Xenharmonic Wiki

Father family: Difference between revisions

← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,262 edits
m {{Technical data page}}<br><br> + edited first sentence of intro so the links had text in between instead of a whole blue sentence, to make it easy to tell where one link ends and one starts
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
Fixed typo (CWE tuning of 11-limit 3&5, mother)
Tags: Mobile edit Mobile web edit
 
(7 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Technical data page}}<br><br>
{{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]].
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]].


Line 14: Line 14:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 737.469
* [[WE]]: ~2 = 1180.875{{c}}, ~3/2 = 732.129{{c}}
: [[error map]]: {{val| 0.000 +35.514 +76.218 }}
: [[error map]]: {{val| -19.125 +11.048 +24.181 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 743.986
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 742.290{{c}}
: error map: {{val| 0.000 +42.031 +69.700 }}
: 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 basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) 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 }}


[[Badness]] (Smith): 0.014884
[[Badness]] (Sintel): 0.349


=== Overview to extensions ===
=== Overview to extensions ===
Line 45: Line 45:


{{Mapping|legend=1| 1 0 4 -2 | 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:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 727.855
* [[WE]]: ~2 = 1180.634{{c}}, ~3/2 = 730.027{{c}}
: [[error map]]: {{val| 0.000 +25.900 +85.831 +14.739 }}
: [[error map]]: {{val| -19.366 +8.706 +25.561 +1.890 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 742.002
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 738.443{{c}}
: error map: {{val| 0.000 +40.047 +71.684 +57.180 }}
: 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 basis|eigenmonzo (unchanged-interval) basis]]: 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
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) 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 }}


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


=== 11-limit ===
=== 11-limit ===
Line 72: Line 70:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 732.209
* WE: ~2 = 1180.894{{c}}, ~3/2 = 735.260{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 747.156
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.387{{c}}


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


Badness (Smith): 0.020589
Badness (Sintel): 0.681


== Mother ==
== Mother ==
Line 85: Line 83:


{{Mapping|legend=1| 1 0 4 6 | 0 1 -1 -2 }}
{{Mapping|legend=1| 1 0 4 6 | 0 1 -1 -2 }}
{{Multival|legend=1| 1 -1 -2 -4 -6 -2 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 727.290
* [[WE]]: ~2 = 1186.985{{c}}, ~3/2 = 713.743{{c}}
: [[error map]]: {{val| 0.000 +25.335 +86.396 -23.406 }}
: [[error map]]: {{val| -13.015 -1.228 +60.898 -48.372 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 721.569
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 722.866{{c}}
: error map: {{val| 0.000 +19.614 +92.118 -11.963 }}
: error map: {{val| 0.000 +20.911 +90.820 -14.559 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 105: Line 101:
Mapping: {{mapping| 1 0 4 6 5 | 0 1 -1 -2 -1 }}
Mapping: {{mapping| 1 0 4 6 5 | 0 1 -1 -2 -1 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 721.699
* WE: ~2 = 1192.016{{c}}, ~3/2 = 712.592{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 717.364
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 718.370{{c}}


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


Badness (Smith): 0.021957
Badness (Sintel): 0.726


== Pater ==
== Pater ==
Line 120: Line 116:
{{Mapping|legend=1| 1 0 4 11 | 0 1 -1 -5 }}
{{Mapping|legend=1| 1 0 4 11 | 0 1 -1 -5 }}


{{Multival|legend=1| 1 -1 -5 -4 -11 -9 }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1179.316{{c}}, ~3/2 = 740.132{{c}}
[[Optimal tuning]]s:
: [[error map]]: {{val| -20.684 +17.493 +11.503 +6.412 }}
* [[CTE]]: ~2 = 1200.000, ~3/2 = 761.765
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 755.156{{c}}
: [[error map]]: {{val| 0.000 +59.810 +51.921 +22.348 }}
: error map: {{val| 0.000 +53.201 +58.530 +55.392 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 753.113
: error map: {{val| 0.000 +51.158 +60.574 +65.610 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 139: Line 133:
Mapping: {{mapping| 1 0 4 11 10 | 0 1 -1 -5 -4 }}
Mapping: {{mapping| 1 0 4 11 10 | 0 1 -1 -5 -4 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 761.872
* WE: ~2 = 1180.290{{c}}, ~3/2 = 739.054{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 751.396
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 753.774{{c}}


{{Optimal ET sequence|legend=0| 3, 5de, 8d }}
{{Optimal ET sequence|legend=0| 3, 5de, 8d }}


Badness (Smith): 0.035487
Badness (Sintel): 1.17


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Father_family"