Archytas family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
m Cleanup
Databoxified
Line 3: Line 3:
= Archytas =
= Archytas =


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


Scales: [[archytas12]], [[archytas12synch]]
</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 =


== Music ==
Period: 1\1


Chris Vaisvil
Optimal ([[POTE]]) generators: ~3/2 = 709.6869, ~5/4 = 391.4507


* [http://micro.soonlabel.com/hobbit_scales/daily201031--hobbit-archytas12.mp3 12 archytas hobbits with ale]
EDO generators: [[15edo|(9, 5)\15]], [[22edo|(13, 7)\22]], [[27edo|(16, 9)\27]]


Joel Taylor
Scales: [[ares12]]


* [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]
<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">


= Ares =
[[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: [{{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: 0.000469
Badness: 0.469 × 10<sup>-3</sup>


Scales: [[ares12]]
</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: 0.000577
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

Technical data

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

Template:Val list

Music

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

Technical data

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

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]
Eigenmonzos: 2, 11/9, 9/7

Template:Val list

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:

Technical data

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

Template:Val list

Badness: 0.577 × 10-3