Hemifamity family

The hemifamity family of rank-3 temperaments tempers out 5120/5103 = [10 -6 1 -1⟩. These temperaments divide an exact or approximate septimal quartertone, 36/35 into two equal steps, each representing 81/80~64/63, the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same chain of fifths inflected by the same comma to the opposite sides. In addition we may identify 10/7 by the augmented fourth (C-F#) and 50/49 by the Pythagorean comma.

It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have 5/4 at the down major third (C-vE) and 7/4 at the down minor seventh (C-vBb).

Hemifamity

Subgroup: 2.3.5.7

Comma list: 5120/5103

Mapping: [⟨1 0 0 10], ⟨0 1 0 -6], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 2 -4], ⟨0 0 1 1]]

Lattice basis:

3/2 length = 0.5670, 10/9 length = 1.8063

Angle (3/2, 10/9) = 82.112 degrees

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.7918, ~5/4 = 386.0144

Minimax tuning: c = 5120/5103

• 7-odd-limit: 3 and 7 1/7c sharp, 5 just

[[1 0 0 0⟩, [10/7 1/7 1/7 -1/7⟩, [0 0 1 0⟩, [10/7 -6/7 1/7 6/7⟩]

eigenmonzo (unchanged-interval) basis: 2.5.7/3

• 9-odd-limit: 3 1/8c sharp, 5 just, 7 1/4c sharp

[[1 0 0 0⟩, [5/4 1/4 1/8 -1/8⟩, [0 0 1 0⟩, [5/2 -3/2 1/4 3/4⟩]

eigenmonzo (unchanged-interval) basis: 2.5.9/7

Optimal ET sequence: 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd

Badness: 0.153 × 10^-3

Projection pairs: 7 5120/729

Music

• Choraled play by Gene Ward Smith
• Hemifamity27 by Chris Vaisvil

Overview to extensions

11- and 13-limit extensions

Strong extensions of hemifamity are pele, laka, akea, and lono. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the 11/8 at the down diminished fifth (C-vGb); laka, up augmented third (C-^E#); akea, double-up fourth (C-^^F); lono, triple-down augmented fourth (C-v³F#). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the 13/11 at the minor third, tempering out 352/351, 847/845, and 2080/2079.

Subgroup extensions

A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.

Counterpyth

Main article: Counterpyth

Developed analogous to parapyth, counterpyth is an extension of hemifamity with an even milder fifth, as it finds 19/15 at the major third (C-E) and 19/10 at the major seventh (C-B). Notice the factorization 5120/5103 = (400/399)(1216/1215). Other important ratios are 21/19 at the diminished third (C-Ebb) and 19/14 at the augmented third (C-E#).

It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.

Subgroup: 2.3.5.7.19

Comma list: 400/399, 1216/1215

Mapping: [⟨1 0 0 10 -6], ⟨0 1 0 -6 5], ⟨0 0 1 1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.6411, ~5/4 = 385.4452

Optimal ET sequence: 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh

Badness: 0.212 × 10^-3

Pele

Main article: Pele

See also: Pentacircle clan

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891

Mapping: [⟨1 0 0 10 17], ⟨0 1 0 -6 -10], ⟨0 0 1 1 1]]

Mapping to lattice: [⟨0 1 4 -2 -6], ⟨0 0 -1 -1 -1]]

Lattice basis:

3/2 length = 0.3812, 56/55 length = 1.5893

Angle(3/2, 56/55) = 90.4578 degrees

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.2829, ~5/4 = 386.5647

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [17/10 0 1/10 0 -1/10⟩, [17/5 -2 6/5 0 -1/5⟩, [16/5 -2 3/5 0 2/5⟩, [17/5 -2 1/5 0 4/5⟩]

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

Optimal ET sequence: 29, 41, 58, 87, 99e, 145, 186e

Badness: 0.648 × 10^-3

Projection pairs: 7 5120/729 11 655360/59049

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363

Mapping: [⟨1 0 0 10 17 22], ⟨0 1 0 -6 -10 -13], ⟨0 0 1 1 1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.4398, ~5/4 = 386.8933

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.13/9
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/9

Optimal ET sequence: 29, 41, 46, 58, 87, 145, 232

Badness: 0.703 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363

Mapping: [⟨1 0 0 10 17 22 8], ⟨0 1 0 -6 -10 -13 -1], ⟨0 0 1 1 1 1 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.5544, ~5/4 = 387.9654

Optimal ET sequence: 29, 41, 46, 58, 87, 99ef, 145

Badness: 0.930 × 10^-3

Laka

Main article: Laka

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5120/5103

Mapping: [⟨1 0 0 10 -18], ⟨0 1 0 -6 15], ⟨0 0 1 1 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5133, ~5/4 = 385.5563

Minimax tuning

• 11-odd-limit

[[1 0 0 0 0⟩, [4/3 0 2/21 -1/21 1/21⟩, [0 0 1 0 0⟩, [2 0 3/7 2/7 -2/7⟩, [2 0 3/7 -5/7 5/7⟩]

eigenmonzo (unchanged-interval) basis: 2.5.11/7

Optimal ET sequence: 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee

Badness: 0.825 × 10^-3

Projection pairs: 5120/729 11 14348907/1310720

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728

Mapping: [⟨1 0 0 10 -18 -13], ⟨0 1 0 -6 15 12], ⟨0 0 1 1 -1 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.4078, ~5/4 = 385.5405

Minimax tuning:

• 13- and 15-odd-limit

[[1 0 0 0 0 0⟩, [13/8 -1/2 1/8 0 0 1/8⟩, [13/4 -3 5/4 0 0 1/4⟩, [7/2 0 1/2 0 0 -1/2⟩, [25/8 -9/2 5/8 0 0 13/8⟩, [13/4 -3 1/4 0 0 5/4⟩]

eigenmonzo (unchanged-interval) basis: 2.11.13/7

Optimal ET sequence: 41, 53, 58, 94, 111, 152f, 415dff*

* optimal patent val: 205

Badness: 0.822 × 10^-3

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 352/351, 400/399, 456/455, 495/494

Mapping: [⟨1 0 0 10 -18 -13 -6], ⟨0 1 0 -6 15 12 5], ⟨0 0 1 1 -1 -1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.4062, ~5/4 = 385.5254

Optimal ET sequence: 41, 53, 58h, 94, 111, 152f, 415dffhh*

* optimal patent val: 205

Badness: 0.661 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 561/560

Mapping: [⟨1 0 0 10 -18 -13 32], ⟨0 1 0 -6 15 12 -22], ⟨0 0 1 1 -1 -1 3]]

Minimax tuning:

• 17-odd-limit

[[1 0 0 0 0 0 0⟩, [13/12 0 0 1/12 1/6 -1/12 0⟩, [-7/4 0 0 5/4 3/2 -5/4 0⟩, [7/4 0 0 3/4 1/2 -3/4 0⟩, [0 0 0 0 1 0 0⟩, [7/4 0 0 -1/4 1/2 1/4 0⟩, [35/12 0 0 23/12 5/6 -23/12 0⟩]

eigenmonzo (unchanged-interval) basis: 2.11.13/7

Optimal ET sequence: 58, 94, 111, 152f, 205, 263df

Badness: 1.19 × 10^-3

Akea

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187

Mapping: [⟨1 0 0 10 -3], ⟨0 1 0 -6 7], ⟨0 0 1 1 -2]]

mapping generators: ~2, ~3, ~5

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.8909, ~5/4 = 385.3273

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [5/3 0 1/6 -1/6 0⟩, [26/9 0 13/18 -7/18 -1/3⟩, [26/9 0 -5/18 11/18 -1/3⟩, [26/9 0 -5/18 -7/18 2/3⟩]

eigenmonzo (unchanged-interval) basis: 2.7/5.11/5

Optimal ET sequence: 34, 41, 53, 87, 140, 181, 321

Badness: 0.998 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384

Mapping: [⟨1 0 0 10 -3 2], ⟨0 1 0 -6 7 4], ⟨0 0 1 1 -2 -2]]

Lattice basis:

3/2 length = 0.5354, 27/20 length = 1.0463

Angle (3/2, 27/20) = 80.5628 degrees

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.9018, ~5/4 = 385.4158

Minimax tuning:

• 13- and 15-odd-limit

[[1 0 0 0 0 0⟩, [5/3 0 1/6 -1/6 0 0⟩, [26/9 0 13/18 -7/18 -1/3 0⟩, [26/9 0 -5/18 11/18 -1/3 0⟩, [26/9 0 -5/18 -7/18 2/3 0⟩, [26/9 0 -7/9 1/9 2/3 0⟩]

eigenmonzo (unchanged-interval) basis: 2.7/5.11/5

Optimal ET sequence: 34, 41, 46, 53, 87, 140, 321, 461e

Badness: 0.822 × 10^-3

Scales: akea46_13

Lono

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103

Mapping: [⟨1 0 0 10 6], ⟨0 1 0 -6 -6], ⟨0 0 1 1 3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.8941, ~5/4 = 388.5932

Optimal ET sequence: 46, 53, 58, 99, 111, 268cd

Badness: 1.18 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845

Mapping: [⟨1 0 0 10 6 11], ⟨0 1 0 -6 -6 -9], ⟨0 0 1 1 3 3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.8670, ~5/4 = 388.6277

Optimal ET sequence: 46, 53, 58, 99, 104c, 111, 268cd

Badness: 0.908 × 10^-3

Kapo

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5120/5103

Mapping: [⟨1 0 0 10 7], ⟨0 1 1 -5 -2], ⟨0 0 2 2 -1]]

mapping generators: ~2, ~3, ~128/99

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.8776, ~128/99 = 441.7516

Minimax tuning:

• 11-odd-limit:

[[1 0 0 0 0⟩, [8/5 2/5 0 -1/15 -2/15⟩, [14/5 6/5 0 7/15 -16/15⟩, [16/5 -6/5 0 13/15 -4/15⟩, [16/5 -6/5 0 -2/15 11/15⟩]

eigenmonzo (unchanged-interval) basis: 2.9/7.11/9

Optimal ET sequence: 41, 87, 111, 152, 239, 391

Badness: 0.994 × 10^-3

Namaka

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 5120/5103

Mapping: [⟨1 0 0 10 -6], ⟨0 2 0 -12 9], ⟨0 0 1 1 1]]

mapping generators: ~2, ~400/231, ~5

Optimal tuning (CTE): ~2 = 1\1, ~400/231 = 951.4956, ~5/4 = 386.7868

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198

Badness: 1.74 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845

Mapping: [⟨1 0 0 10 -6 -1], ⟨0 2 0 -12 9 3], ⟨0 0 1 1 1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.4871, ~5/4 = 386.6606

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198

Badness: 0.781 × 10^-3