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, i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo|-41 26}} comma is not tempered out. Thus the 3/2 is not [[26edo|26EDO]]'s 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments.  
{{Technical data page}}
{{Infobox fractional-octave|26}}
All temperaments on this page have a period that is 1/26th of an octave.  


== Bosonic ==
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]].  
Subgroup: 2.3.5.7


[[Comma list]]: 321489/320000, 589824/588245
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]]).


[[Mapping]]: [{{val|26 0 -22 73}}, {{val|0 1 2 0}}]
Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.


Mapping generators: ~36/35, ~3
== Iron ==
Iron may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.


{{Multival|legend=1|26 52 0 22 -73 -146}}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~3/2 = 701.250
[[Comma list]]: 2460375/2458624, 2147483648/2144153025


{{Val list|legend=1| 26, 130, 546, 676, 806c }}
{{Mapping|legend=1| 26 0 184 73 | 0 1 -3 0 }}
: mapping generators: ~17280/16807, ~3/2


[[Badness]]: 0.155827
[[Optimal tuning]]s:
* [[WE]]: ~17280/16807 = 46.1521{{c}}, ~3/2 = 701.9475{{c}}
: [[error map]]: {{val| -0.047 -0.054 -0.039 +0.274 }}
* [[CWE]]: ~17280/16807 = 46.1538{{c}}, ~3/2 = 701.9779{{c}}
: error map: {{val| 0.000 +0.023 +0.060 +0.405 }}
 
{{Optimal ET sequence|legend=1| 104, 130, 364, 494, 1118, 2730d, 3848dd }}
 
[[Badness]] (Sintel): 2.61


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 8019/8000, 65536/65219
Comma list: 9801/9800, 131072/130977, 759375/758912


Mapping: [{{val|26 0 -22 73 90}}, {{val|0 1 2 0 0}}]
Mapping: {{mapping| 26 0 184 73 296 | 0 1 -3 0 -5 }}


POTE generator: ~3/2 = 701.559
Optimal tunings:  
* WE: ~77/75 = 46.1519{{c}}, ~3/2 = 701.9621{{c}}
* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 701.9967{{c}}


Vals: {{Val list| 26, 104c, 130 }}
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1118, 1612d, 2106d }}


Badness: 0.065219
Badness (Sintel): 1.14


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 364/363, 441/440, 15379/15360
Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
 
Mapping: [{{val|26 0 -22 73 90 55}}, {{val|0 1 2 0 0 1}}]
 
POTE generator: ~3/2 = 701.546
 
Vals: {{Val list| 26, 104c, 130 }}
 
Badness: 0.032946
 
== Fermionic ==
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 78408/78125, 177147/176000
 
Mapping: [{{val|26 0 -22 73 -116}}, {{val|0 1 2 0 5}}]
 
POTE generator: ~3/2 = 701.077
 
Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}
 
Badness: 0.090642
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 540/539, 40656/40625, 142884/142805
Mapping: {{mapping| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}


Mapping: [{{val|26 0 -22 73 -116 55}}, {{val|0 1 2 0 5 1}}]
Optimal tunings:  
* WE: ~77/75 = 46.1525{{c}}, ~3/2 = 701.9820{{c}}
* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 702.0051{{c}}


POTE generator: ~3/2 = 701.038
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1612d, 2106d }}


Vals: {{Val list| 130, 286, 416, 546, 962bf }}
Badness (Sintel): 0.711


Badness: 0.043581
{{Navbox fractional-octave}}


[[Category:26edo]]
[[Category:26edo]]
[[Category:Temperament]]
[[Category:Temperament collection]]

Latest revision as of 16:32, 2 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

← 25th-octave • 26th-octave • 27th-octave →

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

The 26-comma, [-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.

However, 7/4 is frequently inherited, as 26edo is very accurate for 7th harmonic, the 26-7-comma ([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 26-, 130-, 286- or 546edo).

Temperaments discussed elsewhere include bosonic. Considered below is iron.

Iron

Iron may be described as the 130 & 494 temperament. It was named by Eliora in 2023 after the 26th element.

Subgroup: 2.3.5.7

Comma list: 2460375/2458624, 2147483648/2144153025

Mapping: [⟨26 0 184 73], ⟨0 1 -3 0]]

mapping generators: ~17280/16807, ~3/2

Optimal tunings:

  • WE: ~17280/16807 = 46.1521 ¢, ~3/2 = 701.9475 ¢
error map: ⟨-0.047 -0.054 -0.039 +0.274]
  • CWE: ~17280/16807 = 46.1538 ¢, ~3/2 = 701.9779 ¢
error map: ⟨0.000 +0.023 +0.060 +0.405]

Optimal ET sequence: 104, 130, 364, 494, 1118, 2730d, 3848dd

Badness (Sintel): 2.61

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨26 0 184 73 296], ⟨0 1 -3 0 -5]]

Optimal tunings:

  • WE: ~77/75 = 46.1519 ¢, ~3/2 = 701.9621 ¢
  • CWE: ~77/75 = 46.1538 ¢, ~3/2 = 701.9967 ¢

Optimal ET sequence: 104e, 130, 364, 494, 1118, 1612d, 2106d

Badness (Sintel): 1.14

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨26 0 184 73 296 55], ⟨0 1 -3 0 -5 1]]

Optimal tunings:

  • WE: ~77/75 = 46.1525 ¢, ~3/2 = 701.9820 ¢
  • CWE: ~77/75 = 46.1538 ¢, ~3/2 = 702.0051 ¢

Optimal ET sequence: 104e, 130, 364, 494, 1612d, 2106d

Badness (Sintel): 0.711

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