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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


[[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 }}


{{Val list|legend=1| 1, 2, 3, 5, 8, 13c, 21bcc }}
[[Badness]] (Sintel): 0.349


[[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.  


=== 7-limit ===
Temperaments discussed elsewhere include:
{{see also| Trienstonic clan #Father }}
* ''[[Baba]]'' → [[Semaphoresmic clan #Baba|Semaphoresmic clan]]
* ''[[Walid]]'' → [[Jubilismic clan #Walid|Jubilismic clan]]


Subgroup: 2.3.5.7
== Septimal 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


[[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 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1180.634{{c}}, ~3/2 = 730.027{{c}}
[[POTE generator]]: ~3/2 = 742.002
: [[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


== Mother ==
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 16/15, 21/20
Comma list: 16/15, 22/21, 28/27


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


{{Multival|legend=1| 1 -1 -2 -4 -6 -2 }}
Optimal tunings:
* WE: ~2 = 1180.894{{c}}, ~3/2 = 735.260{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.387{{c}}


[[POTE generator]]: ~3/2 = 721.569
{{Optimal ET sequence|legend=0| 2de, 3de, 5, 8d }}


{{Val list|legend=1| 2, 3, 5 }}
Badness (Sintel): 0.681


[[Badness]]: 0.024152
== Mother ==
[[Subgroup]]: 2.3.5.7


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


[[Comma list]]: 16/15, 126/125
{{Mapping|legend=1| 1 0 4 6 | 0 1 -1 -2 }}


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


{{Multival|legend=1| 1 -1 -5 -4 -11 -9 }}
{{Optimal ET sequence|legend=1| 2, 3, 5 }}


[[POTE generator]]: ~3/2 = 753.113
[[Badness]] (Sintel): 0.611


{{Val list|legend=1| 3, 5d, 8d }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.053001
Comma list: 11/10, 16/15, 21/20


== Walid ==
Mapping: {{mapping| 1 0 4 6 5 | 0 1 -1 -2 -1 }}
Subgroup: 2.3.5.7


[[Comma list]]: 16/15, 50/49
Optimal tunings:  
* WE: ~2 = 1192.016{{c}}, ~3/2 = 712.592{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 718.370{{c}}


[[Mapping]]: [{{val| 2 0 8 9 }}, {{val| 0 1 -1 -1 }}]
{{Optimal ET sequence|legend=0| 2, 3, 5 }}


{{Multival|legend=1| 2 -2 -2 -8 -9 1 }}
Badness (Sintel): 0.726


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


{{Val list|legend=1| 2, 6, 8d }}
[[Comma list]]: 16/15, 126/125
 
[[Badness]]: 0.048978
 
== Baba ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 16/15, 49/45
 
[[Mapping]]: [{{val| 1 0 4 2 }}, {{val| 0 2 -2 1 }}]
 
[[POTE generator]]: ~8/7 = 226.704
 
{{Multival|legend=1| 2 -2 1 -8 -4 8 }}
 
{{Val list|legend=1| 5, 11b, 16bc }}
 
[[Badness]]: 0.044321
 
== Quint ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 16/15, 27/25
 
[[Mapping]]: [{{val| 5 8 12 0 }}, {{val| 0 0 0 1 }}]
 
{{Multival|legend=1| 0 0 5 0 8 12 }}


[[POTE generator]]: ~8/7 = 182.097
{{Mapping|legend=1| 1 0 4 11 | 0 1 -1 -5 }}


{{Val list|legend=1| 5, 15ccd }}
[[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]]: 0.048312
{{Optimal ET sequence|legend=1| 3, 5d, 8d, 27bbccdd }}


== Ternary ==
[[Badness]] (Sintel): 1.34
Subgroup: 2.3.5.7


[[Comma list]]: 10/9, 16/15
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 3 5 7 0 }}, {{val| 0 0 0 1 }}]
Comma list: 16/15, 22/21, 100/99


{{Multival|legend=1| 0 0 3 0 5 7 }}
Mapping: {{mapping| 1 0 4 11 10 | 0 1 -1 -5 -4 }}


[[POTE generator]]: ~8/7 = 165.987
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| 3, 6, 15bb }}
{{Optimal ET sequence|legend=0| 3, 5de, 8d }}


[[Badness]]: 0.028673
Badness (Sintel): 1.17


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Father family| ]] <!-- main article -->
[[Category:Father]]
[[Category:Father| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Father_family"