Archytas family: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
- CTE & POTE tunings
 
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The '''archytas family''' of temperaments [[tempering out|tempers out]] [[64/63]], and thereby identifies the [[otonal]] [[tetrad]] with the [[dominant seventh chord]], which is familiar from [[12edo]] and also a feature of [[22edo]] and [[27edo]].
{{Technical data page}}
The '''archytas family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[64/63]]. The head of this family, archytas, tempers out 64/63 alone in the full 7-limit, so it has the same 2.3.7-[[subgroup]] structure as [[archy]] but giving [[prime harmonic|prime]] [[5/1|5]] an independent generator.
 
See [[Archytas clan]] for the rank-2 temperament without the last generator of archytas, and its various extensions.  


== Archytas ==
== Archytas ==
{{Main| Archytas and ares }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[64/63]]
[[Comma list]]: [[64/63]]


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


Mapping generators: ~2, ~3, ~5
: mapping generators: ~2, ~3, ~5


[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Line 16: Line 21:
: Angle (3/2, 5/4) = 90 degrees
: Angle (3/2, 5/4) = 90 degrees


[[Optimal tuning]] ([[POTE]]): ~3/2 = 709.3213, ~5/4 = 393.3747
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}, ~5/4 = 392.3765{{c}}
: [[error map]]: {{val| -3.045 +2.522 -0.027 +3.952 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}, ~5/4 = 391.5995{{c}}
: error map: {{val| 0.000 +7.435 +5.286 +12.394 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}]
: {{monzo list| 1 0 0 0 | 2 1/3 0 -1/3 | 2 -2/3 1 -1/3 | 2 -2/3 0 2/3 }}
: [[Eigenmonzo basis]] (unchanged-intervals): 2.5/3.7/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7/3
* [[9-odd-limit]]
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}]
: {{monzo list| 1 0 0 0 | 3/2 1/2 0 -1/4 | 3/2 -1/2 1 -1/4 | 3 -1 0 1/2 }}
: [[Eigenmonzo basis]] (unchanged-intervals): 2.5/3.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7


{{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }}
{{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }}
[[Badness]] (Sintel): 0.435


Scales: [[archytas12]], [[archytas12synch]]
Scales: [[archytas12]], [[archytas12synch]]
; Music
* Chris Vaisvil – [http://micro.soonlabel.com/hobbit_scales/daily201031--hobbit-archytas12.mp3 12 archytas hobbits with ale]
* Joel Taylor – [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/Archytas12%20sync%20Study%203.mp3 Portrait of a Tragedy (Death of a Son) - Archytas12 sync Study 3]


== Ares ==
== Ares ==
{{Main| Ares }}
{{Main| Archytas and ares }}
{{See also| Ptolemismic clan #Ares }}


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 41: Line 49:
[[Comma list]]: 64/63, 100/99
[[Comma list]]: 64/63, 100/99


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


Mapping generators: ~2, ~3, ~5
[[Mapping to lattice]]: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]
 
Mapping to lattice: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]


Lattice basis:  
Lattice basis:  
Line 51: Line 57:
: Angle (3/2, 6/5) = 75.0475 degrees
: Angle (3/2, 6/5) = 75.0475 degrees


[[Optimal tuning]] ([[POTE]]): ~3/2 = 709.6869, ~5/4 = 391.4507
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.0279{{c}}, ~3/2 = 707.9291{{c}}, ~5/4 = 390.4812{{c}}
: [[error map]]: {{val| -2.972 +3.002 -1.777 +3.427 +1.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.6947{{c}}, ~5/4 = 390.0248{{c}}
: error map: {{val| 0.000 +7.740 +3.711 +11.785 +9.342 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 0 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 -1 0 1/2 0 }}, {{monzo| 3 -1 0 -1/2 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 0 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 -1 0 1/2 0 }}, {{monzo| 3 -1 0 -1/2 1 }}]
: [[Eigenmonzo basis]] (unchanged-intervals): 2.11/9.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11/9.9/7


{{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}
{{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}


[[Badness]]: 0.469 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.563


Scales: [[ares12]]
Scales: [[ares12]]
Line 69: Line 79:
[[Comma list]]: 56/55, 64/63
[[Comma list]]: 56/55, 64/63


[[Mapping]]: [{{val| 1 0 0 6 9 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 0 -1 }}]
{{Mapping|legend=1| 1 0 0 6 9 | 0 1 0 -2 -2 | 0 0 1 0 -1 }}
 
Mapping generators: ~2, ~3, ~5


[[Optimal tuning]] ([[POTE]]): ~3/2 = 710.9720, ~5/4 = 401.2699
[[Optimal tuning]]s:
* [[CWE]]: ~2 = 1195.4265{{c}}, ~3/2 = 708.2623{{c}}, ~5/4 = 399.7405{{c}}
: [[error map]]: {{val| -4.573 +1.734 +4.280 -3.644 +9.550 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5280{{c}}, ~5/4 = 400.4798{{c}}
: error map: {{val| 0.000 +9.573 +14.166 +8.118 +25.146 }}


{{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }}
{{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }}


[[Badness]]: 0.577 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.693


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Archytas family| ]] <!-- main article -->
[[Category:Archytas family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]

Latest revision as of 10:00, 14 July 2025

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 archytas family of rank-3 temperaments tempers out 64/63. The head of this family, archytas, tempers out 64/63 alone in the full 7-limit, so it has the same 2.3.7-subgroup structure as archy but giving prime 5 an independent generator.

See Archytas clan for the rank-2 temperament without the last generator of archytas, and its various extensions.

Archytas

Main article: Archytas and ares

Subgroup: 2.3.5.7

Comma list: 64/63

Mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 0]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 0 -2], ⟨0 0 1 0]]

Lattice basis:

3/2 length = 1.0508, 5/4 length = 2.3219
Angle (3/2, 5/4) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1196.9552 ¢, ~3/2 = 707.5215 ¢, ~5/4 = 392.3765 ¢
error map: ⟨-3.045 +2.522 -0.027 +3.952]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.3901 ¢, ~5/4 = 391.5995 ¢
error map: ⟨0.000 +7.435 +5.286 +12.394]

Minimax tuning:

[[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.7/3
[[1 0 0 0⟩, [3/2 1/2 0 -1/4⟩, [3/2 -1/2 1 -1/4⟩, [3 -1 0 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.9/7

Optimal ET sequence: 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd

Badness (Sintel): 0.435

Scales: archytas12, archytas12synch

Ares

Main article: Archytas and ares
See also: Ptolemismic clan § Ares

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99

Mapping: [⟨1 0 0 6 2], ⟨0 1 0 -2 -2], ⟨0 0 1 0 2]]

Mapping to lattice: [⟨0 1 1 -2 0], ⟨0 0 -1 0 -2]]

Lattice basis:

3/2 length = 0.9878, 6/5 length = 1.4312
Angle (3/2, 6/5) = 75.0475 degrees

Optimal tunings:

  • WE: ~2 = 1197.0279 ¢, ~3/2 = 707.9291 ¢, ~5/4 = 390.4812 ¢
error map: ⟨-2.972 +3.002 -1.777 +3.427 +1.898]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.6947 ¢, ~5/4 = 390.0248 ¢
error map: ⟨0.000 +7.740 +3.711 +11.785 +9.342]

Minimax tuning:

[[1 0 0 0 0⟩, [3/2 1/2 0 -1/4 0⟩, [2 0 0 -1/2 1/2⟩, [3 -1 0 1/2 0⟩, [3 -1 0 -1/2 1⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9.9/7

Optimal ET sequence: 12, 15, 22, 27e, 34d, 49, 181bbcddeeee

Badness (Sintel): 0.563

Scales: ares12

Vulcan

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63

Mapping: [⟨1 0 0 6 9], ⟨0 1 0 -2 -2], ⟨0 0 1 0 -1]]

Optimal tunings:

  • CWE: ~2 = 1195.4265 ¢, ~3/2 = 708.2623 ¢, ~5/4 = 399.7405 ¢
error map: ⟨-4.573 +1.734 +4.280 -3.644 +9.550]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.5280 ¢, ~5/4 = 400.4798 ¢
error map: ⟨0.000 +9.573 +14.166 +8.118 +25.146]

Optimal ET sequence: 5, 10, 12, 15, 27e

Badness (Sintel): 0.693

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