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}}
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|26}}
{{ED intro}}


== Theory ==
== 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 ''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).


With a fifth of 15 steps, it can be equally divided into 3 or 5, supporting [[slendric]] temperament and [[bleu]] temperament respectively.
With a fifth of 15 steps, it can be equally divided into 3 or 5, supporting [[slendric]] temperament and [[bleu]] temperament respectively.
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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 yields interesting but to some unsatisfying results (due mainly to the dissonance of its thirds, and its major second of approximately [[10/9]] instead of [[9/8]]).
# 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]].
# 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 {{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, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant, assuming the relatively common values of ''a'' = 2 and ''s'' = 1.01. 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]].
Thanks to its sevenths, 26edo is an ideal tuning for its size for [[metallic harmony]].
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=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|26}}
{{Harmonics in equal|26}}
=== Subsets and supersets ===
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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! Degrees
! Degrees
! [[Cent]]s
! [[Cent]]s
! Approximate Ratios*
! Approximate ratios<ref group="note">{{sg|limit=13-limit}}</ref>
! Interval<br>Name
! Interval<br>name
! Example<br>in D
! Example<br>in D
![[SKULO interval names|SKULO]]  
! [[SKULO interval names|SKULO]]<br>[[SKULO interval names|Interval name]]
[[SKULO interval names|Interval Name]]
! Example<br>in D
!Example
! colspan="2" | [[Solfege|Solfeges]]
in D
! colspan="2" |[[Solfege|Solfeges]]
|-
|-
| 0
| 0
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| P1
| P1
| D
| D
|P1
| P1
|D
| D
| da
| da
| do
| do
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| A1
| A1
| D#
| D#
|A1, S1
| A1, S1
|D#, SD
| D#, SD
| du
| du
| di
| di
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| d2
| d2
| Ebb
| Ebb
|sm2
| sm2
|sEb
| sEb
| fro
| fro
| rih
| rih
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| m2
| m2
| Eb
| Eb
|m2
| m2
|Eb
| Eb
| fra
| fra
| ru
| ru
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| M2
| M2
| E
| E
|M2
| M2
|E
| E
| ra
| ra
| re
| re
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| A2
| A2
| E#
| E#
|SM2
| SM2
|SE
| SE
| ru
| ru
| ri
| ri
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| d3
| d3
| Fb
| Fb
|sm3
| sm3
|sF
| sF
| no
| no
| ma
| ma
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| m3
| m3
| F
| F
|m3
| m3
|F
| F
| na
| na
| me
| me
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| M3
| M3
| F#
| F#
|M3
| M3
|F#
| F#
| ma
| ma
| muh/mi
| muh/mi
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| A3
| A3
| Fx
| Fx
|SM3
| SM3
|SF#
| SF#
| mu
| mu
| maa
| maa
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| d4
| d4
| Gb
| Gb
|s4
| s4
|sG
| sG
| fo
| fo
| fe
| fe
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| P4
| P4
| G
| G
|P4
| P4
|G
| G
| fa
| fa
| fa
| fa
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| A4
| A4
| G#
| G#
|A4
| A4
|G#
| G#
| fu/pa
| fu/pa
| fu
| fu
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| AA4, dd5
| AA4, dd5
| Gx, Abb
| Gx, Abb
|SA4, sd5
| SA4, sd5
|SG#, sAb
| SG#, sAb
| pu/sho
| pu/sho
| fi/se
| fi/se
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| d5
| d5
| Ab
| Ab
|d5
| d5
|Ab
| Ab
| sha/so
| sha/so
| su
| su
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| P5
| P5
| A
| A
|P5
| P5
|A
| A
| sa
| sa
| sol
| sol
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| A5
| A5
| A#
| A#
|S5
| S5
|SA
| SA
| su
| su
| si
| si
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| d6
| d6
| Bbb
| Bbb
|sm6
| sm6
|sBb
| sBb
| flo
| flo
| leh
| leh
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| m6
| m6
| Bb
| Bb
|m6
| m6
|Bb
| Bb
| fla
| fla
| le/lu
| le/lu
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| M6
| M6
| B
| B
|M6
| M6
|B
| B
| la
| la
| la
| la
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| A6
| A6
| B#
| B#
|SM6
| SM6
|SB
| SB
| lu
| lu
| li
| li
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| d7
| d7
| Cb
| Cb
|sm7
| sm7
|sC
| sC
| tho
| tho
| ta
| ta
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| m7
| m7
| C
| C
|m7
| m7
|C
| C
| tha
| tha
| te
| te
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| M7
| M7
| C#
| C#
|M7
| M7
|C#
| C#
| ta
| ta
| tu/ti
| tu/ti
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| A7
| A7
| Cx
| Cx
|SM7
| SM7
|SC#
| SC#
| tu
| tu
| to
| to
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| d8
| d8
| Db
| Db
|d8, s8
| d8, s8
|Db, sD
| Db, sD
| do
| do
| da
| da
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| P8
| P8
| D
| D
|P8
| P8
|D
| D
| da
| da
| do
| do
|}
|}
* based on treating 26edo as a [[13-limit]] temperament; other approaches are possible.


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
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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 Edoteps
! Notes as Edosteps
! Notes of C Chord
! Notes of C Chord
! Written Name
! Written Name
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== Notation ==
== Notation ==
===Sagittal notation===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[12edo#Sagittal notation|12]], and [[19edo#Sagittal notation|19]], is a subset of the notation for [[52edo#Sagittal notation|52-EDO]], and is a superset of the notation for [[13edo#Sagittal notation|13-EDO]].
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[12edo#Sagittal notation|12]], and [[19edo#Sagittal notation|19]], is a subset of the notation for [[52edo#Sagittal notation|52-EDO]], and is a superset of the notation for [[13edo#Sagittal notation|13-EDO]].
====Evo flavor====


==== Evo flavor ====
<imagemap>
<imagemap>
File:26-EDO_Evo_Sagittal.svg
File:26-EDO_Evo_Sagittal.svg
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Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====


==== Revo flavor ====
<imagemap>
<imagemap>
File:26-EDO_Revo_Sagittal.svg
File:26-EDO_Revo_Sagittal.svg
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default [[File:26-EDO_Revo_Sagittal.svg]]
default [[File:26-EDO_Revo_Sagittal.svg]]
</imagemap>
</imagemap>
===Other===
====MisterShafXen’s notation====
See [[MisterShafXen’s 26edo notation]].


== Approximation to JI ==
== Approximation to JI ==
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== Approximation to irrational intervals ==
== Approximation to irrational intervals ==
After [[13edo #Approximation to irrational intervals|13edo]], the weird coïncidences continue: [[11/7 #Proximity with π/2|acoustic π/2]] (17\26) is just in between the ϕ intervals provided by 13edo (16\26 for [[Logarithmic phi|logarithmic ϕ]]/2, and 18\26 for [[Acoustic phi|acoustic ϕ]]).
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.
 
Not until 1076edo do we find a better edo in terms of relative error on these intervals (which is not a very relevant edo for logarithmic ϕ, since 1076 does not belong to the Fibonacci sequence).
 
However, it should be noted that [[User:Contribution/Logarithmic constants VS acoustic constants (opinion piece article)|from an acoustic perspective]], acoustic π and acoustic ϕ are both better represented on [[23edo]].


{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Direct approximation
|+ style="font-size: 105%;" | Direct approximation
|-
|-
! Interval
! Interval
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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma basis]]
! rowspan="2" | [[Comma basis]]
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* [[List of 26et rank two temperaments by badness]]
* [[List of 26et rank two temperaments by badness]]
* [[List of edo-distinct 26et rank two temperaments]]
* [[List of edo-distinct 26et rank two temperaments]]
Important MOSes include (in addition to ones found in [[13edo]]):


* diatonic ([[flattone]]) 4443443 (15\26, 1\1)
Important mos scales include (in addition to ones found in [[13edo]]):
* chromatic ([[flattone]]) 313131331313 (15\26, 1\1)
* [[Flattone]][7] (diatonic) 4443443 (15\26, 1\1)
* enharmonic ([[flattone]]) 2112112112121121121 (15\26, 1\1)
* [[Flattone]][12] (chromatic) 313131331313 (15\26, 1\1)
* [[orgone]] 5525252 (7\26, 1\1)
* [[Flattone]][19] (enharmonic) 2112112112121121121 (15\26, 1\1)
* [[orgone]] 32322322322 (7\26, 1\1)
* [[Orgone]][7] 5525252 (7\26, 1\1)
* [[orgone]] 212212221222122 (7\26, 1\1)
* [[Orgone]][11] 32322322322 (7\26, 1\1)
* [[lemba]] 553553 (5\26, 1\2)
* [[Orgone]][15] 212212221222122 (7\26, 1\1)
* [[lemba]] 3232332323 (5\26, 1\2)
* [[Lemba]][6] 553553 (5\26, 1\2)
* [[lemba]] 2122122121221221 (5\26, 1\2)
* [[Lemba]][10] 3232332323 (5\26, 1\2)
* [[Lemba]][16] 2122122121221221 (5\26, 1\2)


{| class="wikitable center-all left-3"
{| class="wikitable center-all left-3"
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=== Hendec in 26et ===
=== Hendec in 26et ===
[[Hendec]], the 13-limit 26&amp;46 temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy.
[[Hendec]], the 13-limit {{nowrap|26 & 46}} temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy.


=== Commas ===
=== Commas ===
26et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 26 41 60 73 90 96 }}.)
26et [[tempering out|tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 26 41 60 73 90 96 }}.


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
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| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis, Jubilisma
| Jubilisma, tritonic diesis
|-
|-
| 7
| 7
Line 749: Line 742:
| Animist comma
| Animist comma
|}
|}
<references/>


== Scales ==
== Scales ==
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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.]
Line 820: Line 813:
; [[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 &mdash; contiguous music starts at 08:48)


; [[Herman Miller]]
; [[Herman Miller]]
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* [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)


== See also ==
== Notes ==
* [[Lumatone mapping for 26edo]]
<references group="note" />


[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/26edo"