Father family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Switch to Sintel's badness, WE & CWE tunings, per community consensus
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,019 edits
Fixed typo (CWE tuning of 11-limit 3&5, mother)
Tags: Mobile edit Mobile web edit
 
(2 intermediate revisions by one other user not shown)
Line 18: Line 18:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 742.290{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 742.290{{c}}
: error map: {{val| 0.000 +40.335 +71.396 }}
: error map: {{val| 0.000 +40.335 +71.396 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 737.469{{c}}
: [[error map]]: {{val| 0.000 +35.514 +76.218 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 743.986{{c}}
: error map: {{val| 0.000 +42.031 +69.700 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 55: Line 51:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 738.443{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 738.443{{c}}
: error map: {{val| 0.000 +36.488 +75.244 +46.502 }}
: error map: {{val| 0.000 +36.488 +75.244 +46.502 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 727.855{{c}}
: [[error map]]: {{val| 0.000 +25.900 +85.831 +14.739 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 742.002{{c}}
: error map: {{val| 0.000 +40.047 +71.684 +57.180 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 80: Line 72:
* WE: ~2 = 1180.894{{c}}, ~3/2 = 735.260{{c}}
* WE: ~2 = 1180.894{{c}}, ~3/2 = 735.260{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.387{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.387{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 732.209{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 747.156{{c}} -->


{{Optimal ET sequence|legend=0| 2de, 3de, 5, 8d }}
{{Optimal ET sequence|legend=0| 2de, 3de, 5, 8d }}
Line 99: Line 89:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 722.866{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 722.866{{c}}
: error map: {{val| 0.000 +20.911 +90.820 -14.559 }}
: error map: {{val| 0.000 +20.911 +90.820 -14.559 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 727.290{{c}}
: [[error map]]: {{val| 0.000 +25.335 +86.396 -23.406 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 721.569{{c}}
: error map: {{val| 0.000 +19.614 +92.118 -11.963 }} -->


{{Optimal ET sequence|legend=1| 2, 3, 5 }}
{{Optimal ET sequence|legend=1| 2, 3, 5 }}
Line 117: Line 103:
Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1192.016{{c}}, ~3/2 = 712.592{{c}}
* WE: ~2 = 1192.016{{c}}, ~3/2 = 712.592{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.387{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 718.370{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 718.370{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 717.364{{c}} -->


{{Optimal ET sequence|legend=0| 2, 3, 5 }}
{{Optimal ET sequence|legend=0| 2, 3, 5 }}
Line 137: Line 121:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 755.156{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 755.156{{c}}
: error map: {{val| 0.000 +53.201 +58.530 +55.392 }}
: error map: {{val| 0.000 +53.201 +58.530 +55.392 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 761.765{{c}}
: [[error map]]: {{val| 0.000 +59.810 +51.921 +22.348 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 753.113{{c}}
: 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 }}
Line 156: Line 136:
* WE: ~2 = 1180.290{{c}}, ~3/2 = 739.054{{c}}
* WE: ~2 = 1180.290{{c}}, ~3/2 = 739.054{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 753.774{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 753.774{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 761.872{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 751.396{{c}} -->


{{Optimal ET sequence|legend=0| 3, 5de, 8d }}
{{Optimal ET sequence|legend=0| 3, 5de, 8d }}
Line 164: Line 142:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Father family| ]] <!-- main article -->
[[Category:Father family| ]] <!-- main article -->
[[Category:Father| ]] <!-- key 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

Retrieved from "https://en.xen.wiki/index.php?title=Father_family&oldid=225411"