Keemic family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864).  
The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2 temperament]]s, see [[keemic temperaments]].
 
== Keemic ==
Keemic, also known as ''supermagic'' in earlier materials, has the same lattice structure as [[5-limit]] JI, and [[7/4]] is found by a stack of three [[~]][[6/5]]'s.  


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


Line 16: Line 18:
: Angle (6/5, 3/2) = 77.834
: Angle (6/5, 3/2) = 77.834


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.8733{{c}}, ~5/4 = 380.4684{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0490{{c}}, ~3/2 = 702.4868{{c}}, ~5/4 = 380.8010{{c}}
: [[error map]]: {{val| +1.049 +1.581 -3.415 -1.670 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8060{{c}}, ~5/4 = 381.0536{{c}}
: error map: {{val| 0.000 +0.851 -5.260 -3.569 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 25: Line 31:
{{Optimal ET sequence|legend=1| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}
{{Optimal ET sequence|legend=1| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}


[[Badness]] (Smith): 0.212 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.937


[[Projection pair]]: 7 864/125
[[Projection pair]]: 7 864/125
Line 31: Line 37:
Scales: [[supermagic15]]
Scales: [[supermagic15]]


== Undecimal supermagic ==
== Undecimal keemic ==
{{See also| Ptolemismic clan #Supermagic }}
{{See also| Ptolemismic clan #Keemic }}
 
Keemic is naturally an [[11-limit]] temperament due to the identity 875/864 = ([[100/99]])⋅([[385/384]]). This identifies a stack of two ~6/5's as ~[[16/11]].


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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{{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}
{{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 704.1985{{c}}, ~5/4 = 381.5394{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9868{{c}}, ~3/2 = 704.1908{{c}}, ~5/4 = 381.5352{{c}}
: [[error map]]: {{val| -0.013 +2.223 -4.805 -0.885 +3.318 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1922{{c}}, ~5/4 = 381.5339{{c}}
: error map: {{val| 0.000 +2.237 -4.780 -0.851 +3.365 }}


{{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}
{{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}


[[Badness]] (Smith): 0.641 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.770


Scales: [[supermagic15]]
Scales: [[supermagic15]]


== Supernatural ==
== Supernatural ==
Supernatural uses the same mapping as [[7-limit]] [[magic]] with an independent generator for prime 11.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 56: Line 70:
: mapping generators: ~2, ~5, ~11
: mapping generators: ~2, ~5, ~11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 380.3520{{c}}, ~11/8 = 547.5758{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}, ~11/8 = 548.0690{{c}}
: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 -0.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}, ~11/8 = 548.8962{{c}}
: error map: {{val| 0.000 +0.333 -5.856 -3.335 -2.422 }}


{{Optimal ET sequence|legend=1| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde * }}
{{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]] (Smith): 0.888 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.07


=== 13-limit ===
=== 13-limit ===
Line 71: Line 89:
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 0 18 | 0 0 0 0 1 0 }}
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 0 18 | 0 0 0 0 1 0 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~5/4 = 380.0401{{c}}, ~11/8 = 546.6672{{c}}
Optimal tunings:
* WE: ~2 = 1201.3389{{c}}, ~5/4 = 380.4641{{c}}, ~11/8 = 547.2772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}


{{Optimal ET sequence|legend=1| 19, 22f, 38df, 41, 60e, 79d, 101cd }}
{{Optimal ET sequence|legend=0| 19, 22f, 38df, 41, 60e, 79d, 101cd }}


Badness (Smith): 1.14 × 10<sup>-3</sup>
Badness (Sintel): 1.07


[[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:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 12:27, 27 November 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 (monzo: [-5 -3 3 1⟩, ratio: 875/864). For rank-2 temperaments, see keemic temperaments.

Keemic

Keemic, also known as supermagic in earlier materials, has the same lattice structure as 5-limit JI, and 7/4 is found by a stack of three ~6/5's.

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 tunings:

  • WE: ~2 = 1201.0490 ¢, ~3/2 = 702.4868 ¢, ~5/4 = 380.8010 ¢
error map: ⟨+1.049 +1.581 -3.415 -1.670]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8060 ¢, ~5/4 = 381.0536 ¢
error map: ⟨0.000 +0.851 -5.260 -3.569]

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 (Sintel): 0.937

Projection pair: 7 864/125

Scales: supermagic15

Undecimal keemic

See also: Ptolemismic clan § Keemic

Keemic is naturally an 11-limit temperament due to the identity 875/864 = (100/99)⋅(385/384). This identifies a stack of two ~6/5's as ~16/11.

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 tunings:

  • WE: ~2 = 1199.9868 ¢, ~3/2 = 704.1908 ¢, ~5/4 = 381.5352 ¢
error map: ⟨-0.013 +2.223 -4.805 -0.885 +3.318]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1922 ¢, ~5/4 = 381.5339 ¢
error map: ⟨0.000 +2.237 -4.780 -0.851 +3.365]

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

Badness (Sintel): 0.770

Scales: supermagic15

Supernatural

Supernatural uses the same mapping as 7-limit magic with an independent generator for prime 11.

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 tunings:

  • WE: ~2 = 1201.0786 ¢, ~5/4 = 380.6939 ¢, ~11/8 = 548.0690 ¢
error map: ⟨+1.079 +1.514 -3.463 -1.578 -0.013]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4576 ¢, ~11/8 = 548.8962 ¢
error map: ⟨0.000 +0.333 -5.856 -3.335 -2.422]

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

* optimal patent val: 104

Badness (Sintel): 1.07

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 tunings:

  • WE: ~2 = 1201.3389 ¢, ~5/4 = 380.4641 ¢, ~11/8 = 547.2772 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1532 ¢, ~11/8 = 548.2678 ¢

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

Badness (Sintel): 1.07

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