26th-octave temperaments: Difference between revisions

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-All temperaments on this page have a period that is [[Fractional-octave temperaments|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 that of [[26edo]]. 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. 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.
-edo 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).
+edo 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
+== Iron ==
+Iron may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.
-[[Comma list]]: 321489/320000, 589824/588245
+[[Subgroup]]: 2.3.5.7
-[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]
+[[Comma list]]: 2460375/2458624, 2147483648/2144153025
-Mapping generators: ~36/35, ~3
+{{Mapping|legend=1| 26 0 184 73 | 0 1 -3 0 }}
+: mapping generators: ~17280/16807, ~3/2
-{{Multival|legend=1|26 52 0 22 -73 -146}}
+[[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 }}
-[[POTE generator]]: ~3/2 = 701.250
+{{Optimal ET sequence|legend=1| 104, 130, 364, 494, 1118, 2730d, 3848dd }}
-{{Val list|legend=1| 26, 130, 546, 676, 806c }}
+[[Badness]] (Sintel): 2.61
-[[Badness]]: 0.155827
 === 11-limit ===
 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}}
-Optimal GPV sequence: {{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
-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
-Optimal GPV sequence: {{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
-Optimal GPV sequence: {{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 }}
-Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 962bf }}
+Badness (Sintel): 0.711
-Badness: 0.043581
+{{Navbox fractional-octave}}
 [[Category:26edo]]
-[[Category:Regular temperament theory]]
-[[Category:Temperament collection]]
-[[Category:Rank 2]]