26edo: Difference between revisions
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== Theory == | == Theory == | ||
26edo tempers out [[81/80]] in the [[5-limit]], making it a [[meantone]] tuning | 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). | 26edo's minor sixth (1.6158) is very close to {{nowrap|''φ'' ≈ 1.6180}} (i.e. the golden ratio). | ||
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The structure of 26edo is an interesting beast, with various approaches relating it to various rank-2 temperaments. | The structure of 26edo is an interesting beast, with various approaches relating it to various rank-2 temperaments. | ||
# In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which | # 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]]. | # 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 {{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. | ||
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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. | # 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 | 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]]. | Thanks to its sevenths, 26edo is an ideal tuning for its size for [[metallic harmony]]. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
26edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing harmonics 5 through 23 (including direct approximations) with 26edo. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. | 26edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing harmonics 5 and 9 through 23 (including direct approximations) with 26edo. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. | ||
== Intervals == | == Intervals == | ||
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! [[Kite's color notation|Color of the 3rd]] | ! [[Kite's color notation|Color of the 3rd]] | ||
! JI chord | ! JI chord | ||
! Notes as | ! Notes as Edosteps | ||
! Notes of C Chord | ! Notes of C Chord | ||
! Written Name | ! Written Name | ||
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=== 15-odd-limit interval mappings === | === 15-odd-limit interval mappings === | ||
{{Q-odd-limit intervals|26}} | {{Q-odd-limit intervals|26}} | ||
== Approximation to irrational intervals == | == Approximation to irrational intervals == | ||
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[[File:12072608 10207851395433055 404343132969239728 n.jpg|none|thumb|960x960px]] | [[File:12072608 10207851395433055 404343132969239728 n.jpg|none|thumb|960x960px]] | ||
* [[Lumatone mapping for 26edo]] | |||
== Literature == | == Literature == | ||
[http://www.ronsword.com Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.] | [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]] | ; [[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}} | * [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]] | ; [[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=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) | * [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]] | ; [[Herman Miller]] | ||
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; [[Chris Vaisvil]] | ; [[Chris Vaisvil]] | ||
* [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 ''Morpheous Wing'' in 26 edo] (2016) | * [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 ''Morpheous Wing'' in 26 edo] (2016) | ||
== Notes == | == Notes == | ||