26th-octave temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
{{Infobox fractional-octave|26}}
{{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.
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|546]] EDO).
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 ==
== Bosonic ==
{{See also| Miscellaneous 5-limit temperaments #Tridecatonic }}
{{See also| Miscellaneous 5-limit 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|legend=1| 26 0 -22 73 | 0 1 2 0 }}
: mapping generators: ~36/35, ~3


Mapping generators: ~36/35, ~3
[[Optimal tuning]]s ([[POTE]]): ~36/35 = 46.154{{c}}, ~3/2 = 701.250{{c}}
 
[[POTE generator]]: ~3/2 = 701.250


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


[[Badness]]: 0.155827
[[Badness]] (Smith): 0.155827


=== 11-limit ===
=== 11-limit ===
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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: {{mapping| 26 0 -22 73 90 | 0 1 2 0 0 }}


POTE generator: ~3/2 = 701.559
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.559{{c}}


{{Optimal ET sequence|legend=1| 26, 104c, 130 }}
{{Optimal ET sequence|legend=0| 26, 104c, 130 }}


Badness: 0.065219
Badness (Smith): 0.065219


==== 13-limit ====
==== 13-limit ====
Line 40: Line 39:
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: {{mapping| 26 0 -22 73 90 55 | 0 1 2 0 0 1 }}


POTE generator: ~3/2 = 701.546
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.546{{c}}


{{Optimal ET sequence|legend=1| 26, 104c, 130 }}
{{Optimal ET sequence|legend=0| 26, 104c, 130 }}


Badness: 0.032946
Badness (Smith): 0.032946


=== Fermionic ===
=== Fermionic ===
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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: {{mapping| 26 0 -22 73 -116 | 0 1 2 0 5 }}


POTE generator: ~3/2 = 701.077
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.077{{c}}


{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}
{{Optimal ET sequence|legend=0| 130, 286, 416, 546, 1508bcd }}


Badness: 0.090642
Badness (Smith): 0.090642


==== 13-limit ====
==== 13-limit ====
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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: {{mapping| 26 0 -22 73 -116 55 | 0 1 2 0 5 1 }}


POTE generator: ~3/2 = 701.038
Optimal tuning (POTE): ~36/35 = 46.154{{c}}, ~3/2 = 701.038{{c}}


{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}
{{Optimal ET sequence|legend=0| 130, 286, 416, 546, 962bf }}


Badness: 0.043581
Badness (Smith): 0.043581


== Iron ==
== Iron ==
''Iron'' is named after the 26th element.
Iron is named 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]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]
{{Mapping|legend=1| 26 26 106 73 | 0 1 -3 0 }}
: mapping generators: ~17280/16807, ~3/2


[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017
[[Optimal tuning]]s ([[CTE]]): ~17280/16807 = 46.154{{c}}, ~3/2 = 702.017


[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017
{{Optimal ET sequence|legend=1|130, 364, 494, 624, …}}
 
{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...


=== 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: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]
Mapping: {{mapping| 26 26 106 73 166 | 0 1 -3 0 -5 }}


Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017
Optimal tuning (CTE): ~77/75 = 46.154{{c}}, ~3/2 = 702.017{{c}}


Optimal tuning (CTE):  ~3/2 = 702.017
{{Optimal ET sequence|legend=0| 130, 364, 494, 624, … }}
 
{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...


=== 13-limit ===
=== 13-limit ===
Line 107: Line 103:
Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
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: {{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 tuning (CTE): ~77/75 = 46.154{{c}}, ~3/2 = 702.018{{c}}


{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
{{Optimal ET sequence|legend=0| 130, 364, 494, 624, … }}


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


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

Revision as of 03:15, 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).

Bosonic

See also: Miscellaneous 5-limit 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

Optimal tunings (POTE): ~36/35 = 46.154 ¢, ~3/2 = 701.250 ¢

Optimal ET sequence: 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: [⟨26 0 -22 73 90], ⟨0 1 2 0 0]]

Optimal tuning (POTE): ~36/35 = 46.154 ¢, ~3/2 = 701.559 ¢

Optimal ET sequence: 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: [⟨26 0 -22 73 90 55], ⟨0 1 2 0 0 1]]

Optimal tuning (POTE): ~36/35 = 46.154 ¢, ~3/2 = 701.546 ¢

Optimal ET sequence: 26, 104c, 130

Badness (Smith): 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]]

Optimal tuning (POTE): ~36/35 = 46.154 ¢, ~3/2 = 701.077 ¢

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

Optimal tuning (POTE): ~36/35 = 46.154 ¢, ~3/2 = 701.038 ¢

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

Badness (Smith): 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, ~3/2

Optimal tunings (CTE): ~17280/16807 = 46.154 ¢, ~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]]

Optimal tuning (CTE): ~77/75 = 46.154 ¢, ~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]]

Optimal tuning (CTE): ~77/75 = 46.154 ¢, ~3/2 = 702.018 ¢

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

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