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 tunings:

• CTE: ~4096/3645 = 1\6, ~3/2 = 702.2347
• POTE: ~4096/3645 = 1\6, ~3/2 = 702.212

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 tunings:

• CTE: ~63/50 = 1\3, ~1024/945 = 137.772
• 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 tunings:

• CTE: ~63/50 = 1\3, ~693/640 = 137.773
• POTE: ~63/50 = 1\3, ~693/640 = 137.771

Optimal ET sequence: 87, 183, 270, 1263, 1533, 1803c

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 tunings:

• CTE: ~63/50 = 1\3, ~13/12 = 137.777
• 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 tunings:

• CTE: ~63/50 = 1\3, ~13/12 = 137.758
• POTE: ~63/50 = 1\3, ~13/12 = 137.761

Optimal ET sequence: 87, 183, 270, 453

Badness: 0.0171

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1729/1728, 3025/3024

Mapping: [⟨3 2 8 16 9 8 4 0], ⟨0 8 -3 -22 4 9 24 37]]

Optimal tunings:

• CTE: ~63/50 = 1\3, ~13/12 = 137.763
• POTE: ~63/50 = 1\3, ~13/12 = 137.767

Optimal ET sequence: 87, 183, 270

Badness: 0.0153

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 250/243 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: 10th-octave temperaments § Neon

See also: 60th-octave temperaments § Neodymium

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-53 -12 2 24⟩

Mapping: [⟨30 2 15 66], ⟨0 5 6 2]]

Optimal tunings:

• CTE: ~53747712/52521875 = 1\30, ~216/175 = 364.3896 (~[21 3 1 -10⟩ = 4.3896)
• CWE: ~53747712/52521875 = 1\30, ~216/175 = 364.3879 (~[21 3 1 -10⟩ = 4.3879)

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650

Badness: 0.0742

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250, [-27 -6 4 6 3⟩

Mapping: [⟨30 2 15 66 122], ⟨0 5 6 2 -2]]

Optimal tunings:

• CTE: ~18865/18432 = 1\30, ~216/175 = 364.3886 (~385/384 = 4.3886)
• CWE: ~18865/18432 = 1\30, ~216/175 = 364.3849 (~385/384 = 4.3849)

Optimal ET sequence: 270, 1110, 1380, 1650, 1920, 2190

Badness: 0.0220

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608

Mapping: [⟨30 2 15 66 122 193], ⟨0 5 6 2 -2 -9]]

Optimal tunings:

• CTE: ~351/343 = 1\30, ~216/175 = 364.3879 (~385/384 = 4.3879)
• CWE: ~351/343 = 1\30, ~216/175 = 364.3867 (~385/384 = 4.3867)

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de

Badness: 0.0155

2.3.5.7.11.13.19 subgroup (neozinc)

Subgroup: 2.3.5.7.11.13.19

Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375

Mapping: [⟨30 2 15 66 122 193 91], ⟨0 5 6 2 -2 -9 4]]

Optimal tunings:

• CTE: ~175/171 = 1\30, ~216/175 = 364.3876 (~400/399 = 4.3876)
• CWE: ~175/171 = 1\30, ~216/175 = 364.3864 (~400/399 = 4.3864)

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650, 6840e

Badness: 0.00921