Hemifamity family: Difference between revisions

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The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]].

The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.

It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (C–vBb).

Line 19:

: Angle (3/2, 10/9) = 82.112 degrees

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.7918, ~5/4 = 386.0144

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.7918, ~5/4 = 386.0144

[[Minimax tuning]]: c = 5120/5103

* [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just

: {{monzo list| 1 0 0 0 | 10/7 1/7 1/7 -1/7 | 0 0 1 0 | 10/7 -6/7 1/7 6/7 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.7/3

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3

* [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp

: {{monzo list| 1 0 0 0 | 5/4 1/4 1/8 -1/8 | 0 0 1 0 | 5/2 -3/2 1/4 3/4 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.9/7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7

{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}

[[Badness]]: 0.153 × 10-3

[[Badness]] (Smith): 0.153 × 10-3

[[Projection pair]]s: 7 5120/729

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Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.6411, ~5/4 = 385.4452

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.6411, ~5/4 = 385.4452

~~Optimal ET sequence:~~ {{Optimal ET sequence| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}

{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}

Badness: 0.212 × 10-3

Badness (Smith): 0.212 × 10-3

== Pele ==

Line 81:

: Angle(3/2, 56/55) = 90.4578 degrees

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 703.2829, ~5/4 = 386.5647

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 703.2829, ~5/4 = 386.5647

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5.11/9

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9

{{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}

[[Badness]]: 0.648 × 10-3

[[Badness]] (Smith): 0.648 × 10-3

[[Projection pair]]s: 7 5120/729 11 655360/59049

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Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 703.4398, ~5/4 = 386.8933

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.4398, ~5/4 = 386.8933

Minimax tuning:

* 13-odd-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.9/5.13/9

* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9

* 15-odd-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.5/3.13/9

* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9

~~Optimal ET sequence:~~ {{Optimal ET sequence| 29, 41, 46, 58, 87, 145, 232 }}

Badness: 0.703 × 10-3

Badness (Smith): 0.703 × 10-3

=== 17-limit ===

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Mapping: {{mapping| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 1 1 1 -1 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 703.5544, ~5/4 = 387.9654

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.5544, ~5/4 = 387.9654

~~Optimal ET sequence:~~ {{Optimal ET sequence| 29, 41, 46, 58, 87, 99ef, 145 }}

Badness: 0.930 × 10-3

Badness (Smith): 0.930 × 10-3

== Laka ==

Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]]. [[Gene Ward Smith]] considered it to be a [[17-limit]] temperament, assigning †442/441 ({{nowrap| 41g & 53 & 58 }}) as the main extension. It should be noted that {{nowrap| 41 & 53g & 58 }} also makes for a possible extension.

It's the way the numbers fall. The Laka geometry happens to work reasonably well in the 13-limit but not so well in the 17-limit. There isn't one obvious 17-limit extension and none of them are competitive with other 17-limit temperaments.

</blockquote>

—[[Graham Breed]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref>

It makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, omitting harmonic 17, as the 19 is accurate and easily available in a 24-tone scale.

[[Subgroup]]: 2.3.5.7.11

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.5133, ~5/4 = 385.5563

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.5133, ~5/4 = 385.5563

[[Minimax tuning]]

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.11/7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7

{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}

[[Badness]]: 0.825 × 10-3

[[Badness]] (Smith): 0.825 × 10-3

[[Projection pair]]s: 5120/729 11 14348907/1310720

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Mapping: {{mapping| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.4078, ~5/4 = 385.5405

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4078, ~5/4 = 385.5405

Minimax tuning:

* 13- and 15-odd-limit

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 13/8 -1/2 1/8 0 0 1/8 }}, {{monzo| 13/4 -3 5/4 0 0 1/4 }}, {{monzo| 7/2 0 1/2 0 0 -1/2 }}, {{monzo| 25/8 -9/2 5/8 0 0 13/8 }}, {{monzo| 13/4 -3 1/4 0 0 5/4 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11.13/7

: unchanged-interval (eigenmonzo) basis: 2.11.13/7

{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 111, 152f, 415dff }}*

{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }}*

<nowiki>*</nowiki> optimal patent val: [[205edo|205]]

Badness: 0.822 × 10-3

Badness (Smith): 0.822 × 10-3

=== 2.3.5.7.11.13.19 subgroup ===

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Mapping: {{mapping| 1 0 0 10 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.4062, ~5/4 = 385.5254

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4062, ~5/4 = 385.5254

~~Optimal ET sequence:~~ {{Optimal ET sequence| 41, 53, 58h, 94, 111, 152f, 415dffhh }}*

{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }}*

<nowiki>*</nowiki> optimal patent val: [[205edo|205]]

Badness: 0.661 × 10-3

Badness (Smith): 0.661 × 10-3

=== 17-limit ===

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* 17-odd-limit

: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11.13/7

: unchanged-interval (eigenmonzo) basis: 2.11.13/7

~~Optimal ET sequence:~~ {{Optimal ET sequence| 58, 94, 111, 152f, 205, 263df }}

Badness: 1.19 × 10-3

Badness (Smith): 1.19 × 10-3

== Akea ==

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8909, ~5/4 = 385.3273

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.8909, ~5/4 = 385.3273

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5.11/5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5

[[Badness]]: 0.998 × 10-3

[[Badness]] (Smith): 0.998 × 10-3

=== 13-limit ===

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Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.9018, ~5/4 = 385.4158

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.9018, ~5/4 = 385.4158

Minimax tuning:

* 13- and 15-odd-limit

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.7/5.11/5

: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

{{Optimal ET sequence|legend=1| 34, 41, 46, 53, 87, 140, 321, 461e }}

{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}

Badness: 0.822 × 10-3

Badness (Smith): 0.822 × 10-3

Scales: [[akea46_13]]

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.8941, ~5/4 = 388.5932

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8941, ~5/4 = 388.5932

[[Badness]]: 1.18 × 10-3

[[Badness]] (Smith): 1.18 × 10-3

=== 13-limit ===

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Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.8670, ~5/4 = 388.6277

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.8670, ~5/4 = 388.6277

{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 104c, 111, 268cd }}

{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}

Badness: 0.908 × 10-3

Badness (Smith): 0.908 × 10-3

== Kapo ==

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: mapping generators: ~2, ~3, ~128/99

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.8776, ~128/99 = 441.7516

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8776, ~128/99 = 441.7516

[[Minimax tuning]]:

* [[11-odd-limit]]:

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/7.11/9

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9

[[Badness]]: 0.994 × 10-3

[[Badness]] (Smith): 0.994 × 10-3

== Namaka ==

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: mapping generators: ~2, ~400/231, ~5

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~400/231 = 951.4956, ~5/4 = 386.7868

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~400/231 = 951.4956, ~5/4 = 386.7868

[[Badness]]: 1.74 × 10-3

[[Badness]] (Smith): 1.74 × 10-3

=== 13-limit ===

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Mapping: {{mapping| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~26/15 = 951.4871, ~5/4 = 386.6606

Optimal tuning (CTE): ~2 = 1200.0000, ~26/15 = 951.4871, ~5/4 = 386.6606

Badness (Smith): 0.781 × 10-3

~~Badness: 0.781 × 10-3~~

== Notes ==

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Hemifamity family| ]]

[[Category:Hemifamity| ]]

[[Category:Rank 3]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]].
+The '''hemifamity family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[5120/5103]] ({{monzo|legend=1| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.
 It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (C–vBb).
@@ Line 19: / Line 19: @@
 : Angle (3/2, 10/9) = 82.112 degrees
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.7918, ~5/4 = 386.0144
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.7918, ~5/4 = 386.0144
 [[Minimax tuning]]: c = 5120/5103
 * [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just
 : {{monzo list| 1 0 0 0 | 10/7 1/7 1/7 -1/7 | 0 0 1 0 | 10/7 -6/7 1/7 6/7 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7/3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
 * [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp
 : {{monzo list| 1 0 0 0 | 5/4 1/4 1/8 -1/8 | 0 0 1 0 | 5/2 -3/2 1/4 3/4 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
 {{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}
-[[Badness]]: 0.153 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.153 × 10<sup>-3</sup>
 [[Projection pair]]s: 7 5120/729
@@ Line 59: / Line 59: @@
 Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.6411, ~5/4 = 385.4452
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.6411, ~5/4 = 385.4452
-Optimal ET sequence: {{Optimal ET sequence| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
+{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
-Badness: 0.212 × 10<sup>-3</sup>
+Badness (Smith): 0.212 × 10<sup>-3</sup>
 == Pele ==
@@ Line 81: / Line 81: @@
 : Angle(3/2, 56/55) = 90.4578 degrees
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 703.2829, ~5/4 = 386.5647
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 703.2829, ~5/4 = 386.5647
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11/9
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
 {{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}
-[[Badness]]: 0.648 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.648 × 10<sup>-3</sup>
 [[Projection pair]]s: 7 5120/729 11 655360/59049
@@ Line 101: / Line 101: @@
 Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.4398, ~5/4 = 386.8933
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.4398, ~5/4 = 386.8933
 Minimax tuning:
-* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.13/9
+* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
-* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/9
+* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9
-Optimal ET sequence: {{Optimal ET sequence| 29, 41, 46, 58, 87, 145, 232 }}
+{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}
-Badness: 0.703 × 10<sup>-3</sup>
+Badness (Smith): 0.703 × 10<sup>-3</sup>
 === 17-limit ===
@@ Line 118: / Line 118: @@
 Mapping: {{mapping| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 1 1 1 -1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.5544, ~5/4 = 387.9654
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.5544, ~5/4 = 387.9654
-Optimal ET sequence: {{Optimal ET sequence| 29, 41, 46, 58, 87, 99ef, 145 }}
+{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}
-Badness: 0.930 × 10<sup>-3</sup>
+Badness (Smith): 0.930 × 10<sup>-3</sup>
 == Laka ==
 {{Main| Laka }}
+Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]]. [[Gene Ward Smith]] considered it to be a [[17-limit]] temperament, assigning †442/441 ({{nowrap| 41g & 53 & 58 }}) as the main extension. It should be noted that {{nowrap| 41 & 53g & 58 }} also makes for a possible extension.
+<blockquote>
+It's the way the numbers fall. The Laka geometry happens to work reasonably well in the 13-limit but not so well in the 17-limit. There isn't one obvious 17-limit extension and none of them are competitive with other 17-limit temperaments.
+</blockquote>
+—[[Graham Breed]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref>
+It makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, omitting harmonic 17, as the 19 is accurate and easily available in a 24-tone scale.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 133: / Line 142: @@
 {{Mapping|legend=1| 1 0 0 10 -18 | 0 1 0 -6 15 | 0 0 1 1 -1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.5133, ~5/4 = 385.5563
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.5133, ~5/4 = 385.5563
 [[Minimax tuning]]
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.11/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
 {{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
-[[Badness]]: 0.825 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.825 × 10<sup>-3</sup>
 [[Projection pair]]s: 5120/729 11 14348907/1310720
@@ Line 153: / Line 162: @@
 Mapping: {{mapping| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.4078, ~5/4 = 385.5405
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4078, ~5/4 = 385.5405
 Minimax tuning:
 * 13- and 15-odd-limit
 : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 13/8 -1/2 1/8 0 0 1/8 }}, {{monzo| 13/4 -3 5/4 0 0 1/4 }}, {{monzo| 7/2 0 1/2 0 0 -1/2 }}, {{monzo| 25/8 -9/2 5/8 0 0 13/8 }}, {{monzo| 13/4 -3 1/4 0 0 5/4 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11.13/7
+: unchanged-interval (eigenmonzo) basis: 2.11.13/7
-{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 111, 152f, 415dff }}*
+{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }}*
 <nowiki>*</nowiki> optimal patent val: [[205edo|205]]
-Badness: 0.822 × 10<sup>-3</sup>
+Badness (Smith): 0.822 × 10<sup>-3</sup>
 === 2.3.5.7.11.13.19 subgroup ===
@@ Line 173: / Line 182: @@
 Mapping: {{mapping| 1 0 0 10 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.4062, ~5/4 = 385.5254
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4062, ~5/4 = 385.5254
-Optimal ET sequence: {{Optimal ET sequence| 41, 53, 58h, 94, 111, 152f, 415dffhh }}*
+{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }}*
 <nowiki>*</nowiki> optimal patent val: [[205edo|205]]
-Badness: 0.661 × 10<sup>-3</sup>
+Badness (Smith): 0.661 × 10<sup>-3</sup>
 === 17-limit ===
@@ Line 191: / Line 200: @@
 * 17-odd-limit
 : [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11.13/7
+: unchanged-interval (eigenmonzo) basis: 2.11.13/7
-Optimal ET sequence: {{Optimal ET sequence| 58, 94, 111, 152f, 205, 263df }}
+{{Optimal ET sequence|legend=0| 58, 94, 111, 152f, 205, 263df }}
-Badness: 1.19 × 10<sup>-3</sup>
+Badness (Smith): 1.19 × 10<sup>-3</sup>
 == Akea ==
@@ Line 207: / Line 216: @@
 {{Mapping|legend=1| 1 0 0 10 -3 | 0 1 0 -6 7 | 0 0 1 1 -2 }}
-: mapping generators: ~2, ~3, ~5
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8909, ~5/4 = 385.3273
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.8909, ~5/4 = 385.3273
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5.11/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
 {{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}
-[[Badness]]: 0.998 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.998 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 233: / Line 240: @@
 Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.9018, ~5/4 = 385.4158
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.9018, ~5/4 = 385.4158
 Minimax tuning:
 * 13- and 15-odd-limit
 : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.7/5.11/5
+: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5
-{{Optimal ET sequence|legend=1| 34, 41, 46, 53, 87, 140, 321, 461e }}
+{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}
-Badness: 0.822 × 10<sup>-3</sup>
+Badness (Smith): 0.822 × 10<sup>-3</sup>
 Scales: [[akea46_13]]
@@ Line 253: / Line 260: @@
 {{Mapping|legend=1| 1 0 0 10 6 | 0 1 0 -6 -6 | 0 0 1 1 3 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.8941, ~5/4 = 388.5932
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8941, ~5/4 = 388.5932
 {{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}
-[[Badness]]: 1.18 × 10<sup>-3</sup>
+[[Badness]] (Smith): 1.18 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 266: / Line 273: @@
 Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.8670, ~5/4 = 388.6277
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.8670, ~5/4 = 388.6277
-{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 104c, 111, 268cd }}
+{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}
-Badness: 0.908 × 10<sup>-3</sup>
+Badness (Smith): 0.908 × 10<sup>-3</sup>
 == Kapo ==
@@ Line 281: / Line 288: @@
 : mapping generators: ~2, ~3, ~128/99
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.8776, ~128/99 = 441.7516
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8776, ~128/99 = 441.7516
 [[Minimax tuning]]:
 * [[11-odd-limit]]:
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7.11/9
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9
 {{Optimal ET sequence|legend=1| 41, 87, 111, 152, 239, 391 }}
-[[Badness]]: 0.994 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.994 × 10<sup>-3</sup>
 == Namaka ==
@@ Line 301: / Line 308: @@
 : mapping generators: ~2, ~400/231, ~5
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~400/231 = 951.4956, ~5/4 = 386.7868
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~400/231 = 951.4956, ~5/4 = 386.7868
 {{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
-[[Badness]]: 1.74 × 10<sup>-3</sup>
+[[Badness]] (Smith): 1.74 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 314: / Line 321: @@
 Mapping: {{mapping| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.4871, ~5/4 = 386.6606
+Optimal tuning (CTE): ~2 = 1200.0000, ~26/15 = 951.4871, ~5/4 = 386.6606
-{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
+{{Optimal ET sequence|legend=0| 29, 53, 58, 87, 111, 140, 198 }}
+Badness (Smith): 0.781 × 10<sup>-3</sup>
-Badness: 0.781 × 10<sup>-3</sup>
+== Notes ==
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Hemifamity family| ]] <!-- main article -->
 [[Category:Hemifamity| ]] <!-- key article -->
 [[Category:Rank 3]]
 [[Category:Listen]]