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. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Scales: [[supermagic15]] | Scales: [[supermagic15]] | ||
== Undecimal | == Undecimal keemic == | ||
{{See also| Ptolemismic clan # | {{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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== 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 | ||
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[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Keemic family| ]] <!-- main article --> | [[Category:Keemic family| ]] <!-- main article --> | ||
[[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
- 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]
- 7- and 9-odd-limit
- [[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
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]]
- 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
- 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 *
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