Hemimage 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 hemimage family of rank-3 temperaments tempers out the hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935).

Hemimage

Hemimage is generated by a perfect fifth and a ~28/27 subminor second. Two of the subminor seconds make ~15/14, and three make ~10/9. The ubiquitous presence of this little interval gives rise to the highly intense hemimage bleeding chords.

Subgroup: 2.3.5.7

Comma list: 10976/10935

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

mapping generators: ~2, ~3, ~28/27

Optimal tunings:

• WE: ~2 = 1199.8555 ¢, ~3/2 = 702.6069 ¢, ~28/27 = 60.4665 ¢

error map: ⟨-0.145 +0.507 +0.155 -0.683]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6137 ¢, ~28/27 = 60.4761 ¢

error map: ⟨0.000 +0.659 +0.342 -0.509]

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [5/8 1/8 -1/8 3/8⟩, [5/8 -7/8 7/8 3/8⟩, [0 0 0 1⟩]

unchanged-interval (eigenmonzo) basis: 2.5/3.7

• 9-odd-limit

[[1 0 0 0⟩, [5/9 2/9 -1/9 1/3⟩, [10/9 -14/9 7/9 2/3⟩, [0 0 0 1⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.7

Optimal ET sequence: 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695

Badness (Sintel): 0.878

Projection pair: 5 10976/2187 to 2.3.7

Uni

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891

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

Optimal tunings:

• WE: ~2 = 1199.1497 ¢, ~3/2 = 703.3014 ¢, ~28/27 = 60.0617 ¢

error map: ⟨-0.859 +0.496 -0.376 +0.290 +2.041]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.6475 ¢, ~28/27 = 59.9675 ¢

error map: ⟨0.000 +1.693 +0.884 +2.084 +5.002]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [16/11 6/11 1/11 0 -3/11⟩, [0 0 1 0 0⟩, [19/11 14/11 6/11 0 -7/11⟩, [32/11 -10/11 2/11 0 5/11⟩]

unchanged-interval (eigenmonzo) basis: 2.5.11/9

Optimal ET sequence: 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee

Badness (Sintel): 0.844

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539

Mapping: [⟨1 0 -1 -2 5 10], ⟨0 1 2 3 -1 -4], ⟨0 0 3 1 1 1]]

Optimal tunings:

• WE: ~2 = 1199.0867 ¢, ~3/2 = 703.4048 ¢, ~28/27 = 60.0034 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9457 ¢, ~28/27 = 59.8167 ¢

Optimal ET sequence: 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef

Badness (Sintel): 0.993

Tin

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 729/728

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

Optimal tunings:

• WE: ~2 = 1198.8801 ¢, ~3/2 = 702.5101 ¢, ~28/27 = 60.9480 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8952 ¢, ~28/27 = 60.9129 ¢

Optimal ET sequence: 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff

Badness (Sintel): 1.00

Turan

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935

Mapping: [⟨1 0 -1 -2 -6], ⟨0 1 2 3 6], ⟨0 0 3 1 -1]]

Optimal tunings:

• WE: ~2 = 1200.0289 ¢, ~3/2 = 702.1366 ¢, ~28/27 = 60.8231 ¢

error map: ⟨+0.029 +0.211 +0.458 -1.564 +0.679]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1290 ¢, ~28/27 = 60.8257 ¢

error map: ⟨0.000 +0.174 +0.421 -1.613 +0.630]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [19/12 -2/3 1/12 0 1/4⟩, [19/6 -10/3 7/6 0 1/2⟩, [37/12 -8/3 7/12 0 3/4⟩, [19/6 -10/3 1/6 0 3/2⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.11/9

Optimal ET sequence: 17c, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335

Badness (Sintel): 1.13