26th-octave temperaments
- 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. 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 546edo).
Bosonic
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Tridecatonic.
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
- WE: ~36/35 = 46.1534 ¢, ~3/2 = 701.2433 ¢
- error map: ⟨-0.012 -0.723 +0.775 +0.372]
- CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.2451 ¢
- error map: ⟨0.000 -0.710 +0.792 +0.405]
Optimal ET sequence: 26, …, 104c, 130, 546, 676, 806c
Badness (Sintel): 3.94
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]]
Optimal tunings:
- WE: ~36/35 = 46.1443 ¢, ~3/2 = 701.4143 ¢
- CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.4791 ¢
Optimal ET sequence: 26, …, 104c, 130
Badness (Sintel): 2.16
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]]
Optimal tunings:
- WE: ~36/35 = 46.1469 ¢, ~3/2 = 701.4410 ¢
- CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.4873 ¢
Optimal ET sequence: 26, …, 104c, 130
Badness (Sintel): 1.36
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]]
Optimal tunings:
- WE: ~36/35 = 46.1550 ¢, ~3/2 = 701.0945 ¢
- CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.0853 ¢
Optimal ET sequence: 130, 286, 416, 546
Badness (Sintel): 3.00
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]]
Optimal tunings:
- WE: ~36/35 = 46.1573 ¢, ~3/2 = 701.0904 ¢
- CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.0603 ¢
Optimal ET sequence: 130, 286, 416, 546
Badness (Sintel): 1.80
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