Father family: Difference between revisions
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The '''father family''' of | The '''father family''' of [[Rank-2 temperament|rank-2]] [[temperament]]s [[Tempering out|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 [[semisixth]]s or [[subfourth]]s 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]]. | ||
In a sense, what father is all about is using [[semisixth]]s or [[subfourth]]s and pretending that | |||
== Father == | == Father == | ||
{{Main| Father }} | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[5-odd-limit]]: ~3/2 = {{monzo| 1 1/2 -1/2 }} | * [[5-odd-limit]]: ~3/2 = {{monzo| 1 1/2 -1/2 }} | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3 | ||
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }} | {{Optimal ET sequence|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }} | ||
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== Septimal father == | == Septimal father == | ||
{{Main| Father }} | |||
{{See also| Trienstonic clan #Father }} | {{See also| Trienstonic clan #Father }} | ||
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]]: ~3/2 = {{monzo| 1/2 0 -1/4 1/4 }} | * [[7-odd-limit]]: ~3/2 = {{monzo| 1/2 0 -1/4 1/4 }} | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5 | ||
* [[9-odd-limit]]: ~3/2 = {{monzo| 1 -2 0 1 }} = 14/9 | * [[9-odd-limit]]: ~3/2 = {{monzo| 1 -2 0 1 }} = 14/9 | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7 | ||
{{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }} | {{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }} | ||
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Badness: 0.025592 | Badness: 0.025592 | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Father family| ]] <!-- main article --> | [[Category:Father family| ]] <!-- main article --> | ||
[[Category:Father| ]] <!-- key article --> | |||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Revision as of 08:25, 22 September 2023
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
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
- 5-odd-limit: ~3/2 = [1 1/2 -1/2⟩
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
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
- 7-odd-limit: ~3/2 = [1/2 0 -1/4 1/4⟩
- 9-odd-limit: ~3/2 = [1 -2 0 1⟩ = 14/9
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