26edo: Difference between revisions

'''26edo''' divides the [[octave]] into 26 equal parts of around 46.2 [[cent]]s each.  

26edo tempers out [[81/80]] in the [[5-limit]], making it a [[meantone]] tuning with a very flat fifth.  

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 log₂7.

26edo's minor sixth (1.6158) is very close to ''φ'' ≈ 1.6180 (i.e. the golden ratio).

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 second of approximately [[10/9]] instead of [[9/8]]).

 

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 {{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.

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 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.

! [[Cents|cents]]

! Approximate Ratios*

! Interval<br>Name

|600.00

 

! Monzo format

{| class="wikitable" style="text-align:center;"

! [[Kite's color notation|color of the 3rd]]

! Notes as EDO steps

! Notes of C chord

! Written name

! Spoken name

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

== Selected just intervals approximated ==

=== 15-odd-limit interval mappings ===

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''.  

{| class="wikitable" style="text-align:center;"

|-

| 0.111

| '''[[8/7]], [[7/4]]'''

| '''0.405'''

| [[14/11]], [[11/7]]

| 2.123

|-

| [[10/9]], [[9/5]]

| [[15/13]], [[26/15]]

|-

| '''17.083'''

| [[7/5]], [[10/7]]

| ''[[15/14]], [[28/15]]''

! Interval, complement

| ''29.258''

== Acoustic π just in between the ϕ intervals ==

After [[13edo#Phi vibes|13edo]], the weird coïncidences continue: [[11/7#Proximity with π/2|acoustic π/2]] (17\26) is just in between [[13edo#Phi vibes|the ϕ intervals provided by 13edo]] (16\26 for [[Logarithmic phi|logarithmic ϕ]]/2, and 18\26 for [[Acoustic phi|acoustic ϕ]]).

Not until 1076edo do we find a better EDO in terms of relative error on these intervals (which is not a very relevant EDO for logarithmic ϕ, since 1076 does not belong to the Fibonacci sequence).

However, it should be noted that [[Logarithmic constants VS acoustic constants|from an acoustic perspective]], acoustic π and acoustic ϕ are both better represented on [[23edo]].

|+Direct mapping

| 2**(ϕ) / ϕ

| 2**(ϕ)

| 2**(ϕ) / π

{| class="wikitable"

! rowspan="2" |[[Just intonation subgroup|Subgroup]]

! rowspan="2" |[[Comma basis|Comma List]]

! rowspan="2" |[[Mapping]]

! rowspan="2" |Optimal

8ve Stretch (¢)

! colspan="2" |Tuning Error

![[TE error|Absolute]] (¢)

![[TE simple badness|Relative]] (%)

|2.3

|[-41 26⟩

|[⟨26 41]]

|3.05

|6.61

|2.3.5

|81/80, 78125/73728

|[⟨26 41 60]]

|3.22

|6.98

|2.3.5.7

|50/49, 81/80, 405/392

|[⟨26 41 60 73]]

|3.44

|7.45

|2.3.5.7.11

|45/44, 50/49, 81/80, 99/98

|[⟨26 41 60 73 90]]

|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]]

|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]]

|2.94

|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]]

|2.85

|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.  

* diatonic ([[flattone]]) 4443443 (15\26, 1\1)

* chromatic ([[flattone]]) 313131331313 (15\26, 1\1)

* enharmonic ([[flattone]]) 2112112112121121121 (15\26, 1\1)

* [[orgone]] 5525252 (7\26, 1\1)

* [[orgone]] 32322322322 (7\26, 1\1)

* [[orgone]] 212212221222122 (7\26, 1\1)

* [[lemba]] 553553 (5\26, 1\2)

* [[lemba]] 3232332323 (5\26, 1\2)

* [[lemba]] 2122122121221221 (5\26, 1\2)

! Periods<br>per octave

| [[Quartonic]]/Quarto

| [[Jerome]]/[[Chromatic_pairs|Bleu]]/Secund

| [[Cynder]]/[[Mothra]]

| [[Superkleismic]]/[[Orgone]]/[[Shibboleth]]

| [[Roman]]/Wesley

| [[Meantone]]/[[Flattone]]

| Elvis

| [[Fifive]]/Crepuscular

| [[Unidec]]/[[Gamelismic_clan#Unidec-Hendec|Hendec]]/Dubbla

| | [[Lemba]]

| [[Doublewide]]/Cavalier

| Triskaidekic

[[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.

== Commas ==

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

! [[Harmonic limit|Prime<br>Limit]]

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

| [[Senior]]

| Tritonic diesis, Jubilisma

| Octagar

| Orwellisma, Orwell comma

| Animist

== [[Orgonia|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:

[[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:

{| class="wikitable"

|-

| | 3\11

| | 13\48

|-

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

== Additional Scalar Bases Available in 26-EDO ==

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).

 

* [https://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]]

== See also ==

* [[Lumatone mapping for 26edo]]