Porwell family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>FREEZE
No edit summary
+ intro to hera and semiporwell
 
(30 intermediate revisions by 6 users not shown)
Line 1: Line 1:
__FORCETOC__
{{Technical data page}}
The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.
The '''porwell family''' of [[rank-3 temperament|rank-3]] [[regular temperament]]s [[tempering out|tempers out]] the [[porwell comma]] ({{monzo|legend=1| 11 1 -3 -2 }}, [[ratio]]: 6144/6125).


=Hewuermity=
== Porwell ==
Comma: 6144/6125
Porwell is catalogued as ''hewuermity'' in [[Graham Breed]]'s temperament finder.


7 and 9 limit minimax
[[Subgroup]]: 2.3.5.7


[|1 0 0 0>, |0 1 0 0>, |11/5 1/5 2/5 -2/5>,
[[Comma list]]: [[6144/6125]]
|11/5 1/5 -3/5 3/5>]


Eigenmonzos: 4/3, 7/5
{{Mapping|legend=1| 1 0 1 4 | 0 1 1 -1 | 0 0 -2 3 }}
: mapping generators: ~2, ~3, ~35/32


Map: [<1 0 1 4|, <0 1 1 -1|, <0 0 -2 3|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7101{{c}}, ~3/2 = 702.1785{{c}}, ~35/32 = 157.4296{{c}}
: [[error map]]: {{val| -0.290 -0.066 +0.426 +0.415 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1875{{c}}, ~35/32 = 157.3944{{c}}
: error map: {{val| 0.000 +0.233 +1.085 +1.170 }}


Generators: 2, 3, 35/32
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: {{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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


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


Badness: 0.000142
[[Badness]] (Sintel): 0.628


[[Projection_pair|Projection pair]]s: 3 6125/2048 to 2.5.7
[[Projection pair]]s: <code>3 6125/2048</code> to 2.5.7


=Zeus=
== Zeus ==
Commas: 121/120, 176/175
{{Main| Zeus }}


Minimax tuning
Zeus tempers out [[121/120]], [[176/175]], and [[385/384]]. See [[Biyatismic clan #Zeus]].


[|1 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0&gt;,
== Jupiter ==
|22/9 2/9 1/3 -4/9 0&gt;, |22/9 2/9 -2/3 5/9 0&gt;,
[[Subgroup]]: 2.3.5.7.11
|10/3 2/3 0 -1/3 0&gt;]


Eigenmonzos: 2, 9/7, 7/5
[[Comma list]]: 540/539, 5632/5625


Lattice basis: 11/10, 11/8
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 1 -1 6 | 0 0 -2 3 -8 }}


Map to lattice: [&lt;0 1 -1 2 0|, &lt;0 1 1 -1 1|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7232{{c}}, ~3/2 = 701.9496{{c}}, ~35/32 = 157.4667{{c}}
: [[error map]]: {{val| -0.277 -0.282 +0.149 +0.794 +0.369 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9677{{c}}, ~35/32 = 157.4315{{c}}
: error map: {{val| 0.000 +0.013 +0.791 +1.501 +1.037 }}


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


Map: [&lt;1 0 1 4 2|, &lt;0 1 1 -1 1|, &lt;0 0 -2 3 1|]
[[Badness]] (Sintel): 0.676


Generators: 2, 3, 11/10
== Hera ==
Hera tempers out 8019/8000, the [[trimitone comma]]. It was named by [[Flora Canou]] in 2021 as a more complex counterpart of [[zeus]].


EDOs: 6, 7, 9, 15, 22, 31, 46, 53, 68, 77, 99, 130e, 229e
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000400
[[Comma list]]: 6144/6125, 8019/8000


Projection pairs: 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] [[Hobbits|hobbit]] [[Transversal|transversal]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8410{{c}}, ~3/2 = 701.7105{{c}}, ~35/32 = 157.1441{{c}}
: [[error map]]: {{val| -0.159 -0.403 +0.791 +0.419 -0.268 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8035{{c}}, ~35/32 = 157.1741{{c}}
: error map: {{val| 0.000 -0.153 +1.140 +0.894 +0.232 }}


33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c }}


11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
[[Badness]] (Sintel): 2.68


11/6, 15/8, 64/33, 2
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Zeus11[24] hobbit transversal
Comma list: 351/350, 2080/2079, 4096/4095


33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
Mapping: {{mapping| 1 0 1 4 9 7 | 0 1 1 -1 -3 -2 | 0 0 -2 3 -6 -1 }}


11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
Optimal tunings:
* WE: ~2 = 1199.9170{{c}}, ~3/2 = 701.6155{{c}}, ~35/32 = 157.1791{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6672{{c}}, ~35/32 = 157.1942{{c}}


11/6, 15/8, 64/33, 2
{{Optimal ET sequence|legend=0| 46, 53, 84, 99, 130, 183, 313 }}


[[genus1155zeus|Euler(1155) genus in zeus temperament]]
Badness (Sintel): 1.21


==Tinia==
== Semiporwell ==
Commas: 66/65, 121/120, 176/175
Named by [[Flora Canou]] in 2021, semiporwell tempers out 9801/9800, the [[kalisma]], as well as 16384/16335, the [[semiporwellisma]].


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


EDOs: 7, 9, 15, 24, 31, 77f, 108ef
[[Comma list]]: 6144/6125, 9801/9800


Badness: 0.000808
{{Mapping|legend=1| 2 0 0 11 14 | 0 1 1 -1 -2 | 0 0 2 -3 -1 }}
: mapping generators: ~99/70, ~3, ~128/99


==Tridecimal Zeus==
[[Optimal tuning]]s:
Commas: 121/120, 176/175, 351/350
* [[WE]]: ~99/70 = 599.8385{{c}}, ~3/2 = 702.2722{{c}}, ~128/99 = 442.3712{{c}}
: [[error map]]: {{val| -0.323 -0.006 +0.378 +0.335 +0.152 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.4280{{c}}, ~128/99 = 442.5202{{c}}
: error map: {{val| 0.000 +0.473 +1.155 +1.185 +1.306 }}


13-limit minimax
{{Optimal ET sequence|legend=1| 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee }}


[|1 0 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0 0&gt;,
[[Badness]] (Sintel): 1.51
|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
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


15-limit minimax
Comma list: 1716/1715, 2080/2079, 3584/3575


[|1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;,
Mapping: {{mapping| 2 0 0 11 14 15 | 0 1 1 -1 -2 -1 | 0 0 2 -3 -1 -6 }}
|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
Optimal tunings:  
* WE: ~99/70 = 599.8357{{c}}, ~3/2 = 702.2041{{c}}, ~84/65 = 442.4690{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.3576{{c}}, ~84/65 = 442.6282{{c}}


Lattice basis: 11/10 length 0.7898 11/8 length 1.002
{{Optimal ET sequence|legend=0| 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff }}


Angle(11/10, 11/8) = 106.7439 degrees
Badness (Sintel): 1.14


Map to lattice: [&lt;0 1 -1 2 0 -3|, &lt;0 1 1 -1 1 -2|]
[[Category:Temperament families]]
 
[[Category:Porwell family| ]] <!-- main article -->
Map: [&lt;1 0 1 4 2 7|, &lt;0 1 1 -1 1 -2|, &lt;0 0 -2 3 -1 -1|]
[[Category:Rank 3]]
 
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|]
 
EDOs: 9, 22, 31, 53, 99e, 108, 121, 130, 152, 282, 434de, 465d, 617de, 747def, 899def
 
Badness: 0.000562

Latest revision as of 11:17, 22 April 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The porwell family of rank-3 regular temperaments tempers out the porwell comma (monzo: [11 1 -3 -2⟩, ratio: 6144/6125).

Porwell

Porwell is catalogued as hewuermity in Graham Breed's temperament finder.

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

Optimal tunings:

  • WE: ~2 = 1199.7101 ¢, ~3/2 = 702.1785 ¢, ~35/32 = 157.4296 ¢
error map: ⟨-0.290 -0.066 +0.426 +0.415]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1875 ¢, ~35/32 = 157.3944 ¢
error map: ⟨0.000 +0.233 +1.085 +1.170]

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⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 15, 22, 31, 46, 53, 84, 99, 229, 251, 282, 381

Badness (Sintel): 0.628

Projection pairs: 3 6125/2048 to 2.5.7

Zeus

Main article: Zeus

Zeus tempers out 121/120, 176/175, and 385/384. See Biyatismic clan #Zeus.

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]]

Optimal tunings:

  • WE: ~2 = 1199.7232 ¢, ~3/2 = 701.9496 ¢, ~35/32 = 157.4667 ¢
error map: ⟨-0.277 -0.282 +0.149 +0.794 +0.369]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9677 ¢, ~35/32 = 157.4315 ¢
error map: ⟨0.000 +0.013 +0.791 +1.501 +1.037]

Optimal ET sequence: 22, 31, 53, 84e, 99e, 121, 130, 152, 282, 434de, 465d, 617de, 747cde, 899cdde

Badness (Sintel): 0.676

Hera

Hera tempers out 8019/8000, the trimitone comma. It was named by Flora Canou in 2021 as a more complex counterpart of zeus.

Subgroup: 2.3.5.7.11

Comma list: 6144/6125, 8019/8000

Mapping: [⟨1 0 1 4 9], ⟨0 1 1 -1 -3], ⟨0 0 -2 3 -6]]

Optimal tunings:

  • WE: ~2 = 1199.8410 ¢, ~3/2 = 701.7105 ¢, ~35/32 = 157.1441 ¢
error map: ⟨-0.159 -0.403 +0.791 +0.419 -0.268]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8035 ¢, ~35/32 = 157.1741 ¢
error map: ⟨0.000 -0.153 +1.140 +0.894 +0.232]

Optimal ET sequence: 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c

Badness (Sintel): 2.68

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 4096/4095

Mapping: [⟨1 0 1 4 9 7], ⟨0 1 1 -1 -3 -2], ⟨0 0 -2 3 -6 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9170 ¢, ~3/2 = 701.6155 ¢, ~35/32 = 157.1791 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6672 ¢, ~35/32 = 157.1942 ¢

Optimal ET sequence: 46, 53, 84, 99, 130, 183, 313

Badness (Sintel): 1.21

Semiporwell

Named by Flora Canou in 2021, semiporwell tempers out 9801/9800, the kalisma, as well as 16384/16335, the semiporwellisma.

Subgroup: 2.3.5.7.11

Comma list: 6144/6125, 9801/9800

Mapping: [⟨2 0 0 11 14], ⟨0 1 1 -1 -2], ⟨0 0 2 -3 -1]]

mapping generators: ~99/70, ~3, ~128/99

Optimal tunings:

  • WE: ~99/70 = 599.8385 ¢, ~3/2 = 702.2722 ¢, ~128/99 = 442.3712 ¢
error map: ⟨-0.323 -0.006 +0.378 +0.335 +0.152]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.4280 ¢, ~128/99 = 442.5202 ¢
error map: ⟨0.000 +0.473 +1.155 +1.185 +1.306]

Optimal ET sequence: 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3584/3575

Mapping: [⟨2 0 0 11 14 15], ⟨0 1 1 -1 -2 -1], ⟨0 0 2 -3 -1 -6]]

Optimal tunings:

  • WE: ~99/70 = 599.8357 ¢, ~3/2 = 702.2041 ¢, ~84/65 = 442.4690 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.3576 ¢, ~84/65 = 442.6282 ¢

Optimal ET sequence: 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff

Badness (Sintel): 1.14

Retrieved from "https://en.xen.wiki/index.php?title=Porwell_family&oldid=228251"