Archytas family: Difference between revisions

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Archytas temperament [[tempering_out|tempers out]] [[64/63|64/63]], and thereby identifies the [[otonal|otonal]] [[tetrad|tetrad]] with the [[dominant_seventh_chord|dominant seventh chord]], which is familiar from [[12edo|12edo]] and also a feature of [[22edo|22edo]] and [[27edo|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.  


[[Comma|Comma]]: 64/63
See [[Archytas clan]] for the rank-2 temperament without the last generator of archytas, and its various extensions.


7-limit minimax
== Archytas ==
{{Main| Archytas and ares }}


[|1 0 0 0>, |2 1/3 0 -1/3>,
[[Subgroup]]: 2.3.5.7
|2 -2/3 1 -1/3>, |2 -2/3 0 2/3>]


[[Eigenmonzo|Eigenmonzos]]: 2, 6/5, 7/5
[[Comma list]]: [[64/63]]


9-limit minimax
{{Mapping|legend=1| 1 0 0 6 | 0 1 0 -2 | 0 0 1 0 }}


[|1 0 0 0>, |3/2 1/2 0 -1/4>,  
: mapping generators: ~2, ~3, ~5
|3/2 -1/2 1 -1/4>, |3 -1 0 1/2>]


[[Eigenmonzo|Eigenmonzos]]: 2, 6/5, 9/7
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]


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


Angle(3/2, 5/4) = 90 degrees
[[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 }}


Map to lattice: [<0 1 0 -2|, <0 0 1 0|]
[[Minimax tuning]]:  
* [[7-odd-limit]]
: {{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-interval (eigenmonzo) basis]]: 2.5/3.7/3
* [[9-odd-limit]]
: {{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-interval (eigenmonzo) basis]]: 2.5/3.9/7


Map: [<1 0 0 6|, <0 1 0 -2|, <0 0 1 0|]
{{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }}


[[generator|Generators]]: 2, 3, 5
[[Badness]] (Sintel): 0.435


[[EDO|Edos]]: [[7edo|7]], [[10edo|10]], [[12edo|12]], [[15edo|15]], [[22edo|22]], [[27edo|27]], [[37edo|37]], [[49edo|49]], [[59edo|59]]
Scales: [[archytas12]], [[archytas12synch]]


Scales: [[archytas12|archytas12]], [[archytas12synch|archytas12synch]]
== Ares ==
{{Main| Archytas and ares }}
{{See also| Ptolemismic clan #Ares }}


Music: Chris Vaisvil
[[Subgroup]]: 2.3.5.7.11


[http://micro.soonlabel.com/hobbit_scales/daily201031--hobbit-archytas12.mp3 12 archytas hobbits with ale]
[[Comma list]]: 64/63, 100/99


Joel Taylor
{{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }}


[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]
[[Mapping to lattice]]: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]


==11 limit children==
Lattice basis:
: 3/2 length = 0.9878, 6/5 length = 1.4312
: Angle (3/2, 6/5) = 75.0475 degrees


==Ares==
[[Optimal tuning]]s:
[[Comma|Commas]]: 64/63, 100/99
* [[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]]
[[Minimax tuning]]:
* [[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 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11/9.9/7


[|1 0 0 0 0>, |3/2 1/2 0 -1/4 0>,  
{{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}
|2 0 0 -1/2 1/2>, |3 -1 0 1/2 0>,  
|3 -1 0 -1/2 1>]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 9/7
[[Badness]] (Sintel): 0.563


Lattice basis: 3/2 0.9878 6/5 1.4312
Scales: [[ares12]]


Angle(3/2, 6/5) = 75.0475 degrees
== Vulcan ==
[[Subgroup]]: 2.3.5.7.11


Map to lattice: [<0 1 1 -2 0|, <0 0 -1 0 -2|]
[[Comma list]]: 56/55, 64/63


Map: [<1 0 0 6 2|, <0 1 0 -2 -2|, <0 0 1 0 2|]
{{Mapping|legend=1| 1 0 0 6 9 | 0 1 0 -2 -2 | 0 0 1 0 -1 }}


[[generator|Generators]]: 2, 3, 5
[[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 }}


[[EDO|Edos]]: [[12edo|12]], [[15edo|15]], [[22edo|22]], [[37edo|37]], [[49edo|49]]
{{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }}


Badness: 0.000469
[[Badness]] (Sintel): 0.693


Scales: [[ares12|ares12]]
[[Category:Temperament families]]
 
[[Category:Pages with mostly numerical content]]
==Vulcan==
[[Category:Archytas family| ]] <!-- main article -->
Commas: 56/55, 64/63
 
Map: [&lt;1 0 0 6 9|, &lt;0 1 0 -2 -2|, &lt;0 0 1 0 -1|]
 
Generators: 2, 3, 5
 
EDOs: 5, 10, 12, 15
 
Badness: 0.000577
 
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Archytas]]
[[Category:Rank 3]]
[[Category:Rank 3]]

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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