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 [[26edo]] | 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. | ||
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). | |||
== Bosonic == | == Bosonic == | ||
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POTE generator: ~3/2 = 701.559 | POTE generator: ~3/2 = 701.559 | ||
Optimal GPV sequence: {{Val list| 26, 104c, 130 }} | |||
Badness: 0.065219 | Badness: 0.065219 | ||
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POTE generator: ~3/2 = 701.546 | POTE generator: ~3/2 = 701.546 | ||
Optimal GPV sequence: {{Val list| 26, 104c, 130 }} | |||
Badness: 0.032946 | Badness: 0.032946 | ||
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POTE generator: ~3/2 = 701.077 | POTE generator: ~3/2 = 701.077 | ||
Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 1508bcd }} | |||
Badness: 0.090642 | Badness: 0.090642 | ||
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POTE generator: ~3/2 = 701.038 | POTE generator: ~3/2 = 701.038 | ||
Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 962bf }} | |||
Badness: 0.043581 | Badness: 0.043581 | ||
Revision as of 04:41, 22 February 2022
All temperaments on this page have a period that is 1/26th of an octave, i.e. their pergen is (P8/26, P5). 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 546 EDO).
Bosonic
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
Wedgie: ⟨⟨ 26 52 0 22 -73 -146 ]]
POTE generator: ~3/2 = 701.250
Badness: 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]]
POTE generator: ~3/2 = 701.559
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~3/2 = 701.546
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~3/2 = 701.077
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~3/2 = 701.038
Optimal GPV sequence: Template:Val list
Badness: 0.043581