Didymus rank-3 family

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.

The didymus rank-3 family of rank-3 temperaments tempers out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit rank-3 temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Most of these temperaments were named by Gene Ward Smith in 2010.

Temperaments discussed elsewhere include:

• Urania (+243/242 or 121/120) → Rastmic rank-3 clan

Didymus

Subgroup: 2.3.5.7

Comma list: 81/80

Mapping: [⟨1 0 -4 0], ⟨0 1 4 0], ⟨0 0 0 1]]

Optimal tunings:

• WE: ~2 = 1201.3906 ¢, ~3/2 = 697.0455 ¢, ~7/4 = 966.0272 ¢

error map: ⟨+1.391 -3.519 +1.868 -0.018]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6512 ¢, ~7/4 = 966.5654 ¢

error map: ⟨0.000 -5.304 +0.291 -2.261]

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness (Sintel): 0.419

Euterpe

Euterpe tempers out 99/98 and is related to huygens.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98

Mapping: [⟨1 0 -4 0 1], ⟨0 1 4 0 -2], ⟨0 0 0 1 2]]

Optimal tunings:

• WE: ~2 = 1201.2743 ¢, ~3/2 = 696.9375 ¢, ~7/4 = 969.4564 ¢

error map: ⟨+1.274 -3.743 +1.436 +3.179 -2.457]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.5797 ¢, ~7/4 = 969.7942 ¢

error map: ⟨0.000 -5.375 +0.005 +0.968 -4.889]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-1 0 -1/2 2 0⟩]

eigenmonzo (unchanged-interval) basis: 2.5.7

Optimal ET sequence: 12, 17c, 19e, 26, 31, 88, 119be, 150be

Badness (Sintel): 0.644

Clio

Clio tempers out 176/175. It is another temperament related to huygens.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175

Mapping: [⟨1 0 -4 0 -12], ⟨0 1 4 0 8], ⟨0 0 0 1 1]]

Optimal tunings:

• WE: ~2 = 1200.6376 ¢, ~3/2 = 697.6207 ¢, ~7/4 = 969.1442 ¢

error map: ⟨+0.638 -3.697 +4.169 +1.593 -2.483]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3350 ¢, ~7/4 = 969.0152 ¢

error map: ⟨0.000 -4.620 +3.026 +0.189 -3.622]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-4 0 2 1 0⟩]

eigenmonzo (unchanged-interval) basis: 2.5.7

Optimal ET sequence: 7, 12, 19e, 24, 31, 105, 129, 136b

Badness (Sintel): 0.886

Terpsichore

Terpsichore tempers out 540/539 and is related to meanpop.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 540/539

Mapping: [⟨1 0 -4 0 -2], ⟨0 1 4 0 7], ⟨0 0 0 1 -2]]

Optimal tunings:

• WE: ~2 = 1201.5055 ¢, ~3/2 = 697.1093 ¢, ~7/4 = 965.2101 ¢

error map: ⟨+1.506 -3.340 +2.123 -0.605 -0.468]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.7166 ¢, ~7/4 = 964.9479 ¢

error map: ⟨0.000 -5.238 +0.553 -3.878 -4.197]

Optimal ET sequence: 12e, 14c, 17c, 19, 31, 81, 112b

Badness (Sintel): 1.02

Complexity spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11

Polyhymnia

Polyhymnia tempers out 385/384. It is another temperament related to meanpop.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384

Mapping: [⟨1 0 -4 0 11], ⟨0 1 4 0 -3], ⟨0 0 0 1 -1]]

Optimal tunings:

• WE: ~2 = 1201.4003 ¢, ~3/2 = 697.0429 ¢, ~7/4 = 965.9954 ¢

error map: ⟨+1.400 -3.512 +1.858 -0.030 -0.040]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.3248 ¢, ~7/4 = 964.8389 ¢

error map: ⟨0.000 -5.630 -1.014 -3.987 -5.131]

Optimal ET sequence: 7, 12e, 19, 24, 26, 31, 81, 112b

Badness (Sintel): 1.07

Scales: polyhymnia12

Calliope

Calliope tempers out 45/44 and is related to flattone.

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80

Mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6], ⟨0 0 0 1 0]]

Optimal tunings:

• WE: ~2 = 1202.0621 ¢, ~3/2 = 694.5448 ¢, ~7/4 = 964.5866 ¢

error map: ⟨+2.062 -5.348 -8.135 -0.115 +15.951]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.9085 ¢, ~7/4 = 965.3621 ¢

error map: ⟨0.000 -8.047 -10.680 -3.464 +12.133]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [1 0 0 0 1/6⟩, [0 0 0 0 2/3⟩, [1 -1 0 1 1/6⟩, [0 0 0 0 1⟩]

Eigenmonzo (unchanged-interval) basis: 2.7/3.11

Optimal ET sequence: 7d, 12, 19, 26, 45

Badness (Sintel): 0.637

Thalia

Subgroup: 2.3.5.7.11

Comma list: 33/32, 55/54

Mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1], ⟨0 0 0 1 0]]

Optimal tunings:

• WE: ~2 = 1206.5832 ¢, ~3/2 = 695.8763 ¢, ~7/4 = 955.4695 ¢

error map: ⟨+6.583 +0.504 -2.809 -0.190 -20.862]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.1206 ¢, ~7/4 = 955.1393 ¢

error map: ⟨0.000 -8.834 -13.831 -13.687 -44.439]

Optimal ET sequence: 5, 7, 12e, 14c, 19e, 33cdee, 52cdeee

Badness (Sintel): 0.898

Melpomene

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80

Mapping: [⟨1 0 -4 0 7], ⟨0 1 4 0 -4], ⟨0 0 0 1 1]]

Optimal tunings:

• WE: ~2 = 1198.7942 ¢, ~3/2 = 698.5204 ¢, ~7/4 = 963.2674 ¢

error map: ⟨-1.206 -4.640 +7.768 -7.970 +11.839]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.9964 ¢, ~7/4 = 962.5723 ¢

error map: ⟨0.000 -2.959 +9.672 -6.254 +15.269]

Optimal ET sequence: 12, 17c, 19, 24, 31e, 36

Badness (Sintel): 0.929

Erato

Erato uses the same mapping for the 7-limit as septimal meantone, but adds an independent generator for prime 11.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13 0], ⟨0 1 4 10 0], ⟨0 0 0 0 1]]

Optimal tunings:

• WE: ~2 = 1201.2358 ¢, ~3/2 = 697.2122 ¢, ~11/8 = 547.5886 ¢

error map: ⟨+1.236 -3.507 +2.535 -0.412 -0.022]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6562 ¢, ~11/8 = 548.5396 ¢

error map: ⟨0.000 -5.299 +0.311 -2.264 -2.778]

Optimal ET sequence: 12, 19, 31, 50, 81, 112b

Badness (Sintel): 0.670

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125

Mapping: [⟨1 0 -4 -13 0 -20], ⟨0 1 4 10 0 15], ⟨0 0 0 0 1 0]]

Optimal tunings:

• WE: ~2 = 1201.6213 ¢, ~3/2 = 696.9287 ¢, ~11/8 = 546.4190 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1573 ¢, ~11/8 = 547.5865 ¢

Optimal ET sequence: 12f, 19, 31, 50, 69de, 81, 119bdde

Badness (Sintel): 0.823