Archytas family: Difference between revisions

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+ WE tunings
- CTE & POTE tunings
 
Line 26: Line 26:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}, ~5/4 = 391.5995{{c}}
* [[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 }}
: error map: {{val| 0.000 +7.435 +5.286 +12.394 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.5949{{c}}, ~5/4 = 386.3137{{c}}
: [[error map]]: {{val| 0.000 +7.640 0.000 +11.984 }} -->
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3213{{c}}, ~5/4 = 393.3747{{c}}
: error map: {{val| 0.000 +7.366 +7.061 +12.532 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 66: Line 62:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.6947{{c}}, ~5/4 = 390.0248{{c}}
* [[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 }}
: error map: {{val| 0.000 +7.740 +3.711 +11.785 +9.342 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.7182{{c}}, ~5/4 = 385.7114{{c}}
: [[error map]]: {{val| 0.000 +7.763 -0.602 +11.738 +0.668 }} -->
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.6869{{c}}, ~5/4 = 391.4507{{c}}
: error map: {{val| 0.000 +7.732 +5.137 +11.800 +12.210 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 94: Line 86:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5280{{c}}, ~5/4 = 400.4798{{c}}
* [[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 }}
: error map: {{val| 0.000 +9.573 +14.166 +8.118 +25.146 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.9739{{c}}, ~5/4 = 397.0040{{c}}
: [[error map]]: {{val| 0.000 +12.019 +10.690 +3.226 +23.730 }} -->
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.9720{{c}}, ~5/4 = 401.2699{{c}}
: error map: {{val| 0.000 +9.017 +14.956 +9.230 +25.468 }} -->


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

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

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 sequence5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd

Badness (Sintel): 0.435

Scales: archytas12, archytas12synch

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 sequence12, 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 sequence5, 10, 12, 15, 27e

Badness (Sintel): 0.693