Father family: Difference between revisions

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== Father ==
== Father ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 16/15
[[Comma list]]: 16/15


[[Mapping]]: [{{val| 1 0 4 }}, {{val| 0 1 -1 }}]
{{Mapping|legend=1| 1 0 4 | 0 1 -1 }}


Mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[POTE generator]]: ~3/2 = 743.986
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 743.986


[[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]]s (unchanged-intervals): 2, 6/5
: [[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 ==
{{see also| Trienstonic clan #Father }}
{{See also| Trienstonic clan #Father }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16/15, 28/27
[[Comma list]]: 16/15, 28/27


[[Mapping]]: [{{val| 1 0 4 -2 }}, {{val| 0 1 -1 3 }}]
{{Mapping|legend=1| 1 0 4 -2 | 0 1 -1 3 }}


{{Multival|legend=1| 1 -1 3 -4 2 10 }}
{{Multival|legend=1| 1 -1 3 -4 2 10 }}


[[POTE generator]]: ~3/2 = 742.002
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 742.002


[[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]]s (unchanged-intervals): 2, 7/5
: [[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
: Eigenmonzos (unchanged-intervals): 2, 9/7
: [[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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Comma list: 16/15, 22/21, 28/27
Comma list: 16/15, 22/21, 28/27


Mapping: [{{val| 1 0 4 -2 -3 }}, {{val| 0 1 -1 3 4 }}]
Mapping: {{mapping| 1 0 4 -2 -3 | 0 1 -1 3 4 }}


POTE generator: ~3/2 = 747.156
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 747.156


{{Optimal ET sequence|legend=1| 2de, 3de, 5, 8d }}
{{Optimal ET sequence|legend=1| 2de, 3de, 5, 8d }}
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== Mother ==
== Mother ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16/15, 21/20
[[Comma list]]: 16/15, 21/20


[[Mapping]]: [{{val| 1 0 4 6 }}, {{val| 0 1 -1 -2 }}]
{{Mapping|legend=1| 1 0 4 6 | 0 1 -1 -2 }}


{{Multival|legend=1| 1 -1 -2 -4 -6 -2 }}
{{Multival|legend=1| 1 -1 -2 -4 -6 -2 }}


[[POTE generator]]: ~3/2 = 721.569
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 721.569


{{Optimal ET sequence|legend=1| 2, 3, 5 }}
{{Optimal ET sequence|legend=1| 2, 3, 5 }}
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Comma list: 11/10, 16/15, 21/20
Comma list: 11/10, 16/15, 21/20


Mapping: [{{val| 1 0 4 6 5 }}, {{val| 0 1 -1 -2 -1 }}]
Mapping: {{mapping| 1 0 4 6 5 | 0 1 -1 -2 -1 }}


POTE generator: ~3/2 = 717.364
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 717.364


{{Optimal ET sequence|legend=1| 2, 3, 5 }}
{{Optimal ET sequence|legend=1| 2, 3, 5 }}
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== Pater ==
== Pater ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16/15, 126/125
[[Comma list]]: 16/15, 126/125


[[Mapping]]: [{{val| 1 0 4 11 }}, {{val| 0 1 -1 -5 }}]
{{Mapping|legend=1| 1 0 4 11 | 0 1 -1 -5 }}


{{Multival|legend=1| 1 -1 -5 -4 -11 -9 }}
{{Multival|legend=1| 1 -1 -5 -4 -11 -9 }}


[[POTE generator]]: ~3/2 = 753.113
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 753.113


{{Optimal ET sequence|legend=1| 3, 5d, 8d }}
{{Optimal ET sequence|legend=1| 3, 5d, 8d }}
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Comma list: 16/15, 22/21, 100/99
Comma list: 16/15, 22/21, 100/99


Mapping: [{{val| 1 0 4 11 10 }}, {{val| 0 1 -1 -5 -4 }}]
Mapping: {{mapping| 1 0 4 11 10 | 0 1 -1 -5 -4 }}


POTE generator: ~3/2 = 751.396
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 751.396


{{Optimal ET sequence|legend=1| 3, 5de, 8d }}
{{Optimal ET sequence|legend=1| 3, 5de, 8d }}
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== Baba ==
== Baba ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16/15, 49/45
[[Comma list]]: 16/15, 49/45


[[Mapping]]: [{{val| 1 0 4 2 }}, {{val| 0 2 -2 1 }}]
{{Mapping|legend=1| 1 0 4 2 | 0 2 -2 1 }}


Mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


[[POTE generator]]: ~7/4 = 973.296
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/4 = 973.296


{{Multival|legend=1| 2 -2 1 -8 -4 8 }}
{{Multival|legend=1| 2 -2 1 -8 -4 8 }}
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Comma list: 16/15, 22/21, 49/45
Comma list: 16/15, 22/21, 49/45


Mapping: [{{val| 1 0 4 2 1 }}, {{val| 0 2 -2 1 3 }}]
Mapping: {{mapping| 1 0 4 2 1 | 0 2 -2 1 3 }}


POTE generator: ~7/4 = 978.164
Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 978.164


{{Optimal ET sequence|legend=1| 5, 11b, 16bc, 27bbcc }}
{{Optimal ET sequence|legend=1| 5, 11b, 16bc, 27bbcc }}
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== Walid ==
== Walid ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16/15, 50/49
[[Comma list]]: 16/15, 50/49


[[Mapping]]: [{{val| 2 0 8 9 }}, {{val| 0 1 -1 -1 }}]
{{Mapping|legend=1| 2 0 8 9 | 0 1 -1 -1 }}


Mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3


{{Multival|legend=1| 2 -2 -2 -8 -9 1 }}
{{Multival|legend=1| 2 -2 -2 -8 -9 1 }}


[[POTE generator]]: ~3/2 = 749.415
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 749.415


{{Optimal ET sequence|legend=1| 2, 6, 8d }}
{{Optimal ET sequence|legend=1| 2, 6, 8d }}
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Comma list: 16/15, 22/21, 50/49
Comma list: 16/15, 22/21, 50/49


Mapping: [{{val| 2 0 8 9 7 }}, {{val| 0 1 -1 -1 0 }}]
Mapping: {{mapping| 2 0 8 9 7 | 0 1 -1 -1 0 }}


POTE generator: ~3/2 = 749.756
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 749.756


{{Optimal ET sequence|legend=1| 2, 6, 8d }}
{{Optimal ET sequence|legend=1| 2, 6, 8d }}
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== Ternary ==
== Ternary ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10/9, 16/15
[[Comma list]]: 10/9, 16/15


[[Mapping]]: [{{val| 3 5 7 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 3 5 7 0 | 0 0 0 1 }}


Mapping generators: ~5/4, ~7
: mapping generators: ~5/4, ~7


{{Multival|legend=1| 0 0 3 0 5 7 }}
{{Multival|legend=1| 0 0 3 0 5 7 }}


[[POTE generator]]: ~7/4 = 1034.013
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~7/4 = 1034.013


{{Optimal ET sequence|legend=1| 3d, 6, 9bd, 15bbd }}
{{Optimal ET sequence|legend=1| 3d, 6, 9bd, 15bbd }}
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Comma list: 10/9, 16/15, 22/21
Comma list: 10/9, 16/15, 22/21


Mapping: [{{val| 3 5 7 0 2 }}, {{val| 0 0 0 1 1 }}]
Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}


POTE generator: ~7/4 = 1033.153
Optimal tuning (POTE): ~5/4 = 1\3, ~7/4 = 1033.153


{{Optimal ET sequence|legend=1| 3de, 6, 15bbdee }}
{{Optimal ET sequence|legend=1| 3de, 6, 15bbdee }}
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== Quint ==
== Quint ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16/15, 27/25
[[Comma list]]: 16/15, 27/25


[[Mapping]]: [{{val| 5 8 12 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}


Mapping generators: ~9/8, ~7
: mapping generators: ~9/8, ~7


{{Multival|legend=1| 0 0 5 0 8 12 }}
{{Multival|legend=1| 0 0 5 0 8 12 }}


[[POTE generator]]: ~7/4 = 1017.903
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\5, ~7/4 = 1017.903


{{Optimal ET sequence|legend=1| 5, 15ccd }}
{{Optimal ET sequence|legend=1| 5, 15ccd }}

Revision as of 11:09, 4 August 2023

The father family of 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's 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

Minimax tuning:

Eigenmonzo (unchanged-interval) basis: 2.5/3

Optimal ET sequence: 1, 2, 3, 5, 8, 13c, 21bcc

Badness: 0.014884

Septimal 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 tuning (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 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

Baba

Subgroup: 2.3.5.7

Comma list: 16/15, 49/45

Mapping: [⟨1 0 4 2], ⟨0 2 -2 1]]

mapping generators: ~2, ~7/4

Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 973.296

Wedgie: ⟨⟨ 2 -2 1 -8 -4 8 ]]

Optimal ET sequence: 5, 11b, 16bc

Badness: 0.044321

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 49/45

Mapping: [⟨1 0 4 2 1], ⟨0 2 -2 1 3]]

Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 978.164

Optimal ET sequence: 5, 11b, 16bc, 27bbcc

Badness: 0.036538

Walid

Subgroup: 2.3.5.7

Comma list: 16/15, 50/49

Mapping: [⟨2 0 8 9], ⟨0 1 -1 -1]]

mapping generators: ~7/5, ~3

Wedgie: ⟨⟨ 2 -2 -2 -8 -9 1 ]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 749.415

Optimal ET sequence: 2, 6, 8d

Badness: 0.048978

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 50/49

Mapping: [⟨2 0 8 9 7], ⟨0 1 -1 -1 0]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 749.756

Optimal ET sequence: 2, 6, 8d

Badness: 0.029193

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

Quint

Subgroup: 2.3.5.7

Comma list: 16/15, 27/25

Mapping: [⟨5 8 12 0], ⟨0 0 0 1]]

mapping generators: ~9/8, ~7

Wedgie: ⟨⟨ 0 0 5 0 8 12 ]]

Optimal tuning (POTE): ~9/8 = 1\5, ~7/4 = 1017.903

Optimal ET sequence: 5, 15ccd

Badness: 0.048312

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