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. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the bosonic temperaments.
All temperaments on this page have a period that is 1/26th of an octave. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the bosonic temperaments.


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]]).
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]]).  


== Bosonic ==
Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.  
{{See also| Miscellaneous 5-limit temperaments #Tridecatonic }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 321489/320000, 589824/588245
 
{{Mapping|legend=1| 26 0 -22 73 | 0 1 2 0 }}
: mapping generators: ~36/35, ~3
 
[[Optimal tuning]]s ([[POTE]]): ~36/35 = 46.154{{c}}, ~3/2 = 701.250{{c}}
 
{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}
 
[[Badness]] (Smith): 0.155827
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 8019/8000, 65536/65219
 
Mapping: {{mapping| 26 0 -22 73 90 | 0 1 2 0 0 }}
 
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.559{{c}}
 
{{Optimal ET sequence|legend=0| 26, 104c, 130 }}
 
Badness (Smith): 0.065219
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 441/440, 15379/15360
 
Mapping: {{mapping| 26 0 -22 73 90 55 | 0 1 2 0 0 1 }}
 
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.546{{c}}
 
{{Optimal ET sequence|legend=0| 26, 104c, 130 }}
 
Badness (Smith): 0.032946
 
=== Fermionic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 78408/78125, 177147/176000
 
Mapping: {{mapping| 26 0 -22 73 -116 | 0 1 2 0 5 }}
 
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.077{{c}}
 
{{Optimal ET sequence|legend=0| 130, 286, 416, 546, 1508bcd }}
 
Badness (Smith): 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 -22 73 -116 55 | 0 1 2 0 5 1 }}
 
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.038{{c}}
 
{{Optimal ET sequence|legend=0| 130, 286, 416, 546, 962bf }}
 
Badness (Smith): 0.043581


== Iron ==
== Iron ==
Iron is named 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
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 2460375/2458624, 2147483648/2144153025
[[Comma list]]: 2460375/2458624, 2147483648/2144153025


{{Mapping|legend=1| 26 26 106 73 | 0 1 -3 0 }}
{{Mapping|legend=1| 26 0 184 73 | 0 1 -3 0 }}
: mapping generators: ~17280/16807, ~3/2
: mapping generators: ~17280/16807, ~3/2


[[Optimal tuning]]s ([[CTE]]): ~17280/16807 = 46.154{{c}}, ~3/2 = 702.017
[[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 }}


{{Optimal ET sequence|legend=1|130, 364, 494, 624, …}}
[[Badness]] (Sintel): 2.61


=== 11-limit ===
=== 11-limit ===
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Comma list: 9801/9800, 131072/130977, 759375/758912
Comma list: 9801/9800, 131072/130977, 759375/758912


Mapping: {{mapping| 26 26 106 73 166 | 0 1 -3 0 -5 }}
Mapping: {{mapping| 26 0 184 73 296 | 0 1 -3 0 -5 }}
 
Optimal tunings:
* WE: ~77/75 = 46.1519{{c}}, ~3/2 = 701.9621{{c}}
* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 701.9967{{c}}


Optimal tuning (CTE): ~77/75 = 46.154{{c}}, ~3/2 = 702.017{{c}}
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1118, 1612d, 2106d }}


{{Optimal ET sequence|legend=0| 130, 364, 494, 624, … }}
Badness (Sintel): 1.14


=== 13-limit ===
=== 13-limit ===
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Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091


Mapping: {{mapping| 26 26 106 73 166 81 | 0 1 -3 0 -5 1 }}
Mapping: {{mapping| 26 0 184 73 296 55 | 0 1 -3 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}}


Optimal tuning (CTE): ~77/75 = 46.154{{c}}, ~3/2 = 702.018{{c}}
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1612d, 2106d }}


{{Optimal ET sequence|legend=0| 130, 364, 494, 624, … }}
Badness (Sintel): 0.711


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}


[[Category:26edo]]
[[Category:26edo]]

Latest revision as of 04:18, 21 July 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. However, the [-41 26⟩ comma is not tempered out. Thus the 3/2 is not that of 26edo. However, 7/4 is, as is 11/8 in the bosonic temperaments.

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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