Father family

The father family of temperaments tempers out 16/15, the just diatonic semitone. This equates 4/3 with 5/4, so these are "fourth-thirds", or father, temperaments. In a sense, what father is all about is using semisixths or subfourths and pretending that's 5-limit, and like any temperament which seems to involve pretending, father is at the edge of what can sensibly be called a temperament at all. In other words, it is an exotemperament.

Father

Subgroup: 2.3.5

Comma list: 16/15

Mapping: [⟨1 0 4], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 743.986

Minimax tuning:

• 5-odd-limit: ~3/2 = [1 1/2 -1/2⟩

Eigenmonzo (unchanged-interval) basis: 2.5/3

Optimal ET sequence: 1, 2, 3, 5, 8, 13c, 21bcc

Badness: 0.014884

Overview to extensions

Strong extensions of father to include an approximation of harmonic 7 are septimal father (5 & 8d), mother (2 & 3), and pater (3 & 5d), all considered below.

Temperaments discussed elsewhere include:

• Baba → Slendro clan
• Walid → Jubilismic clan

Septimal father

See also: Trienstonic clan § Father

Subgroup: 2.3.5.7

Comma list: 16/15, 28/27

Mapping: [⟨1 0 4 -2], ⟨0 1 -1 3]]

Wedgie: ⟨⟨ 1 -1 3 -4 2 10 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 742.002

Minimax tuning:

• 7-odd-limit: ~3/2 = [1/2 0 -1/4 1/4⟩

Eigenmonzo (unchanged-interval) basis: 2.7/5

• 9-odd-limit: ~3/2 = [1 -2 0 1⟩ = 14/9

Eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 2d, 3d, 5, 8d, 13cd, 21bccdd

Badness: 0.021312

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 28/27

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 747.156

Optimal ET sequence: 2de, 3de, 5, 8d

Badness: 0.020589

Mother

Subgroup: 2.3.5.7

Comma list: 16/15, 21/20

Mapping: [⟨1 0 4 6], ⟨0 1 -1 -2]]

Wedgie: ⟨⟨ 1 -1 -2 -4 -6 -2 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 721.569

Optimal ET sequence: 2, 3, 5

Badness: 0.024152

11-limit

Subgroup: 2.3.5.7.11

Comma list: 11/10, 16/15, 21/20

Mapping: [⟨1 0 4 6 5], ⟨0 1 -1 -2 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 717.364

Optimal ET sequence: 2, 3, 5

Badness: 0.021957

Pater

Subgroup: 2.3.5.7

Comma list: 16/15, 126/125

Mapping: [⟨1 0 4 11], ⟨0 1 -1 -5]]

Wedgie: ⟨⟨ 1 -1 -5 -4 -11 -9 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 753.113

Optimal ET sequence: 3, 5d, 8d

Badness: 0.053001

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 100/99

Mapping: [⟨1 0 4 11 10], ⟨0 1 -1 -5 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 751.396

Optimal ET sequence: 3, 5de, 8d

Badness: 0.035487

Ternary

Subgroup: 2.3.5.7

Comma list: 10/9, 16/15

Mapping: [⟨3 5 7 0], ⟨0 0 0 1]]

mapping generators: ~5/4, ~7

Wedgie: ⟨⟨ 0 0 3 0 5 7 ]]

Optimal tuning (POTE): ~5/4 = 1\3, ~7/4 = 1034.013

Optimal ET sequence: 3d, 6, 9bd, 15bbd

Badness: 0.028673

11-limit

Subgroup: 2.3.5.7.11

Comma list: 10/9, 16/15, 22/21

Mapping: [⟨3 5 7 0 2], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~5/4 = 1\3, ~7/4 = 1033.153

Optimal ET sequence: 3de, 6, 15bbdee

Badness: 0.025592

Quint

Subgroup: 2.3.5.7

Comma list: 16/15, 27/25

Mapping: [⟨5 8 12 0], ⟨0 0 0 1]]

mapping generators: ~9/8, ~7

Wedgie: ⟨⟨ 0 0 5 0 8 12 ]]

Optimal tuning (POTE): ~9/8 = 1\5, ~7/4 = 1017.903

Optimal ET sequence: 5, 15ccd

Badness: 0.048312