Horwell family: Difference between revisions

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The '''horwell family''' of rank-3 ~~temperaments~~ tempers out the [[horwell comma]]~~, 65625/65536 =~~ {{monzo| -16 1 5 1 }}.

The '''horwell family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536).

== Horwell ==

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 65625/65536

~~[[Mapping]]: [~~{{~~val~~| 1 0 0 16 ~~}}, {{val~~| 0 1 0 -1 ~~}}, {{val~~| 0 0 1 -5 }}]

: mapping generators: ~2, ~3, ~5

~~Mapping generators: ~2, ~3, ~5~~

[[Mapping to lattice]]: [{{val| 0 -1 1 -4 }}, {{val| 0 -1 0 -1 }}]

Mapping to lattice: [{{val| 0 -1 1 -4 }}, {{val| 0 -1 0 -1 }}]

Lattice basis:

: 5/4 length = 0.6213, 16/15 length = 1.3863

: Angle (5/4, 16/15) = 64.6379

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0937{{c}}, ~3/2 = 701.8466{{c}}, ~5/4 = 385.9684{{c}}

: [[error map]]: {{val| +0.094 -0.015 -0.158 -0.046 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8243{{c}}, ~5/4 = 385.9221{{c}}

: error map: {{val| 0.000 -0.131 -0.392 -0.261 }}

[[Minimax tuning]]:

* [[7-odd-limit]]

: [{{monzo| 1 0 0 0 ~~}}, {{monzo~~| 16/7 6/7 -5/7 -1/7 ~~}}, {{monzo~~| 16/7 -1/7 2/7 -1/7 ~~}}, {{monzo~~| 16/7 -1/7 -5/7 6/7 }}]

: {{monzo list| 1 0 0 0 | 16/7 6/7 -5/7 -1/7 | 16/7 -1/7 2/7 -1/7 | 16/7 -1/7 -5/7 6/7 }}

: [[Eigenmonzo]]s: 2, 7/~~6, 6/5~~

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7/3

* [[9-odd-limit]]

: [{{monzo| 1 0 0 0 ~~}}, {{monzo~~| 16/13 12/13 -5/13 -1/13 ~~}}, {{monzo~~| 32/13 -2/13 3/13 -2/13 ~~}}, {{monzo~~| 32/13 -2/13 -10/13 11/13 }}]

: {{monzo list| 1 0 0 0 | 16/13 12/13 -5/13 -1/13 | 32/13 -2/13 3/13 -2/13 | 32/13 -2/13 -10/13 11/13 }}

: [[Eigenmonzo]]s: 2, 9/7~~, 7/5~~

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.9/7

{{~~Val list~~|legend=1| 22, 31, 53, 84, 87, 118, 140, 171, 1592c, 1763c, 1934c, 2074c, 2105c, 2245cd, 2416cd, ~~2587cd, 2758cd, 2929cd, 3100cd, 3271ccd~~, 3442ccd, 3613ccd, 5034bcccdd, 5205bcccdd }}

{{Optimal ET sequence|legend=1| 22, 31, 53, 84, 87, 118, 140, 171, 1592c, 1763c, 1934c, 2074c, 2105c, 2245cd, 2416cd, …, 3442ccd, 3613ccd, 5034bcccdd, 5205bcccdd }}

[[Badness]]: 0.~~173 × 10-3~~

[[Badness]] (Sintel): 0.761

[[Projection pair]]: 7 65536/9375

[[Projection pair]]: <code>7 65536/9375</code>

== Zelda ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 385/384, 3388/3375

~~[[Mapping]]: [~~{{~~val~~| 1 0 0 16 -9 }}, {{val| 0 1 0 -1 2 }}, {{val| 0 0 1 -~~5 4~~ }}]

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.2765{{c}}, ~3/2 = 702.1662{{c}}, ~5/4 = 386.2362{{c}}

: [[error map]]: {{val| +0.276 +0.488 +0.475 -0.791 -1.764 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1206{{c}}, ~5/4 = 386.1124{{c}}

: error map: {{val| 0.000 +0.166 -0.201 -1.508 -2.627 }}

{{~~Val list~~|legend=1| 22, 31, 53, 84, 87, 118, 258e, 376de, ~~547de~~ }}

{{Optimal ET sequence|legend=1| 22, 31, 53, 84e, 87, 118, 258e, 376de, 547dee }}

[[Badness]]: 0.~~642 × 10-3~~

[[Badness]] (Sintel): 0.771

== Horwellic ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 3025/3024, 65625/65536

[[~~Mapping~~]]: [{{~~val| 1 0 0 16 10~~ }}, {{val| 0 1 0 -~~1 1~~ }}, {{val| 0 0 2 -10 -7 }}]

: mapping generators: ~2, ~3, ~352/315

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0937{{c}}, ~3/2 = 701.8440{{c}}, ~352/315 = 192.9848{{c}}

: [[error map]]: {{val| +0.094 -0.017 -0.157 -0.049 +0.007 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8212{{c}}, ~352/315 = 192.9618{{c}}

: error map: {{val| 0.000 -0.134 -0.390 -0.265 -0.229 }}

{{~~Val list~~|legend=1| 31, 75e, 87, 118, 193, 224, 311, 342, 1592c, 1934ce, 2276cde, 2587cde, 2929cde, 3271ccde, 3613ccde, 5205bcccddee }}

{{Optimal ET sequence|legend=1| 25e, 31, 75e, 87, 118, 193, 224, 311, 342, 1592c, 1934ce, 2276cde, 2587cde, 2929cde, 3271ccde, 3613ccde, 5205bcccddee }}

[[Badness]]: 0.~~505 × 10-3~~

[[Badness]] (Sintel): 0.607

~~[[Category:Regular temperament theory]]~~

[[Category:Temperament families]]

[[Category:Temperament ~~family~~]]

[[Category:Horwell family| ]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-The '''horwell family''' of rank-3 temperaments tempers out the [[horwell comma]], 65625/65536 = {{monzo| -16 1 5 1 }}.
+{{Technical data page}}
+The '''horwell family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536).
 == Horwell ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 65625/65536
-[[Mapping]]: [{{val| 1 0 0 16 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -5 }}]
+{{Mapping|legend=1| 1 0 0 16 | 0 1 0 -1 | 0 0 1 -5 }}
+: mapping generators: ~2, ~3, ~5
-Mapping generators: ~2, ~3, ~5
+[[Mapping to lattice]]: [{{val| 0 -1 1 -4 }}, {{val| 0 -1 0 -1 }}]
-Mapping to lattice: [{{val| 0 -1 1 -4 }}, {{val| 0 -1 0 -1 }}]
 Lattice basis:
 : 5/4 length = 0.6213, 16/15 length = 1.3863
 : Angle (5/4, 16/15) = 64.6379
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0937{{c}}, ~3/2 = 701.8466{{c}}, ~5/4 = 385.9684{{c}}
+: [[error map]]: {{val| +0.094 -0.015 -0.158 -0.046 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8243{{c}}, ~5/4 = 385.9221{{c}}
+: error map: {{val| 0.000 -0.131 -0.392 -0.261 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 16/7 6/7 -5/7 -1/7 }}, {{monzo| 16/7 -1/7 2/7 -1/7 }}, {{monzo| 16/7 -1/7 -5/7 6/7 }}]
+: {{monzo list| 1 0 0 0 | 16/7 6/7 -5/7 -1/7 | 16/7 -1/7 2/7 -1/7 | 16/7 -1/7 -5/7 6/7 }}
-: [[Eigenmonzo]]s: 2, 7/6, 6/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7/3
 * [[9-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 16/13 12/13 -5/13 -1/13 }}, {{monzo| 32/13 -2/13 3/13 -2/13 }}, {{monzo| 32/13 -2/13 -10/13 11/13 }}]
+: {{monzo list| 1 0 0 0 | 16/13 12/13 -5/13 -1/13 | 32/13 -2/13 3/13 -2/13 | 32/13 -2/13 -10/13 11/13 }}
-: [[Eigenmonzo]]s: 2, 9/7, 7/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.9/7
-{{Val list|legend=1| 22, 31, 53, 84, 87, 118, 140, 171, 1592c, 1763c, 1934c, 2074c, 2105c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd, 5034bcccdd, 5205bcccdd }}
+{{Optimal ET sequence|legend=1| 22, 31, 53, 84, 87, 118, 140, 171, 1592c, 1763c, 1934c, 2074c, 2105c, 2245cd, 2416cd, …, 3442ccd, 3613ccd, 5034bcccdd, 5205bcccdd }}
-[[Badness]]: 0.173 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.761
-[[Projection pair]]: 7 65536/9375
+[[Projection pair]]: <code>7 65536/9375</code>
 == Zelda ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 385/384, 3388/3375
-[[Mapping]]: [{{val| 1 0 0 16 -9 }}, {{val| 0 1 0 -1 2 }}, {{val| 0 0 1 -5 4 }}]
+{{Mapping|legend=1| 1 0 0 16 -9 | 0 1 0 -1 2 | 0 0 1 -5 4 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2765{{c}}, ~3/2 = 702.1662{{c}}, ~5/4 = 386.2362{{c}}
+: [[error map]]: {{val| +0.276 +0.488 +0.475 -0.791 -1.764 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1206{{c}}, ~5/4 = 386.1124{{c}}
+: error map: {{val| 0.000 +0.166 -0.201 -1.508 -2.627 }}
-{{Val list|legend=1| 22, 31, 53, 84, 87, 118, 258e, 376de, 547de }}
+{{Optimal ET sequence|legend=1| 22, 31, 53, 84e, 87, 118, 258e, 376de, 547dee }}
-[[Badness]]: 0.642 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.771
 == Horwellic ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 3025/3024, 65625/65536
-[[Mapping]]: [{{val| 1 0 0 16 10 }}, {{val| 0 1 0 -1 1 }}, {{val| 0 0 2 -10 -7 }}]
+{{Mapping|legend=1| 1 0 0 16 10 | 0 1 0 -1 1 | 0 0 2 -10 -7 }}
+: mapping generators: ~2, ~3, ~352/315
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0937{{c}}, ~3/2 = 701.8440{{c}}, ~352/315 = 192.9848{{c}}
+: [[error map]]: {{val| +0.094 -0.017 -0.157 -0.049 +0.007 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8212{{c}}, ~352/315 = 192.9618{{c}}
+: error map: {{val| 0.000 -0.134 -0.390 -0.265 -0.229 }}
-{{Val list|legend=1| 31, 75e, 87, 118, 193, 224, 311, 342, 1592c, 1934ce, 2276cde, 2587cde, 2929cde, 3271ccde, 3613ccde, 5205bcccddee }}
+{{Optimal ET sequence|legend=1| 25e, 31, 75e, 87, 118, 193, 224, 311, 342, 1592c, 1934ce, 2276cde, 2587cde, 2929cde, 3271ccde, 3613ccde, 5205bcccddee }}
-[[Badness]]: 0.505 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.607
-[[Category:Regular temperament theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
 [[Category:Horwell family| ]] <!-- main article -->
 [[Category:Rank 3]]