26th-octave temperaments: Difference between revisions

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All temperaments on this page have a period that is 1/26th of an octave.

~~26edo is very accurate for 7th harmonic, the~~ [[26-7-comma]] ({{monzo| ~~73 0 0~~ -26 }}, ~~the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves)~~ is ~~tempered out~~ by ~~26-fold multiple edos up to 1456 (such as [[~~26edo~~|26-]],~~ [[~~130edo|130-~~]], ~~[[286edo|286~~-~~]] or~~ [[~~546edo|546edo~~]]).

The [[26-comma]], {{monzo| -41 26 }}, is technically usable for regular temperament purposes, however it makes a very imprecise temperament, only supported by 26edo itself and [[52edo]]. Thus, 26th-octave temperaments which are of musical interest typically do not inherit the flattone approximation of 3/2 which is found in [[26edo]].

~~Temperaments discussed elsewhere include~~ [[~~varunismic temperaments #Bosonic|bosonic~~]].

However, 7/4 is frequently inherited, as 26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo| 73 0 0 -26 }}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple edos up to 1456 (such as [[26edo|26-]], [[130edo|130-]], [[286edo|286-]] or [[546edo|546edo]]).

~~== 26-commatic ==~~

~~The ''26-commatic'' temperament tempers out the~~ [[26-comma]], {{monzo|-41 26}}. 26edo is notable for having a flattone, even if innacurate, fifth, which has seen some attention in music theory. Due to large error, 26edo and its double, [[52edo]], are the only temperaments ~~tempering out this comma. Hence it primarily serves as an object of mathematical interest.~~

Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.

~~[[Subgroup]]: 2.3~~

~~[[Comma list]]: {{monzo|-41 26}}~~

~~{{mapping|legend=2~~| ~~37 28 }}~~

~~: mapping generators: ~2187/2048 = 1\26~~

~~[[Support~~]]~~ing [[ET]]s: {{EDOs|26, 52}}~~

~~== Fynn ==~~

~~{{Main|Fynn's comma}}~~

~~''Fynn'', or systematically, 26-7-commatic, tempers out the [[Fynn's comma]] or the 26-7-comma~~.

~~[[Subgroup]]: 2~~.3

~~[[Comma list]]: {{monzo| 73 -26 }}~~

~~{{mapping|legend=2| 26 73 }}~~

~~: mapping generators: ~ 16807/16384 = 1\26~~

~~[[Support]]ing [[ET]]s: 26N, N = 1 to 56~~

== Iron ==

''Iron'' may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.

Iron may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.

[[Subgroup]]: 2.3.5.7

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Comma list: 9801/9800, 131072/130977, 759375/758912

~~Mapping:~~ {{~~mapping~~| 26 0 184 73 296 | 0 1 -3 0 -5 }}

Optimal tunings:

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Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

~~Mapping:~~ {{~~mapping~~| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}

Optimal tunings:

@@ Line 1: / Line 1: @@
 {{Technical data page}}
 {{Infobox fractional-octave|26}}
 All temperaments on this page have a period that is 1/26th of an octave.
-edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo| 73 0 0 -26 }}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple edos up to 1456 (such as [[26edo|26-]], [[130edo|130-]], [[286edo|286-]] or [[546edo|546edo]]).
+The [[26-comma]], {{monzo| -41 26 }}, is technically usable for regular temperament purposes, however it makes a very imprecise temperament, only supported by 26edo itself and [[52edo]]. Thus, 26th-octave temperaments which are of musical interest typically do not inherit the flattone approximation of 3/2 which is found in [[26edo]].
-Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]].
+However, 7/4 is frequently inherited, as 26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo| 73 0 0 -26 }}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple edos up to 1456 (such as [[26edo|26-]], [[130edo|130-]], [[286edo|286-]] or [[546edo|546edo]]).
-== 26-commatic ==
-{{Mathematical interest}}
-The ''26-commatic'' temperament tempers out the [[26-comma]], {{monzo|-41 26}}. 26edo is notable for having a flattone, even if innacurate, fifth, which has seen some attention in music theory. Due to large error, 26edo and its double, [[52edo]], are the only temperaments tempering out this comma. Hence it primarily serves as an object of mathematical interest.
+Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.
-[[Subgroup]]: 2.3
-[[Comma list]]: {{monzo|-41 26}}
-{{mapping|legend=2| 37 28 }}
-: mapping generators: ~2187/2048 = 1\26
-[[Support]]ing [[ET]]s: {{EDOs|26, 52}}
-== Fynn ==
-{{Main|Fynn's comma}}
-''Fynn'', or systematically, 26-7-commatic, tempers out the [[Fynn's comma]] or the 26-7-comma.
-[[Subgroup]]: 2.3
-[[Comma list]]: {{monzo| 73 -26 }}
-{{mapping|legend=2| 26 73 }}
-: mapping generators: ~	16807/16384 = 1\26
-[[Support]]ing [[ET]]s: 26N, N = 1 to 56
 == Iron ==
-''Iron'' may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.
+Iron may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.
 [[Subgroup]]: 2.3.5.7
@@ Line 60: / Line 34: @@
 Comma list: 9801/9800, 131072/130977, 759375/758912
-Mapping: {{mapping| 26 0 184 73 296 | 0 1 -3 0 -5 }}
+{{Mapping|legend=0| 26 0 184 73 296 | 0 1 -3 0 -5 }}
 Optimal tunings:
@@ Line 75: / Line 49: @@
 Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
-Mapping: {{mapping| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}
+{{Mapping|legend=0| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}
 Optimal tunings:

26th-octave temperaments: Difference between revisions

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