Landscape microtemperaments: Difference between revisions

This is a collection of [[rank-2 temperament|rank-2]] '''landscape microtemperaments''', which [[tempering out|temper out]] the [[landscape comma]] ({{monzo|legend=1| -4 6 -6 3 }}, [[ratio]]: 250047/250000). For the [[rank-3 temperament]], see [[Landscape family #Landscape]].  

Temperaments discussed elsewhere are:  

* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]

* ''[[Sextile]]'' (+33554432/33480783) → [[Garischismic clan #Sextile|Garischismic clan]]

* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]

* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]

* ''[[Chromat]]'' (10976/10935) → [[Hemimage temperaments #Chromat|Hemimage temperaments]]

* [[Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]

* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]

* ''[[Domain (temperament)|Domain]]'' (+645700815/645657712) → [[Minortonic family #Domain|Minortonic family]]

* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments #Aemilic|159th-octave temperaments]]

Considered below are septichrome, akjayland, pnict, atomic, slendscape, avicenna, zinc, magnesium, poe, and chromium in the order of increasing [[badness]].

== Septichrome ==

[[Comma list]]: 250047/250000, 2460375/2458624

{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}

[[Optimal tuning]]s:  

* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}

{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}

[[Badness]] (Sintel): 0.426

=== Semiseptichrome ===

Comma list: 9801/9800, 151263/151250, 234375/234256

Mapping: {{mapping| 6 1 -15 -24 -32 | 0 5 17 24 31 }}

Optimal tunings:

* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})

* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})

{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}

Badness (Sintel): 0.642

==== 13-limit ====

Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184

Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}

Optimal tunings:

* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})

* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})

{{Optimal ET sequence|legend=0| 282, 342f, 624 }}

Badness (Sintel): 1.64

Subgroup: 2.3.5.7.11.13

Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952

Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}

Optimal tunings:

* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})

* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})

{{Optimal ET sequence|legend=0| 282, 342f, 624 }}

Badness (Sintel): 1.39

Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408

 

 

* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{c}})

 

 

 

 

Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375

 

 

* WE: ~55/49 = 199.9829{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})

 

 

 

 

 

 

 

 

 

 

 

 

 

Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}

Optimal tunings:

* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}

{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}

Badness (Sintel): 3.14

==== 13-limit ====

Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683

Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}

Optimal tunings:

* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}

{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}

Badness (Sintel): 2.28

Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625

Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}

Optimal tunings:

* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}

{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}

Badness (Sintel): 1.62

Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095

Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}

Optimal tunings:

* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}

{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}

Badness (Sintel): 1.64

==== 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: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 6 2 -8 3 -6 -7 15 0 }}

Optimal tunings:

* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8980{{c}}

{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}

Badness (Sintel): 1.43

{{Mapping|legend=1| 3 -3 1 12 | 0 13 10 -6 }}

* [[WE]]: ~63/50 = 400.0312{{c}}, ~147/128 = 238.6196{{c}} (~192/175 = 161.4116{{c}})

: [[error map]]: {{val| +0.094 +0.006 -0.087 -0.169 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~147/128 = 238.6038{{c}} (~192/175 = 161.3962{{c}})

: error map: {{val| 0.000 -0.106 -0.276 -0.449 }}

{{Optimal ET sequence|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}

[[Badness]] (Sintel): 1.16

: ''For the 5-limit version, see [[Very high accuracy temperaments #Atomic]].''

Atomic tempers out the [[atom]], {{monzo| 161 -84 -12 }}, and in the [[7-limit]] the [[nommisma]], {{monzo| -55 30 2 1 }}, so that a stack of two [[schisma]]s gives the [[garischisma]], from which intervals of [[7/1|7]] can be derived. It may be described as the {{nowrap| 12 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid monocot.

[[Comma list]]: 250047/250000, {{monzo| -55 30 2 1 }}

 

 

* [[WE]]: ~30375/28672 = 99.999866{{c}}, ~3/2 = 701.948670{{c}} (~32805/32768 = 1.949605{{c}})

 

 

 

 

 

 

 

 

==== Minutes ====

Minutes (600e & 2460) splits the 1/12-octave period into five 1/60-octave parts.

Comma list: 9801/9800, 151263/151250, 371293/371250, 184549376/184528125

Mapping: {{mapping| 60 0 805 1690 2585 1173 | 0 1 7 -16 -25 -10 }}

Optimal tunings:  

* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.948111{{c}}

 

 

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 12376/12375, 28561/28560, 151263/151250, 11275335/11275264

Mapping: {{mapping| 60 0 805 1690 2585 1173 1957 | 0 1 7 -16 -25 -10 -18 }}

* WE: ~2704/2673 = 19.999974{{c}}, ~3/2 = 701.946956{{c}}

* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.947926{{c}}

{{Optimal ET sequence|legend=0| 600e, …, 1860, 2460, 6780, 9240 }}

Badness (Sintel): 1.67

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 12376/12375, 12636/12635, 23409/23408, 28561/28560, 151263/151250

Mapping: {{mapping| 60 60 385 730 1085 573 877 -61 | 0 1 7 -16 -25 -10 -18 9 }}

Optimal tunings:  

* WE: ~2704/2673 = 19.999962{{c}}, ~3/2 = 701.946354{{c}}

* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.947751{{c}}

{{Optimal ET sequence|legend=0| 600e, 1860, 2460, 4320, 6780, 9240 }}

Badness (Sintel): 1.50

Hafnium (1224 & 4320), named after the 72nd element, splits the 1/12-octave period into six 1/72-octave parts. Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these primes are also prescribed. In the add-37 version, [[37/22]] is mapped to exact 3\4.  

Comma list: 9801/9800, 151263/151250, 184549376/184528125, 308915776/308828625

Mapping: {{mapping| 72 0 966 2028 3102 2777 | 0 1 -7 -16 -25 -22 }}

Optimal tunings:

* WE: ~105/104 = 16.666629{{c}}, ~3/2 = 701.945668{{c}}

* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947388{{c}}

{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}

Badness (Sintel): 4.76

===== 17-limit =====

Comma list: 9801/9800, 12376/12375, 151263/151250, 1713660/1713481, 97144749/97140736

Mapping: {{mapping| 72 0 966 2028 3102 2777 979 | 0 1 -7 -16 -25 -22 -6 }}

 

* WE: ~105/104 = 16.666615{{c}}, ~3/2 = 701.944968{{c}}

* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947293{{c}}

 

{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}

 

 

 

 

Subgroup-val mapping: {{mapping| 72 0 966 2028 3102 2777 979 -328 | 0 1 -7 -16 -25 -22 -6 6 }}

Optimal tunings:

* WE: ~105/104 = 16.666643{{c}}, ~3/2 = 701.946542{{c}}

* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947581{{c}}

{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544 }}

Badness (Sintel): 2.36

===== 2.3.5.7.11.13.17.31.37 subgroup =====

Subgroup: 2.3.5.7.11.13.17.31.37

Comma list: 9801/9800, 10881/10880, 12376/12375, 16576/16575, 57629/57624, 93093/93092, 179712/179707

Subgroup-val mapping: {{mapping| 72 0 966 2028 3102 2777 979 -328 3228 | 0 1 -7 -16 -25 -22 -6 6 -25 }}

Optimal tunings:

* WE: ~105/104 = 16.666648{{c}}, ~3/2 = 701.946549{{c}}

* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947386{{c}}

{{Optimal ET sequence|legend=0| 1224, 3096el, 4320, 5544 }}

Badness (Sintel): 1.98

== Slendscape ==

[[Comma list]]: 250047/250000, 12884901888/12867859375

{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}

[[Optimal tuning]]s:

* [[CWE]]: ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4962{{c}}

: error map: {{val| 0.000 +0.030 +0.175 +0.182 }}

{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}

[[Badness]] (Sintel): 1.47

Comma list: 3025/3024, 102487/102400, 180224/180075

Mapping: {{mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}

Optimal tunings:

* CWE: ~22/21 = 80.0000{{c}}, ~968/735 = 475.4955{{c}}

{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}

Badness (Sintel): 0.868

Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625

 

 

 

 

 

 

 

 

* [[WE]]: ~63/50 = 399.9681{{c}}, ~1024/945 = 137.7570{{c}}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~1024/945 = 137.7689{{c}}

 

[[Badness]] (Sintel): 1.57

 

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 102487/102400

Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}

Optimal tunings:  

* CWE: ~63/50 = 400.0000{{c}}, ~693/640 = 137.7716{{c}}

{{Optimal ET sequence|legend=0| 87, 183, 270, 1263, 1533, 1803c }}

Badness (Sintel): 0.763

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095

Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}

Optimal tunings:  

* CWE: ~63/50 = 400.0000{{c}}, ~13/12 = 137.7770{{c}}

{{Optimal ET sequence|legend=0| 87, 183, 270 }}

Badness (Sintel): 0.643

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095

Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}

Optimal tunings:  

* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7608{{c}}

{{Optimal ET sequence|legend=0| 87, 183, 270, 453 }}

Badness (Sintel): 0.869

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728

Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}

Optimal tunings:  

* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7664{{c}}

{{Optimal ET sequence|legend=0| 87, 183, 270 }}

Badness (Sintel): 0.928

Zinc maybe described as the {{nowrap| 270 & 2190 }} temperament. It was named by [[Eliora]] in 2023 after the 30th element for having a 30th-octave period.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}

: mapping generators: ~53747712/52521875, ~216/175

: [[error map]]: {{val| +0.007 -0.009 +0.024 -0.032 }}

* [[CWE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3879{{c}})

: error map: {{val| 0.000 -0.015 +0.014 -0.050 }}

[[Badness]] (Sintel): 1.88

Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}

Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}

Optimal tunings:  

* WE: ~18865/18432 = 40.0005{{c}}, ~216/175 = 364.3881{{c}} (~385/384 = 4.3837{{c}})

{{Optimal ET sequence|legend=0| 270, 1110, 1380, 1650, 1920, 2190 }}

Subgroup: 2.3.5.7.11.13

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

Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}

Optimal tunings:  

* WE: ~351/343 = 40.0003{{c}}, ~216/175 = 364.3894{{c}} (~385/384 = 4.3865{{c}})

* CWE: ~351/343 = 40.0000{{c}}, ~216/175 = 364.3867{{c}} (~385/384 = 4.3867{{c}})

{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}

Subgroup: 2.3.5.7.11.13.19

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

Mapping: {{mapping| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}

Optimal tunings:  

* WE: ~175/171 = 40.0002{{c}}, ~216/175 = 364.3885{{c}} (~400/399 = 4.3862{{c}})

* CWE: ~175/171 = 40.0000{{c}}, ~216/175 = 364.3864{{c}} (~400/399 = 4.3864{{c}})

{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}

Neodymium (540 & 1920) splits the period into 1/60-octave halves for prime 17.  

Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 10648/10647, 31213/31212, 105644/105625, 196625/196608

Mapping: {{mapping| 60 4 30 132 244 386 391 | 0 5 6 2 -2 -9 -8 }}

Optimal tunings:

* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3864{{c}}

{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}

Badness (Sintel): 1.23

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 23409/23408, 89376/89375

Mapping: {{mapping| 60 4 30 132 244 386 391 182 | 0 5 6 2 -2 -9 -8 4 }}

Optimal tunings:

* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3860{{c}}

{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}

Badness (Sintel): 1.02

: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].

Magnesium is named by [[Eliora]] in 2023 after the 12th element for having a 1/12-octave period; however, it is not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit, and 7 generator steps together with two [[12edo]] semitones reach the [[3/1|3rd]] [[harmonic]]. It may be described as the {{nowrap| 84 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid gamma-heptacot.  

[[Optimal tuning]]s:  

* [[WE]]: ~138915/131072 = 100.0021{{c}}, ~3145728/2734375 = 243.1333{{c}}

* [[CWE]]: ~138915/131072 = 100.0000{{c}}, ~3145728/2734375 = 243.1285{{c}}

Badness (Sintel): 2.44

== Poe ==

Named by [[Tristan Bay]] in 2025, poe may be described as the {{nowrap| 60 & 270 }} temperament.  

[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}

{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2240/2187 = 39.9982{{c}}, ~3/2 = 702.1533{{c}}

{{Optimal ET sequence|legend=1| 60, 150cd, 210, 270, 1950, 2220, 2490, 2760b, 3030bc, 3300bc }}

[[Badness]] (Sintel): 2.90

Comma list: 9801/9800, 19712/19683, 151263/151250

Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}

Optimal tunings:

* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.2129{{c}}

{{Optimal ET sequence|legend=0| 60e, …, 210e, 270 }}

Badness (Sintel): 1.31

Comma list: 1001/1000, 4225/4224, 4459/4455, 19712/19683

 

Optimal tunings:  

* WE: ~45/44 = 40.0008{{c}}, ~3/2 = 702.1778{{c}}

* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.1699{{c}}

{{Optimal ET sequence|legend=0| 60e, 210e, 270, 1410ef, 1680ef }}

Badness (Sintel): 1.19

: ''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, and may be described as the {{nowrap| 72 & 624 }} temperament. It was named by [[Eliora]] in 2022 after the 24th element for having a 24th-octave period.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 250047/250000, 49589822592/49433168575

: mapping generators: ~250/243, ~10/7

* [[WE]]: ~250/243 = 49.9992{{c}}, ~10/7 = 617.2714{{c}}

: [[error map]]: {{val| -0.019 -0.142 +0.048 +0.246 }}

* [[CWE]]: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2762{{c}}

: error map: {{val| 0.000 -0.126 +0.067 +0.279 }}

[[Badness]] (Sintel): 3.52

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 46656/46585, 151263/151250

Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}

Optimal tunings:

* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2823{{c}}

{{Optimal ET sequence|legend=0| 72, …, 480, 552, 624 }}

Badness (Sintel): 1.32

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325

Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}

Optimal tunings:

* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.3161{{c}}

{{Optimal ET sequence|legend=0| 72, …, 480f, 552, 624, 1176de, 1800cdee }}

Badness (Sintel): 1.21

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011

Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}

Optimal tunings:

* CWE: ~35/34 = 50.0000{{c}}, ~10/7 = 617.3015{{c}}

{{Optimal ET sequence|legend=0| 72, …, 480fgg, 552g, 624 }}

Badness (Sintel): 1.06