26th-octave temperaments: Difference between revisions

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 {{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.
+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 ==
 {{See also| Miscellaneous 5-limit temperaments #Tridecatonic }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}
-[[Badness]]: 0.155827
+[[Badness]] (Smith): 0.155827
 === 11-limit ===
@@ Line 27: / Line 26: @@
 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 ====
@@ Line 40: / Line 39: @@
 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 ===
@@ Line 53: / Line 52: @@
 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 ====
@@ Line 66: / Line 65: @@
 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'' is named after the 26th element.
+Iron is named after the 26th element.
 [[Subgroup]]: 2.3.5.7
@@ Line 81: / Line 80: @@
 [[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 ===
@@ Line 94: / Line 92: @@
 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 ===
@@ Line 107: / Line 103: @@
 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}}
 [[Category:26edo]]