Father family: Difference between revisions

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If we're moving quint then may as well move ternary to Augmented clan
+CTE tunings as this is a family where they tend to differ a lot from POTE
Line 12: Line 12:
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 743.986
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 737.4686
* [[POTE]]: ~2 = 1\1, ~3/2 = 743.986


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 41: Line 43:
{{Multival|legend=1| 1 -1 3 -4 2 10 }}
{{Multival|legend=1| 1 -1 3 -4 2 10 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 742.002
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.8550
* [[POTE]]: ~2 = 1\1, ~3/2 = 742.002


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 60: Line 64:
Mapping: {{mapping| 1 0 4 -2 -3 | 0 1 -1 3 4 }}
Mapping: {{mapping| 1 0 4 -2 -3 | 0 1 -1 3 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 747.156
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 732.2094
* POTE: ~2 = 1\1, ~3/2 = 747.156


{{Optimal ET sequence|legend=1| 2de, 3de, 5, 8d }}
Optimal ET sequence: {{Optimal ET sequence| 2de, 3de, 5, 8d }}


Badness: 0.020589
Badness: 0.020589
Line 75: Line 81:
{{Multival|legend=1| 1 -1 -2 -4 -6 -2 }}
{{Multival|legend=1| 1 -1 -2 -4 -6 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 721.569
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.2902
* [[POTE]]: ~2 = 1\1, ~3/2 = 721.569


{{Optimal ET sequence|legend=1| 2, 3, 5 }}
{{Optimal ET sequence|legend=1| 2, 3, 5 }}
Line 88: Line 96:
Mapping: {{mapping| 1 0 4 6 5 | 0 1 -1 -2 -1 }}
Mapping: {{mapping| 1 0 4 6 5 | 0 1 -1 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 717.364
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 721.6985
* POTE: ~2 = 1\1, ~3/2 = 717.364


{{Optimal ET sequence|legend=1| 2, 3, 5 }}
Optimal ET sequence: {{Optimal ET sequence| 2, 3, 5 }}


Badness: 0.021957
Badness: 0.021957
Line 103: Line 113:
{{Multival|legend=1| 1 -1 -5 -4 -11 -9 }}
{{Multival|legend=1| 1 -1 -5 -4 -11 -9 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 753.113
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 761.7652
* [[POTE]]: ~2 = 1\1, ~3/2 = 753.113


{{Optimal ET sequence|legend=1| 3, 5d, 8d }}
{{Optimal ET sequence|legend=1| 3, 5d, 8d, 27bbccdd }}


[[Badness]]: 0.053001
[[Badness]]: 0.053001
Line 116: Line 128:
Mapping: {{mapping| 1 0 4 11 10 | 0 1 -1 -5 -4 }}
Mapping: {{mapping| 1 0 4 11 10 | 0 1 -1 -5 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 751.396
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 761.8715
* POTE: ~2 = 1\1, ~3/2 = 751.396


{{Optimal ET sequence|legend=1| 3, 5de, 8d }}
Optimal ET sequence: {{Optimal ET sequence| 3, 5de, 8d }}


Badness: 0.035487
Badness: 0.035487

Revision as of 09:40, 4 May 2024

The father family of rank-2 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 is 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

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 = 1\1, ~3/2 = 737.4686
  • POTE: ~2 = 1\1, ~3/2 = 743.986

Minimax tuning:

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:

Septimal father

Main article: 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 tunings:

  • CTE: ~2 = 1\1, ~3/2 = 727.8550
  • POTE: ~2 = 1\1, ~3/2 = 742.002

Minimax tuning:

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

  • CTE: ~2 = 1\1, ~3/2 = 732.2094
  • 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 tunings:

  • CTE: ~2 = 1\1, ~3/2 = 727.2902
  • 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 tunings:

  • CTE: ~2 = 1\1, ~3/2 = 721.6985
  • 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 tunings:

  • CTE: ~2 = 1\1, ~3/2 = 761.7652
  • POTE: ~2 = 1\1, ~3/2 = 753.113

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

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 tunings:

  • CTE: ~2 = 1\1, ~3/2 = 761.8715
  • POTE: ~2 = 1\1, ~3/2 = 751.396

Optimal ET sequence: 3, 5de, 8d

Badness: 0.035487

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