/864]], and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.

26edo's minor sixth (1.6158) is very close to {{nowrap|''φ'' ≈ 1.6180}} (i.e. the golden ratio).

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# 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.

Its step of 46.2{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible. ~~This~~ is ~~because the harmonic entropy model is usually tuned to reflect the~~ common perception of ~~quarter-tones~~ as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

Its step of 46.2{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible. In other words, there is a common perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

Thanks to its sevenths, 26edo is an ideal tuning for its size for [[metallic harmony]].

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! [[Kite's color notation|Color of the 3rd]]

! JI chord

! Notes as ~~Edoteps~~

! Notes as Edosteps

! Notes of C Chord

! Written Name

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

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

~~{{ZPI~~

~~| zpi = 100~~

~~| steps = 25.9356996537225~~

~~| step size = 46.2682717652372~~

~~| tempered height = 5.545073~~

~~| pure height = 4.164318~~

~~| integral = 1.031155~~

~~| gap = 14.793013~~

~~| octave = 1202.97506589617~~

~~| consistent = 14~~

~~| distinct = 9~~

}}

== Approximation to irrational intervals ==

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[[File:12072608 10207851395433055 404343132969239728 n.jpg|none|thumb|960x960px]]

* [[Lumatone mapping for 26edo]]

== Literature ==

[http://www.ronsword.com Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.]

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; [[Zach Curley]]

* [http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Guitar%20Serenade%20in%20Q%20Major.mp3 Guitar Serenade in Q Major]{{dead link}}

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/FxTxQ0ayDpg ''Microtonal Improvisation in 26edo''] (2023)

; [[User:Eboone|Ebooone]]

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* [https://www.youtube.com/watch?v=r0jCdHEZpzM Claudi Meneghin - Suite (Prelude, Variations, Fugue) in 26edo, for Synth & Baroque Bassoon] (2023)

* [https://www.youtube.com/watch?v=rjo3X1-D57Y Canon 3-in-1 on a Ground for Baroque Ensemble] (2023)

; [[Microtonal Maverick]] (formerly The Xen Zone)

* [https://www.youtube.com/watch?v=qm_k9xjXRf0 ''The Microtonal Magic of 26EDO (with 13-limit jam)''] (2024)

* [https://www.youtube.com/watch?v=im2097HVqgA ''The Blues but with 26 Notes per Octave''] (2024) (explanatory video — contiguous music starts at 08:48)

; [[Herman Miller]]

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; [[Chris Vaisvil]]

* [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 ''Morpheous Wing'' in 26 edo] (2016)

~~== See also ==~~

* [[Lumatone mapping for 26edo]]

== Notes ==

@@ Line 9: / Line 9: @@
 == Theory ==
-edo tempers out [[81/80]] in the [[5-limit]], making it a [[meantone]] tuning with a very flat fifth (0.088957¢ flat of the [[4/9-comma meantone]] fifth).
+edo has a [[3/2|perfect fifth]] of about 692 cents and [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a very flat [[meantone]] tuning (0.088957{{c}} flat of the [[4/9-comma meantone]] fifth) with a very soft [[5L 2s|diatonic scale]].
-In the [[7-limit]], it tempers out 50/49, 525/512, and 875/864, and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.
+In the [[7-limit]], it tempers out [[50/49]], [[525/512]], and [[875/864]], and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.
 edo's minor sixth (1.6158) is very close to {{nowrap|''φ'' ≈ 1.6180}} (i.e. the golden ratio).
@@ Line 25: / Line 25: @@
 # 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.
-Its step of 46.2{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible. This is because the harmonic entropy model is usually tuned to reflect the common perception of quarter-tones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
+Its step of 46.2{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible. In other words, there is a common perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
 Thanks to its sevenths, 26edo is an ideal tuning for its size for [[metallic harmony]].
@@ Line 365: / Line 365: @@
 ! [[Kite's color notation|Color of the 3rd]]
 ! JI chord
-! Notes as Edoteps
+! Notes as Edosteps
 ! Notes of C Chord
 ! Written Name
@@ Line 428: / Line 428: @@
 === 15-odd-limit interval mappings ===
 {{Q-odd-limit intervals|26}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 100
-| steps = 25.9356996537225
-| step size = 46.2682717652372
-| tempered height = 5.545073
-| pure height = 4.164318
-| integral = 1.031155
-| gap = 14.793013
-| octave = 1202.97506589617
-| consistent = 14
-| distinct = 9
-}}
 == Approximation to irrational intervals ==
@@ Line 785: / Line 771: @@
 [[File:12072608 10207851395433055 404343132969239728 n.jpg|none|thumb|960x960px]]
+* [[Lumatone mapping for 26edo]]
 == Literature ==
 [http://www.ronsword.com Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.]
@@ Line 826: / Line 813: @@
 ; [[Zach Curley]]
 * [http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Guitar%20Serenade%20in%20Q%20Major.mp3 Guitar Serenade in Q Major]{{dead link}}
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/FxTxQ0ayDpg ''Microtonal Improvisation in 26edo''] (2023)
 ; [[User:Eboone|Ebooone]]
@@ Line 849: / Line 839: @@
 * [https://www.youtube.com/watch?v=r0jCdHEZpzM Claudi Meneghin - Suite (Prelude, Variations, Fugue) in 26edo, for Synth & Baroque Bassoon] (2023)
 * [https://www.youtube.com/watch?v=rjo3X1-D57Y Canon 3-in-1 on a Ground for Baroque Ensemble] (2023)
+; [[Microtonal Maverick]] (formerly The Xen Zone)
+* [https://www.youtube.com/watch?v=qm_k9xjXRf0 ''The Microtonal Magic of 26EDO (with 13-limit jam)''] (2024)
+* [https://www.youtube.com/watch?v=im2097HVqgA ''The Blues but with 26 Notes per Octave''] (2024) (explanatory video &mdash; contiguous music starts at 08:48)
 ; [[Herman Miller]]
@@ Line 884: / Line 878: @@
 ; [[Chris Vaisvil]]
 * [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 ''Morpheous Wing'' in 26 edo] (2016)
-== See also ==
-* [[Lumatone mapping for 26edo]]
 == Notes ==