26th-octave temperaments: Difference between revisions
nonetheless expand whithin limits of "not cataloging edos" - [-41 26] is technically a 26th-octave temperament, just a really bad one, and precisely clarify what it means that "3/2 is not that of 26edo" |
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All temperaments on this page have a period that is 1/26th of an octave. | All temperaments on this page have a period that is 1/26th of an octave. | ||
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]] | 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]]. | ||
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]]). | 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]]). | ||
Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron. | Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron. | ||
Latest revision as of 16:32, 2 August 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
All temperaments on this page have a period that is 1/26th of an octave.
The 26-comma, [-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.
However, 7/4 is frequently inherited, as 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 546edo).
Temperaments discussed elsewhere include bosonic. Considered below is iron.
Iron
Iron may be described as the 130 & 494 temperament. It was named by Eliora in 2023 after the 26th element.
Subgroup: 2.3.5.7
Comma list: 2460375/2458624, 2147483648/2144153025
Mapping: [⟨26 0 184 73], ⟨0 1 -3 0]]
- mapping generators: ~17280/16807, ~3/2
- WE: ~17280/16807 = 46.1521 ¢, ~3/2 = 701.9475 ¢
- error map: ⟨-0.047 -0.054 -0.039 +0.274]
- CWE: ~17280/16807 = 46.1538 ¢, ~3/2 = 701.9779 ¢
- error map: ⟨0.000 +0.023 +0.060 +0.405]
Optimal ET sequence: 104, 130, 364, 494, 1118, 2730d, 3848dd
Badness (Sintel): 2.61
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 131072/130977, 759375/758912
Mapping: [⟨26 0 184 73 296], ⟨0 1 -3 0 -5]]
Optimal tunings:
- WE: ~77/75 = 46.1519 ¢, ~3/2 = 701.9621 ¢
- CWE: ~77/75 = 46.1538 ¢, ~3/2 = 701.9967 ¢
Optimal ET sequence: 104e, 130, 364, 494, 1118, 1612d, 2106d
Badness (Sintel): 1.14
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091
Mapping: [⟨26 0 184 73 296 55], ⟨0 1 -3 0 -5 1]]
Optimal tunings:
- WE: ~77/75 = 46.1525 ¢, ~3/2 = 701.9820 ¢
- CWE: ~77/75 = 46.1538 ¢, ~3/2 = 702.0051 ¢
Optimal ET sequence: 104e, 130, 364, 494, 1612d, 2106d
Badness (Sintel): 0.711