Father family

The father family of rank-2 temperaments tempers out 16/15, the just diatonic semitone. This equates 4/3 with 5/4, so the generator is a "fourth-third". In a sense, what father is all about is using semisixths or subfourths and pretending that this is 5-limit, and like any temperament which seems to involve this level of "pretending", father is close to the edge of what can sensibly be called a temperament at all. In other words, it is an exotemperament.

Father

Main article: Father

Subgroup: 2.3.5

Comma list: 16/15

Mapping: [⟨1 0 4], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 737.469

error map: ⟨0.000 +35.514 +76.218]

• POTE: ~2 = 1200.000, ~3/2 = 743.986

error map: ⟨0.000 +42.031 +69.700]

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 (Smith): 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 → Semaphoresmic clan
• Walid → Jubilismic clan

Septimal father

Main article: Father

See also: Trienstonic clan

Septimal father tempers out 28/27, making it a strong extension of trienstonian.

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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 727.855

error map: ⟨0.000 +25.900 +85.831 +14.739]

• POTE: ~2 = 1200.000, ~3/2 = 742.002

error map: ⟨0.000 +40.047 +71.684 +57.180]

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 (Smith): 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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 732.209
• POTE: ~2 = 1200.000, ~3/2 = 747.156

Optimal ET sequence: 2de, 3de, 5, 8d

Badness (Smith): 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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 727.290

error map: ⟨0.000 +25.335 +86.396 -23.406]

• POTE: ~2 = 1200.000, ~3/2 = 721.569

error map: ⟨0.000 +19.614 +92.118 -11.963]

Optimal ET sequence: 2, 3, 5

Badness (Smith): 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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 721.699
• POTE: ~2 = 1200.000, ~3/2 = 717.364

Optimal ET sequence: 2, 3, 5

Badness (Smith): 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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 761.765

error map: ⟨0.000 +59.810 +51.921 +22.348]

• POTE: ~2 = 1200.000, ~3/2 = 753.113

error map: ⟨0.000 +51.158 +60.574 +65.610]

Optimal ET sequence: 3, 5d, 8d, 27bbccdd

Badness (Smith): 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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 761.872
• POTE: ~2 = 1200.000, ~3/2 = 751.396

Optimal ET sequence: 3, 5de, 8d

Badness (Smith): 0.035487