Porwell family: Difference between revisions

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The '''porwell family''' of rank three temperaments tempers out the porwell comma, [[6144/6125]].
The '''porwell family''' of rank three temperaments tempers out the porwell comma, [[6144/6125]].


== Hewuermity ==
== Hewuermity ==
Comma: 6144/6125
Subgroup: 2.3.5.7


7 and 9 limit minimax
[[Comma list]]: [[6144/6125]]


[|1 0 0 0>, |0 1 0 0>, |11/5 1/5 2/5 -2/5>,
[[Mapping]]: [<1 0 1 4|, <0 1 1 -1|, <0 0 -2 3|]
|11/5 1/5 -3/5 3/5>]


Eigenmonzos: 4/3, 7/5
Mapping generators: ~2, ~3, ~35/32


Map: [<1 0 1 4|, <0 1 1 -1|, <0 0 -2 3|]
[[Minimax tuning]]:  
* 7- and [[9-odd-limit ]]
: [|1 0 0 0>, |0 1 0 0>, |11/5 1/5 2/5 -2/5>, |11/5 1/5 -3/5 3/5>]
: Eigenmonzos: 4/3, 7/5


Generators: 2, 3, 35/32
{{Val list|legend=1| 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 229, 251, 282, 381 }}


Edos: 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 229, 251, 282, 381
[[Badness]]: 0.142 × 10<sup>-3</sup>
 
Badness: 0.000142


[[Projection pair]]s: 3 6125/2048 to 2.5.7
[[Projection pair]]s: 3 6125/2048 to 2.5.7


== Zeus ==
== Zeus ==
Commas: 121/120, 176/175
Subgroup: 2.3.5.7.11


Minimax tuning
[[Comma list]]: 121/120, 176/175


[|1 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0&gt;,
[[Mapping]]: [&lt;1 0 1 4 2|, &lt;0 1 1 -1 1|, &lt;0 0 -2 3 1|]
|22/9 2/9 1/3 -4/9 0&gt;, |22/9 2/9 -2/3 5/9 0&gt;,
|10/3 2/3 0 -1/3 0&gt;]


Eigenmonzos: 2, 9/7, 7/5
Mapping generators: 2, 3, 11/10
 
Lattice basis: 11/10, 11/8


Map to lattice: [&lt;0 1 -1 2 0|, &lt;0 1 1 -1 1|]
Map to lattice: [&lt;0 1 -1 2 0|, &lt;0 1 1 -1 1|]


Angle(11/10, 11/8) = 87.464 degrees
Lattice basis:
 
: 11/10, 11/8
Map: [&lt;1 0 1 4 2|, &lt;0 1 1 -1 1|, &lt;0 0 -2 3 1|]
: Angle (11/10, 11/8) = 87.464 degrees
 
Generators: 2, 3, 11/10


EDOs: 6, 7, 9, 15, 22, 31, 46, 53, 68, 77, 99, 130e, 229e
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0&gt;, |22/9 2/9 1/3 -4/9 0&gt;, |22/9 2/9 -2/3 5/9 0&gt;, |10/3 2/3 0 -1/3 0&gt;]
: [[Eigenmonzo]]s: 2, 9/7, 7/5


Badness: 0.000400
{{Val list|legend=1| 6, 7, 9, 15, 22, 31, 46, 53, 68, 77, 99, 130e, 229e }}


Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
[[Badness]]: 0.400 × 10<sup>-3</sup>


Zeus11[22] [[Hobbits|hobbit]] [[transversal]]
[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5


33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
Zeus11[22] [[hobbit]] [[transversal]]
 
: 33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
: 11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
 
: 11/6, 15/8, 64/33, 2
11/6, 15/8, 64/33, 2


Zeus11[24] hobbit transversal
Zeus11[24] hobbit transversal
: 33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
: 11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
: 11/6, 15/8, 64/33, 2


33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
Scales:
* [[genus1155zeus|Euler(1155) genus in zeus temperament]]


11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


11/6, 15/8, 64/33, 2
Comma list: 121/120, 176/175, 351/350


[[genus1155zeus|Euler(1155) genus in zeus temperament]]
Mapping: [&lt;1 0 1 4 2 7|, &lt;0 1 1 -1 1 -2|, &lt;0 0 -2 3 -1 -1|]


=== Tinia  ===
Mapping generators: 2, 3, 11/10
Commas: 66/65, 121/120, 176/175


Map: [&lt;1 0 1 4 2 2|, &lt;0 1 1 -1 1 1|, &lt;0 0 -2 3 -1 -1|]
Map to lattice: [&lt;0 1 -1 2 0 -3|, &lt;0 1 1 -1 1 -2|]


EDOs: 7, 9, 15, 24, 31, 77f, 108ef
Lattice basis:  
: 11/10 length = 0.7898, 11/8 length = 1.002
: Angle (11/10, 11/8) = 106.7439 degrees


Badness: 0.000808
Minimax tuning:  
* 13-odd-limit
: [|1 0 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0 0&gt;, |22/9 2/9 1/3 -4/9 0 0&gt;, |22/9 2/9 -2/3 5/9 0 0&gt;, |10/3 2/3 0 -1/3 0 0&gt;, |14/3 -8/3 1 1/3 0 0&gt;]
: Eigenmonzos: 2, 9/7, 7/5
* 15-odd-limit
: [|1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;, |11/5 1/5 2/5 -2/5 0 0&gt;,|11/5 1/5 -3/5 3/5 0 0&gt;, |13/5 3/5 1/5 -1/5 0 0&gt;, |38/5 -12/5 1/5 -1/5 0 0&gt;]
: Eigenmonzos: 2, 4/3, 7/5


=== Tridecimal Zeus  ===
Vals: {{Val list| 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 130e }}
Commas: 121/120, 176/175, 351/350


13-limit minimax
Badness: 0.934 × 10<sup>-3</sup>


[|1 0 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0 0&gt;,
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
|22/9 2/9 1/3 -4/9 0 0&gt;, |22/9 2/9 -2/3 5/9 0 0&gt;,
|10/3 2/3 0 -1/3 0 0&gt;, |14/3 -8/3 1 1/3 0 0&gt;]


Eigenmonzos: 2, 9/7, 7/5
Zeus13[22] hobbit transversal
: 260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2


15-limit minimax
=== Tinia ===
Subgroup: 2.3.5.7.11.13


[|1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;,
Comma list: 66/65, 121/120, 176/175
|11/5 1/5 2/5 -2/5 0 0&gt;,|11/5 1/5 -3/5 3/5 0 0&gt;,
|13/5 3/5 1/5 -1/5 0 0&gt;, |38/5 -12/5 1/5 -1/5 0 0&gt;]


Eigenmonzos: 2, 4/3, 7/5
Mapping: [&lt;1 0 1 4 2 2|, &lt;0 1 1 -1 1 1|, &lt;0 0 -2 3 -1 -1|]


Lattice basis: 11/10 length 0.7898 11/8 length 1.002
Vals: {{Val list| 7, 9, 15, 24, 31, 77f, 108ef }}


Angle(11/10, 11/8) = 106.7439 degrees
Badness: 0.808 × 10<sup>-3</sup>


Map to lattice: [&lt;0 1 -1 2 0 -3|, &lt;0 1 1 -1 1 -2|]
== Jupiter  ==
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 1 4 2 7|, &lt;0 1 1 -1 1 -2|, &lt;0 0 -2 3 -1 -1|]
[[Comma list]]: 540/539, 5632/5625
 
Generators: 2, 3, 11/10
 
Edos: 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 130e
 
Badness: 0.000934
 
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
 
Zeus13[22] hobbit transversal
 
260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2
 
== Jupiter  ==
Commas: 540/539, 5632/5625


Map: [&lt;1 0 1 4 -5|, &lt;0 1 1 -1 6|, &lt;0 0 2 -3 8|]
[[Mapping]]: [&lt;1 0 1 4 -5|, &lt;0 1 1 -1 6|, &lt;0 0 2 -3 8|]


EDOs: 9, 22, 31, 53, 99e, 108, 121, 130, 152, 282, 434de, 465d, 617de, 747def, 899def
{{Val list|legend=1| 9, 22, 31, 53, 99e, 108, 121, 130, 152, 282, 434de, 465d, 617de, 747def, 899def }}


Badness: 0.000562
[[Badness]]: 0.562 × 10<sup>-3</sup>


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Porwell]]
[[Category:Porwell family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 12:48, 14 June 2021

The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.

Hewuermity

Subgroup: 2.3.5.7

Comma list: 6144/6125

Mapping: [<1 0 1 4|, <0 1 1 -1|, <0 0 -2 3|]

Mapping generators: ~2, ~3, ~35/32

Minimax tuning:

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

Template:Val list

Badness: 0.142 × 10-3

Projection pairs: 3 6125/2048 to 2.5.7

Zeus

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175

Mapping: [<1 0 1 4 2|, <0 1 1 -1 1|, <0 0 -2 3 1|]

Mapping generators: 2, 3, 11/10

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

Lattice basis:

11/10, 11/8
Angle (11/10, 11/8) = 87.464 degrees

Minimax tuning:

[|1 0 0 0 0>, |11/9 10/9 -1/3 -2/9 0>, |22/9 2/9 1/3 -4/9 0>, |22/9 2/9 -2/3 5/9 0>, |10/3 2/3 0 -1/3 0>]
Eigenmonzos: 2, 9/7, 7/5

Template:Val list

Badness: 0.400 × 10-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5

Zeus11[22] hobbit transversal

33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
11/6, 15/8, 64/33, 2

Zeus11[24] hobbit transversal

33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
11/6, 15/8, 64/33, 2

Scales:

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350

Mapping: [<1 0 1 4 2 7|, <0 1 1 -1 1 -2|, <0 0 -2 3 -1 -1|]

Mapping generators: 2, 3, 11/10

Map to lattice: [<0 1 -1 2 0 -3|, <0 1 1 -1 1 -2|]

Lattice basis:

11/10 length = 0.7898, 11/8 length = 1.002
Angle (11/10, 11/8) = 106.7439 degrees

Minimax tuning:

  • 13-odd-limit
[|1 0 0 0 0 0>, |11/9 10/9 -1/3 -2/9 0 0>, |22/9 2/9 1/3 -4/9 0 0>, |22/9 2/9 -2/3 5/9 0 0>, |10/3 2/3 0 -1/3 0 0>, |14/3 -8/3 1 1/3 0 0>]
Eigenmonzos: 2, 9/7, 7/5
  • 15-odd-limit
[|1 0 0 0 0 0>, |0 1 0 0 0 0>, |11/5 1/5 2/5 -2/5 0 0>,|11/5 1/5 -3/5 3/5 0 0>, |13/5 3/5 1/5 -1/5 0 0>, |38/5 -12/5 1/5 -1/5 0 0>]
Eigenmonzos: 2, 4/3, 7/5

Vals: Template:Val list

Badness: 0.934 × 10-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5

Zeus13[22] hobbit transversal

260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2

Tinia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 176/175

Mapping: [<1 0 1 4 2 2|, <0 1 1 -1 1 1|, <0 0 -2 3 -1 -1|]

Vals: Template:Val list

Badness: 0.808 × 10-3

Jupiter

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625

Mapping: [<1 0 1 4 -5|, <0 1 1 -1 6|, <0 0 2 -3 8|]

Template:Val list

Badness: 0.562 × 10-3

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