Alpharabian temperaments

From Xenharmonic Wiki
Jump to navigation Jump to search

This is a collection of rank-3 temperaments that temper out the Alpharabian comma, [-17 2 0 0 4 = 131769/131072, but we can begin with the rank-4 temperament.

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

Vals22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee

Badness: 1.26 × 10-6

Betarabian

Whenever the rastma is added to the list of commas tempered out among Alpharabian temperaments, you end up with Betarabian, 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, and although there are technically other Betarabian temperaments, they have different names.

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

Vals24, 48, 72, 264, 336b, 408b, 480bcde

Badness: 6.07 × 10-3