Xenharmonic Wiki

26edo: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>hstraub
**Imported revision 156243101 - Original comment: MOS names added**
Sintel (talk | contribs)
autopatrolled
1,547 edits
→Approximation to JI: -zeta peak index
 
(246 intermediate revisions by 52 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 26-EDO
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-08-12 08:22:12 UTC</tt>.<br>
| en = 26edo
: The original revision id was <tt>156243101</tt>.<br>
| es =
: The revision comment was: <tt>MOS names added</tt><br>
| ja =
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Compositions=
{{ED intro}}


[[http://soundclick.com/share?songid=5683791|A Time-Yellowed Photo of the Cliffs Hangs on the Wall ]] by Igliashon Jones
== Theory ==
[[http://www.io.com/%7Ehmiller/midi/26tet.mid|Etude in 26-tone equal tuning]] by Herman Miller
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]].


=**Structure**=
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.


The structure of 26-EDO is an interesting beast, and I find three particular manners of understanding the chords and chordal relationships in it;
26edo's minor sixth (1.6158) is very close to {{nowrap|''φ'' ≈ 1.6180}} (i.e. the golden ratio).
1. In terms of traditional chord types, that is, as a variant of meantone stretched too far, which yields interesting but perhaps 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). This is not meant to say that this approach is without merit.
2. As two parallel 13-EDO scales (And as I've considered, if these two chains be shifted together or apart slightly but appropriately, interesting variants may be constructed) which is suitable for more atonal melodies. In this way its internal dynamics quite resemble those of 14-EDO.
3. 26-EDO 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. The resulting scale is jagged, and it perverts common musical rules and conventions, but this particular organisation has its charm -perhaps because of this- and leads to some intriguing music.


- K.
With a fifth of 15 steps, it can be equally divided into 3 or 5, supporting [[slendric]] temperament and [[bleu]] temperament respectively.


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


[[user:Andrew_Heathwaite|1281204709]] proposes a temperament family which takes advantage of 26edo's excellent 11 &amp; 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:
# 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.
 
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]].
 
=== Odd harmonics ===
{{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 ==
{| class="wikitable center-all right-2 left-3"
|-
! Degrees
! [[Cent]]s
! Approximate ratios<ref group="note">{{sg|limit=13-limit}}</ref>
! Interval<br>name
! Example<br>in D
! [[SKULO interval names|SKULO]]<br>[[SKULO interval names|Interval name]]
! Example<br>in D
! colspan="2" | [[Solfege|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:
 
{| class="wikitable" style="text-align:center;"
|-
! Quality
! Color
! Monzo Format
! Examples
|-
| diminished
| zo
| {a, b, 0, 1}
| 7/6, 7/4
|-
| minor
| fourthward wa
| {a, b}, b &lt; -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 &gt; 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:
 
{| class="wikitable center-all"
|-
! [[Kite's color notation|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 [[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 ====
<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 ===
{{Q-odd-limit intervals|26}}
 
== 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 center-all"
|+ style="font-size: 105%;" | Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
|-
| 2<sup>ϕ</sup> / ϕ
| 0.858
|-
| ϕ
| 2.321
|-
| π
| 2.820
|-
| 2<sup>ϕ</sup>
| 3.179
|-
| π/ϕ
| 5.141
|-
| 2<sup>ϕ</sup> / π
| 5.999
|}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma basis]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -41 26 }}
| {{mapping| 26 41 }}
| +3.043
| 3.05
| 6.61
|-
| 2.3.5
| 81/80, 78125/73728
| {{mapping| 26 41 60 }}
| +4.489
| 3.22
| 6.98
|-
| 2.3.5.7
| 50/49, 81/80, 405/392
| {{mapping| 26 41 60 73 }}
| +3.324
| 3.44
| 7.45
|-
| 2.3.5.7.11
| 45/44, 50/49, 81/80, 99/98
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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-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-2 Temperaments ===
* [[List of 26et rank two temperaments by badness]]
* [[List of edo-distinct 26et rank two 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)
 
{| class="wikitable center-all left-3"
|-
! Periods<br>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]]<br>[[Unidec]] / [[hendec]]
|-
| 2
| 5\26
| [[Lemba]]
|-
| 2
| 6\26
| [[Doublewide]] / [[cavalier]]
|-
| 13
| 1\26
| [[Triskaidekic]]
|}
 
=== Hendec in 26et ===
[[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 ===
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"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name(s)
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymos comma, meantone comma
|-
| 5
| <abbr title="381520424476945831628649898809/381469726562500000000000000000">(60 digits)</abbr>
| {{monzo| -17 62 -35 }}
| 0.23
| Quadla-sepquingu
| [[Senior comma]]
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Sarurutriyo
| Octagar comma
|-
| 7
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| <abbr title="321489/320000">(12 digits)</abbr>
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| <abbr title="201768035/201326592">(18 digits)</abbr>
| {{monzo| -26 -1 1 9 }}
| 3.79
| Latritrizo-ayo
| [[Wadisma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[105/104]]
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| Thuzoyo
| Animist comma
|}
 
== Scales ==
=== 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:
 
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)]].
The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.
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. [[MOSScales|MOS]] of type [[4L 7s]].
The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. [[MOSScales|MOS]] of type [[4L_7s|4L 7s]].
The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108, 1200.
 
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|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.
 
[[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|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.
[[File:12072608 10207851395433055 404343132969239728 n.jpg|none|thumb|960x960px]]
 
* [[Lumatone mapping for 26edo]]
== Literature ==
[http://www.ronsword.com Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.]
 
== 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)


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.
; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=OjW8dgooG9Q ''Spring''] (2024)


[[37edo]] is another excellent Orgone tuning. [[11edo]] is passable, but barely. If we take 11 and 26 to be the edges of the Orgone Spectrum, that suggests some other edos which can play Orgone: 11, 48, 37,63, 26.
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/26edo/20161224_26edo_wing.mp3 ''Morpheous Wing'' in 26 edo] (2016)


If a name already exists for this temperament, I'd be interested to know about it. As far as I know, temperaments which ignore lower primes (in this case 3 and 5) in favor of higher ones (in this case 7 and 11) are still largely uncharted, but I am interested in finding out if someone has walked this path before.
== Notes ==
<references group="note" />


[[image:orgone_heptatonic.jpg]]</pre></div>
[[Category:Listen]]
<h4>Original HTML content:</h4>
[[Category:Twentuning]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;26edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Compositions&lt;/h1&gt;
[[Category:Meantone]]
&lt;br /&gt;
[[Category:Flattone]]
&lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683791" rel="nofollow"&gt;A Time-Yellowed Photo of the Cliffs Hangs on the Wall &lt;/a&gt; by Igliashon Jones&lt;br /&gt;
[[Category:Mothra]]
&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/26tet.mid" rel="nofollow"&gt;Etude in 26-tone equal tuning&lt;/a&gt; by Herman Miller&lt;br /&gt;
[[Category:Unidec]]
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Structure"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;strong&gt;Structure&lt;/strong&gt;&lt;/h1&gt;
&lt;br /&gt;
The structure of 26-EDO is an interesting beast, and I find three particular manners of understanding the chords and chordal relationships in it;&lt;br /&gt;
1. In terms of traditional chord types, that is, as a variant of meantone stretched too far, which yields interesting but perhaps 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). This is not meant to say that this approach is without merit.&lt;br /&gt;
2. As two parallel 13-EDO scales (And as I've considered, if these two chains be shifted together or apart slightly but appropriately, interesting variants may be constructed) which is suitable for more atonal melodies. In this way its internal dynamics quite resemble those of 14-EDO.&lt;br /&gt;
3. 26-EDO 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. The resulting scale is jagged, and it perverts common musical rules and conventions, but this particular organisation has its charm -perhaps because of this- and leads to some intriguing music.&lt;br /&gt;
&lt;br /&gt;
- K.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Orgone Temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Orgone Temperament&lt;/h1&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextUserlinkRule:00:[[user:Andrew_Heathwaite|1281204709]] --&gt;&lt;span class="membersnap"&gt;- &lt;a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;"&gt;&lt;img src="http://www.wikispaces.com/user/pic/Andrew_Heathwaite-lg.jpg" width="16" height="16" alt="Andrew_Heathwaite" class="userPicture" /&gt;&lt;/a&gt; &lt;a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;"&gt;Andrew_Heathwaite&lt;/a&gt; &lt;small&gt;Aug 7, 2010&lt;/small&gt;&lt;/span&gt;&lt;!-- ws:end:WikiTextUserlinkRule:00 --&gt; proposes a temperament family which takes advantage of 26edo's excellent 11 &amp;amp; 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:&lt;br /&gt;
&lt;br /&gt;
The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; of type &lt;a class="wiki_link" href="/4L%203s"&gt;4L 3s (mish)&lt;/a&gt;.&lt;br /&gt;
The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.&lt;br /&gt;
&lt;br /&gt;
The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; of type &lt;a class="wiki_link" href="/4L%207s"&gt;4L 7s&lt;/a&gt;.&lt;br /&gt;
The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108, 1200.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/37edo"&gt;37edo&lt;/a&gt; is another excellent Orgone tuning. &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt; is passable, but barely. If we take 11 and 26 to be the edges of the Orgone Spectrum, that suggests some other edos which can play Orgone: 11, 48, 37,63, 26.&lt;br /&gt;
&lt;br /&gt;
If a name already exists for this temperament, I'd be interested to know about it. As far as I know, temperaments which ignore lower primes (in this case 3 and 5) in favor of higher ones (in this case 7 and 11) are still largely uncharted, but I am interested in finding out if someone has walked this path before.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:6:&amp;lt;img src=&amp;quot;/file/view/orgone_heptatonic.jpg/155606933/orgone_heptatonic.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/orgone_heptatonic.jpg/155606933/orgone_heptatonic.jpg" alt="orgone_heptatonic.jpg" title="orgone_heptatonic.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:6 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/26edo"