Landscape microtemperaments
Landscape microtemperaments are rank-2 temperaments tempering out the landscape comma ([-4 6 -6 3⟩ = 250047/250000).
Landscape rank-2 temperaments include sextile, septichrome, pnict, and domain. Considered elsewhere are:
- Augene, {64/63, 126/125} → Augmented family
- Waage, {225/224, 250047/250000} → Compton family
- Tritikleismic, {1029/1024, 15625/15552} → Kleismic family
- Trisensory, {1728/1715, 78732/78125} → Sensipent family
- Ennealimmal, {2401/2400, 4375/4374} → Ragismic microtemperaments
- Misty, {3136/3125, 5120/5103} → Misty family
- Nessafof, {6144/6125, 250047/250000} → Porwell temperaments
- Chromat, {10976/10935, 235298/234375} → Hemimage temperaments
- Term, {32805/32768, 250047/250000} → Schismatic family
- Caleb, {33075/32768, 250047/250000} → Mirwomo temperaments
- Mutt, {65625/65536, 250047/250000} → Horwell temperaments
- Stearnscape, {118098/117649, 250047/250000} → Stearnsmic clan
- Tritricot, {250047/250000, 11785390260224/11767897353375} → Tricot family
Sextile
Subgroup: 2.3.5.7
Comma list: 250047/250000, 33554432/33480783
Mapping: [⟨6 0 71 150], ⟨0 1 -6 -14]]
Wedgie: ⟨⟨ 6 -36 -84 -71 -150 -94 ]]
POTE generator: ~3/2 = 702.212
Badness: 0.070097
11-limit
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 9801/9800, 151263/151250
Mapping: [⟨6 0 71 150 230], ⟨0 1 -6 -14 -22]]
POTE generator: ~3/2 = 702.202
Optimal GPV sequence: Template:Val list
Badness: 0.029677
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 5632/5625, 9801/9800, 151263/151250
Mapping: [⟨6 0 71 150 230 ?], ⟨0 1 -6 -14 -22 -27]]
POTE generator: ~3/2 = 702.202
Optimal GPV sequence: Template:Val list
Badness: 0.029677
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 1716/1715, 5632/5625, 9801/9800, 151263/151250
Mapping: [⟨6 0 71 150 230 ? ?], ⟨0 1 -6 -14 -22 -27 3]]
POTE generator: ~3/2 = 702.202
Optimal GPV sequence: Template:Val list
Badness: 0.029677
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 936/935, 1216/1215, 1716/1715, 5632/5625, 9801/9800, 151263/151250
Mapping: [⟨6 0 71 150 230 ? M], ⟨0 1 -6 -14 -22 -27 3 -1]]
POTE generator: ~3/2 = 702.202
Optimal GPV sequence: Template:Val list
Badness: 0.029677
Sextilia
1001/1000
17-limit
715/714
19-limit
1216/1215
Septichrome
Subgroup: 2.3.5.7
Comma list: 250047/250000, 2460375/2458624
Mapping: [⟨3 3 1 0], ⟨0 5 17 24]]
Wedgie: ⟨⟨ 15 51 72 46 72 24 ]]
POTE generator: ~243/224 = 140.367
Badness: 0.016814
Pnict
Subgroup: 2.3.5.7
Comma list: 250047/250000, 2100875/2097152
Mapping: [⟨3 10 11 6], ⟨0 -13 -10 6]]
Wedgie: ⟨⟨ 39 30 -18 -43 -138 -126 ]]
POTE generator: ~192/175 = 161.399
Badness: 0.045660
Domain
Subgroup: 2.3.5.7
Comma list: 250047/250000, 645700815/645657712
Mapping: [⟨3 -3 -9 -8], ⟨0 17 35 36]]
Wedgie: ⟨⟨ 51 105 108 48 28 -44 ]]
POTE generator: ~10/9 = 182.467
Badness: 0.013979
Avicenna
Subgroup: 2.3.5.7
Comma list: 250047/250000, 29360128/29296875
Mapping: [⟨3 2 8 16], ⟨0 8 -3 -22]]
Wedgie: ⟨⟨ 24 -9 -66 -70 -172 -128 ]]
POTE generator: ~1024/945 = 137.768
Badness: 0.062187
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 5632/5625, 102487/102400
Mapping: [⟨3 2 8 16 9], ⟨0 8 -3 -22 4]]
POTE generator: ~693/640 = 137.771
Optimal GPV sequence: Template:Val list
Badness: 0.023085
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
Mapping: [⟨3 2 8 16 9 8], ⟨0 8 -3 -22 4 9]]
POTE generator: ~13/12 = 137.777
Optimal GPV sequence: Template:Val list
Badness: 0.015557
Terture
Subgroup: 2.3.5.7
Comma list: 250047/250000, 359661568/358722675
Mapping: [⟨3 4 3 2], ⟨0 4 21 34]]
POTE generator: ~392/375 = 75.555
Badness: 0.087156
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 19712/19683, 102487/102400
Mapping: [⟨3 4 3 2 10], ⟨0 4 21 34 2]]
POTE generator: ~392/375 = 75.550
Optimal GPV sequence: Template:Val list
Badness: 0.029326
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
Mapping: [⟨3 4 3 2 10 6], ⟨0 4 21 34 2 27]]
POTE generator: ~117/112 = 75.553
Optimal GPV sequence: Template:Val list
Badness: 0.018647
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
Mapping: [⟨3 4 3 2 10 6 10], ⟨0 4 21 34 2 27 12]]
POTE generator: ~117/112 = 75.560
Optimal GPV sequence: Template:Val list
Badness: 0.018705
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
Mapping: [⟨3 4 3 2 10 6 10 5], ⟨0 4 21 34 2 27 12 41]]
POTE generator: ~95/91 = 75.560
Optimal GPV sequence: Template:Val list
Badness: 0.013902
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215
Mapping: [⟨3 4 3 2 10 6 10 5 13], ⟨0 4 21 34 2 27 12 41 3]]
POTE generator: ~24/23 = 75.548
Optimal GPV sequence: Template:Val list
Badness: 0.014915
Akjayland
Akjayland tempers out the akjaysma in addition to landscape comma, and thereby features a period of 1\21.
Subgroup: 2.3.5.7
Comma list: 250047/250000, [43 -1 -13 -4⟩
Mapping: [⟨21 1 38 102], ⟨0 3 1 -4]]
Mapping generators: ~1323/1280, ~131072/91875
Optimal tuning (CTE): ~131072/91875 = 614.9354
Optimal GPV sequence: Template:Val list
Badness: 0.0309
Vasca
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the [95 0 0 0 0 0 0 0 -21⟩, and sets a stack of 21 23/16's equal with 11 octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 102487/102400, [39 -4 -11 -5 2⟩
Mapping: [⟨21 4 39 98 58], ⟨0 6 2 -8 3 3]]
Mapping generators: ~1323/1280, ~6615/5632
Optimal tuning (CTE): ~6615/5632 = 278.8998
Optimal GPV sequence: Template:Val list
Badness: 0.0949
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
Mapping: [⟨21 4 39 98 58 107], ⟨0 6 2 -8 3 -6]]
Optimal tuning (CTE): ~168/143 = 278.9058
Optimal GPV sequence: Template:Val list
Badness: 0.0551
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
Mapping: [⟨21 4 39 98 58 107 120], ⟨0 6 2 -8 3 -6 -7]]
Optimal tuning (CTE): ~168/143 = 278.9036
Optimal GPV sequence: Template:Val list
Badness: 0.0319
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
Mapping: [⟨21 4 39 98 58 107 120 16], ⟨0 6 2 -8 3 -6 -7 15]]
Optimal tuning (CTE): ~168/143 = 278.9036
Optimal GPV sequence: Template:Val list
Badness: 0.0270
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
Mapping: [⟨21 34 49 58 73 77 85 91 95], ⟨0 -6 -2 8 -3 6 7 -15 0]]
Optimal tuning (CTE): ~168/143 = 278.8971
Optimal GPV sequence: Template:Val list
Badness: 0.0199
Chromium
Chromium is described as the 72 & 624 temperament, and named after the 24th element for being period 24.
Subgroup: 2.3.5.7
Comma list: 250047/250000, 49589822592/49433168575
Mapping: [⟨24 1 -6 18], ⟨0 3 5 4]]
Mapping generators: ~250/243, ~10/7
Optimal tuning (CTE): ~10/7 = 617.2710
Badness: 0.139
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 46656/46585, 250047/250000
Mapping: [⟨24 1 -6 18 46], ⟨0 3 5 4 3]]
Optimal tuning (CTE): ~10/7 = 617.2597
Optimal GPV sequence: Template:Val list
Badness: 0.0398
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325
Mapping: [⟨24 1 -6 18 46 -47 -13], ⟨0 3 5 4 3 11]]
Optimal tuning (CTE): ~10/7 = 617.2869
Optimal GPV sequence: Template:Val list
Badness: 0.0293
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011
Mapping: [⟨24 1 -6 18 46 -47 -13], ⟨0 3 5 4 3 11 9]]
Optimal tuning (CTE): ~10/7 = 617.2732
Optimal GPV sequence: Template:Val list
Badness: 0.0209