Very high accuracy temperaments: Difference between revisions

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Line 18:

* [[CWE]]: ~94921875/67108864 = 600.000000{{c}}, ~1125/1024 = 162.742801{{c}}

: error map: {{val| 0.0000 -0.0126 -0.0277 }}

~~<!-- * [[CTE]]: ~94921875/67108864 = 600.000000{{c}}, ~1125/1024 = 162.743279{{c}}~~

* [[POTE]]: ~94921875/67108864 = 600.000000{{c}}, ~1125/1024 = 162.742626{{c}} -->

{{Optimal ET sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.805278{{c}}

: error map: {{val| 0.0000 +0.0086 +0.0230 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.805146{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.805332{{c}} -->

{{Optimal ET sequence|legend=1| 9, …, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}

[[Badness]] (Sintel): 1.47

[[Badness]] (Sintel): 1.473

== Gaster ==

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| -18 19 -5 }} = 605.365482{{c}}

: error map: {{val| 0.0000 -0.0108 +0.0010 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| -18 19 -5 }} = 605.365544{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| -18 19 -5 }} = 605.365435{{c}} -->

{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 2.077

== Whoosh ==

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~625/432 = 639.453117{{c}}

: error map: {{val| 0.0000 -0.0022 +0.0142 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~625/432 = 639.453048{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~625/432 = 639.453174{{c}} -->

{{Optimal ET sequence|legend=1| 15, …, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 27 -18 1 }} = 950.981535{{c}}

: error map: {{val| 0.0000 +0.0081 +0.0031 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 27 -18 1 }} = 950.981427{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 27 -18 1 }} = 950.981594{{c}} -->

{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~5/3 = 844.352090{{c}}

: error map: {{val| 0.0000 +0.0016 -0.0050 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~5/3 = 884.352101{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~5/3 = 884.352081{{c}} -->

{{Optimal ET sequence|legend=1| 19, …, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 0.981

=== Hemiegads ===

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* WE: ~2 = 1200.000308{{c}}, ~67108864/36756909 = 1042.176313{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~67108864/36756909 = 1042.176046{{c}}

~~~~

{{Optimal ET sequence|legend=0| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}

Badness (Sintel): ~~TBD~~

Badness (Sintel): 0.313

=== Catabolic ===

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* WE: ~2 = 1200.000308{{c}}, ~5/3 = 884.354064{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~5/3 = 884.351835{{c}}

~~<!-- * CTE: ~2 = 1200.0031{{c}}, ~5/3 = 884.3510{{c}}~~

* POTE: ~2 = 1200.0000{{c}}, ~5/3 = 884.3518{{c}} -->

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.567889{{c}}

: error map: {{val| 0.0000 -0.0046 -0.0045 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.567861{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.567899{{c}} -->

{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 0.654

=== 7-limit ===

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* WE: ~2 = 1200.001970{{c}}, ~8388608/7381125 = 221.568395{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568026{{c}}

~~<!-- * CTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568013{{c}}~~

* POTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568031{{c}} -->

{{Optimal ET sequence|legend=0| 65d, …, 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}

Badness (Sintel): ~~TBD~~

Badness (Sintel): 0.600

=== 11-limit ===

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* WE: ~2 = 1200.003854{{c}}, ~8388608/7381125 = 221.568850{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568133{{c}}

~~<!-- * CTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568116{{c}}~~

* POTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568139{{c}} -->

{{Optimal ET sequence|legend=0| 65dee, …, 612, 2513, 3125, 3737, 8086, 19909d }}

Badness (Sintel): ~~TBD~~

Badness (Sintel): 1.211

== Senior ==

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 7 -23 13 }} = 877.198647{{c}}

: error map: {{val| 0.0000 -0.0024 -0.0024 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 7 -23 13 }} = 877.198651{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 7 -23 13 }} = 877.198642{{c}} -->

{{Optimal ET sequence|legend=1| 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 0.498

== Gross ==

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* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 46 -7 -15 }} = 91.530995{{c}}

: error map: {{val| 0.0000 +0.0018 +0.0044 }}

~~<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 46 -7 -15 }} = 91.530979{{c}}~~

* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 46 -7 -15 }} = 91.531001{{c}} -->

{{Optimal ET sequence|legend=1| 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 1.086

; Music

* ''Gross temperament EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/02Fz6IIJ2x9k7bEvFylZ2A Spotify] | [https://francium223.bandcamp.com/album/gross-temperament-ep Bandcamp] | [https://youtube.com/playlist?list=PLLZE7hMjEXRZy_JWYbZxbsxPsXahiEJA2&si=RttlCFpkeN4Rl7Tj YouTube] – 2-piece extended play

== ~~Quectismic~~ ==

== Septarium ==

Septarium splits the octave in seven, and tempers out the [[septarium comma]], {{monzo| 73 77 -84 }}. It can be described as {{nowrap| 441 & 2513 }} temperament, which tempers out 78125000/78121827 and {{monzo| 47 -7 -7 -7 }} ([[akjaysma]]) in the 7-limit.

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| ~~554~~ -~~351 1~~ }}

[[Comma list]]: {{monzo| 73 77 -84 }}

: mapping generators: ~2, ~3

: mapping generators: ~{{monzo| 21 22 -24 }}, ~{{monzo| -20 -21 23 }}

[[Optimal tuning]]s:

* [[WE]]: ~~~2 = 1199.999902567~~{{c}}~~, ~3/2~~ = ~~701~~.~~955253373~~{{c}}

* [[WE]]: ~{{monzo| 21 22 -24 }} = 171.429{{c}}, ~{{monzo| -20 -21 23 }} = 144.210{{c}}

~~: [[error map]]:~~ {{~~val~~| -~~0.000097 +0.000155~~ -0.~~000001~~ }}

* [[CWE]]: ~{{monzo| 21 22 -24 }} = 171.429{{c}}, ~{{monzo| -20 -21 23 }} = 144.210{{c}}

* [[CWE]]: ~~~2 = 1200.000000000~~{{c}}~~, ~3/2~~ = ~~701~~.~~955310152~~{{c}}

~~: error map:~~ {{~~val~~| ~~0.000000 +0.000309 +0.000149 }}~~

{{Optimal ET sequence|legend=1| ~~306~~, ~~359c~~, ~~665~~, ~~2301~~, ~~2966~~, ~~3631~~, ~~4296~~, ~~46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551~~ }}

{{Optimal ET sequence|legend=1| 133, 308, 441, 1631, 2072, 2513, 5467, 7980 }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 1.395

=== ~~Nanoquectismic~~ ===

=== 7-limit ===

Subgroup: 2.3.5.7

Comma list: ~~43046721~~/~~43025920~~, ~~{{monzo| 82 -19 -3 -16 }}~~

Comma list: 78125000/78121827, 140737488355328/140710042265625

Mapping: {{~~mapping~~| 1 ~~0 -554 109~~ | 0 ~~1 351~~ -67 }}

Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}

Optimal tunings:

* WE: ~2 = ~~1199~~.~~979407~~{{c}}, ~3/2 = ~~701~~.~~943437~~{{c}}

* WE: ~1157625/1048576 = 171.428{{c}}, ~2097152/1929375 = 144.211{{c}}

* CWE: ~2 = ~~1200~~.~~000000~~{{c}}, ~3/2 = ~~701~~.~~955457~~{{c}}

* CWE: ~1157625/1048576 = 171.429{{c}}, ~2097152/1929375 = 144.212{{c}}

{{Optimal ET sequence|legend=0| 133, 308, 441, 1631, 2072, 2513, 2954 }}

Badness (Sintel): 0.777

== Raider ==

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 71 -99 37 }}

: mapping generators: ~2, ~8000/6561

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.999880537{{c}}, ~8000/6561 = 343.296063356{{c}}

: [[error map]]: {{val| -0.000119 +0.000418 -0.000336 }}

* [[CWE]]: ~2 = 1200.000000000{{c}}, ~8000/6561 = 343.296095985{{c}}

: error map: {{val| 0.000000 +0.000551 -0.000211 }}

{{Optimal ET sequence|legend=1| 7, …, 381, 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}

~~{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}~~

[[Badness]] (Sintel): 0.302

~~Badness (Sintel): TBD~~

== Pirate ==

~~=== Conanoquectismic ===~~

[[Subgroup]]: 2.3.5

Subgroup: 2.3.5.7

Comma list: {{monzo| 33 -~~26 12~~ -~~7 }}, {{monzo| -32 13 17 -10~~ }}

[[Comma list]]: {{monzo| -90 -15 49 }}

~~Mapping:~~ {{~~mapping~~| 1 0 ~~-554 -945~~ | 0 ~~1 351 598~~ }}

: mapping generators: ~2, ~{{monzo| 24 4 -13 }}

Optimal ~~tunings~~:

[[Optimal tuning]]s:

* WE: ~2 = 1200.~~010750~~{{c}}, ~~~3/2~~ = ~~701~~.~~961041~~{{c}}

* [[WE]]: ~2 = 1200.000195604{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}

* CWE: ~2 = 1200.~~000000~~{{c}}, ~~~3/2 = 701.954771~~{{c}}

: [[error map]]: {{val| -0.000196 +0.000418 -0.000336 }}

~~~~

* [[CWE]]: ~2 = 1200.000000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}

: error map: {{val| 0.000000 +0.000082 -0.000574 }}

{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}

Badness (Sintel): ~~TBD~~

[[Badness]] (Sintel): 0.133

== Counterquectismic ==

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* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.954415845{{c}}

: error map: {{val| 0.000000 -0.000585 -0.000289 }}

~~~~

{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 11.971

== ~~Raider~~ ==

== Quectismic ==

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 71 -~~99 37~~ }}

[[Comma list]]: {{monzo| 554 -351 1 }}

: mapping generators: ~2, ~~~6561/4000~~

: mapping generators: ~2, ~3

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000000~~{{c}}, ~~~8000~~/~~6561~~ = ~~343~~.~~296094~~{{c}}

* [[WE]]: ~2 = 1199.999902567{{c}}, ~3/2 = 701.955253373{{c}}

* [[~~POTE~~]]: ~2 = 1200.~~000000~~{{c}}, ~~~8000~~/~~6561~~ = ~~343~~.~~296098~~{{c}}

: [[error map]]: {{val| -0.000097 +0.000155 -0.000001 }}

* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.955310152{{c}}

: error map: {{val| 0.000000 +0.000309 +0.000149 }}

{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}

~~{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}~~

[[Badness]] (Sintel): 8.736

~~[[Badness]] (Sintel)~~: ~~TBD~~

=== Nanoquectismic ===

Subgroup: 2.3.5.7

Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}

Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}

Optimal tunings:

* WE: ~2 = 1199.979407{{c}}, ~3/2 = 701.943437{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955457{{c}}

~~== Pirate ==~~

Badness (Sintel): 15.285

~~{{See also| Hemfiness temperaments #Piracy }}~~

[[Subgroup]]: 2.3.5

=== Conanoquectismic ===

Subgroup: 2.3.5.7

[[Comma list]]: {{monzo| -90 -~~15 49~~ }}

Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}

{{~~Mapping|legend=1~~| 1 ~~43 15 |~~ 0 -49 -~~15 }}~~

Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}

~~: Mapping generators: ~2, ~{{monzo~~| ~~-23 -4 13~~ }}

[[Optimal ~~tuning]]s~~:

Optimal tunings:

* ~~[[CTE]]~~: ~2 = 1200.~~000000~~{{c}}, ~~~{{monzo| 24 4 -13 }}~~ = ~~185~~.~~754186~~{{c}}

* WE: ~2 = 1200.010750{{c}}, ~3/2 = 701.961041{{c}}

* ~~[[POTE]]~~: ~2 = 1200.000000{{c}}, ~~~{{monzo| 24 4 -13 }}~~ = ~~185~~.~~754179~~{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954771{{c}}

{{Optimal ET sequence|legend=1| 84, ~~239~~, ~~323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 …~~ }}

[[Badness]] (Sintel): ~~TBD~~

Badness (Sintel): 54.974

== Atomic ==

~~{{See also | Kirnberger~~'~~s atom }}~~

: ''For extensions, see [[Landscape microtemperaments #Atomic]].''

~~Atomic observes the [[schisma]]~~, ~~and the 3/2 is one schisma sharp of its~~ [[~~12edo~~]] value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the [[Pythagorean comma]] is twelve schismas, and hence [[81/80]], the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.

Atomic ~~extensions discussed elsewhere include~~ [[~~minutes~~]] and [[~~72nd-octave temperaments#Hafnium~~|~~hafnium~~]].

Atomic tempers out [[Kirnberger's atom]], the microscopic interval separating eleven [[schisma]]s and a [[syntonic comma]], or twelfth schismas and a [[Pythagorean comma]]. It is a member of the [[schismic–commatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 12 }}. In this extremely accurate temperament, the [[3/2|perfect fifth]] is one [[schisma]] sharp of its [[12edo]] value.

[[Subgroup]]: 2.3.5

Line 396:

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[[Optimal tuning]]s:

* [[~~CTE~~]]: ~{{monzo| -67 35 5 }} = ~~100~~.~~000000~~{{c}}, ~3/2 = 701.~~955176~~{{c}} (~32805/32768 = 1.~~955176~~{{c}})

* [[WE]]: ~{{monzo| -67 35 5 }} = 99.999995361{{c}}, ~3/2 = 701.955129505{{c}} (~32805/32768 = 1.955161980{{c}})

* [[~~POTE~~]]: ~{{monzo| -67 35 5 }} = 100.~~000000~~{{c}}, ~3/2 = 701.~~955162~~{{c}} (~32805/32768 = 1.~~955162~~{{c}})

: [[error map]]: {{val| -0.000056 +0.000072 +0.000022 }}

* [[CWE]]: ~{{monzo| -67 35 5 }} = 100.000000000{{c}}, ~3/2 = 701.955166184{{c}} (~32805/32768 = 1.955166184{{c}})

: error map: {{val| 0.000000 +0.000165 +0.000123 }}

{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}

[[Badness]] (Sintel): ~~TBD~~

[[Badness]] (Sintel): 0.0892

~~=== 7-limit ===~~

~~Subgroup: 2.3.5.7~~

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

~~Mapping: {{mapping| 12~~ 0 ~~161 338 | 0 1 -7 -16 }}~~

~~Optimal tunings:~~

* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949917{{c}} (or ~32805/32768 = 1.949917{{c}})

* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949608{{c}} (or ~32805/32768 = 1.~~949608{{c}})~~

~~{{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}~~

== Revopentic ==

~~Badness (Sintel): 1.16~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 9801/9800, 151263/151250, 184549376/184528125~~

~~Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}~~

~~Optimal tunings:~~

* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948456{{c}} (or ~32805/32768 = 1.948456{{c}})

* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.947988{{c}} (or ~32805/32768 = 1.947988{{c}})

~~{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}~~

~~Badness (Sintel): 0.530~~

== Revopentic ~~(2.3.7)~~ ==

Subgroup: 2.3.7

Comma list: ~~11399736556781568/11398895185373143~~

Comma list: {{monzo| 47 4 -19 }}

: mapping generators: ~2, ~~~24576~~/~~16807~~

: mapping generators: ~2, ~16807/12288

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000000~~{{c}}, ~~~24576~~/~~16807~~ = ~~657~~.~~791865~~{{c}}

* [[WE]]: ~2 = 1199.998821{{c}}, ~16807/12288 = 542.207710{{c}}

* [[CWE]]: ~2 = 1200.000000{{c}}, ~~~24576~~/~~16807~~ = ~~657~~.~~791796~~{{c}}

: [[error map]]: {{val| -0.0012 -0.0003 +0.0038 }}

* [[CWE]]: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.208204{{c}}

: error map: {{val| 0.0000 +0.0009 +0.0069 }}

{{Optimal ET sequence|legend=1| 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, … }}

Line 458:

Line 438:

Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}

Mapping: {{mapping| 1 12 -~~155 5~~ | 0 -19 287 -4 }}

Mapping: {{mapping| 1 -7 132 1 | 0 19 -287 4 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000000~~{{c}}, ~~~24576~~/~~16807~~ = ~~657~~.~~792032~~{{c}}

* WE: ~2 = 1200.000110{{c}}, ~16807/12288 = 542.208018{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~~~24576~~/~~16807~~ = ~~657~~.~~792032~~{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.207968{{c}}

{{Optimal ET sequence|legend=0| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}

Badness (Sintel): ~~TBD~~

Badness (Sintel): 0.665

== Nanic ==

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Line 459:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000000~~{{c}}, ~3/2 = 701.~~957820~~{{c}}

* [[WE]]: ~2 = 1199.999111{{c}}, ~3/2 = 701.957264{{c}}

* [[~~POTE~~]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.~~957784~~{{c}}

: [[error map]]: {{val| -0.0009 +0.0014 +0.0001 }}

* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.957796{{c}}

: error map: {{val| 0.0000 +0.0028 +0.0018 }}

{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}

Line 493:

Line 475:

[[Comma list]]: {{monzo| -32 -64 71 }}

: mapping generators: ~3, ~{{monzo| -4 -9 10 }}

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~3 = 1901.~~955~~{{c}}, ~{{monzo| 5 9 -10 }} = ~~898~~.~~340~~{{c}}

* [[WE]]: ~3 = 1901.955003{{c}}, ~{{monzo| -4 -9 10 }} = 1003.615406{{c}}

* [[CWE]]: ~3 = 1901.~~955~~{{c}}, ~{{monzo| 5 9 -10 }} = ~~898~~.~~340~~{{c}}

: [[error map]]: {{val| +0.0000 +0.0000 -0.0000 }}

* [[CWE]]: ~3 = 1901.955001{{c}}, ~{{monzo| -4 -9 10 }} = 1003.615405{{c}}

: error map: {{val| 0.0000 -0.0000 -0.0000 }}

Line 504:

Line 489:

= Rank-3 temperaments =

== Akjaysmic ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: {{monzo| 47 -7 -7 -7 }}

: mapping generators: ~1157625/1048576, ~3, ~5

[[Optimal tuning]]s:

* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}

: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}

* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}

: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}

{{Optimal ET sequence|legend=1| 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}

[[Badness]] (Sintel): 2.22

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 184549376/184528125, 199297406/199290375

Mapping: {{mapping| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}

: mapping generators: ~29160/26411, ~3, ~5

Optimal tunings:

* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}

* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}

{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}

Badness (Sintel): 1.32

== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==

[[Subgroup]]: 2.3.5.7

Line 517:

Line 538:

[[Badness]] (Sintel): 1.24

== ~~Saquadtrizo~~-~~asepgu (171~~ & ~~2019~~ & ~~3566) ==~~

== Izarismic ==

This temperament is identical to the [[izar]] in the no-twos subgroup, but has an independent generator for prime 2. It can be described as {{nowrap| 41 & 49 & 130 }} temperament, and tempers out the [[izar comma]], {{monzo| 0 -11 -7 12 }}.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: ~~{{monzo| 0 -11 -7 12 }} (~~13841287201/13839609375)

[[Comma list]]: 13841287201/13839609375

: mapping generators: ~2, ~3, ~16807/10125

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}

[[Optimal tuning]]s:

* [[TE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}

* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}

{{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}

@@ Line 18: / Line 18: @@
 * [[CWE]]: ~94921875/67108864 = 600.000000{{c}}, ~1125/1024 = 162.742801{{c}}
 : error map: {{val| 0.0000 -0.0126 -0.0277 }}
-<!-- * [[CTE]]: ~94921875/67108864 = 600.000000{{c}}, ~1125/1024 = 162.743279{{c}}
-* [[POTE]]: ~94921875/67108864 = 600.000000{{c}}, ~1125/1024 = 162.742626{{c}} -->
 {{Optimal ET sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}
@@ Line 40: / Line 38: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.805278{{c}}
 : error map: {{val| 0.0000 +0.0086 +0.0230 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.805146{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.805332{{c}} -->
 {{Optimal ET sequence|legend=1| 9, …, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}
-[[Badness]] (Sintel): 1.47
+[[Badness]] (Sintel): 1.473
 == Gaster ==
@@ Line 63: / Line 59: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| -18 19 -5 }} = 605.365482{{c}}
 : error map: {{val| 0.0000 -0.0108 +0.0010 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| -18 19 -5 }} = 605.365544{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| -18 19 -5 }} = 605.365435{{c}} -->
 {{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 2.077
 == Whoosh ==
@@ Line 85: / Line 79: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~625/432 = 639.453117{{c}}
 : error map: {{val| 0.0000 -0.0022 +0.0142 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~625/432 = 639.453048{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~625/432 = 639.453174{{c}} -->
 {{Optimal ET sequence|legend=1| 15, …, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}
@@ Line 107: / Line 99: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 27 -18 1 }} = 950.981535{{c}}
 : error map: {{val| 0.0000 +0.0081 +0.0031 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 27 -18 1 }} = 950.981427{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 27 -18 1 }} = 950.981594{{c}} -->
 {{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}
@@ Line 127: / Line 117: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~5/3 = 844.352090{{c}}
 : error map: {{val| 0.0000 +0.0016 -0.0050 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~5/3 = 884.352101{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~5/3 = 884.352081{{c}} -->
 {{Optimal ET sequence|legend=1| 19, …, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 0.981
 === Hemiegads ===
@@ Line 145: / Line 133: @@
 * WE: ~2 = 1200.000308{{c}}, ~67108864/36756909 = 1042.176313{{c}}
 * CWE: ~2 = 1200.000000{{c}}, ~67108864/36756909 = 1042.176046{{c}}
-<!-- * CTE: ~2 = 1200.000000{{c}}, ~67108864/36756909 = 1042.176048{{c}} -->
 {{Optimal ET sequence|legend=0| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}
-Badness (Sintel): TBD
+Badness (Sintel): 0.313
 === Catabolic ===
@@ Line 161: / Line 148: @@
 * WE: ~2 = 1200.000308{{c}}, ~5/3 = 884.354064{{c}}
 * CWE: ~2 = 1200.000000{{c}}, ~5/3 = 884.351835{{c}}
-<!-- * CTE: ~2 = 1200.0031{{c}}, ~5/3 = 884.3510{{c}}
-* POTE: ~2 = 1200.0000{{c}}, ~5/3 = 884.3518{{c}} -->
 {{Optimal ET sequence|legend=0| 19, …, 441, 460, 901, 1342 }}
@@ Line 183: / Line 168: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.567889{{c}}
 : error map: {{val| 0.0000 -0.0046 -0.0045 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.567861{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.567899{{c}} -->
 {{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 0.654
 === 7-limit ===
@@ Line 200: / Line 183: @@
 * WE: ~2 = 1200.001970{{c}}, ~8388608/7381125 = 221.568395{{c}}
 * CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568026{{c}}
-<!-- * CTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568013{{c}}
-* POTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568031{{c}} -->
 {{Optimal ET sequence|legend=0| 65d, …, 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}
-Badness (Sintel): TBD
+Badness (Sintel): 0.600
 === 11-limit ===
@@ Line 217: / Line 198: @@
 * WE: ~2 = 1200.003854{{c}}, ~8388608/7381125 = 221.568850{{c}}
 * CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568133{{c}}
-<!-- * CTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568116{{c}}
-* POTE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568139{{c}} -->
 {{Optimal ET sequence|legend=0| 65dee, …, 612, 2513, 3125, 3737, 8086, 19909d }}
-Badness (Sintel): TBD
+Badness (Sintel): 1.211
 == Senior ==
@@ Line 239: / Line 218: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 7 -23 13 }} = 877.198647{{c}}
 : error map: {{val| 0.0000 -0.0024 -0.0024 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 7 -23 13 }} = 877.198651{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 7 -23 13 }} = 877.198642{{c}} -->
 {{Optimal ET sequence|legend=1| 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 0.498
 == Gross ==
@@ Line 261: / Line 238: @@
 * [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 46 -7 -15 }} = 91.530995{{c}}
 : error map: {{val| 0.0000 +0.0018 +0.0044 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 46 -7 -15 }} = 91.530979{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 46 -7 -15 }} = 91.531001{{c}} -->
 {{Optimal ET sequence|legend=1| 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 1.086
 ; Music
 * ''Gross temperament EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/02Fz6IIJ2x9k7bEvFylZ2A Spotify] | [https://francium223.bandcamp.com/album/gross-temperament-ep Bandcamp] | [https://youtube.com/playlist?list=PLLZE7hMjEXRZy_JWYbZxbsxPsXahiEJA2&si=RttlCFpkeN4Rl7Tj YouTube] – 2-piece extended play
-== Quectismic ==
+== Septarium ==
-{{See also| Quectisma }}
+Septarium splits the octave in seven, and tempers out the [[septarium comma]], {{monzo| 73 77 -84 }}. It can be described as {{nowrap| 441 & 2513 }} temperament, which tempers out 78125000/78121827 and {{monzo| 47 -7 -7 -7 }} ([[akjaysma]]) in the 7-limit.
 [[Subgroup]]: 2.3.5
-[[Comma list]]: {{monzo| 554 -351 1 }}
+[[Comma list]]: {{monzo| 73 77 -84 }}
-{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}
+{{Mapping|legend=1| 7 1 7 | 0 12 11 }}
-: mapping generators: ~2, ~3
+: mapping generators: ~{{monzo| 21 22 -24 }}, ~{{monzo| -20 -21 23 }}
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.999902567{{c}}, ~3/2 = 701.955253373{{c}}
+* [[WE]]: ~{{monzo| 21 22 -24 }} = 171.429{{c}}, ~{{monzo| -20 -21 23 }} = 144.210{{c}}
-: [[error map]]: {{val| -0.000097 +0.000155 -0.000001 }}
+* [[CWE]]: ~{{monzo| 21 22 -24 }} = 171.429{{c}}, ~{{monzo| -20 -21 23 }} = 144.210{{c}}
-* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.955310152{{c}}
-: error map: {{val| 0.000000 +0.000309 +0.000149 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.955310{{c}} -->
-{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}
+{{Optimal ET sequence|legend=1| 133, 308, 441, 1631, 2072, 2513, 5467, 7980 }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 1.395
-=== Nanoquectismic ===
+=== 7-limit ===
 Subgroup: 2.3.5.7
-Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}
+Comma list: 78125000/78121827, 140737488355328/140710042265625
-Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}
+Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}
 Optimal tunings:
-* WE: ~2 = 1199.979407{{c}}, ~3/2 = 701.943437{{c}}
+* WE: ~1157625/1048576 = 171.428{{c}}, ~2097152/1929375 = 144.211{{c}}
-* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955457{{c}}
+* CWE: ~1157625/1048576 = 171.429{{c}}, ~2097152/1929375 = 144.212{{c}}
-<!-- * CTE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955371{{c}} -->
+{{Optimal ET sequence|legend=0| 133, 308, 441, 1631, 2072, 2513, 2954 }}
+Badness (Sintel): 0.777
+== Raider ==
+{{See also| Syntonic–chromatic equivalence continuum }}
+[[Subgroup]]: 2.3.5
+[[Comma list]]: {{monzo| 71 -99 37 }}
+{{Mapping|legend=1| 1 -9 -26 | 0 37 99 }}
+: mapping generators: ~2, ~8000/6561
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.999880537{{c}}, ~8000/6561 = 343.296063356{{c}}
+: [[error map]]: {{val| -0.000119 +0.000418 -0.000336 }}
+* [[CWE]]: ~2 = 1200.000000000{{c}}, ~8000/6561 = 343.296095985{{c}}
+: error map: {{val| 0.000000 +0.000551 -0.000211 }}
+{{Optimal ET sequence|legend=1| 7, …, 381, 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
-{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
+[[Badness]] (Sintel): 0.302
-Badness (Sintel): TBD
+== Pirate ==
+{{See also| Hemfiness temperaments #Piracy }}
-=== Conanoquectismic ===
+[[Subgroup]]: 2.3.5
-Subgroup: 2.3.5.7
-Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
+[[Comma list]]: {{monzo| -90 -15 49 }}
-Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}
+{{Mapping|legend=1| 1 -6 0 | 0 49 15 }}
+: mapping generators: ~2, ~{{monzo| 24 4 -13 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* WE: ~2 = 1200.010750{{c}}, ~3/2 = 701.961041{{c}}
+* [[WE]]: ~2 = 1200.000195604{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}
-* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954771{{c}}
+: [[error map]]: {{val| -0.000196 +0.000418 -0.000336 }}
-<!-- * CTE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954810{{c}} -->
+* [[CWE]]: ~2 = 1200.000000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}
+: error map: {{val| 0.000000 +0.000082 -0.000574 }}
-{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}
+{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
-Badness (Sintel): TBD
+[[Badness]] (Sintel): 0.133
 == Counterquectismic ==
@@ Line 339: / Line 334: @@
 * [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.954415845{{c}}
 : error map: {{val| 0.000000 -0.000585 -0.000289 }}
-<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.954415{{c}} -->
 {{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 11.971
-== Raider ==
+== Quectismic ==
-{{See also| Syntonic–chromatic equivalence continuum }}
+{{See also| Quectisma }}
 [[Subgroup]]: 2.3.5
-[[Comma list]]: {{monzo| 71 -99 37 }}
+[[Comma list]]: {{monzo| 554 -351 1 }}
-{{Mapping|legend=1| 1 28 73 | 0 -37 -99 }}
+{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}
-: mapping generators: ~2, ~6561/4000
+: mapping generators: ~2, ~3
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296094{{c}}
+* [[WE]]: ~2 = 1199.999902567{{c}}, ~3/2 = 701.955253373{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296098{{c}}
+: [[error map]]: {{val| -0.000097 +0.000155 -0.000001 }}
+* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.955310152{{c}}
+: error map: {{val| 0.000000 +0.000309 +0.000149 }}
+{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}
-{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
+[[Badness]] (Sintel): 8.736
-[[Badness]] (Sintel): TBD
+=== Nanoquectismic ===
+Subgroup: 2.3.5.7
+Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}
+Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}
+Optimal tunings:
+* WE: ~2 = 1199.979407{{c}}, ~3/2 = 701.943437{{c}}
+* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955457{{c}}
+{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
-== Pirate ==
+Badness (Sintel): 15.285
-{{See also| Hemfiness temperaments #Piracy }}
-[[Subgroup]]: 2.3.5
+=== Conanoquectismic ===
+Subgroup: 2.3.5.7
-[[Comma list]]: {{monzo| -90 -15 49 }}
+Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
-{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}
+Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}
-: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754186{{c}}
+* WE: ~2 = 1200.010750{{c}}, ~3/2 = 701.961041{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754179{{c}}
+* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954771{{c}}
-{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
+{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}
-[[Badness]] (Sintel): TBD
+Badness (Sintel): 54.974
 == Atomic ==
-{{See also | Kirnberger's atom }}
+: ''For extensions, see [[Landscape microtemperaments #Atomic]].''
-Atomic observes the [[schisma]], and the 3/2 is one schisma sharp of its [[12edo]] value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the [[Pythagorean comma]] is twelve schismas, and hence [[81/80]], the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.
-Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].
+Atomic tempers out [[Kirnberger's atom]], the microscopic interval separating eleven [[schisma]]s and a [[syntonic comma]], or twelfth schismas and a [[Pythagorean comma]]. It is a member of the [[schismic–commatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 12 }}. In this extremely accurate temperament, the [[3/2|perfect fifth]] is one [[schisma]] sharp of its [[12edo]] value.
 [[Subgroup]]: 2.3.5
@@ Line 396: / Line 402: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955176{{c}} (~32805/32768 = 1.955176{{c}})
+* [[WE]]: ~{{monzo| -67 35 5 }} = 99.999995361{{c}}, ~3/2 = 701.955129505{{c}} (~32805/32768 = 1.955161980{{c}})
-* [[POTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955162{{c}} (~32805/32768 = 1.955162{{c}})
+: [[error map]]: {{val| -0.000056 +0.000072 +0.000022 }}
+* [[CWE]]: ~{{monzo| -67 35 5 }} = 100.000000000{{c}}, ~3/2 = 701.955166184{{c}} (~32805/32768 = 1.955166184{{c}})
+: error map: {{val| 0.000000 +0.000165 +0.000123 }}
 {{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
-[[Badness]] (Sintel): TBD
+[[Badness]] (Sintel): 0.0892
-=== 7-limit ===
-Subgroup: 2.3.5.7
-Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
-Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}
-Optimal tunings:
-* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949917{{c}} (or ~32805/32768 = 1.949917{{c}})
-* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949608{{c}} (or ~32805/32768 = 1.949608{{c}})
-{{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
+== Revopentic ==
-Badness (Sintel): 1.16
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 9801/9800, 151263/151250, 184549376/184528125
-Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}
-Optimal tunings:
-* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948456{{c}} (or ~32805/32768 = 1.948456{{c}})
-* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.947988{{c}} (or ~32805/32768 = 1.947988{{c}})
-{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}
-Badness (Sintel): 0.530
-== Revopentic (2.3.7) ==
 {{See also| Revopentisma }}
 Subgroup: 2.3.7
-Comma list: 11399736556781568/11398895185373143
+Comma list: {{monzo| 47 4 -19 }}
-{{Mapping|legend=2| 1 12 5 | 0 -19 -4 }}
+{{Mapping|legend=2| 1 -7 1 | 0 19 4 }}
-: mapping generators: ~2, ~24576/16807
+: mapping generators: ~2, ~16807/12288
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.791865{{c}}
+* [[WE]]: ~2 = 1199.998821{{c}}, ~16807/12288 = 542.207710{{c}}
-* [[CWE]]: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.791796{{c}}
+: [[error map]]: {{val| -0.0012 -0.0003 +0.0038 }}
+* [[CWE]]: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.208204{{c}}
+: error map: {{val| 0.0000 +0.0009 +0.0069 }}
 {{Optimal ET sequence|legend=1| 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, … }}
@@ Line 458: / Line 438: @@
 Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}
-Mapping: {{mapping| 1 12 -155 5 | 0 -19 287 -4 }}
+Mapping: {{mapping| 1 -7 132 1 | 0 19 -287 4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.792032{{c}}
+* WE: ~2 = 1200.000110{{c}}, ~16807/12288 = 542.208018{{c}}
-* CWE: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.792032{{c}}
+* CWE: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.207968{{c}}
 {{Optimal ET sequence|legend=0| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}
-Badness (Sintel): TBD
+Badness (Sintel): 0.665
 == Nanic ==
@@ Line 479: / Line 459: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.957820{{c}}
+* [[WE]]: ~2 = 1199.999111{{c}}, ~3/2 = 701.957264{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.957784{{c}}
+: [[error map]]: {{val| -0.0009 +0.0014 +0.0001 }}
+* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.957796{{c}}
+: error map: {{val| 0.0000 +0.0028 +0.0018 }}
 {{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}
@@ Line 493: / Line 475: @@
 [[Comma list]]: {{monzo| -32 -64 71 }}
-{{Mapping|legend=1| 1 35 32 | 0 -71 -64 }}
+{{Mapping|legend=2| 1 -36 -32 | 0 71 64 }}
+: mapping generators: ~3, ~{{monzo| -4 -9 10 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~3 = 1901.955{{c}}, ~{{monzo| 5 9 -10 }} = 898.340{{c}}
+* [[WE]]: ~3 = 1901.955003{{c}}, ~{{monzo| -4 -9 10 }} = 1003.615406{{c}}
-* [[CWE]]: ~3 = 1901.955{{c}}, ~{{monzo| 5 9 -10 }} = 898.340{{c}}
+: [[error map]]: {{val| +0.0000 +0.0000 -0.0000 }}
+* [[CWE]]: ~3 = 1901.955001{{c}}, ~{{monzo| -4 -9 10 }} = 1003.615405{{c}}
+: error map: {{val| 0.0000 -0.0000 -0.0000 }}
 [[Optimal ET sequence]]: [[163edt|b163cd]], [[199edt|b199c]], [[235edt|b235]], [[271edt|b271]], [[4643edt|b4643]], [[4914edt|b4914]], …, [[8708edt|b8708]], [[8979edt|b8979]], [[18229edt|b18229]], [[27208edt|b27208]], [[99853edt|b99853]], [[127061edt|b127061]], [[154269edt|b154269]], [[181477edt|b181477]], [[208685edt|b208685]], [[390162edt|b390162]]
@@ Line 504: / Line 489: @@
 = Rank-3 temperaments =
+== Akjaysmic ==
+{{See also| Akjaysma }}
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: {{monzo| 47 -7 -7 -7 }}
+{{Mapping|legend=1| 7 0 0 47 | 0 1 0 -1 | 0 0 1 -1 }}
+: mapping generators: ~1157625/1048576, ~3, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}
+: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}
+* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}
+: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}
+{{Optimal ET sequence|legend=1| 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}
+[[Badness]] (Sintel): 2.22
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 184549376/184528125, 199297406/199290375
+Mapping: {{mapping| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}
+: mapping generators: ~29160/26411, ~3, ~5
+Optimal tunings:
+* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}
+* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}
+{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}
+Badness (Sintel): 1.32
 == Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
 [[Subgroup]]: 2.3.5.7
@@ Line 517: / Line 538: @@
 [[Badness]] (Sintel): 1.24
-== Saquadtrizo-asepgu (171 & 2019 & 3566) ==
+== Izarismic ==
+This temperament is identical to the [[izar]] in the no-twos subgroup, but has an independent generator for prime 2. It can be described as {{nowrap| 41 & 49 & 130 }} temperament, and tempers out the [[izar comma]], {{monzo| 0 -11 -7 12 }}.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: {{monzo| 0 -11 -7 12 }} (13841287201/13839609375)
+[[Comma list]]: 13841287201/13839609375
 {{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
+: mapping generators: ~2, ~3, ~16807/10125
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}
+[[Optimal tuning]]s:
+* [[TE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}
+* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}
 {{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}