Keemic family

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).

Supermagic

Supermagic 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:

• 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 supermagic

See also: Ptolemismic clan § Supermagic

Supermagic 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

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