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The '''keemic family''' of rank-3 temperaments are [[planar temperament]]s tempering out the keema, [[875/864]].
{{Technical data page}}
The '''keemic family''' is a family of [[rank-3 temperament]]s which [[temper out]] the [[keema]], 875/864 = {{monzo| -5 -3 3 1 }}.


== Supermagic ==
== Supermagic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864
[[Comma list]]: 875/864


[[Mapping]]: [{{val| 1 0 0 5 }}, {{val| 0 1 0 3 }}, {{val| 0 0 1 -3 }}]
{{Mapping|legend=1| 1 0 0 5 | 0 1 0 3 | 0 0 1 -3 }}


Mapping generators: ~2, ~3, ~5
: mapping generators: ~2, ~3, ~5


[[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}]
[[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}]
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: Angle (6/5, 3/2) = 77.834
: Angle (6/5, 3/2) = 77.834


[[POTE generator]]s: ~3/2 = 701.8733, ~5/4 = 380.4684
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.8733, ~5/4 = 380.4684


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/4 3/4 1/4 -1/4 }}, {{monzo| 5/4 3/4 -3/4 3/4 }}]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/4 3/4 1/4 -1/4 | 5/4 3/4 -3/4 3/4 }}
: Eigenmonzos: 2, 4/3, 7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


{{Val list|legend=1| 7, 15, 19, 41 }}
{{Optimal ET sequence|legend=1| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}
 
[[Badness]]: 0.212 × 10<sup>-3</sup>


[[Projection pair]]: 7 864/125
[[Projection pair]]: 7 864/125
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== Undecimal supermagic ==
== Undecimal supermagic ==
Subgroup: 2.3.5.7.11
{{See also| Ptolemismic clan #Supermagic }}
 
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 100/99, 385/384
[[Comma list]]: 100/99, 385/384


[[Mapping]]: [{{val| 1 0 0 5 2 }}, {{val| 0 1 0 3 -2 }}, {{val| 0 0 1 -3 2 }}]
{{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}


[[POTE generator]]s: ~3/2 = 704.1985, ~5/4 = 381.5394
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394


{{Val list|legend=1| 7, 15, 19, 22, 37, 41, 104, 145c, 201ce, 264bce, 305bcce }}
{{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}


[[Badness]]: 0.641 × 10<sup>-3</sup>
[[Badness]]: 0.641 × 10<sup>-3</sup>
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== Supernatural ==
== Supernatural ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 225/224, 245/243
[[Comma list]]: 225/224, 245/243


[[Mapping]]: [{{val| 1 0 2 -1 0 }}, {{val| 0 5 1 12 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=1| 1 0 2 -1 0 | 0 5 1 12 0 | 0 0 0 0 1 }}


[[POTE generator]]s: ~5/4 = 380.3520, ~11/8 = 547.5758
: mapping generators: ~2, ~5, ~11


{{Val list|legend=1| 19, 22, 41, 82e, 101cd, 224ccde, 265ccde * }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 380.3520, ~11/8 = 547.5758
 
{{Optimal ET sequence|legend=1| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde * }}


<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]


[[Badness]]: 0.888 × 10<sup>-3</sup>
[[Badness]]: 0.888 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 245/243
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 0 18 | 0 0 0 0 1 0 }}
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.0401, ~11/8 = 546.6672
{{Optimal ET sequence|legend=1| 19, 22f, 38df, 41, 60e, 79d, 101cd }}
Badness: 1.14 × 10<sup>-3</sup>


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Keemic family| ]] <!-- main article -->
[[Category:Keemic family| ]] <!-- main article -->
[[Category:Supermagic]]
[[Category:Supermagic]]
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:36, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The keemic family is a family of rank-3 temperaments which temper out the keema, 875/864 = [-5 -3 3 1⟩.

Supermagic

Subgroup: 2.3.5.7

Comma list: 875/864

Mapping: [⟨1 0 0 5], ⟨0 1 0 3], ⟨0 0 1 -3]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 0 -1 3], ⟨0 1 1 0]]

Lattice basis:

6/5 length = 0.8879, 3/2 length = 1.3391
Angle (6/5, 3/2) = 77.834

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8733, ~5/4 = 380.4684

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [5/4 3/4 1/4 -1/4⟩, [5/4 3/4 -3/4 3/4⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd

Badness: 0.212 × 10-3

Projection pair: 7 864/125

Scales: supermagic15

Undecimal supermagic

See also: Ptolemismic clan § Supermagic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384

Mapping: [⟨1 0 0 5 2], ⟨0 1 0 3 -2], ⟨0 0 1 -3 2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394

Optimal ET sequence: 7, 15, 19, 22, 37, 41, 104

Badness: 0.641 × 10-3

Scales: supermagic15

Supernatural

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243

Mapping: [⟨1 0 2 -1 0], ⟨0 5 1 12 0], ⟨0 0 0 0 1]]

mapping generators: ~2, ~5, ~11

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.3520, ~11/8 = 547.5758

Optimal ET sequence: 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde *

* optimal patent val: 104

Badness: 0.888 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243

Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 0 18], ⟨0 0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.0401, ~11/8 = 546.6672

Optimal ET sequence: 19, 22f, 38df, 41, 60e, 79d, 101cd

Badness: 1.14 × 10-3

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