Father family: Difference between revisions

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→Septimal father: note the relation to trienstonic clan
Spell out 1\1 in decimal and note the badness metric being used
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 737.469
* [[CTE]]: ~2 = 1200.000, ~3/2 = 737.469
: [[error map]]: {{val| 0.000 +35.514 +76.218 }}
: [[error map]]: {{val| 0.000 +35.514 +76.218 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 743.986
* [[POTE]]: ~2 = 1200.000, ~3/2 = 743.986
: error map: {{val| 0.000 +42.031 +69.700 }}
: error map: {{val| 0.000 +42.031 +69.700 }}


Line 24: Line 24:
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }}
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }}


[[Badness]]: 0.014884
[[Badness]] (Smith): 0.014884


=== Overview to extensions ===
=== Overview to extensions ===
Line 48: Line 48:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.855
* [[CTE]]: ~2 = 1200.000, ~3/2 = 727.855
: [[error map]]: {{val| 0.000 +25.900 +85.831 +14.739 }}
: [[error map]]: {{val| 0.000 +25.900 +85.831 +14.739 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 742.002
* [[POTE]]: ~2 = 1200.000, ~3/2 = 742.002
: error map: {{val| 0.000 +40.047 +71.684 +57.180 }}
: error map: {{val| 0.000 +40.047 +71.684 +57.180 }}


Line 61: Line 61:
{{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
{{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}


[[Badness]]: 0.021312
[[Badness]] (Smith): 0.021312


=== 11-limit ===
=== 11-limit ===
Line 71: Line 71:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 732.209
* CTE: ~2 = 1200.000, ~3/2 = 732.209
* POTE: ~2 = 1\1, ~3/2 = 747.156
* POTE: ~2 = 1200.000, ~3/2 = 747.156


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


Badness: 0.020589
Badness (Smith): 0.020589


== Mother ==
== Mother ==
Line 88: Line 88:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 727.290
* [[CTE]]: ~2 = 1200.000, ~3/2 = 727.290
: [[error map]]: {{val| 0.000 +25.335 +86.396 -23.406 }}
: [[error map]]: {{val| 0.000 +25.335 +86.396 -23.406 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 721.569
* [[POTE]]: ~2 = 1200.000, ~3/2 = 721.569
: error map: {{val| 0.000 +19.614 +92.118 -11.963 }}
: 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 }}


[[Badness]]: 0.024152
[[Badness]] (Smith): 0.024152


=== 11-limit ===
=== 11-limit ===
Line 105: Line 105:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 721.699
* CTE: ~2 = 1200.000, ~3/2 = 721.699
* POTE: ~2 = 1\1, ~3/2 = 717.364
* POTE: ~2 = 1200.000, ~3/2 = 717.364


Optimal ET sequence: {{Optimal ET sequence| 2, 3, 5 }}
Optimal ET sequence: {{Optimal ET sequence| 2, 3, 5 }}


Badness: 0.021957
Badness (Smith): 0.021957


== Pater ==
== Pater ==
Line 122: Line 122:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 761.765
* [[CTE]]: ~2 = 1200.000, ~3/2 = 761.765
: [[error map]]: {{val| 0.000 +59.810 +51.921 +22.348 }}
: [[error map]]: {{val| 0.000 +59.810 +51.921 +22.348 }}
* [[POTE]]: ~2 = 1\1, ~3/2 = 753.113
* [[POTE]]: ~2 = 1200.000, ~3/2 = 753.113
: error map: {{val| 0.000 +51.158 +60.574 +65.610 }}
: 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 }}


[[Badness]]: 0.053001
[[Badness]] (Smith): 0.053001


=== 11-limit ===
=== 11-limit ===
Line 139: Line 139:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 761.872
* CTE: ~2 = 1200.000, ~3/2 = 761.872
* POTE: ~2 = 1\1, ~3/2 = 751.396
* POTE: ~2 = 1200.000, ~3/2 = 751.396


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


Badness: 0.035487
Badness (Smith): 0.035487


[[Category:Temperament families]]
[[Category:Temperament families]]

Revision as of 15:57, 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 = 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:

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:

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:

eigenmonzo (unchanged-interval) basis: 2.7/5
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

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