26th-octave temperaments: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
-__FORCETOC__
+{{Technical data page}}
-=Bosonic=
+{{Infobox fractional-octave|26}}
-Commas: 321489/320000, 589824/588245
+All temperaments on this page have a period that is 1/26th of an octave.
-POTE generator: ~3/2 = 701.250
+The [[26-comma]], {{monzo| -41 26 }}, is technically usable for regular temperament purposes, however it makes a very imprecise temperament, only supported by 26edo itself and [[52edo]]. Thus, 26th-octave temperaments which are of musical interest typically do not inherit the flattone approximation of 3/2 which is found in [[26edo]].
-Map: [&lt;26 0 -22 73|, &lt;0 1 2 0|]
+However, 7/4 is frequently inherited, as 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]]).
-Wedgie: &lt;&lt;26 52 0 22 -73 -146||
+Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.
-EDOs: 26, 130, 546, 676, 806c
+== Iron ==
+Iron may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.
-Badness: 0.1558
+[[Subgroup]]: 2.3.5.7
-==11-limit==
+[[Comma list]]: 2460375/2458624, 2147483648/2144153025
-Commas: 441/440, 8019/8000, 65536/65219
-POTE generator: ~3/2 = 701.559
+{{Mapping|legend=1| 26 0 184 73 | 0 1 -3 0 }}
+: mapping generators: ~17280/16807, ~3/2
-Map: [&lt;26 0 -22 73 90|, &lt;0 1 2 0 0|]
+[[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 }}
-EDOs: 26, 104c, 130
+{{Optimal ET sequence|legend=1| 104, 130, 364, 494, 1118, 2730d, 3848dd }}
-Badness: 0.0652
+[[Badness]] (Sintel): 2.61
-==13-limit==
+=== 11-limit ===
-Commas: 351/350, 364/363, 441/440, 15379/15360
+Subgroup: 2.3.5.7.11
-POTE generator: ~3/2 = 701.546
+Comma list: 9801/9800, 131072/130977, 759375/758912
-Map: [&lt;26 0 -22 73 90 55|, &lt;0 1 2 0 0 1|]
+Mapping: {{mapping| 26 0 184 73 296 | 0 1 -3 0 -5 }}
-EDOs: 26, 104c, 130
+Optimal tunings:
+* WE: ~77/75 = 46.1519{{c}}, ~3/2 = 701.9621{{c}}
+* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 701.9967{{c}}
-Badness: 0.0329
+{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1118, 1612d, 2106d }}
-=Fermionic=
+Badness (Sintel): 1.14
-Commas: 540/539, 78408/78125, 177147/176000
-POTE generator: ~3/2 = 701.077
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Map: [&lt;26 0 -22 73 -116|, &lt;0 1 2 0 5|]
+Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
-EDOs: 130, 286, 416, 546, 1508bcd
+Mapping: {{mapping| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}
-Badness: 0.0906
+Optimal tunings:
+* WE: ~77/75 = 46.1525{{c}}, ~3/2 = 701.9820{{c}}
+* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 702.0051{{c}}
-==13-limit==
+{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1612d, 2106d }}
-Commas: 351/350, 540/539, 40656/40625, 142884/142805
-POTE generator: ~3/2 = 701.038
+Badness (Sintel): 0.711
-Map: [&lt;26 0 -22 73 -116 55|, &lt;0 1 2 0 5 1|]
+{{Navbox fractional-octave}}
-EDOs: 130, 286, 416, 546, 962bf
+[[Category:26edo]]
-Badness: 0.0436