21st-octave temperaments

Fractional-octave temperaments

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← 20th-octave • 21st-octave • 22nd-octave →

This page collects temperaments with a period of 1/21 of an octave.

Although 21edo itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as 441, 1407, and 1848 are. 441 and 1848 are also members of the zeta edo list.

Temperaments discussed elsewhere include

• Akjayland → Landscape microtemperaments

21-23-commatic

Subgroup: 2.23

Comma list: [95 0 0 0 0 0 0 0 -21⟩

Subgroup-val mapping: [⟨21 95]]

Mapping generator: ~529/512 = 1\21

Supporting ETs: 21N, N = 1 to 96, largest: 2016edo

Scandium

Described as the 525 & 1911 temperament, and named after the 21st element for splitting the octave into 21 parts. Coincidentally, Encyclopaedia Britannica entry for scandium was written in the year 1911 which was used as the reason for the naming. Remarkably, unlike akjayland or many temperaments in the thousands which contain 3edo as a subset, it is not a landscape system. 39/32 is mapped into 6\21 and 23/16 is, as usual, mapped into 11\21.

Subgroup: 2.3.5.7

Comma list: [47 -7 -7 -7⟩, [-29 0 27 -12⟩

Mapping: [⟨21 0 59 82], ⟨0 13 -4 -9]]

Mapping generators: ~403368/390625 = 1\21, ~160/147

Optimal tuning (CTE): ~160/147 = 146.305

Supporting ETs: 189b, 525, 861, 1050, 1386, 1911, 2436

23-limit

Subgroup: 2.3.5.7.11.13.17.19.21.23

Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092

Mapping: [⟨21 0 59 82 24 111 114 38 95], ⟨0 13 -4 -9 19 -13 -11 20 0]]

Mapping generators: ~216/209 = 1\21, ~160/147

Optimal tuning (CTE): ~160/147 = 146.308 ¢

Supporting ETs: 525, 861h, 1050f, 1911

Blackmagic

Blackmagic is the 63 & 84 temperament, merging two systems which cover many large primes.

Subgroup: 2.3.5.7

Comma list: 225/224, [27 1 1 -11⟩

Mapping: [⟨21 0 82 59], ⟨0 1 -1 0]]

Mapping generators: ~16807/16384 = 1\21, ~3

Optimal tuning (CWE): ~3/2 = 701.120 ¢

Optimal ET sequence: 21, 63, 84, 147

Badness (Sintel): 5.605

2.3.5.7.11.13.23.29.31.43 subgroup

Subgroup: 2.3.5.7.11.13.23.29.31.43

Comma list: 155/154, 225/224, 232/231, 300/299, 364/363, 560/559, 640/637, 1716/1715

Mapping: [⟨21 0 82 59 106 111 95 102 104 114], ⟨0 1 -1 0 -1 -1 0 0 0 0]]

Mapping generators: ~16807/16384 = 1\21, ~3

Optimal tuning (CWE): ~3/2 = 701.742 ¢

Optimal ET sequence: 21, 63, 84, 147

Badness (Sintel): 1.317