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.

Fractional-octave temperaments

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← 25th-octave • 26th-octave • 27th-octave →

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.

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

Optimal tunings:

• 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