26edo: Difference between revisions

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'''26edo''' divides the [[octave]] into 26 equal parts of around 46.2 [[cent]]s each~~. It tempers out [[81/80]] in the [[5-limit]], making it a meantone tuning with a very flat fifth~~.

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

== Theory ==

~~{{Odd harmonics~~ in ~~edo|edo=26}}~~

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 [[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]]), as it is the denominator of a convergent to log<sub>2</sub>7.

In the [[7-limit]], it tempers out 50/49, 525/512 and 875/864, and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]] and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.

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

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

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]]).

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.

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 [[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~~>~~ comma leads to a half-octave period and an approximate 49/44 generator of 4\26, leading to ~~MOS~~ of size 8 and 14.

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 ~~26-EDO~~ as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower ~~EDOs~~).

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.

=== Odd harmonics ===

== Intervals ==

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 }}
-'''26edo''' divides the [[octave]] into 26 equal parts of around 46.2 [[cent]]s each. It tempers out [[81/80]] in the [[5-limit]], making it a meantone tuning with a very flat fifth.
+'''26edo''' divides the [[octave]] into 26 equal parts of around 46.2 [[cent]]s each.
 == Theory ==
-{{Odd harmonics in edo|edo=26}}
+edo 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 [[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]]), as it is the denominator of a convergent to log<sub>2</sub>7.
+In the [[7-limit]], it tempers out 50/49, 525/512 and 875/864, and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]] and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.
-edo's "minor sixth" (1.6158) is very close to φ ≈ 1.6180 (i. e., the golden ratio).
+edo's minor sixth (1.6158) is very close to ''φ'' ≈ 1.6180 (i.e. the golden ratio).
-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.
-. 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 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]]).
-. As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, 38edo) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&amp;26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of 14edo.
+. As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, [[38edo]]) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&amp;26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of [[14edo]].
-. 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas 65536/65219 and | -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.
+. 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 26-EDO as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower EDOs).
+. We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).
 . It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three 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.
+=== Odd harmonics ===
+{{Harmonics in equal|26}}
 == Intervals ==