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Fixed typo (CWE tuning of 11-limit 3&5, mother)
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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.
{{Technical data page}}
In a sense, what father is all about is using [[semisixth]]s or [[subfourth]]s 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]].
The '''father family''' is a family of [[rank-2 temperament]]s which [[temper out]] the classic diatonic semitone, [[16/15]]. This equates [[4/3]] with [[5/4]] and [[8/5]] with [[3/2]], so the generator is a "fourth-third" (or "fifth-sixth"), hence the name ''father''. In a sense, what father is all about is using [[semisixth]]s or [[subfourth]]s and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", father is close to the edge of what can be sensibly called a temperament at all. In other words, it is an [[exotemperament]].


== Father ==
== Father ==
Subgroup: 2.3.5
{{Main| Father }}
 
[[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]]s:  
* [[WE]]: ~2 = 1180.875{{c}}, ~3/2 = 732.129{{c}}
: [[error map]]: {{val| -19.125 +11.048 +24.181 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 742.290{{c}}
: error map: {{val| 0.000 +40.335 +71.396 }}


[[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|unchanged-interval (eigenmonzo) basis]]: 2.5/3
 
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }}
 
[[Badness]] (Sintel): 0.349


{{Val list|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }}
=== 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.


[[Badness]]: 0.014884
Temperaments discussed elsewhere include:
* ''[[Baba]]'' → [[Semaphoresmic clan #Baba|Semaphoresmic clan]]
* ''[[Walid]]'' → [[Jubilismic clan #Walid|Jubilismic clan]]


== Septimal father ==
== Septimal father ==
{{see also| Trienstonic clan #Father }}
{{Main| Father }}
{{See also| Trienstonic clan }}
 
Septimal father tempers out [[28/27]], making it a strong extension of [[trienstonian]].


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 }}


[[POTE generator]]: ~3/2 = 742.002
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1180.634{{c}}, ~3/2 = 730.027{{c}}
: [[error map]]: {{val| -19.366 +8.706 +25.561 +1.890 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 738.443{{c}}
: error map: {{val| 0.000 +36.488 +75.244 +46.502 }}


[[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|unchanged-interval (eigenmonzo) 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|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Val list|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
{{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}


[[Badness]]: 0.021312
[[Badness]] (Sintel): 0.539


=== 11-limit ===
=== 11-limit ===
Line 49: Line 67:
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 tunings:
* WE: ~2 = 1180.894{{c}}, ~3/2 = 735.260{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.387{{c}}


Optimal GPV sequence: {{Val list| 2de, 3de, 5, 8d }}
{{Optimal ET sequence|legend=0| 2de, 3de, 5, 8d }}


Badness: 0.020589
Badness (Sintel): 0.681


== 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 }}


[[POTE generator]]: ~3/2 = 721.569
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1186.985{{c}}, ~3/2 = 713.743{{c}}
: [[error map]]: {{val| -13.015 -1.228 +60.898 -48.372 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 722.866{{c}}
: error map: {{val| 0.000 +20.911 +90.820 -14.559 }}


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


[[Badness]]: 0.024152
[[Badness]] (Sintel): 0.611


=== 11-limit ===
=== 11-limit ===
Line 77: Line 99:
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 tunings:  
* WE: ~2 = 1192.016{{c}}, ~3/2 = 712.592{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 718.370{{c}}


Optimal GPV sequence: {{Val list| 2, 3, 5 }}
{{Optimal ET sequence|legend=0| 2, 3, 5 }}


Badness: 0.021957
Badness (Sintel): 0.726


== 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 }}


[[POTE generator]]: ~3/2 = 753.113
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1179.316{{c}}, ~3/2 = 740.132{{c}}
: [[error map]]: {{val| -20.684 +17.493 +11.503 +6.412 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 755.156{{c}}
: error map: {{val| 0.000 +53.201 +58.530 +55.392 }}


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


[[Badness]]: 0.053001
[[Badness]] (Sintel): 1.34


=== 11-limit ===
=== 11-limit ===
Line 105: Line 131:
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 GPV sequence: {{Val list| 3, 5de, 8d }}
 
Badness: 0.035487
 
== Baba ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 16/15, 49/45
 
[[Mapping]]: [{{val| 1 0 4 2 }}, {{val| 0 2 -2 1 }}]
 
Mapping generators: ~2, ~7/4
 
[[POTE generator]]: ~7/4 = 973.296
 
{{Multival|legend=1| 2 -2 1 -8 -4 8 }}
 
{{Val list|legend=1| 5, 11b, 16bc }}
 
[[Badness]]: 0.044321
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 16/15, 22/21, 49/45
 
Mapping: [{{val| 1 0 4 2 1 }}, {{val| 0 2 -2 1 3 }}]
 
POTE generator: ~7/4 = 978.164
 
Optimal GPV sequence: {{Val list| 5, 11b, 16bc, 27bbcc }}
 
Badness: 0.036538
 
== Walid ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 16/15, 50/49
 
[[Mapping]]: [{{val| 2 0 8 9 }}, {{val| 0 1 -1 -1 }}]
 
Mapping generators: ~7/5, ~3
 
{{Multival|legend=1| 2 -2 -2 -8 -9 1 }}
 
[[POTE generator]]: ~3/2 = 749.415
 
{{Val list|legend=1| 2, 6, 8d }}
 
[[Badness]]: 0.048978
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 16/15, 22/21, 50/49
 
Mapping: [{{val| 2 0 8 9 7 }}, {{val| 0 1 -1 -1 0 }}]
 
POTE generator: ~3/2 = 749.756
 
Optimal GPV sequence: {{Val list| 2, 6, 8d }}
 
Badness: 0.029193
 
== Ternary ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 10/9, 16/15
 
[[Mapping]]: [{{val| 3 5 7 0 }}, {{val| 0 0 0 1 }}]
 
Mapping generators: ~5/4, ~7
 
{{Multival|legend=1| 0 0 3 0 5 7 }}
 
[[POTE generator]]: ~7/4 = 1034.013
 
{{Val list|legend=1| 3d, 6, 9bd, 15bbd }}
 
[[Badness]]: 0.028673
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 10/9, 16/15, 22/21
 
Mapping: [{{val| 3 5 7 0 2 }}, {{val| 0 0 0 1 1 }}]
 
POTE generator: ~7/4 = 1033.153
 
Optimal GPV sequence: {{Val list| 3de, 6, 15bbdee }}
 
Badness: 0.025592
 
== Quint ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 16/15, 27/25
 
[[Mapping]]: [{{val| 5 8 12 0 }}, {{val| 0 0 0 1 }}]
 
Mapping generators: ~9/8, ~7
 
{{Multival|legend=1| 0 0 5 0 8 12 }}


[[POTE generator]]: ~7/4 = 1017.903
Optimal tunings:  
* WE: ~2 = 1180.290{{c}}, ~3/2 = 739.054{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 753.774{{c}}


{{Val list|legend=1| 5, 15ccd }}
{{Optimal ET sequence|legend=0| 3, 5de, 8d }}


[[Badness]]: 0.048312
Badness (Sintel): 1.17


[[Category:Regular temperament theory]]
[[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]]

Latest revision as of 02:20, 8 March 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The father family is a family of rank-2 temperaments which temper out the classic diatonic semitone, 16/15. This equates 4/3 with 5/4 and 8/5 with 3/2, so the generator is a "fourth-third" (or "fifth-sixth"), hence the name father. In a sense, what father is all about is using semisixths or subfourths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", father is close to the edge of what can be sensibly 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:

  • WE: ~2 = 1180.875 ¢, ~3/2 = 732.129 ¢
error map: ⟨-19.125 +11.048 +24.181]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 742.290 ¢
error map: ⟨0.000 +40.335 +71.396]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5/3

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

Badness (Sintel): 0.349

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]]

Optimal tunings:

  • WE: ~2 = 1180.634 ¢, ~3/2 = 730.027 ¢
error map: ⟨-19.366 +8.706 +25.561 +1.890]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 738.443 ¢
error map: ⟨0.000 +36.488 +75.244 +46.502]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.7/5
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 2d, 3d, 5, 8d, 13cd, 21bccdd

Badness (Sintel): 0.539

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:

  • WE: ~2 = 1180.894 ¢, ~3/2 = 735.260 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 743.387 ¢

Optimal ET sequence: 2de, 3de, 5, 8d

Badness (Sintel): 0.681

Mother

Subgroup: 2.3.5.7

Comma list: 16/15, 21/20

Mapping: [⟨1 0 4 6], ⟨0 1 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1186.985 ¢, ~3/2 = 713.743 ¢
error map: ⟨-13.015 -1.228 +60.898 -48.372]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 722.866 ¢
error map: ⟨0.000 +20.911 +90.820 -14.559]

Optimal ET sequence: 2, 3, 5

Badness (Sintel): 0.611

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:

  • WE: ~2 = 1192.016 ¢, ~3/2 = 712.592 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 718.370 ¢

Optimal ET sequence: 2, 3, 5

Badness (Sintel): 0.726

Pater

Subgroup: 2.3.5.7

Comma list: 16/15, 126/125

Mapping: [⟨1 0 4 11], ⟨0 1 -1 -5]]

Optimal tunings:

  • WE: ~2 = 1179.316 ¢, ~3/2 = 740.132 ¢
error map: ⟨-20.684 +17.493 +11.503 +6.412]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 755.156 ¢
error map: ⟨0.000 +53.201 +58.530 +55.392]

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

Badness (Sintel): 1.34

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:

  • WE: ~2 = 1180.290 ¢, ~3/2 = 739.054 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 753.774 ¢

Optimal ET sequence: 3, 5de, 8d

Badness (Sintel): 1.17

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