Archytas family: Difference between revisions
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= Archytas = | = Archytas = | ||
[[Comma]] | Period: 1\1 | ||
Optimal ([[POTE]]) generators: ~3/2 = 709.3213, ~5/4 = 393.3747 | |||
EDO generators: [[15edo|(9, 5)\15]], [[22edo|(13, 7)\22]], [[27edo|(16, 9)\27]] | |||
Scales: [[archytas12]], [[archytas12synch]] | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | |||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
[[Comma list]]: 64/63 | |||
Mapping: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}] | Mapping: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}] | ||
Mapping [[generator]]s: 2, 3, 5 | Mapping [[generator]]s: ~2, ~3, ~5 | ||
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}] | Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}] | ||
| Line 23: | Line 35: | ||
: [[Eigenmonzo]]s: 2, 6/5, 9/7 | : [[Eigenmonzo]]s: 2, 6/5, 9/7 | ||
{{Val list|legend=1| 5, 7, 10, 12, 22, 27, 49 }} | {{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }} | ||
</div></div> | |||
; 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 = | |||
Period: 1\1 | |||
Optimal ([[POTE]]) generators: ~3/2 = 709.6869, ~5/4 = 391.4507 | |||
EDO generators: [[15edo|(9, 5)\15]], [[22edo|(13, 7)\22]], [[27edo|(16, 9)\27]] | |||
Scales: [[ares12]] | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | |||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
[[Comma list]]: 64/63, 100/99 | |||
[[Comma]] | |||
Mapping: [{{val| 1 0 0 6 2 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 0 2 }}] | Mapping: [{{val| 1 0 0 6 2 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 0 2 }}] | ||
Mapping [[generator]]s: 2, 3, 5 | Mapping [[generator]]s: ~2, ~3, ~5 | ||
Map to lattice: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}] | Map to lattice: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}] | ||
Lattice basis: | Lattice basis: | ||
: 3/2 0.9878, 6/5 1.4312 | : 3/2 length = 0.9878, 6/5 length = 1.4312 | ||
: Angle (3/2, 6/5) = 75.0475 degrees | : Angle (3/2, 6/5) = 75.0475 degrees | ||
| Line 57: | Line 76: | ||
{{Val list|legend=1| 12, 15, 22, 27e, 34d, 49 }} | {{Val list|legend=1| 12, 15, 22, 27e, 34d, 49 }} | ||
Badness: 0. | Badness: 0.469 × 10<sup>-3</sup> | ||
</div></div> | |||
= Vulcan = | = Vulcan = | ||
Period: 1\1 | |||
Optimal ([[POTE]]) generators: ~3/2 = 710.9720, ~5/4 = 401.2699 | |||
EDO generators: [[10edo|(6, 3)\10]], [[15edo|(9, 5)\15]], [[17edo|(10, 6)\17]] | |||
Scales: | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | |||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
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: [{{val| 1 0 0 6 9 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 0 -1 }}] | ||
Mapping generators: 2, 3, 5 | Mapping generators: ~2, ~3, ~5 | ||
{{Val list|legend=1| 5, 10, 12, 15, 27e }} | {{Val list|legend=1| 5, 10, 12, 15, 27e }} | ||
Badness: 0. | Badness: 0.577 × 10<sup>-3</sup> | ||
</div></div> | |||
[[Category:Theory]] | [[Category:Theory]] | ||
Revision as of 07:15, 20 February 2021
The archytas family of temperaments 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.
Archytas
Period: 1\1
Optimal (POTE) generators: ~3/2 = 709.3213, ~5/4 = 393.3747
EDO generators: (9, 5)\15, (13, 7)\22, (16, 9)\27
Scales: archytas12, archytas12synch
Comma list: 64/63
Mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 0]]
Mapping generators: ~2, ~3, ~5
Map 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
Minimax tuning:
- [[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]
- Eigenmonzos: 2, 6/5, 7/5
- [[1 0 0 0⟩, [3/2 1/2 0 -1/4⟩, [3/2 -1/2 1 -1/4⟩, [3 -1 0 1/2⟩]
- Eigenmonzos: 2, 6/5, 9/7
- Music
- Chris Vaisvil - 12 archytas hobbits with ale
- Joel Taylor - Portrait of a Tragedy (Death of a Son) - Archytas12 sync Study 3
Ares
Period: 1\1
Optimal (POTE) generators: ~3/2 = 709.6869, ~5/4 = 391.4507
EDO generators: (9, 5)\15, (13, 7)\22, (16, 9)\27
Scales: ares12
Comma list: 64/63, 100/99
Mapping: [⟨1 0 0 6 2], ⟨0 1 0 -2 -2], ⟨0 0 1 0 2]]
Mapping generators: ~2, ~3, ~5
Map 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
- [[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⟩]
- Eigenmonzos: 2, 11/9, 9/7
Badness: 0.469 × 10-3
Vulcan
Period: 1\1
Optimal (POTE) generators: ~3/2 = 710.9720, ~5/4 = 401.2699
EDO generators: (6, 3)\10, (9, 5)\15, (10, 6)\17
Scales:
Comma list: 56/55, 64/63
Mapping: [⟨1 0 0 6 9], ⟨0 1 0 -2 -2], ⟨0 0 1 0 -1]]
Mapping generators: ~2, ~3, ~5
Badness: 0.577 × 10-3