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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}}
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 ==
Comma list: 16/15
{{Main| Father }}


POTE generator: ~3/2 = 743.986
[[Subgroup]]: 2.3.5


Mapping: [{{val| 1 0 4 }}, {{val| 0 1 -1 }}]
[[Comma list]]: 16/15


{{Val list|legend=1| 1, 2, 3, 5, 8, 10c, 11b, 13c, 21bc }}
{{Mapping|legend=1| 1 0 4 | 0 1 -1 }}


Badness: 0.0149
: mapping generators: ~2, ~3


= 7-limit =
[[Optimal tuning]]s:
{{see also| Trienstonic clan #Father }}
* [[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 }}


Comma list: 16/15, 28/27
[[Minimax tuning]]:  
* [[5-odd-limit]]: ~3/2 = {{monzo| 1 1/2 -1/2 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3


[[POTE generator]]: ~3/2 = 742.002
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }}


Mapping: [{{val| 1 0 4 -2 }}, {{val| 0 1 -1 3 }}]
[[Badness]] (Sintel): 0.349


Wedgie: {{wedgie| 1 -1 3 -4 2 10 }}
=== 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.


Minimax tuning:  
Temperaments discussed elsewhere include:  
* 7-odd-limit
* ''[[Baba]]'' → [[Semaphoresmic clan #Baba|Semaphoresmic clan]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 0 -1/4 1/4 }}, {{monzo| 5/2 0 1/4 -1/4 }}, {{monzo| 5/2 0 -3/4 3/4 }}]
* ''[[Walid]]'' → [[Jubilismic clan #Walid|Jubilismic clan]]
: [[Eigenmonzo subgroup]]: 2.7/5
* 9-odd-limit
: Eigenmonzo subgroup: 2.9/5


{{Val list|legend=1| 5, 8d, 13cd, 21bcd }}
== Septimal father ==
{{Main| Father }}
{{See also| Trienstonic clan }}


Badness: 0.0213
Septimal father tempers out [[28/27]], making it a strong extension of [[trienstonian]].  


= Mother =
[[Subgroup]]: 2.3.5.7
Comma list: 16/15, 21/20


[[POTE generator]]: ~3/2 = 721.569
[[Comma list]]: 16/15, 28/27


Mapping: [{{val| 1 0 4 6 }}, {{val| 0 1 -1 -2 }}]
{{Mapping|legend=1| 1 0 4 -2 | 0 1 -1 3 }}


Wedgie: {{wedgie| 1 -1 -2 -4 -6 -2 }}
[[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 }}


{{Val list|legend=1| 5, 148, 153 }}
[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 1/2 0 -1/4 1/4 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1 -2 0 1 }} = 14/9
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


Badness: 0.0242
{{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}


= Pater =
[[Badness]] (Sintel): 0.539
Commas: 16/15, 126/125


POTE generator: ~3/2 = 753.113
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<1 0 4 11|, <0 1 -1 -5|]
Comma list: 16/15, 22/21, 28/27


Wedgie: <<1 -1 -5 -4 -11 -9||
Mapping: {{mapping| 1 0 4 -2 -3 | 0 1 -1 3 4 }}


EDOs: 3, 11b, 16bcd, 27bcd
Optimal tunings:
* WE: ~2 = 1180.894{{c}}, ~3/2 = 735.260{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.387{{c}}


Badness: 0.0530
{{Optimal ET sequence|legend=0| 2de, 3de, 5, 8d }}


= Walid =
Badness (Sintel): 0.681
Commas: 16/15, 50/49


POTE generator: ~3/2 = 749.415
== Mother ==
[[Subgroup]]: 2.3.5.7


Map: [<2 0 8 9|, <0 1 -1 -1|]
[[Comma list]]: 16/15, 21/20


Wedgie: <<2 -2 -2 -8 -9 1||
{{Mapping|legend=1| 1 0 4 6 | 0 1 -1 -2 }}


EDOs: 2, 6, 10cd, 14bd, 16bcd
[[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 }}


Badness: 0.0490
{{Optimal ET sequence|legend=1| 2, 3, 5 }}


= Baba =
[[Badness]] (Sintel): 0.611
Commas: 16/15, 49/45


POTE generator: ~8/7 = 226.704
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<1 0 4 2|, <0 2 -2 1|]
Comma list: 11/10, 16/15, 21/20


Wedgie: <<2 -2 1 -8 -4 8||
Mapping: {{mapping| 1 0 4 6 5 | 0 1 -1 -2 -1 }}


EDOs: 1, 5, 11b, 16bc
Optimal tunings:  
* WE: ~2 = 1192.016{{c}}, ~3/2 = 712.592{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 718.370{{c}}


Badness: 0.0443
{{Optimal ET sequence|legend=0| 2, 3, 5 }}


= Quint =
Badness (Sintel): 0.726
Commas: 16/15, 27/25


POTE generator: ~8/7 = 182.097
== Pater ==
[[Subgroup]]: 2.3.5.7


Map: [<5 8 12 0|, <0 0 0 -1|]
[[Comma list]]: 16/15, 126/125


Wedgie: <<0 0 5 0 8 12||
{{Mapping|legend=1| 1 0 4 11 | 0 1 -1 -5 }}


EDOs: 5, 15cd
[[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 }}


Badness: 0.0483
{{Optimal ET sequence|legend=1| 3, 5d, 8d, 27bbccdd }}


[[Category:Theory]]
[[Badness]] (Sintel): 1.34
[[Category:Temperament family]]
 
[[Category:Father]]
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 16/15, 22/21, 100/99
 
Mapping: {{mapping| 1 0 4 11 10 | 0 1 -1 -5 -4 }}
 
Optimal tunings:
* WE: ~2 = 1180.290{{c}}, ~3/2 = 739.054{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 753.774{{c}}
 
{{Optimal ET sequence|legend=0| 3, 5de, 8d }}
 
Badness (Sintel): 1.17
 
[[Category:Temperament families]]
[[Category:Father family| ]] <!-- main article -->
[[Category:Father| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
{{todo|review|improve readability}}

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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