Diaschismic rank three family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Wikispaces>FREEZE
No edit summary
m Cleanup (1/2)
Line 1: Line 1:
These are the rank three temperaments tempering out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
These are the rank-3 temperaments tempering out the [[diaschisma]], 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.


==Julius==
== Julius ==
Commas: 176/175, 896/891
Subgroup: 2.3.5.7.11


Minimax tuning
[[Comma list]]: 176/175, 896/891


[|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>,
[[Mapping]]: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1 |]
|-1/8 3/2 -5/4 0 1>, |11/8 -1/2 -1/4 0 1>]
 
Eigenmonzos: 2, 12/11, 10/9
 
Lattice basis: 3/2, 9/7


Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|]
Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|]


Angle(3/2, 9/7) = 82.289 degrees
Lattice basis:
: 3/2, 9/7
: Angle (3/2, 9/7) = 82.289 degrees


Map: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1 |]
[[Minimax tuning]]:
* [[11-odd-limit]]
: [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |-1/8 3/2 -5/4 0 1>, |11/8 -1/2 -1/4 0 1>]
: Eigenmonzos: 2, 12/11, 10/9


EDOs: 58, 68, 80, 126
{{Val list|legend=1| 58, 68, 80, 126 }}


Badness: 0.000728
[[Badness]]: 0.728 × 10<sup>-3</sup>


Julius[22] [[Hobbits|hobbit]] [[Transversal|transversal]]:
Julius[22] [[hobbit]] [[transversal]]:
 
: 28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2
28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2


Julius[24] hobbit transversal
Julius[24] hobbit transversal
: 28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2


28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
===13-limit===
Commas: 176/175, 325/324, 896/891
 
Map: [&lt;2 0 11 0 14 -18|, &lt;0 1 -2 0 -4 8|, &lt;0 0 0 1 1 0|]


EDOs: 12, 46, 68, 80
Comma list: 176/175, 325/324, 896/891


Badness: 0.00178
Mapping: [&lt;2 0 11 0 14 -18|, &lt;0 1 -2 0 -4 8|, &lt;0 0 0 1 1 0|]


==Augustus==
Vals: {{Val list| 12, 46, 68, 80 }}
Commas: 385/384, 1232/1215


Minimax tuning
Badness: 1.78 × 10<sup>-3</sup>


[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,
== Augustus ==
|45/16 3/4 -3/8 1/2 -1/2&gt;, |45/16 3/4 -3/8 -1/2 1/2&gt;]
Subgroup: 2.3.5.7.11


Eigenmonzos: 2, 10/9, 14/11
[[Comma list]]: 385/384, 1232/1215


Map: [&lt;2 0 11 0 3|, &lt;0 1 -2 0 3|, &lt;0 0 0 1 -1|]
[[Mapping]]: [&lt;2 0 11 0 3|, &lt;0 1 -2 0 3|, &lt;0 0 0 1 -1|]


EDOs: 46, 68, 114, 124
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;, |45/16 3/4 -3/8 1/2 -1/2&gt;, |45/16 3/4 -3/8 -1/2 1/2&gt;]
: Eigenmonzos: 2, 10/9, 14/11


Badness: 0.00158
{{Val list|legend=1| 46, 68, 114, 124 }}


==Tiberius==
[[Badness]]: 1.58 × 10<sup>-3</sup>
Commas: 2048/2025, 441/440


Minimax tuning
== Tiberius ==
Subgroup: 2.3.5.7.11


[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,
[[Comma list]]: 441/440, 2048/2025
|0 0 0 1 0&gt;, |-3 2 -1 2 0&gt;]


Eigenmonzos: 2, 9/5, 7
[[Mapping]]: [&lt;2 0 11 0 -17|, &lt;0 1 -2 0 4|, &lt;0 0 0 1 2|]


Map: [&lt;2 0 11 0 -17|, &lt;0 1 -2 0 4|, &lt;0 0 0 1 2|]
[[Minimax tuning]]:
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;, |0 0 0 1 0&gt;, |-3 2 -1 2 0&gt;]
: Eigenmonzos: 2, 9/5, 7


EDOs: 58, 148, 160
{{Val list|legend=1| 58, 148, 160 }}


Badness: 0.00171
[[Badness]]: 1.71 × 10<sup>-3</sup>


==Claudius==
== Claudius ==
Commas: 2048/2025, 540/539
Subgroup: 2.3.5.7.11


Minimax tuning
[[Comma list]]: 540/539, 2048/2025


[|1 0 0 0 0&gt;, |9/7 0 -3/7 2/7 1/7&gt;, |41/14 0 6/7 -4/7 -2/7&gt;,
[[Mapping]]: [&lt;2 0 11 0 15|, &lt;0 1 -2 0 1|, &lt;0 0 0 1 -2|]
|41/14 0 -1/7 3/7 -2/7&gt;, |41/14 0 -1/7 -4/7 5/7&gt;]


Eigenmonzos: 2, 7/5, 11/5
[[Minimax tuning]]:
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |9/7 0 -3/7 2/7 1/7&gt;, |41/14 0 6/7 -4/7 -2/7&gt;, |41/14 0 -1/7 3/7 -2/7&gt;, |41/14 0 -1/7 -4/7 5/7&gt;]
: Eigenmonzos: 2, 7/5, 11/5


Map: [&lt;2 0 11 0 15|, &lt;0 1 -2 0 1|, &lt;0 0 0 1 -2|]
{{Val list|legend=1| 58, 80 }}


EDOs: 58, 80
[[Badness]]: 1.97 × 10<sup>-3</sup>


Badness: 0.00197
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Diaschismic rank-3 family| ]] <!-- main article -->
[[Category:Rank 3]]

Revision as of 12:03, 14 June 2021

These are the rank-3 temperaments tempering out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Julius

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891

Mapping: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1 |]

Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|]

Lattice basis:

3/2, 9/7
Angle (3/2, 9/7) = 82.289 degrees

Minimax tuning:

[|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |-1/8 3/2 -5/4 0 1>, |11/8 -1/2 -1/4 0 1>]
Eigenmonzos: 2, 12/11, 10/9

Template:Val list

Badness: 0.728 × 10-3

Julius[22] hobbit transversal:

28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2

Julius[24] hobbit transversal

28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 325/324, 896/891

Mapping: [<2 0 11 0 14 -18|, <0 1 -2 0 -4 8|, <0 0 0 1 1 0|]

Vals: Template:Val list

Badness: 1.78 × 10-3

Augustus

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1232/1215

Mapping: [<2 0 11 0 3|, <0 1 -2 0 3|, <0 0 0 1 -1|]

Minimax tuning:

[|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |45/16 3/4 -3/8 1/2 -1/2>, |45/16 3/4 -3/8 -1/2 1/2>]
Eigenmonzos: 2, 10/9, 14/11

Template:Val list

Badness: 1.58 × 10-3

Tiberius

Subgroup: 2.3.5.7.11

Comma list: 441/440, 2048/2025

Mapping: [<2 0 11 0 -17|, <0 1 -2 0 4|, <0 0 0 1 2|]

Minimax tuning:

[|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |0 0 0 1 0>, |-3 2 -1 2 0>]
Eigenmonzos: 2, 9/5, 7

Template:Val list

Badness: 1.71 × 10-3

Claudius

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2048/2025

Mapping: [<2 0 11 0 15|, <0 1 -2 0 1|, <0 0 0 1 -2|]

Minimax tuning:

[|1 0 0 0 0>, |9/7 0 -3/7 2/7 1/7>, |41/14 0 6/7 -4/7 -2/7>, |41/14 0 -1/7 3/7 -2/7>, |41/14 0 -1/7 -4/7 5/7>]
Eigenmonzos: 2, 7/5, 11/5

Template:Val list

Badness: 1.97 × 10-3

Retrieved from "https://en.xen.wiki/index.php?title=Diaschismic_rank_three_family&oldid=72587"