Father family: Difference between revisions

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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 737.4686
* [[CTE]]: ~2 = 1\1, ~3/2 = 737.469
: [[error map]]: {{val| 0.000 +35.514 +76.218 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 743.986
* [[POTE]]: ~2 = 1\1, ~3/2 = 743.986
: error map: {{val| 0.000 +42.031 +69.700 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Baba]]'' → [[Slendro clan #Baba|Slendro clan]]
* ''[[Baba]]'' → [[Semaphoresmic clan #Baba|Semaphoresmic clan]]
* ''[[Walid]]'' → [[Jubilismic clan #Walid|Jubilismic clan]]
* ''[[Walid]]'' → [[Jubilismic clan #Walid|Jubilismic clan]]


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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.8550
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.855
: [[error map]]: {{val| 0.000 +25.900 +85.831 +14.739 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 742.002
* [[POTE]]: ~2 = 1\1, ~3/2 = 742.002
: error map: {{val| 0.000 +40.047 +71.684 +57.180 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 732.2094
* CTE: ~2 = 1\1, ~3/2 = 732.209
* POTE: ~2 = 1\1, ~3/2 = 747.156
* POTE: ~2 = 1\1, ~3/2 = 747.156


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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.2902
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.290
: [[error map]]: {{val| 0.000 +25.335 +86.396 -23.406 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 721.569
* [[POTE]]: ~2 = 1\1, ~3/2 = 721.569
: error map: {{val| 0.000 +19.614 +92.118 -11.963 }}


{{Optimal ET sequence|legend=1| 2, 3, 5 }}
{{Optimal ET sequence|legend=1| 2, 3, 5 }}
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 721.6985
* CTE: ~2 = 1\1, ~3/2 = 721.699
* POTE: ~2 = 1\1, ~3/2 = 717.364
* POTE: ~2 = 1\1, ~3/2 = 717.364


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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 761.7652
* [[CTE]]: ~2 = 1\1, ~3/2 = 761.765
: [[error map]]: {{val| 0.000 +59.810 +51.921 +22.348 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 753.113
* [[POTE]]: ~2 = 1\1, ~3/2 = 753.113
: error map: {{val| 0.000 +51.158 +60.574 +65.610 }}


{{Optimal ET sequence|legend=1| 3, 5d, 8d, 27bbccdd }}
{{Optimal ET sequence|legend=1| 3, 5d, 8d, 27bbccdd }}
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 761.8715
* CTE: ~2 = 1\1, ~3/2 = 761.872
* POTE: ~2 = 1\1, ~3/2 = 751.396
* POTE: ~2 = 1\1, ~3/2 = 751.396


Revision as of 15:49, 20 December 2024

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 = 1\1, ~3/2 = 737.469
error map: ⟨0.000 +35.514 +76.218]
  • POTE: ~2 = 1\1, ~3/2 = 743.986
error map: ⟨0.000 +42.031 +69.700]

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 = 727.855
error map: ⟨0.000 +25.900 +85.831 +14.739]
  • POTE: ~2 = 1\1, ~3/2 = 742.002
error map: ⟨0.000 +40.047 +71.684 +57.180]

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 = 732.209
  • 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 = 727.290
error map: ⟨0.000 +25.335 +86.396 -23.406]
  • POTE: ~2 = 1\1, ~3/2 = 721.569
error map: ⟨0.000 +19.614 +92.118 -11.963]

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 = 721.699
  • 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 = 761.765
error map: ⟨0.000 +59.810 +51.921 +22.348]
  • POTE: ~2 = 1\1, ~3/2 = 753.113
error map: ⟨0.000 +51.158 +60.574 +65.610]

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 = 761.872
  • POTE: ~2 = 1\1, ~3/2 = 751.396

Optimal ET sequence: 3, 5de, 8d

Badness: 0.035487

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