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Porwell family: Difference between revisions

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{{Technical data page}}
The '''porwell family''' of rank three temperaments tempers out the porwell comma, [[6144/6125]].
The '''porwell family''' of [[rank-3 temperament]]s tempers out the porwell comma, {{monzo| 11 1 -3 -2 }} = [[6144/6125]].


== Hewuermity ==
== Porwell a.k.a. hewuermity ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[6144/6125]]
[[Comma list]]: [[6144/6125]]


[[Mapping]]: [<1 0 1 4|, <0 1 1 -1|, <0 0 -2 3|]
{{Mapping|legend=1| 1 0 1 4 | 0 1 1 -1 | 0 0 -2 3 }}


Mapping generators: ~2, ~3, ~35/32
: mapping generators: ~2, ~3, ~35/32
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.3482, ~35/32 = 157.4677


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit ]]
* 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>]
: {{monzo list| 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
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


{{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 }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 84, 99, 229, 251, 282, 381 }}


[[Badness]]: 0.142 × 10<sup>-3</sup>
[[Badness]]: 0.142 × 10<sup>-3</sup>
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== Zeus ==
== Zeus ==
Subgroup: 2.3.5.7.11
{{Main| Zeus }}


[[Comma list]]: 121/120, 176/175
Zeus tempers out [[121/120]], [[176/175]], and [[385/384]]. See [[Biyatismic clan #Zeus]].


[[Mapping]]: [&lt;1 0 1 4 2|, &lt;0 1 1 -1 1|, &lt;0 0 -2 3 1|]
== Jupiter ==
[[Subgroup]]: 2.3.5.7.11


Mapping generators: 2, 3, 11/10
[[Comma list]]: 540/539, 5632/5625


Map to lattice: [&lt;0 1 -1 2 0|, &lt;0 1 1 -1 1|]
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 1 -1 6 | 0 0 -2 3 -8 }}


Lattice basis:
{{Optimal ET sequence|legend=1| 22, 31, 53, 84e, 99e, 121, 130, 152, 282, 434de, 465d, 617de, 747cde, 899cdde }}
: 11/10, 11/8
: Angle (11/10, 11/8) = 87.464 degrees


[[Minimax tuning]]:  
[[Badness]]: 0.562 × 10<sup>-3</sup>
* [[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


{{Val list|legend=1| 6, 7, 9, 15, 22, 31, 46, 53, 68, 77, 99, 130e, 229e }}
== Hera ==
[[Subgroup]]: 2.3.5.7.11


[[Badness]]: 0.400 × 10<sup>-3</sup>
[[Comma list]]: 6144/6125, 8019/8000


[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
{{Mapping|legend=1| 1 0 1 4 9 | 0 1 1 -1 -3 | 0 0 -2 3 -6 }}


Zeus11[22] [[hobbit]] [[transversal]]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.8035, ~35/32 = 157.1649
: 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
{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c }}
: 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:
[[Badness]]: 2.23 × 10<sup>-3</sup>
* [[genus1155zeus|Euler(1155) genus in zeus temperament]]


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 351/350
Comma list: 351/350, 2080/2079, 4096/4095


Mapping: [&lt;1 0 1 4 2 7|, &lt;0 1 1 -1 1 -2|, &lt;0 0 -2 3 -1 -1|]
Mapping: {{mapping| 1 0 1 4 9 7 | 0 1 1 -1 -3 -2 | 0 0 -2 3 -6 -1 }}


Mapping generators: 2, 3, 11/10
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.6641, ~35/32 = 157.1899


Map to lattice: [&lt;0 1 -1 2 0 -3|, &lt;0 1 1 -1 1 -2|]
{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313 }}


Lattice basis:  
Badness: 1.30 × 10<sup>-3</sup>
: 11/10 length = 0.7898, 11/8 length = 1.002
: Angle (11/10, 11/8) = 106.7439 degrees


Minimax tuning:
== Semiporwell ==
* 13-odd-limit
[[Subgroup]]: 2.3.5.7.11
: [|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


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 }}
[[Comma list]]: 6144/6125, 9801/9800


Badness: 0.934 × 10<sup>-3</sup>
{{Mapping|legend=1| 2 0 0 11 14 | 0 1 1 -1 -2 | 0 0 2 -3 -1 }}


Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
: mapping generators: ~99/70, ~3, ~128/99


Zeus13[22] hobbit transversal
[[Optimal tuning]] ([[POTE]]): ~99/70 = 1\2, ~3/2 = 702.4613, ~128/99 = 442.4903
: 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 ===
{{Optimal ET sequence|legend=1| 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee }}
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 121/120, 176/175
[[Badness]]: 1.25 × 10<sup>-3</sup>


Mapping: [&lt;1 0 1 4 2 2|, &lt;0 1 1 -1 1 1|, &lt;0 0 -2 3 -1 -1|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 7, 9, 15, 24, 31, 77f, 108ef }}
Comma list: 1716/1715, 2080/2079, 3584/3575


Badness: 0.808 × 10<sup>-3</sup>
Mapping: {{mapping| 2 0 0 11 14 15 | 0 1 1 -1 -2 -1 | 0 0 2 -3 -1 -6 }}


== Jupiter  ==
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 702.3964, ~84/65 = 442.5902
Subgroup: 2.3.5.7.11


[[Comma list]]: 540/539, 5632/5625
{{Optimal ET sequence|legend=1| 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff }}


[[Mapping]]: [&lt;1 0 1 4 -5|, &lt;0 1 1 -1 6|, &lt;0 0 2 -3 8|]
Badness: 1.22 × 10<sup>-3</sup>
 
{{Val list|legend=1| 9, 22, 31, 53, 99e, 108, 121, 130, 152, 282, 434de, 465d, 617de, 747def, 899def }}
 
[[Badness]]: 0.562 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Porwell family| ]] <!-- main article -->
[[Category:Porwell family| ]] <!-- main article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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