Rastmic rank three clan

The rastmic rank-3 clan of temperaments tempers out the rastma, 243/242. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an 11/9 neutral third.

Temperaments not listed include:

• Urania, {81/80, 121/120} → Didymus rank-3 family
• Mandos, {176/175, 243/242} → Valinorsmic temperaments
• Jove, {243/242, 441/440} → Breed family
• Hagrid, {243/242, 9801/9800} → Cataharry family
• Betarabian, {243/242, 131769/131072} → Alpharabian rank-3 clan

Spectacle

Spectacle, named by Gene Ward Smith in 2010, can be described as the 31 & 34 & 41 temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 4 5], ⟨0 0 1 2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.0570, ~5/4 = 383.9323

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [1/5 0 0 0 2/5⟩, [2/5 -2 1 0 4/5⟩, [-19/5 -4 2 0 12/5⟩, [0 0 0 0 1⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.11

Optimal ET sequence: 31, 41, 72, 247c, 281, 353c, 425bc, 497bc

Badness: 0.499 × 10^-3

Projection pairs: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9

Scales: spectacle31

Associated temperament: marvo

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 4 5 -2], ⟨0 0 1 2 0 4]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 349.9247, ~5/4 = 384.3505

Optimal ET sequence: 31, 72, 103, 175f *

* optimal patent val: 240

Badness: 1.009 × 10^-3

Cuckoo

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 -4 5], ⟨0 0 1 3 0]]

mapping generators: ~2, ~11/9, ~5

Optimal ET sequence: 24d, 27e, 31, 58, 89, 154, 185

Badness: 0.933 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]

Optimal ET sequence: 27e, 31, 58, 96d, 154

Mirwomo

The mirwomo temperament tunes ~36/35 to exactly half of an apotome ~2187/2048, and tunes ~128/105 to exactly half of a perfect fifth ~3/2.

7-limit

Subgroup: 2.3.5.7

Comma list: 33075/32768

Mapping: [⟨1 1 0 6], ⟨0 2 0 -3], ⟨0 0 1 -1]]

Mapping generators: ~2, ~128/105, ~5

POTE generators: ~128/105 = 350.1375, ~5/4 = 383.8665

Optimal ET sequence: 17, 21, 24, 31, 41, 72, 281d

Badness: 0.770 × 10^-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 -3 5], ⟨0 0 1 -1 0]]

POTE generators: ~11/9 = 349.9554, ~5/4 = 384.1048

Optimal ET sequence: 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde

Badness: 0.641 × 10^-3

Mirage

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

mapping generators: ~2, ~15/14, ~13

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.6327, ~13/8 = 837.7040

Optimal ET sequence: 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef

Badness: 0.738 × 10^-3

Parahemif

7-limit

Subgroup: 2.3.5.7

Comma list: 1605632/1594323

Mapping: [⟨1 1 0 -1], ⟨0 2 0 13], ⟨0 0 1 0]]

Mapping generators: ~2, ~896/729, ~5

POTE generators: ~896/729 = 351.4846, ~5/4 = 386.9396

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99, 239, 338

Badness: 0.607 × 10^-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891

Mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5], ⟨0 0 1 0 0]]

POTE generators: ~11/9 = 351.5347, ~5/4 = 388.0217

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99e *

* optimal patent val: 123

Badness: 1.34547 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 243/242, 364/363

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1], ⟨0 0 1 0 0 0]]

POTE generators: ~11/9 = 351.6908, ~5/4 = 389.1892

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff

Badness: 1.19366 × 10^-3