Alphaxenic rank three clan

The alpharabian rank-3 clan of temperaments tempers out the Alpharabian comma, [-17 2 0 0 4⟩ = 131769/131072. Both no-5 alpharabian and no-7 alpharabian can be the head of this clan, but we can begin with the rank-4 temperament.

Temperaments discussed elsewhere include:

• Uniwiz, {385/384, 9801/9800} → Keenanismic temperaments

Alpharabian

Not to be confused with Alpharabian tuning.

Subgroup: 2.3.5.7.11

Comma list: 131769/131072

Mapping: [⟨2 1 0 0 8], ⟨0 2 0 0 -1], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

Mapping generators: ~363/256, ~16/11, ~5, ~7

POTE generators: ~16/11 = 650.6784, ~5/4 = 385.5751, ~7/4 = 967.9328

Optimal ET sequence: 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee

Badness: 1.26 × 10^-6

Rabic

Whenever the rastma is added to the list of commas tempered out among Alpharabian temperaments, you end up with Rabic, which splits the octave into 24 equal parts. This temperament is named as such because tempering out both the Alpharabian comma and the rastma automatically tempers out the Betarabian comma.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 131769/131072

Mapping: [⟨24 38 0 0 83], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

Mapping generators: ~33/32, ~5, ~7

Wedgie: ⟨⟨⟨ 0 0 0 24 0 0 38 0 0 83 ]]]

POTE generators: ~5/4 = 385.0647, ~7/4 = 967.3158

Optimal ET sequence: 24, 48, 72, 264, 336b, 408b, 480bcde

Badness: 6.07 × 10^-3