26edo: Difference between revisions

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Move Fynn's comma to subsets and supersets section
 
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{{interwiki
{{Interwiki
| en = 26edo
| de = 26-EDO
| de = 26-EDO
| en = 26edo
| es =  
| es =  
| ja =
| 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.
# 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 [[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).
# 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.
# 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 ===
=== 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.
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 ==
== Intervals ==
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| 8
| 8
| 369.23
| 369.23
| [[5/4]], [[11/9]], [[16/13]]
| [[5/4]], [[11/9]], [[16/13]], [[26/21]]
| M3
| M3
| F#
| F#
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| 10
| 10
| 461.54
| 461.54
| [[21/16]], [[13/10]]
| [[21/16]], [[13/10]], [[64/49]]
| d4
| d4
| Gb
| Gb
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| 16
| 16
| 738.46
| 738.46
| [[32/21]], [[20/13]]
| [[32/21]], [[20/13]], [[49/32]]
| A5
| A5
| A#
| A#
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| 18
| 18
| 830.77
| 830.77
| [[13/8]], [[8/5]]
| [[8/5]], [[13/8]], [[21/13]]
| m6
| m6
| Bb
| Bb
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| do
| do
|}
|}
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
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== Notation ==
== Notation ==
=== Standard notation ===
Because the chromatic semitone is 1 step, only sharps and flats are needed to notate 26edo.
{{sharpness-sharp1}}
=== 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]].
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== Approximation to irrational intervals ==
== 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"
{| class="wikitable center-all"
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|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 45/44, 50/49, 65/64 78/77, 81/80, 85/84
| 45/44, 50/49, 65/64, 78/77, 81/80, 85/84
| {{mapping| 26 41 60 73 90 96 106 }}
| {{mapping| 26 41 60 73 90 96 106 }}
| +2.613
| +2.613
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| 2
| 2
| 6\26
| 6\26
| [[Doublewide]] / [[cavalier]]
| [[Doublewide]] / [[Jubilismic_clan#Cavalier|cavalier]]
|-
|-
| 13
| 13
| 1\26
| 1\26
| [[Triskaidekic]]
| [[Orwellismic_temperaments#Triskaidekic|Triskaidekic]]
|}
|}


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| Animist comma
| Animist comma
|}
|}
<references group="note" />


== Octave stretch or compression ==
== Octave stretch or compression ==
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{{main|List of MOS scales in 26edo}}
{{main|List of MOS scales in 26edo}}


Important mos scales include (in addition to ones found in [[13edo]]):
; Most important [[mos scale]]s
* [[Flattone]][7] (diatonic) 4 4 4 3 4 4 3 (15\26, 1\1) (quasi-[[equiheptatonic]])
* [[Flattone]][7] (diatonic) 4 4 4 3 4 4 3 (15\26, 1\1) (quasi-[[equiheptatonic]])
* [[Flattone]][12] (chromatic) 3 1 3 1 3 1 3 3 1 3 1 3 (15\26, 1\1)
* [[Flattone]][12] (chromatic) 3 1 3 1 3 1 3 3 1 3 1 3 (15\26, 1\1)
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* [[Lemba]][16] 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 (5\26, 1\2)
* [[Lemba]][16] 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 (5\26, 1\2)


==== Orgone temperament ====
; Additional mos scales
Since the perfect 5th in 26edo spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12).
 
The former approach produces MOS at:
* 1L+4s (5 5 5 5 6) ([[mothra]][5])
* 5L+1s (5 5 5 5 5 1) ([[mothra]][6])
* 5L+6s (4 1 4 1 4 1 4 1 4 1 1) ([[mothra]][11])
* 5L+11s (1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1) ([[mothra]][16])
and is excellent for 4:6:7 triads.
 
The latter produces MOS at:
* 1L+7s (3 3 3 3 3 3 3 5) ([[secund]][8])
* 8L+1s (3 3 3 3 3 3 3 3 2) ([[secund]][9])
and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (though further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).
 
=== Orgone temperament ===
[[Andrew Heathwaite]] first proposed [[Orgonia|orgone]] temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:
[[Andrew Heathwaite]] first proposed [[Orgonia|orgone]] temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:
* The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L_3s|4L 3s (mish)]].
* The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L_3s|4L 3s (mish)]].
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[[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|orgone_heptatonic.jpg]]
[[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|orgone_heptatonic.jpg]]


==== Additional scalar bases available ====
=== Other scales ===
Since the perfect 5th in 26edo spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12).
* Approximate [[5afdo]]: 4 4 7 6 5
 
* Approximate [[6afdo]]: 6 5 4 4 4 3
The former approach produces MOS at:
* Free range octatonic{{idio}} ([[modmos]] of [[hendec]][8]): 2 7 2 2 2 7 2 2
* 1L+4s (5 5 5 5 6)
* Free range 14-tonic{{idio}} ([[modmos]] of [[hendec]][14]): 1 1 1 7 1 1 1 1 1 1 7 1 1 1
* 5L+1s (5 5 5 5 5 1)
* Pseudo-[[equipentatonic]]: 5 6 4 6 5 or 6 5 4 5 6
* 5L+6s (4 1 4 1 4 1 4 1 4 1 1)
and is excellent for 4:6:7 triads.
 
The latter produces MOS at:
* 1L+7s (3 3 3 3 3 3 3 5)
* 8L+1s (3 3 3 3 3 3 3 3 2)
and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (though further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).


== Instruments ==
== Instruments ==
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{{Catrel|26edo tracks}}
{{Catrel|26edo tracks}}


=== Modern renderings ===
=== 26 equal divisions of the octave (26edo proper) ===
==== Modern renderings ====
; {{W|Johann Sebastian Bach}}
; {{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)
* [https://www.youtube.com/watch?v=LUNOFjiyZ0Y ''Contrapunctus 4'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=dlXFoIoc_uk ''Contrapunctus 11'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* ''Contrapunctus 11'' from ''The Art of Fugue'', BWV 1080 (1742–1749) – rendered by Claudi Meneghin
** [https://www.youtube.com/watch?v=dlXFoIoc_uk organ rendition] (2024)
** [https://www.youtube.com/watch?v=NUo1uvUuOlo harpsichord rendition] (2025)
* [https://www.youtube.com/watch?v=PcQIojtN-lk ''"Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025)


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


=== 21st century ===
==== 21st century ====
; [[Abnormality]]
; [[Abnormality]]
* [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)
* [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)
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* [https://www.youtube.com/shorts/FxTxQ0ayDpg ''Microtonal Improvisation in 26edo''] (2023)
* [https://www.youtube.com/shorts/FxTxQ0ayDpg ''Microtonal Improvisation in 26edo''] (2023)
* [https://www.youtube.com/shorts/Gw9Fu5MAozs ''Waltz in 26edo''] (2025)
* [https://www.youtube.com/shorts/Gw9Fu5MAozs ''Waltz in 26edo''] (2025)
* [https://www.youtube.com/shorts/OFSK_9QjebE ''Change of Generation - Unlucky Morpheus (microtonal cover in 26edo)''] (2025)
* [https://www.youtube.com/watch?v=cXdhuyibzQs ''What Is This Diddy Blud Doing On The Calculator (26edo microtonal Lumatone cover)''] (2025)
* [https://www.youtube.com/watch?v=n6v7xxM9mmc ''Harpy Hare - Yaelokre (microtonal cover in 26edo)''] (2025)
* [https://www.youtube.com/shorts/yr35O0hnedM ''26edo lullaby''] (2025)
* [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/L2JzCNj6jak <nowiki>[short 2]</nowiki>'']
* [https://www.youtube.com/shorts/wHGLOaeAkt8 ''26edo groove''] (2026)


; [[User:Eboone|Ebooone]]
; [[User:Eboone|Ebooone]]
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; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=2ziAZx03KF8 ''Lemba Suite, for two organs''] (2022)
* [https://www.youtube.com/watch?v=siNzE3d_WsQ ''Canon 3-in-1 on a ground "The Tempest", in 26edo''] (2023)
* [https://www.youtube.com/watch?v=siNzE3d_WsQ ''Canon 3-in-1 on a ground "The Tempest", in 26edo''] (2023)
* [https://youtube.com/shorts/uh7auNGakk4 ''Greensleeves for three soprano saxes and baroque bassoon, in 26edo''] (2023)
* [https://youtube.com/shorts/uh7auNGakk4 ''Greensleeves for three soprano saxes and baroque bassoon, in 26edo''] (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=r0jCdHEZpzM ''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)
* [https://www.youtube.com/shorts/hPW4hXlu8cc ''THE TEMPEST - CANON in 26edo, 3-in-1 for 3 Baroque Violins and Continuo''] (2025)


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


== Notes ==
; [[Stephen Weigel]]
<references group="note" />
* [https://www.youtube.com/watch?v=rfIbSZh7Iuw ''When She Loved Me (Toy Story 2)'' - microtonal cover] (2023)
 
; [[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]]
[[Category:Listen]]

Latest revision as of 07:44, 11 April 2026

← 25edo 26edo 27edo →
Prime factorization 2 × 13
Step size 46.1538 ¢ 
Fifth 15\26 (692.308 ¢)
Semitones (A1:m2) 1:3 (46.15 ¢ : 138.5 ¢)
Consistency limit 13
Distinct consistency limit 5

26 equal divisions of the octave (abbreviated 26edo or 26ed2), also called 26-tone equal temperament (26tet) or 26 equal temperament (26et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 26 equal parts of about 46.2 ¢ each. Each step represents a frequency ratio of 21/26, or the 26th root of 2.

Theory

26edo has a perfect fifth of about 692 cents and tempers out 81/80 in the 5-limit, making it a very flat meantone tuning (0.088957 ¢ flat of the 4/9-comma meantone fifth) with a very soft diatonic scale.

In the 7-limit, it tempers out 50/49, 525/512, and 875/864, and supports 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 consistently. 26edo has a very good approximation of the harmonic seventh (7/4), as it is the denominator of a convergent to log27.

26edo's minor sixth (1.6158) is very close to 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.

The structure of 26edo is an interesting beast, with various approaches relating it to various rank-2 temperaments.

  1. 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.
  2. 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.
  3. 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 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.
  4. We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).
  5. 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 ¢, 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.

Odd harmonics

Approximation of odd harmonics in 26edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -9.6 -17.1 +0.4 -19.3 +2.5 -9.8 +19.4 -12.6 -20.6 -9.2 +17.9
Relative (%) -20.9 -37.0 +0.9 -41.8 +5.5 -21.1 +42.1 -27.4 -44.6 -20.0 +38.7
Steps
(reduced)
41
(15)
60
(8)
73
(21)
82
(4)
90
(12)
96
(18)
102
(24)
106
(2)
110
(6)
114
(10)
118
(14)

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.

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

Degrees Cents Approximate ratios[note 1] Interval
name
Example
in D
SKULO
Interval name
Example
in D
Solfeges
0 0.00 1/1 P1 D P1 D da do
1 46.15 33/32, 49/48, 36/35, 25/24 A1 D# A1, S1 D#, SD du di
2 92.31 21/20, 22/21, 26/25 d2 Ebb sm2 sEb fro rih
3 138.46 12/11, 13/12, 14/13, 16/15 m2 Eb m2 Eb fra ru
4 184.62 9/8, 10/9, 11/10 M2 E M2 E ra re
5 230.77 8/7, 15/13 A2 E# SM2 SE ru ri
6 276.92 7/6, 13/11, 33/28 d3 Fb sm3 sF no ma
7 323.08 135/112, 6/5 m3 F m3 F na me
8 369.23 5/4, 11/9, 16/13, 26/21 M3 F# M3 F# ma muh/mi
9 415.38 9/7, 14/11, 33/26 A3 Fx SM3 SF# mu maa
10 461.54 21/16, 13/10, 64/49 d4 Gb s4 sG fo fe
11 507.69 75/56, 4/3 P4 G P4 G fa fa
12 553.85 11/8, 18/13 A4 G# A4 G# fu/pa fu
13 600.00 7/5, 10/7 AA4, dd5 Gx, Abb SA4, sd5 SG#, sAb pu/sho fi/se
14 646.15 16/11, 13/9 d5 Ab d5 Ab sha/so su
15 692.31 112/75, 3/2 P5 A P5 A sa sol
16 738.46 32/21, 20/13, 49/32 A5 A# S5 SA su si
17 784.62 11/7, 14/9 d6 Bbb sm6 sBb flo leh
18 830.77 8/5, 13/8, 21/13 m6 Bb m6 Bb fla le/lu
19 876.92 5/3, 224/135 M6 B M6 B la la
20 923.08 12/7, 22/13 A6 B# SM6 SB lu li
21 969.23 7/4, 26/15 d7 Cb sm7 sC tho ta
22 1015.38 9/5, 16/9, 20/11 m7 C m7 C tha te
23 1061.54 11/6, 13/7, 15/8, 24/13 M7 C# M7 C# ta tu/ti
24 1107.69 21/11, 25/13, 40/21 A7 Cx SM7 SC# tu to
25 1153.85 64/33, 96/49, 35/18, 48/25 d8 Db d8, s8 Db, sD do da
26 1200.00 2/1 P8 D P8 D da do
  1. Based on treating 26edo as a 13-limit temperament; other approaches are also possible.

Interval quality and chord names in color notation

Using color notation, qualities can be loosely associated with colors:

Quality Color Monzo Format Examples
diminished zo {a, b, 0, 1} 7/6, 7/4
minor fourthward wa {a, b}, b < -1 32/27, 16/9
" gu {a, b, -1} 6/5, 9/5
major yo {a, b, 1} 5/4, 5/3
" fifthward wa {a, b}, b > 1 9/8, 27/16
augmented ru {a, b, 0, -1} 9/7, 12/7

All 26edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Spelling certain chords properly may require triple sharps and flats, especially if the tonic is anything other than the 11 keys in the Eb-C# range. Here are the zo, gu, yo and ru triads:

Color of the 3rd JI chord Notes as Edosteps Notes of C Chord Written Name Spoken Name
zo 6:7:9 0-6-15 C Ebb G C(b3) or C(d3) C flat-three or C dim-three
gu 10:12:15 0-7-15 C Eb G Cm C minor
yo 4:5:6 0-8-15 C E G C C major or C
ru 14:18:21 0-9-15 C E# G C(#3) or C(A3) C sharp-three or C aug-three

For a more complete list, see Ups and downs notation #Chord names in other EDOs.

Notation

Standard notation

Because the chromatic semitone is 1 step, only sharps and flats are needed to notate 26edo.

Step offset −2 −1 0 +1 +2
Symbol

Sagittal notation

This notation uses the same sagittal sequence as EDOs 5, 12, and 19, is a subset of the notation for 52-EDO, and is a superset of the notation for 13-EDO.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation

Approximation to JI

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 26edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 26edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/12, 24/13 0.111 0.2
7/4, 8/7 0.405 0.9
11/7, 14/11 2.123 4.6
9/5, 10/9 2.212 4.8
11/8, 16/11 2.528 5.5
13/10, 20/13 7.325 15.9
5/3, 6/5 7.436 16.1
13/9, 18/13 9.536 20.7
3/2, 4/3 9.647 20.9
13/8, 16/13 9.758 21.1
7/6, 12/7 10.052 21.8
13/7, 14/13 10.163 22.0
11/6, 12/11 12.176 26.4
13/11, 22/13 12.287 26.6
15/11, 22/15 16.895 36.6
15/13, 26/15 16.972 36.8
5/4, 8/5 17.083 37.0
7/5, 10/7 17.488 37.9
15/14, 28/15 19.019 41.2
9/8, 16/9 19.295 41.8
15/8, 16/15 19.424 42.1
11/10, 20/11 19.611 42.5
9/7, 14/9 19.699 42.7
11/9, 18/11 21.823 47.3
15-odd-limit intervals in 26edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/12, 24/13 0.111 0.2
7/4, 8/7 0.405 0.9
11/7, 14/11 2.123 4.6
9/5, 10/9 2.212 4.8
11/8, 16/11 2.528 5.5
13/10, 20/13 7.325 15.9
5/3, 6/5 7.436 16.1
13/9, 18/13 9.536 20.7
3/2, 4/3 9.647 20.9
13/8, 16/13 9.758 21.1
7/6, 12/7 10.052 21.8
13/7, 14/13 10.163 22.0
11/6, 12/11 12.176 26.4
13/11, 22/13 12.287 26.6
15/13, 26/15 16.972 36.8
5/4, 8/5 17.083 37.0
7/5, 10/7 17.488 37.9
9/8, 16/9 19.295 41.8
11/10, 20/11 19.611 42.5
9/7, 14/9 19.699 42.7
11/9, 18/11 21.823 47.3
15/8, 16/15 26.730 57.9
15/14, 28/15 27.135 58.8
15/11, 22/15 29.258 63.4

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[clarification needed].

Direct approximation
Interval Error (abs, ¢)
2ϕ / ϕ 0.858
ϕ 2.321
π 2.820
2ϕ 3.179
π/ϕ 5.141
2ϕ / π 5.999

Regular temperament properties

Subgroup Comma basis Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.3 [-41 26 [26 41]] +3.043 3.05 6.61
2.3.5 81/80, 78125/73728 [26 41 60]] +4.489 3.22 6.98
2.3.5.7 50/49, 81/80, 405/392 [26 41 60 73]] +3.324 3.44 7.45
2.3.5.7.11 45/44, 50/49, 81/80, 99/98 [26 41 60 73 90]] +2.509 3.48 7.53
2.3.5.7.11.13 45/44, 50/49, 65/64, 78/77, 81/80 [26 41 60 73 90 96]] +2.531 3.17 6.87
2.3.5.7.11.13.17 45/44, 50/49, 65/64, 78/77, 81/80, 85/84 [26 41 60 73 90 96 106]] +2.613 2.94 6.38
2.3.5.7.11.13.17.19 45/44, 50/49, 57/56, 65/64, 78/77, 81/80, 85/84 [26 41 60 73 90 96 106 110]] +2.894 2.85 6.18
  • 26et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-, and 29-limit (using the 26i val for the 23- and 29-limit). The next equal temperaments performing better in those subgroups are 27eg, 27eg, 29g, and 46, respectively.

Rank-2 Temperaments

Periods
per 8ve
Generator Temperaments
1 1\26 Quartonic / quarto
1 3\26 Glacier / bleu / jerome / secund
1 5\26 Cynder / mothra
1 7\26 Orgone / superkleismic
1 9\26 Wesley / roman
1 11\26 Flattone / flattertone
2 1\26 Elvis
2 2\26 Injera
2 3\26 Fifive / crepuscular
2 4\26 Dubbla
Unidec / hendec
2 5\26 Lemba
2 6\26 Doublewide / cavalier
13 1\26 Triskaidekic

Hendec in 26et

Hendec, the 13-limit 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

26et tempers out the following commas. This assumes the val 26 41 60 73 90 96].

Prime
limit
Ratio[note 1] Monzo Cents Color name Name(s)
5 81/80 [-4 4 -1 21.51 Gu Syntonic comma, Didymos comma, meantone comma
5 (60 digits) [-17 62 -35 0.23 Quadla-sepquingu Senior comma
7 525/512 [-9 1 2 1 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 50/49 [1 0 2 -2 34.98 Biruyo Jubilisma, tritonic diesis
7 875/864 [-5 -3 3 1 21.90 Zotriyo Keema
7 4000/3969 [5 -4 3 -2 13.47 Sarurutriyo Octagar comma
7 1728/1715 [6 3 -1 -3 13.07 Triru-agu Orwellisma
7 1029/1024 [-10 1 0 3 8.43 Latrizo Gamelisma
7 (12 digits) [-9 8 -4 2 8.04 Labizogugu Varunisma
7 (18 digits) [-26 -1 1 9 3.79 Latritrizo-ayo Wadisma
7 4375/4374 [-1 -7 4 1 0.40 Zoquadyo Ragisma
11 99/98 [-1 2 0 -2 1 17.58 Loruru Mothwellsma
11 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
11 65536/65219 [16 0 0 -2 -3 8.39 Satrilu-aruru Orgonisma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1 3.93 Luzozogu Werckisma
11 3025/3024 [-4 -3 2 -1 2 0.57 Loloruyoyo Lehmerisma
11 9801/9800 [-3 4 -2 -2 2 0.18 Bilorugu Kalisma, Gauss' comma
13 105/104 [-3 1 1 1 0 -1 16.57 Thuzoyo Animist comma
  1. Ratios longer than 10 digits are presented by placeholders with informative hints.

Octave stretch or compression

26edo's simple primes with the most error - 3, 5 and 13 - are all tuned flat, so it can benefit from octave stretching. Some suitable stretched-octave 26edo tunings include 93ed12 or 100zpi.

Scales

MOS scales

Most important mos scales
  • Flattone[7] (diatonic) 4 4 4 3 4 4 3 (15\26, 1\1) (quasi-equiheptatonic)
  • Flattone[12] (chromatic) 3 1 3 1 3 1 3 3 1 3 1 3 (15\26, 1\1)
  • Flattone[19] (enharmonic) 2 1 1 2 1 1 2 1 1 2 1 2 1 1 2 1 1 2 1 (15\26, 1\1)
  • Orgone[7] 5 5 2 5 2 5 2 (7\26, 1\1)
  • Orgone[11] 3 2 3 2 2 3 2 2 3 2 2 (7\26, 1\1)
  • Orgone[15] 2 1 2 2 1 2 2 2 1 2 2 2 1 2 2 (7\26, 1\1)
  • Lemba[6] 5 5 3 5 5 3 (5\26, 1\2)
  • Lemba[10] 3 2 3 2 3 3 2 3 2 3 (5\26, 1\2)
  • Lemba[16] 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 (5\26, 1\2)
Additional mos scales

Since the perfect 5th in 26edo spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12).

The former approach produces MOS at:

  • 1L+4s (5 5 5 5 6) (mothra[5])
  • 5L+1s (5 5 5 5 5 1) (mothra[6])
  • 5L+6s (4 1 4 1 4 1 4 1 4 1 1) (mothra[11])
  • 5L+11s (1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1) (mothra[16])

and is excellent for 4:6:7 triads.

The latter produces MOS at:

  • 1L+7s (3 3 3 3 3 3 3 5) (secund[8])
  • 8L+1s (3 3 3 3 3 3 3 3 2) (secund[9])

and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (though further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).

Orgone temperament

Andrew Heathwaite first proposed orgone temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:

  • The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. MOS of type 4L 3s (mish).
  • The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.
  • The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. MOS of type 4L 7s.
  • The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108 1200.

The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates 16:11 and 3g approximates 7:4 (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.

orgone_heptatonic.jpg

Other scales

Instruments

James Fenn's midi keyboard.

Literature

Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.

Music

See also: Category:26edo tracks

26 equal divisions of the octave (26edo proper)

Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns

21st century

Abnormality
Jim Aikin
Beheld
benyamind
Cameron Bobro
CellularAutomaton
City of the Asleep
Zach Curley
Bryan Deister
Ebooone
Francium
Igliashon Jones
Budjarn Lambeth
Melopœia
Claudi Meneghin
Microtonal Maverick (formerly The Xen Zone)
Herman Miller
Shaahin Mohajeri
Mundoworld
NullPointerException Music
Ray Perlner
Sevish
Jon Lyle Smith
Tapeworm Saga
Uncreative Name
Chris Vaisvil
Stephen Weigel
YoVariable

Unequal Derivatives of 26edo

Bryan Deister