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 [[Fractional-octave temperaments|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|546]] EDO).


== Bosonic ==
== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val|26 0 -22 73}}, {{val|0 1 2 0}}]
[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]


Mapping generators: ~36/35, ~3
Mapping generators: ~36/35, ~3
{{Multival|legend=1|26 52 0 22 -73 -146}}


[[POTE generator]]: ~3/2 = 701.250
[[POTE generator]]: ~3/2 = 701.250


{{Val list|legend=1| 26, 130, 546, 676, 806c }}
{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}


[[Badness]]: 0.155827
[[Badness]]: 0.155827
Line 23: Line 27:
Comma list: 441/440, 8019/8000, 65536/65219
Comma list: 441/440, 8019/8000, 65536/65219


Mapping: [{{val|26 0 -22 73 90}}, {{val|0 1 2 0 0}}]
Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]


POTE generator: ~3/2 = 701.559
POTE generator: ~3/2 = 701.559


Vals: {{Val list| 26, 104c, 130 }}
{{Optimal ET sequence|legend=1| 26, 104c, 130 }}


Badness: 0.065219
Badness: 0.065219


=== 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: 351/350, 364/363, 441/440, 15379/15360


Mapping: [{{val|26 0 -22 73 90 55}}, {{val|0 1 2 0 0 1}}]
Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]


POTE generator: ~3/2 = 701.546
POTE generator: ~3/2 = 701.546


Vals: {{Val list| 26, 104c, 130 }}
{{Optimal ET sequence|legend=1| 26, 104c, 130 }}


Badness: 0.032946
Badness: 0.032946


== Fermionic ==
=== Fermionic ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 78408/78125, 177147/176000
Comma list: 540/539, 78408/78125, 177147/176000


Mapping: [{{val|26 0 -22 73 -116}}, {{val|0 1 2 0 5}}]
Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]


POTE generator: ~3/2 = 701.077
POTE generator: ~3/2 = 701.077


Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}
{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}


Badness: 0.090642
Badness: 0.090642


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


Comma list: 351/350, 540/539, 40656/40625, 142884/142805
Comma list: 351/350, 540/539, 40656/40625, 142884/142805


Mapping: [{{val|26 0 -22 73 -116 55}}, {{val|0 1 2 0 5 1}}]
Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]


POTE generator: ~3/2 = 701.038
POTE generator: ~3/2 = 701.038


Vals: {{Val list| 130, 286, 416, 546, 962bf }}
{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}


Badness: 0.043581
Badness: 0.043581
== Iron ==
''Iron'' is named after the 26th element.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2460375/2458624, 2147483648/2144153025
[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]
[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017
[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017
{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 131072/130977, 759375/758912
Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]
Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017
Optimal tuning (CTE):  ~3/2 = 702.017
{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]
Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018
Optimal tuning (CTE):  ~3/2 = 702.018
{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
{{Navbox fractional-octave}}


[[Category:26edo]]
[[Category:26edo]]
[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Rank 2]]

Latest revision as of 05:13, 12 June 2025

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 546 EDO).

Bosonic

See also: High badness temperaments § Tridecatonic

Subgroup: 2.3.5.7

Comma list: 321489/320000, 589824/588245

Mapping: [⟨26 0 -22 73], ⟨0 1 2 0]]

Mapping generators: ~36/35, ~3

POTE generator: ~3/2 = 701.250

Optimal ET sequence: 26, 130, 546, 676, 806c

Badness: 0.155827

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [⟨26 0 -22 73 90], ⟨0 1 2 0 0]]

POTE generator: ~3/2 = 701.559

Optimal ET sequence: 26, 104c, 130

Badness: 0.065219

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [⟨26 0 -22 73 90 55], ⟨0 1 2 0 0 1]]

POTE generator: ~3/2 = 701.546

Optimal ET sequence: 26, 104c, 130

Badness: 0.032946

Fermionic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [⟨26 0 -22 73 -116], ⟨0 1 2 0 5]]

POTE generator: ~3/2 = 701.077

Optimal ET sequence: 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: [⟨26 0 -22 73 -116 55], ⟨0 1 2 0 5 1]]

POTE generator: ~3/2 = 701.038

Optimal ET sequence: 130, 286, 416, 546, 962bf

Badness: 0.043581

Iron

Iron is named after the 26th element.

Subgroup: 2.3.5.7

Comma list: 2460375/2458624, 2147483648/2144153025

Mapping: [⟨26 26 106 73], ⟨0 1 -3 0]]

Mapping generators: ~17280/16807 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE): ~3/2 = 702.017

Optimal ET sequence: 130, 364, 494, 624, ...

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨26 26 106 73 166], ⟨0 1 -3 0 -5]]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE): ~3/2 = 702.017

Optimal ET sequence: 130, 364, 494, 624, ...

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨26 26 106 73 166 81]], ⟨0 1 -3 0 -5 1]]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE): ~3/2 = 702.018

Optimal ET sequence: 130, 364, 494, 624, ...


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