Landscape microtemperaments
Landscape microtemperaments are rank-2 temperaments tempering out the landscape comma ([-4 6 -6 3⟩ = 250047/250000).
Landscape rank-2 temperaments discussed elsewhere are:
- Augene (+64/63 or 126/125) → Augmented family
- Compton (+225/224) → Compton family
- Tritikleismic (+1029/1024) → Kleismic family
- Trisensory (+1728/1715) → Sensipent family
- Ennealimmal (+2401/2400 or 4375/4374) → Ragismic microtemperaments
- Misty (+3136/3125 or 5120/5103) → Misty family
- Nessafof (+6144/6125) → Porwell temperaments
- Chromat (10976/10935) → Hemimage temperaments
- Term (+32805/32768) → Schismatic family
- Caleb (+33075/32768) → Mirwomo temperaments
- Mutt (+65625/65536) → Horwell temperaments
- Triquart (+117649/116640) → Quartonic family
- Stearnscape (+118098/117649) → Stearnsmic clan
- Tritricot (+[35 -23 -3 3⟩) → Tricot family
- Aemilic (+[-84 53⟩) → 159th-octave temperaments
Considered below are sextile, septichrome, pnict, domain, avicenna, terture, akjayland and terture.
Sextile
Sextile tempers out the garischisma with a 1/6-octave period and is the 12 & 270 temperament.
Subgroup: 2.3.5.7
Comma list: 250047/250000, 33554432/33480783
Mapping: [⟨6 0 71 150], ⟨0 1 -6 -14]]
- mapping generators: ~4096/3645, ~3
Wedgie: ⟨⟨6 -36 -84 -71 -150 -94]]
Optimal ET sequence: 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc
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]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 702.2225
- POTE: ~55/49 = 1\6, ~3/2 = 702.202
Optimal ET sequence: 12, 246dee, 258e, 270, 822, 1092, 1362c
Badness: 0.029677
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647
Mapping: [⟨6 0 71 150 230 279], ⟨0 1 -6 -14 -22 -27]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 702.2141
- CWE: ~55/49 = 1\6, ~3/2 = 702.2001
Optimal ET sequence: 12f, 258ef, 270, 552, 822, 1092, 1914cde
Badness: 0.0191
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735
Mapping: [⟨6 0 71 150 230 279 -4], ⟨0 1 -6 -14 -22 -27 3]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 702.2117
- CWE: ~55/49 = 1\6, ~3/2 = 702.1869
Optimal ET sequence: 12f, 270, 552g
Badness: 0.0209
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 936/935, 1216/1215, 1701/1700, 1716/1715, 2376/2375, 4200/4199
Mapping: [⟨6 0 71 150 230 279 -4 35], ⟨0 1 -6 -14 -22 -27 3 -1]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 702.2118
- CWE: ~55/49 = 1\6, ~3/2 = 702.1890
Optimal ET sequence: 12f, 270, 552g, 822gg
Badness: 0.0156
Sextilia
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412
Mapping: [⟨6 0 71 150 230 -149], ⟨0 1 -6 -14 -22 18]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 702.2296
- CWE: ~55/49 = 1\6, ~3/2 = 702.2285
Optimal ET sequence: 12, 258e, 270
Badness: 0.0391
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455
Mapping: [⟨6 0 71 150 230 -149 -4], ⟨0 1 -6 -14 -22 18 3]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 702.2264
- CWE: ~55/49 = 1\6, ~3/2 = 702.2207
Optimal ET sequence: 12, 258e, 270
Badness: 0.0384
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907
Mapping: [⟨6 0 71 150 230 -149 -4 35], ⟨0 1 -6 -14 -22 18 3 -1]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 702.2266
- CWE: ~55/49 = 1\6, ~3/2 = 702.2208
Optimal ET sequence: 12, 258e, 270
Badness: 0.0252
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~243/224 = 140.367
Optimal ET sequence: 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc
Badness: 0.016814
Semiseptichrome
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 151263/151250, 234375/234256
Mapping: [⟨6 1 -15 -24 -32], ⟨0 5 17 24 31]]
- mapping generators: ~55/49, ~375/308
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3687 (~1760/1701 = 59.6313)
Optimal ET sequence: 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc
Badness: 0.0194
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184
Mapping: [⟨6 1 -15 -24 -32 -68], ⟨0 5 17 24 31 53]]
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3778 (~121/117 = 59.6222)
Optimal ET sequence: 282, 342f, 624
Badness: 0.0397
17-limit
Subgroup: 2.3.5.7.11.13
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952
Mapping: [⟨6 1 -15 -24 -32 -68 -1], ⟨0 5 17 24 31 53 15]]
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3763 (~88/85 = 59.6237)
Optimal ET sequence: 282, 342f, 624
Badness: 0.0272
19-limit
Subgroup: 2.3.5.7.11.13
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
Mapping: [⟨6 1 -15 -24 -32 -68 -1 34], ⟨0 5 17 24 31 53 15 -5]]
Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3767 (~88/85 = 59.6233)
Optimal ET sequence: 282, 342f, 624
Badness: 0.0222
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375
Mapping: [⟨6 1 -15 -24 -32 -68 -1 34 -12], ⟨0 5 17 24 31 53 15 -5 23]]
Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3757 (~88/85 = 59.6243)
Optimal ET sequence: 282, 342f, 624
Badness: 0.0160
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~192/175 = 161.399
Optimal ET sequence: 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd
Badness: 0.045660
Domain
- See also: Minortonic family
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~10/9 = 182.467
Optimal ET sequence: 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~1024/945 = 137.768
Optimal ET sequence: 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~693/640 = 137.771
Optimal ET sequence: 87, 183, 270, 1263, 1533, 1803c, 2073c
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal ET sequence: 87, 183, 270
Badness: 0.015557
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 1001/1000, 3025/3024, 4096/4095
Mapping: [⟨3 2 8 16 9 8 4], ⟨0 8 -3 -22 4 9 24]]
Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal ET sequence: 87, 183, 270
Badness: 0.015557
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 3025/3024, 4096/4095
Mapping: [⟨3 2 8 16 9 8 4 0], ⟨0 8 -3 -22 4 9 24 37]]
Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal ET sequence: 87, 183, 270
Badness: 0.015557
Terture
- See also: Vulture family
Subgroup: 2.3.5.7
Comma list: 250047/250000, 359661568/358722675
Mapping: [⟨3 4 3 2], ⟨0 4 21 34]]
Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.555
Optimal ET sequence: 111, 159, 270
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550
Optimal ET sequence: 111, 159, 270, 1239, 1509, 1779, 2049, 2319
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553
Optimal ET sequence: 111, 159, 270
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560
Optimal ET sequence: 111, 159, 270
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560
Optimal ET sequence: 111, 159, 270
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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548
Optimal ET sequence: 111, 159, 270
Badness: 0.014915
Akjayland
- See also: 21st-octave temperaments
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): ~1323/1280 = 1\21, ~131072/91875 = 614.9354
Optimal ET sequence: 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c
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 twenty-one 23/16's equal with eleven 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): ~1323/1280 = 1\21, ~6615/5632 = 278.8998
Optimal ET sequence: 168, 357, 525, 882, 1407, 2289e
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): ~336/325 = 1\21, ~168/143 = 278.9058
Optimal ET sequence: 168, 357, 525, 882
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): ~336/325 = 1\21, ~168/143 = 278.9036
Optimal ET sequence: 168, 357, 525, 882
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): ~336/325 = 1\21, ~168/143 = 278.9036
Optimal ET sequence: 168h, 357, 525, 882, 1407
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 4 39 98 58 107 120 16 95], ⟨0 -6 -2 8 -3 6 7 -15 0]]
Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.8971
Optimal ET sequence: 168h, 357, 525, 882, 1407
Badness: 0.0199
Magnesium
- For the 5-limit version, see 12th-octave temperaments#Magnesium (5-limit).
Magnesium is named after element 12 for being period 12, however, it's not an extension of the atomic – the associated comma is [-157 -24 84⟩ in the 5-limit and the 7 generators together with 12edo major second reach the just perfect fifth, 3/2.
Subgroup: 2.3.5.7
Comma list: 250047/250000, [-59 2 18 5⟩
Mapping: [⟨12 2 23 58], ⟨0 7 2 -10]]
- mapping generators: ~138915/131072, 3145728/2734375
Optimal tuning (CTE): ~138915/131072 = 1\12, ~3145728/2734375 = 243.130
Optimal ET sequence: 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd
Badness: 0.0964
Chromium
- For the 5-limit version, see 24th-octave temperaments#Chromium (5-limit).
Chromium is defined by associating the porcupine comma 256/243 to the 24th of an octave. 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): ~250/243 = 1\24, ~10/7 = 617.2710
Optimal ET sequence: 72, …, 480, 552, 624, 1320, 1944d, 3264d
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): ~250/243 = 1\24, ~10/7 = 617.2597
Optimal ET sequence: 72, …, 480, 552, 624
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): ~250/243 = 1\24, ~10/7 = 617.2869
Optimal ET sequence: 72, …, 480f, 552, 624, 1176de, 1800cdee
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): ~35/34 = 1\24, ~10/7 = 617.2732
Optimal ET sequence: 72, …, 480fgg, 552g, 624
Badness: 0.0209
Zinc
- See also: 30th-octave temperaments#Zinc
Subgroup: 2.3.5.7
Comma list: 250047/250000, [-45 -24 14 18⟩
Mapping: [⟨30 2 15 66], ⟨0 5 6 2]]
Optimal tuning (CTE): ~53747712/52521875 = 1\30, ~216/175 = 364.389
Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650
Badness: 0.0742
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 250047/250000, 97869261875/97844723712
Mapping: [⟨30 2 15 66 122], ⟨0 5 6 2 -2]]
Optimal tuning (CTE): ~18865/18432 = 1\30, ~216/175 = 364.389
Optimal ET sequence: 270, 1110, 1380, 1650, 1920, 2190
Badness: 0.0220
13-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 123201/123200, 250047/250000, 1990656/1990625
Mapping: [⟨30 2 15 66 122 193], ⟨0 5 6 2 -2 -9]]
Optimal tuning (CTE): ~351/343 = 1\30, ~216/175 = 364.389
Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de
Badness: 0.0150