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{{Infobox ET
{{interwiki
| Prime factorization = 2 × 13
| de = 26-EDO
| Step size = 46.154¢
| en = 26edo
| Fifth = 15\26 = 692.31¢
| es =  
| Major 2nd = 4\26 = 185¢
| ja =
| Minor 2nd = 3\26 = 138¢
| Augmented 1sn = 1\26 = 46¢
}}
}}
 
{{Infobox ET}}
'''26edo''' divides the [[octave]] into 26 equal parts of 46.154 [[Cent|cents]] each. It tempers out 81/80 in the [[5-limit]], making it a meantone tuning with a very flat fifth.
{{ED intro}}


== Theory ==
== Theory ==
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]].


{| class="wikitable center-all"
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.
|-
|+ Approximation of [[primary interval]]s in 26 EDO
|-
! colspan="2" | Prime interval
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
! 23
|-
! rowspan="2" | Error
! absolute ([[cent|¢]])
| 0
| -9.6
| -17.1
| +0.4
| +2.5
| -9.8
| -12.6
| -20.6
| +17.9
|-
! [[Relative error|relative]] (%)
| 0
| -21
| -37
| +1
| +5
| -21
| -27
| -45
| +39
|-
! colspan="2" | [[nearest edomapping]]
| 26
| 15
| 8
| 21
| 12
| 18
| 2
| 6
| 14
|-
!
! [[fifthspan]]
| 0
| +1
| +4
| -9
| +6
| -4
| -12
| -10
| -6
|}


In the [[7-limit]], it tempers out 50/49, 525/512 and 875/864, and supports [[injera]], [[flattone]], [[Jubilismic clan#Lemba|lemba]] and [[Jubilismic clan#Doublewide|doublewide]] temperaments. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13 odd limit]] [[consistent|consistently]]. 26edo has a very good approximation of the harmonic seventh ([[7/4]]).
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 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 two temperaments.
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 yields interesting but to some unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately [[10/9]] or [[8/7]], but ''not'' [[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]].
# 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.
# 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.


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&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.
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.


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 | -3 0 0 6 -4&gt;. The 65536/65219 comma, the orgonisma, leads to [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with MOS scales of size 7, 11 and 15. The | -3 0 0 6 -4&gt; comma leads to a half-octave period and an approximate 49/44 generator of 4\26, leading to MOS of size 8 and 14.
Thanks to its sevenths, 26edo is an ideal tuning for its size for [[metallic harmony]].


4. We can also treat 26-EDO as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower EDOs).
=== Odd harmonics ===
{{Harmonics in equal|26}}


5. It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fifths gives a 17:14 and four gives 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.
=== 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 ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! Degrees
! [[Cents|cents]]
! [[Cent]]s
! Approximate Ratios*
! Approximate ratios<ref group="note">{{sg|limit=13-limit}}</ref>
! Solfege
! Interval<br>name
! Interval<br>Name
! Example<br>in D
! [[SKULO interval names|SKULO]]<br>[[SKULO interval names|Interval name]]
! Example<br>in D
! Example<br>in D
! colspan="2" | [[Solfege|Solfeges]]
|-
|-
| 0
| 0
| 0.00
| 0.00
| 1/1
| 1/1
| do
| P1
| P1
| D
| D
| P1
| D
| da
| do
|-
|-
| 1
| 1
| 46.15
| 46.15
| [[33/32]], [[49/48]], [[36/35]], [[25/24]]
| [[33/32]], [[49/48]], [[36/35]], [[25/24]]
| di
| A1
| A1
| D#
| D#
| A1, S1
| D#, SD
| du
| di
|-
|-
| 2
| 2
| 92.31
| 92.31
| [[21/20]], [[22/21]], [[26/25]]
| [[21/20]], [[22/21]], [[26/25]]
| rih
| d2
| d2
| Ebb
| Ebb
| sm2
| sEb
| fro
| rih
|-
|-
| 3
| 3
| 138.46
| 138.46
| [[12/11]], [[13/12]], [[14/13]], [[16/15]]
| [[12/11]], [[13/12]], [[14/13]], [[16/15]]
| ru
| m2
| m2
| Eb
| Eb
| m2
| Eb
| fra
| ru
|-
|-
| 4
| 4
| 184.62
| 184.62
| [[9/8]], [[10/9]], [[11/10]]
| [[9/8]], [[10/9]], [[11/10]]
| re
| M2
| M2
| E
| E
| M2
| E
| ra
| re
|-
|-
| 5
| 5
| 230.77
| 230.77
| [[8/7]], 15/13
| [[8/7]], [[15/13]]
| ri
| A2
| A2
| E#
| E#
| SM2
| SE
| ru
| ri
|-
|-
| 6
| 6
| 276.92
| 276.92
| [[7/6]], [[13/11]], [[33/28]]
| [[7/6]], [[13/11]], [[33/28]]
| ma
| d3
| d3
| Fb
| Fb
| sm3
| sF
| no
| ma
|-
|-
| 7
| 7
| 323.08
| 323.08
| [[6/5]]
| [[135/112]], [[6/5]]
| me
| m3
| F
| m3
| m3
| F
| F
| na
| me
|-
|-
| 8
| 8
| 369.23
| 369.23
| [[5/4]], [[11/9]], [[16/13]]
| [[5/4]], [[11/9]], [[16/13]]
| muh/mi
| M3
| M3
| F#
| F#
| M3
| F#
| ma
| muh/mi
|-
|-
| 9
| 9
| 415.38
| 415.38
| [[9/7]], [[14/11]], [[33/26]]
| [[9/7]], [[14/11]], [[33/26]]
| maa
| A3
| A3
| Fx
| Fx
| SM3
| SF#
| mu
| maa
|-
|-
| 10
| 10
| 461.54
| 461.54
| [[21/16]], [[13/10]]
| [[21/16]], [[13/10]]
| fe
| d4
| d4
| Gb
| Gb
| s4
| sG
| fo
| fe
|-
|-
| 11
| 11
| 507.69
| 507.69
| [[75/56]], [[4/3]]
| [[75/56]], [[4/3]]
| fa
| P4
| P4
| G
| G
| P4
| G
| fa
| fa
|-
|-
| 12
| 12
| 553.85
| 553.85
| [[11/8]], [[18/13]]
| [[11/8]], [[18/13]]
| fu
| A4
| A4
| G#
| G#
| A4
| G#
| fu/pa
| fu
|-
|-
| 13
| 13
|600.00
| 600.00
| [[7/5]], [[10/7]]
| [[7/5]], [[10/7]]
| fi/se
| AA4, dd5
| AA4, dd5
| Gx, Abb
| Gx, Abb
| SA4, sd5
| SG#, sAb
| pu/sho
| fi/se
|-
|-
| 14
| 14
| 646.15
| 646.15
| [[16/11]], [[13/9]]
| [[16/11]], [[13/9]]
| su
| d5
| d5
| Ab
| Ab
| d5
| Ab
| sha/so
| su
|-
|-
| 15
| 15
| 692.31
| 692.31
| [[112/75]], [[3/2]]
| [[112/75]], [[3/2]]
| sol
| P5
| P5
| A
| A
| P5
| A
| sa
| sol
|-
|-
| 16
| 16
| 738.46
| 738.46
| [[32/21]], [[20/13]]
| [[32/21]], [[20/13]]
| si
| A5
| A5
| A#
| A#
| S5
| SA
| su
| si
|-
|-
| 17
| 17
| 784.62
| 784.62
| [[11/7]], [[14/9]]
| [[11/7]], [[14/9]]
| leh
| d6
| d6
| Bbb
| Bbb
| sm6
| sBb
| flo
| leh
|-
|-
| 18
| 18
| 830.77
| 830.77
| [[13/8]], [[8/5]]
| [[13/8]], [[8/5]]
| le/lu
| m6
| m6
| Bb
| Bb
| m6
| Bb
| fla
| le/lu
|-
|-
| 19
| 19
| 876.92
| 876.92
| [[5/3]]
| [[5/3]], [[224/135]]
| la
| M6
| B
| M6
| M6
| B
| B
| la
| la
|-
|-
| 20
| 20
| 923.08
| 923.08
| [[12/7]], [[22/13]]
| [[12/7]], [[22/13]]
| li
| A6
| A6
| B#
| B#
| SM6
| SB
| lu
| li
|-
|-
| 21
| 21
| 969.23
| 969.23
| [[7/4]]
| [[7/4]], [[26/15]]
| ta
| d7
| d7
| Cb
| Cb
| sm7
| sC
| tho
| ta
|-
|-
| 22
| 22
| 1015.38
| 1015.38
| [[9/5]], [[16/9]], [[20/11]]
| [[9/5]], [[16/9]], [[20/11]]
| te
| m7
| m7
| C
| C
| m7
| C
| tha
| te
|-
|-
| 23
| 23
| 1061.54
| 1061.54
| [[11/6]], [[13/7]], [[15/8]], [[24/13]]
| [[11/6]], [[13/7]], [[15/8]], [[24/13]]
| tu/ti
| M7
| M7
| C#
| C#
| M7
| C#
| ta
| tu/ti
|-
|-
| 24
| 24
| 1107.69
| 1107.69
| [[21/11]], [[25/13]], [[40/21]]
| [[21/11]], [[25/13]], [[40/21]]
| to
| A7
| A7
| Cx
| Cx
| SM7
| SC#
| tu
| to
|-
|-
| 25
| 25
| 1153.85
| 1153.85
| [[64/33]], [[96/49]], [[35/18]], [[48/25]]
| [[64/33]], [[96/49]], [[35/18]], [[48/25]]
| da
| d8
| d8
| Db
| Db
| d8, s8
| Db, sD
| do
| da
|-
|-
| 26
| 26
| 1200.00
| 1200.00
| 2/1
| 2/1
| do
| P8
| P8
| D
| D
| P8
| D
| da
| do
|}
|}


*based on treating 26-EDO as a [[13-limit]] temperament; other approaches are possible.
=== Interval quality and chord names in color notation ===
 
Using [[color notation]], qualities can be loosely associated with colors:
Using [[color notation]], qualities can be loosely associated with colors:


Line 299: Line 325:
! Quality
! Quality
! Color
! Color
! Monzo format
! Monzo Format
! Examples
! Examples
|-
|-
Line 335: Line 361:
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:
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:


{| class="wikitable" style="text-align:center;"
{| class="wikitable center-all"
|-
|-
! [[Kite's color notation|color of the 3rd]]
! [[Kite's color notation|Color of the 3rd]]
! JI chord
! JI chord
! Notes as EDO steps
! Notes as Edosteps
! Notes of C chord
! Notes of C Chord
! Written name
! Written Name
! Spoken name
! Spoken Name
|-
|-
| zo
| zo
Line 373: Line 399:
|}
|}


For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].
 
== 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]].


== Selected just intervals approximated ==
==== Evo flavor ====
<imagemap>
File:26-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 463 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:26-EDO_Evo_Sagittal.svg]]
</imagemap>


Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
==== Revo flavor ====
<imagemap>
File:26-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 511 0 671 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:26-EDO_Revo_Sagittal.svg]]
</imagemap>
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|26}}


The following table shows how [[15-odd-limit intervals]] are represented in 26edo. Prime harmonics are in '''bold'''; intervals with a non-[[consistent]] mapping are in ''italic''.  
== 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.


{| class="wikitable" style="text-align:center;"
{| class="wikitable center-all"
|+Direct mapping (even if inconsistent)
|+ style="font-size: 105%;" | Direct approximation
|-
|-
! Interval, complement
! Interval
! Error (abs, [[Cent|¢]])
! Error (abs, [[Cent|¢]])
|-
|-
| [[13/12]], [[24/13]]
| 2<sup>ϕ</sup> / ϕ
| 0.111
| 0.858
|-
|-
| '''[[8/7]], [[7/4]]'''
| ϕ
| '''0.405'''
| 2.321
|-
|-
| [[14/11]], [[11/7]]
| π
| 2.123
| 2.820
|-
|-
| [[10/9]], [[9/5]]
| 2<sup>ϕ</sup>
| 2.212
| 3.179
|-
|-
| '''[[11/8]], [[16/11]]'''
| π/ϕ
| '''2.528'''
| 5.141
|-
|-
| [[13/10]], [[20/13]]
| 2<sup>ϕ</sup> / π
| 7.325
| 5.999
|-
| [[6/5]], [[5/3]]
| 7.436
|-
| [[18/13]], [[13/9]]
| 9.536
|-
| '''[[4/3]], [[3/2]]'''
| '''9.647'''
|-
| '''[[16/13]], [[13/8]]'''
| '''9.758'''
|-
| [[7/6]], [[12/7]]
| 10.052
|-
| [[14/13]], [[13/7]]
| 10.163
|-
| [[12/11]], [[11/6]]
| 12.176
|-
| [[13/11]], [[22/13]]
| 12.287
|-
| ''[[15/11]], [[22/15]]''
| ''16.895''
|-
| [[15/13]], [[26/15]]
| 16.972
|-
| '''[[5/4]], [[8/5]]'''
| '''17.083'''
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| ''[[15/14]], [[28/15]]''
| ''19.019''
|-
| [[9/8]], [[16/9]]
| 19.295
|-
| ''[[16/15]], [[15/8]]''
| ''19.424''
|-
| [[11/10]], [[20/11]]
| 19.611
|-
| [[9/7]], [[14/9]]
| 19.699
|-
| [[11/9]], [[18/11]]
| 21.823
|}
|}


{| class="wikitable" style="text-align:center;"
== Regular temperament properties ==
|+Patent val mapping
{| class="wikitable center-4 center-5 center-6"
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
|-
| [[13/12]], [[24/13]]
| 0.111
|-
|-
| '''[[8/7]], [[7/4]]'''
! rowspan="2" | [[Subgroup]]
| '''0.405'''
! rowspan="2" | [[Comma basis]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| [[14/11]], [[11/7]]
! [[TE error|Absolute]] (¢)
| 2.123
! [[TE simple badness|Relative]] (%)
|-
|-
| [[10/9]], [[9/5]]
| 2.3
| 2.212
| {{monzo| -41 26 }}
| {{mapping| 26 41 }}
| +3.043
| 3.05
| 6.61
|-
|-
| '''[[11/8]], [[16/11]]'''
| 2.3.5
| '''2.528'''
| 81/80, 78125/73728
| {{mapping| 26 41 60 }}
| +4.489
| 3.22
| 6.98
|-
|-
| [[13/10]], [[20/13]]
| 2.3.5.7
| 7.325
| 50/49, 81/80, 405/392
| {{mapping| 26 41 60 73 }}
| +3.324
| 3.44
| 7.45
|-
|-
| [[6/5]], [[5/3]]
| 2.3.5.7.11
| 7.436
| 45/44, 50/49, 81/80, 99/98
| {{mapping| 26 41 60 73 90 }}
| +2.509
| 3.48
| 7.53
|-
|-
| [[18/13]], [[13/9]]
| 2.3.5.7.11.13
| 9.536
| 45/44, 50/49, 65/64, 78/77, 81/80
| {{mapping| 26 41 60 73 90 96 }}
| +2.531
| 3.17
| 6.87
|-
|-
| '''[[4/3]], [[3/2]]'''
| 2.3.5.7.11.13.17
| '''9.647'''
| 45/44, 50/49, 65/64 78/77, 81/80, 85/84
| {{mapping| 26 41 60 73 90 96 106 }}
| +2.613
| 2.94
| 6.38
|-
|-
| '''[[16/13]], [[13/8]]'''
| 2.3.5.7.11.13.17.19
| '''9.758'''
| 45/44, 50/49, 57/56, 65/64, 78/77, 81/80, 85/84
|-
| {{mapping| 26 41 60 73 90 96 106 110 }}
| [[7/6]], [[12/7]]
| +2.894
| 10.052
| 2.85
|-
| 6.18
| [[14/13]], [[13/7]]
| 10.163
|-
| [[12/11]], [[11/6]]
| 12.176
|-
| [[13/11]], [[22/13]]
| 12.287
|-
| [[15/13]], [[26/15]]
| 16.972
|-
| '''[[5/4]], [[8/5]]'''
| '''17.083'''
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[9/8]], [[16/9]]
| 19.295
|-
| [[11/10]], [[20/11]]
| 19.611
|-
| [[9/7]], [[14/9]]
| 19.699
|-
| [[11/9]], [[18/11]]
| 21.823
|-
| ''[[16/15]], [[15/8]]''
| ''26.730''
|-
| ''[[15/14]], [[28/15]]''
| ''27.135''
|-
| ''[[15/11]], [[22/15]]''
| ''29.258''
|}
|}
* 26et is lower in relative error than any previous equal temperaments in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]] (using the 26i val for the 23- and 29-limit). The next equal temperaments performing better in those subgroups are [[27edo|27eg]], 27eg, [[29edo|29g]], and [[46edo|46]], respectively.


== Rank two temperaments ==
=== Rank-2 Temperaments ===
* [[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:


* diatonic ([[flattone]]) 5*4-2*3 (15\26, 1\1)
Important mos scales include (in addition to ones found in [[13edo]]):
* [[orgone]] 4*5-3*2 (7\26, 1\1)
* [[Flattone]][7] (diatonic) 4443443 (15\26, 1\1)
* [[lemba]] 4*5-2*3 (5\26, 1\2)
* [[Flattone]][12] (chromatic) 313131331313 (15\26, 1\1)
* [[Flattone]][19] (enharmonic) 2112112112121121121 (15\26, 1\1)
* [[Orgone]][7] 5525252 (7\26, 1\1)
* [[Orgone]][11] 32322322322 (7\26, 1\1)
* [[Orgone]][15] 212212221222122 (7\26, 1\1)
* [[Lemba]][6] 553553 (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"
|-
|-
! Periods<br>per octave
! Periods<br>per 8ve
! Generator
! Generator
! Temperaments
! Temperaments
Line 556: Line 543:
| 1
| 1
| 1\26
| 1\26
| [[Quartonic]]/Quarto
| [[Quartonic]] / [[quarto]]
|-
|-
| 1
| 1
| 3\26
| 3\26
| [[Jerome]]/[[Chromatic_pairs|Bleu]]/Secund
| [[Glacier]] / [[bleu]] / [[jerome]] / [[secund]]
|-
|-
| 1
| 1
| 5\26
| 5\26
| [[Cynder]]/[[Mothra]]
| [[Cynder]] / [[mothra]]
|-
|-
| 1
| 1
| 7\26
| 7\26
| [[Superkleismic]]/[[Orgone]]/[[Shibboleth]]
| [[Orgone]] / [[superkleismic]]
|-
|-
| 1
| 1
| 9\26
| 9\26
| [[Roman]]/Wesley
| [[Wesley]] / [[roman]]
|-
|-
| 1
| 1
| 11\26
| 11\26
| [[Meantone]]/[[Flattone]]
| [[Flattone]] / [[flattertone]]
|-
|-
| 2
| 2
| 1\26
| 1\26
| Elvis
| [[Elvis]]
|-
|-
| 2
| 2
Line 588: Line 575:
| 2
| 2
| 3\26
| 3\26
| [[Fifive]]/Crepuscular
| [[Fifive]] / [[crepuscular]]
|-
|-
| 2
| 2
| 4\26
| 4\26
| [[Unidec]]/[[Gamelismic_clan#Unidec-Hendec|Hendec]]/Dubbla
| [[Dubbla]]<br>[[Unidec]] / [[hendec]]
|-
|-
| 2
| 2
| 5\26
| 5\26
| | [[Lemba]]
| [[Lemba]]
|-
|-
| 2
| 2
| 6\26
| 6\26
| [[Doublewide]]/Cavalier
| [[Doublewide]] / [[cavalier]]
|-
|-
| 13
| 13
| 1\26
| 1\26
| Triskaidekic
| [[Triskaidekic]]
|}
|}


=== Hendec in 26et ===
=== Hendec in 26et ===
[[Gamelismic_clan#Unidec-Hendec|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="wikitable center-all left-3"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
! [[Color notation/Temperament Names|Color Name]]
! [[Color name]]
! Name 1
! Name(s)
! Name 2
! Name 3
|-
|-
| 5
| 5
| 81/80
| [[81/80]]
| |<nowiki> | -4 4 -1 </nowiki>&gt;
| {{monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Gu
| Syntonic comma
| Syntonic comma, Didymos comma, meantone comma
| Didymos comma
| Meantone comma
|-
|-
| "
| 5
| 5696703/5695946
| <abbr title="381520424476945831628649898809/381469726562500000000000000000">(60 digits)</abbr>
| |<nowiki> | -17 62 -35 </nowiki>&gt;
| {{monzo| -17 62 -35 }}
| 0.23
| 0.23
| Quadla-sepquingu
| Quadla-sepquingu
| Senior
| [[Senior comma]]
|
|
|-
|-
| 7
| 7
| 525/512
| [[525/512]]
| |<nowiki> | -9 1 2 1 </nowiki>&gt;
| {{monzo| -9 1 2 1 }}
| 43.41
| 43.41
| Zoyoyo
| Lazoyoyo
| Avicennma
| Avicennma, Avicenna's enharmonic diesis
| Avicenna's enharmonic diesis
|
|-
|-
| "
| 7
| 50/49
| [[50/49]]
| |<nowiki> | 1 0 2 -2 </nowiki>&gt;
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic diesis
| Jubilisma, tritonic diesis
| Jubilisma
|
|-
|-
| "
| 7
| 875/864
| [[875/864]]
| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| {{monzo| -5 -3 3 1 }}
| 21.90
| 21.90
| Zotriyo
| Zotriyo
| Keema
| Keema
|
|
|-
|-
| "
| 7
| 4000/3969
| [[4000/3969]]
| |<nowiki> | 5 -4 3 -2 </nowiki>&gt;
| {{monzo| 5 -4 3 -2 }}
| 13.47
| 13.47
| Sarurutriyo
| Sarurutriyo
| Octagar
| Octagar comma
|
|
|-
|-
| "
| 7
| 1728/1715
| [[1728/1715]]
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| {{monzo| 6 3 -1 -3 }}
| 13.07
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma
| Orwellisma
| Orwell comma
|
|-
|-
| "
| 7
| 1029/1024
| [[1029/1024]]
| |<nowiki> | -10 1 0 3 </nowiki>&gt;
| {{monzo| -10 1 0 3 }}
| 8.43
| 8.43
| Latrizo
| Latrizo
| Gamelisma
| Gamelisma
|
|
|-
|-
| "
| 7
| 321489/320000
| <abbr title="321489/320000">(12 digits)</abbr>
| |<nowiki> | -9 8 -4 2 </nowiki>&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| 8.04
| Labizogugu
| Labizogugu
| Varunisma
| [[Varunisma]]
|
|
|-
|-
| "
| 7
| 1065875/1063543
| <abbr title="201768035/201326592">(18 digits)</abbr>
| |<nowiki> | -26 -1 1 9 </nowiki>&gt;
| {{monzo| -26 -1 1 9 }}
| 3.79
| 3.79
| Latritrizo-ayo
| Latritrizo-ayo
| Wadisma
| [[Wadisma]]
|
|
|-
|-
| "
| 7
| 4375/4374
| [[4375/4374]]
| |<nowiki> | -1 -7 4 1 </nowiki>&gt;
| {{monzo| -1 -7 4 1 }}
| 0.40
| 0.40
| Zoquadyo
| Zoquadyo
| Ragisma
| Ragisma
|
|
|-
|-
| 11
| 11
| 99/98
| [[99/98]]
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruru
| Mothwellsma
| Mothwellsma
|
|
|-
|-
| "
| 11
| 100/99
| [[100/99]]
| |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyo
| Ptolemisma
| Ptolemisma
|
|
|-
|-
| "
| 11
| 65536/65219
| [[65536/65219]]
| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| 8.39
| Satrilu-aruru
| Satrilu-aruru
| Orgonisma
| Orgonisma
|
|
|-
|-
| "
| 11
| 385/384
| [[385/384]]
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|
|
|-
|-
| "
| 11
| 441/440
| [[441/440]]
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozogu
| Werckisma
| Werckisma
|
|
|-
|-
| "
| 11
| 3025/3024
| [[3025/3024]]
| |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| 0.57
| Loloruyoyo
| Loloruyoyo
| Lehmerisma
| Lehmerisma
|
|
|-
|-
| "
| 11
| 9801/9800
| [[9801/9800]]
| |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Bilorugu
| Kalisma
| Kalisma, Gauss' comma
| Gauss' comma
|
|-
|-
| 13
| 13
| 105/104
| [[105/104]]
| |<nowiki> | -3 1 1 1 0 -1 </nowiki>&gt;
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| 16.57
| Thuzoyo
| Thuzoyo
| Animist
| Animist comma
|
|
|}
|}


== [[Orgonia|Orgone Temperament]] ==
== Scales ==
 
=== Orgone temperament ===
[[Andrew_Heathwaite|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:
[[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)]].
Line 810: Line 757:
The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16/11|16:11]] and 3g approximates [[7/4|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.
The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16/11|16:11]] and 3g approximates [[7/4|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.


[[37edo]] is another orgone tuning, and [[89edo]] is better even than 26. If we take 11 and 26 to be the edges of the Orgone Spectrum, we may fill in the rest of the spectrum thus:
[[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|orgone_heptatonic.jpg]]
 
{| class="wikitable"
|-
| | 3\11
| |
| |
| |
| |
|-
| |
| |
| |
| |
| | 19\70
|-
| |
| |
| |
| | 16\59
| |
|-
| |
| |
| |
| |
| | 29\107
|-
| |
| |
| | 13\48
| |
| |
|-
| |
| |
| |
| |
| | 36\133
|-
| |
| |
| |
| | 23\85
| |
|-
| |
| |
| |
| |
| | 33\122
|-
| |
| | 10\37
| |
| |
| |
|-
| |
| |
| |
| |
| | 37\137
|-
| |
| |
| |
| | 27\100
| |
|-
| |
| |
| |
| |
| | 44\163
|-
| |
| |
| | 17\63
| |
| |
|-
| |
| |
| |
| |
| | 41\152
|-
| |
| |
| |
| | 24\89
| |
|-
| |
| |
| |
| |
| | 31\115
|-
| | 7\26
| |
| |
| |
| |
|}


Orgone has a minimax tuning which sharpens both 7 and 11 by 1/5 of an orgonisma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgonisma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.
=== Additional scalar bases available ===
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, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), 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).


[[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|orgone_heptatonic.jpg]]
-Igs


== Additional Scalar Bases Available in 26-EDO ==
=== MOS scales ===
Since the perfect 5th in 26-EDO 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, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), 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).
''See [[List of MOS scales in 26edo]]''


-Igs
== Instruments ==
James Fenn's midi keyboard.
[[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.]


== Music ==
== Music ==
{{Catrel|26edo tracks}}
=== 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)
* [https://www.youtube.com/watch?v=dlXFoIoc_uk "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=K7oTEXgmdKY ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=-EVO5ntuoSM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
=== 21st century ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)
* [https://www.youtube.com/watch?v=f5eYIH3TO4o ''Moondust''] (2024)
; [[Jim Aikin]]
* [http://midiguru.wordpress.com/2013/01/06/public-rituals/ Public Rituals « Jim Aikin's Oblong Blob ''The Triumphal Procession of Nebuchadnezzar''] (2013)
; [[Beheld]]
* [https://www.youtube.com/watch?v=0WbLTtDZUms ''Damp vibe''] (2022)
; [[benyamind]]
* [https://www.youtube.com/watch?v=H1hYI2hBcEU ''Cinematic music in 26-tone equal temperament''] (2024)
; [[Cameron Bobro]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/LittleFugueIn26_CBobro.mp3 Little Fugue in 26]{{dead link}}
; [[User:CellularAutomaton|CellularAutomaton]]
* [https://cellularautomaton.bandcamp.com/track/innerstate ''Innerstate''] (2024)
; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/two-pairs-of-socks-26edo Two Pairs of Socks (26edo)]{{dead link}}
* [https://cityoftheasleep.bandcamp.com/track/between-the-branes-26edo Between the Branes (26edo)]{{dead link}}
; [[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]]
* [https://www.youtube.com/watch?v=KvaEyzCuBwA ''26-EDO Nocturne No. 1 in F♯ Minor''] (2024)
; [[Francium]]
* [https://www.youtube.com/watch?v=4i6no5-zwKQ ''Eskalation''] (2022)
* [https://www.youtube.com/watch?v=tIZjchfF2Iw ''Dark Forest''] (2023)
* [https://www.youtube.com/watch?v=FRd_sLuTpQQ ''Lembone''] (2024)
* [https://www.youtube.com/watch?v=XzQ09i6RBsg ''Happy Birthday in 26edo''] (2024)
; [[IgliashonJones|Igliashon Jones]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20A%20Time-Yellowed%20Photograph%20of%20Cliffs%20Hangs%20in%20the%20Hall.mp3 A Time-Yellowed Photo of the Cliffs Hangs on the Wall]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks]{{dead link}}
; [[Melopœia]]
* [https://melopoeia.bandcamp.com/album/ainulindal Ainulindalë] (2016) – A text to music translation of Tolkien's Silmarillion using 26edo.
* [https://melopoeia.bandcamp.com/album/valaquenta Valaquenta] (2023)
; [[Claudi Meneghin]]
* [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://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]]
* [https://sites.google.com/site/teamouse/26-et-Pianoteq-Bechstein.mp3 Etude in 26-tone equal temperament]{{dead link}}
; [[Shaahin Mohajeri]]
* [http://www.96edo.com/music/micro900607.mp3 Microtonal music in 26-EDO]{{dead link}}
; [[Mundoworld]]
* [https://www.youtube.com/watch?v=43WoQd_UO7w ''Getting to 100'' (with David Sinclair)] (2021)
* [https://www.youtube.com/watch?v=TRloI6ONcmw ''Primitive Mountain''] (2022)
* "Into Thin Air" from ''The Vanishing Bus'' (2024) – [https://open.spotify.com/track/4R2cBwidLwfy5QsdjcFRvi Spotify] | [https://mundoworld.bandcamp.com/track/into-thin-air Bandcamp] | [https://www.youtube.com/watch?v=cekBZ3Ql69U YouTube]
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=37KM1roKOMQ ''Edolian - Riemann''] (2020)
* [https://www.youtube.com/watch?v=Y0-Bf2ePfzY ''Redvault''] (2021)
; [[Ray Perlner]]
* [https://youtu.be/rivfU8Rw4IM Scherzo in 26 EDO for Oboe, Horn, and Organ] (2020)
* [https://www.youtube.com/watch?v=hsd00wrSJnE Octatonic Groove] (2021)
* [https://www.youtube.com/watch?v=xhn5lz2cB-4 A Little Prog Rock in 26EDO] (2023)
; [[Sevish]]
* [https://www.youtube.com/watch?v=Kh_C5Va5jDE ''Yeah Groove''] (2022)
; [[Jon Lyle Smith]]
* [http://archive.org/details/UnderTheHeatdome under the heatDome]{{dead link}} [http://archive.org/download/UnderTheHeatdome/under_the_heatDome.mp3 play]{{dead link}}
; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=pJOlZ9sHCjk ''Languor Study''] (2022)
; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=OjW8dgooG9Q ''Spring''] (2024)
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 ''Morpheous Wing'' in 26 edo] (2016)


* [http://soonlabel.com/xenharmonic/archives/3335 Canon 3-in-1 on a ground ‘The tempest’, by Claudi Meneghin] {{dead link}} &rarr; [https://www.youtube.com/watch?v=OCh7ZvKBSnQ Version on YouTube ''Claudi Meneghin: Canon 3-in-1 on a ground "The Tempest" (2017), in 26edo'']
== Notes ==
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20A%20Time-Yellowed%20Photograph%20of%20Cliffs%20Hangs%20in%20the%20Hall.mp3 A Time-Yellowed Photo of the Cliffs Hangs on the Wall] by [[IgliashonJones|Igliashon Jones]]
<references group="note" />
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] by [[IgliashonJones|Igliashon Jones]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/26tet.mp3 Etude in 26-tone equal temperament] by [[Herman Miller]]
* [http://archive.org/details/UnderTheHeatdome under the heatDome] [http://archive.org/download/UnderTheHeatdome/under_the_heatDome.mp3 play] by [[Jon Lyle Smith]]
* [http://danielthompson.blogspot.com/2007/04/new-version-of-organ-study-1.html A New Recording of Organ Study #1] [http://www.microtonalmusic.net/audio/organstudyremix26edo.mp3 play] {{dead link}} by [[Daniel Thompson]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/LittleFugueIn26_CBobro.mp3 Little Fugue in 26] by [[Cameron Bobro]]
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/26-tone-fugue.mp3 26 tone fugue (unfinished)] {{dead link}} by Peter Kosmorsky (based on the melody he was singing in the shower, in orgone, to the presumed confusion of those in earshot)
* [http://www.96edo.com/music/micro900607.mp3 Microtonal music in 26-EDO] by [[Shaahin Mohajeri]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Guitar%20Serenade%20in%20Q%20Major.mp3 Guitar Serenade in Q Major] by [[Zach Curley]]
* [http://midiguru.wordpress.com/2013/01/06/public-rituals/ Public Rituals « Jim Aikin's Oblong Blob] “The Triumphal Procession of Nebuchadnezzar“
* [http://www.unfretted.com/microtonal/melopoeia-project-26-edo-album-based-on-tolkeins-silmarillion-ainulindale/ Melopœia Project – 26 EDO album based on Tolkein’s Silmarillion – Ainulindalë – Unfretted]
* [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 Morpheous Wing in 26 edo] by [[Chris Vaisvil]]
* [https://youtu.be/rivfU8Rw4IM Scherzo in 26 EDO for Oboe, Horn, and Organ] by [[Ray Perlner]]
* [https://cityoftheasleep.bandcamp.com/track/two-pairs-of-socks-26edo Two Pairs of Socks (26edo)] by [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/between-the-branes-26edo Between the Branes (26edo) &#124; City of the Asleep]


[[Category:26edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Theory]]
[[Category:Twentuning]]
[[Category:Twentuning]]
[[Category:Meantone]]
[[Category:Meantone]]

Latest revision as of 00:00, 16 August 2025

← 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 [-3 0 0 6 -4⟩. 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 [-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.
  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 harmonics 5 and 9 through 23 (including direct approximations) with 26edo. Multiplying 26edo by 3 yields 78edo, which corrects several harmonics.

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 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 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 A5 A# S5 SA su si
17 784.62 11/7, 14/9 d6 Bbb sm6 sBb flo leh
18 830.77 13/8, 8/5 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

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

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.

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

Important mos scales include (in addition to ones found in 13edo):

  • Flattone[7] (diatonic) 4443443 (15\26, 1\1)
  • Flattone[12] (chromatic) 313131331313 (15\26, 1\1)
  • Flattone[19] (enharmonic) 2112112112121121121 (15\26, 1\1)
  • Orgone[7] 5525252 (7\26, 1\1)
  • Orgone[11] 32322322322 (7\26, 1\1)
  • Orgone[15] 212212221222122 (7\26, 1\1)
  • Lemba[6] 553553 (5\26, 1\2)
  • Lemba[10] 3232332323 (5\26, 1\2)
  • Lemba[16] 2122122121221221 (5\26, 1\2)
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 2] 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

Scales

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

Additional scalar bases available

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, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), 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).

-Igs

MOS scales

See List of MOS scales in 26edo

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

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
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

Notes

  1. ↑ Based on treating 26edo as a 13-limit temperament; other approaches are also possible.
  2. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.
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