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.

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 are 26-commatic and iron.

26-commatic

This page presents a topic of primarily mathematical interest. While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

The 26-commatic temperament tempers out the 26-comma, [-41 26⟩. 26edo is notable for having a flattone fifth, and due to large error, 26edo and its double, 52edo, are the only temperaments tempering out this comma. Hence it primarily serves as an object of mathematical interest.

Subgroup: 2.3

Comma list: [-41 26⟩

Subgroup-val mapping: [⟨37 28]]

mapping generators: ~2187/2048 = 1\26

Supporting ETs: 26, 52

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