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26th-octave temperaments: Difference between revisions

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{{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 ==
Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.  
{{See also| High badness temperaments #Tridecatonic }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 321489/320000, 589824/588245
 
[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]
 
Mapping generators: ~36/35, ~3
 
{{Multival|legend=1|26 52 0 22 -73 -146}}
 
[[POTE generator]]: ~3/2 = 701.250
 
{{Optimal ET sequence|legend=1| 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: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]
 
POTE generator: ~3/2 = 701.559
 
{{Optimal ET sequence|legend=1| 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: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]
 
POTE generator: ~3/2 = 701.546
 
{{Optimal ET sequence|legend=1| 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
 
{{Optimal ET sequence|legend=1| 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: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]
 
POTE generator: ~3/2 = 701.038
 
{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}
 
Badness: 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
Line 82: Line 14:
[[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 0 184 73 | 0 1 -3 0 }}
: mapping generators: ~17280/16807, ~3/2


[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~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 tuning]] ([[CTE]]):  ~3/2 = 702.017
{{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: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]
Mapping: {{mapping| 26 0 184 73 296 | 0 1 -3 0 -5 }}


Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017
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):  ~3/2 = 702.017
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1118, 1612d, 2106d }}


{{Optimal ET sequence|legend=1|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: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]
Mapping: {{mapping| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}


Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018
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):  ~3/2 = 702.018
{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1612d, 2106d }}


{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
Badness (Sintel): 0.711


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


[[Category:26edo]]
[[Category:26edo]]
Retrieved from "https://en.xen.wiki/w/26th-octave_temperaments"