Hemifamity family: Difference between revisions

Latest revision as of 00:41, 24 June 2025

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 hemifamity family of rank-3 temperaments tempers out 5120/5103 (monzo: [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).

Hemifamity

Subgroup: 2.3.5.7

Comma list: 5120/5103

Mapping: [⟨1 0 0 10], ⟨0 1 0 -6], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 2 -4], ⟨0 0 1 1]]

Lattice basis:

3/2 length = 0.5670, 10/9 length = 1.8063

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

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

[[1 0 0 0⟩, [10/7 1/7 1/7 -1/7⟩, [0 0 1 0⟩, [10/7 -6/7 1/7 6/7⟩]

unchanged-interval (eigenmonzo) basis: 2.5.7/3

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

[[1 0 0 0⟩, [5/4 1/4 1/8 -1/8⟩, [0 0 1 0⟩, [5/2 -3/2 1/4 3/4⟩]

unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd

Badness (Smith): 0.153 × 10^-3

Projection pairs: 7 5120/729

Music

Choraled play by Gene Ward Smith
Hemifamity27 by Chris Vaisvil

Overview to extensions

11- and 13-limit extensions

Strong extensions of hemifamity are pele, laka, akea, and lono. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the 11/8 at the down diminished fifth (C–vGb); laka, up augmented third (C–^E#); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v³F#). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the 13/11 at the minor third (C–Eb), tempering out 352/351, 847/845, and 2080/2079.

Subgroup extensions

A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.

Counterpyth

Main article: Counterpyth

Developed analogous to parapyth, counterpyth is an extension of hemifamity with an even milder fifth, as it finds 19/15 at the major third (C–E) and 19/10 at the major seventh (C–B). Notice the factorization 5120/5103 = (400/399)⋅(1216/1215). Other important ratios are 21/19 at the diminished third (C–Ebb) and 19/14 at the augmented third (C–E#).

It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.

Subgroup: 2.3.5.7.19

Comma list: 400/399, 1216/1215

Mapping: [⟨1 0 0 10 -6], ⟨0 1 0 -6 5], ⟨0 0 1 1 1]]

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

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

Badness (Smith): 0.212 × 10^-3

Pele

Main article: Pele

See also: Pentacircle clan

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891

Mapping: [⟨1 0 0 10 17], ⟨0 1 0 -6 -10], ⟨0 0 1 1 1]]

Mapping to lattice: [⟨0 1 4 -2 -6], ⟨0 0 -1 -1 -1]]

Lattice basis:

3/2 length = 0.3812, 56/55 length = 1.5893

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

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

Minimax tuning:

11-odd-limit

[[1 0 0 0 0⟩, [17/10 0 1/10 0 -1/10⟩, [17/5 -2 6/5 0 -1/5⟩, [16/5 -2 3/5 0 2/5⟩, [17/5 -2 1/5 0 4/5⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.11/9

Optimal ET sequence: 29, 41, 58, 87, 99e, 145, 186e

Badness (Smith): 0.648 × 10^-3

Projection pairs: 7 5120/729 11 655360/59049

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363

Mapping: [⟨1 0 0 10 17 22], ⟨0 1 0 -6 -10 -13], ⟨0 0 1 1 1 1]]

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

Minimax tuning:

13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9

Optimal ET sequence: 29, 41, 46, 58, 87, 145, 232

Badness (Smith): 0.703 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363

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 = 1200.0000, ~3/2 = 703.5544, ~5/4 = 387.9654

Optimal ET sequence: 29, 41, 46, 58, 87, 99ef, 145

Badness (Smith): 0.930 × 10^-3

Laka

Main article: Laka

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

—Graham Breed^[1]

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

Comma list: 540/539, 5120/5103

Mapping: [⟨1 0 0 10 -18], ⟨0 1 0 -6 15], ⟨0 0 1 1 -1]]

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

Minimax tuning

11-odd-limit

[[1 0 0 0 0⟩, [4/3 0 2/21 -1/21 1/21⟩, [0 0 1 0 0⟩, [2 0 3/7 2/7 -2/7⟩, [2 0 3/7 -5/7 5/7⟩]

unchanged-interval (eigenmonzo) basis: 2.5.11/7

Optimal ET sequence: 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee

Badness (Smith): 0.825 × 10^-3

Projection pairs: 5120/729 11 14348907/1310720

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728

Mapping: [⟨1 0 0 10 -18 -13], ⟨0 1 0 -6 15 12], ⟨0 0 1 1 -1 -1]]

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

Minimax tuning:

13- and 15-odd-limit

[[1 0 0 0 0 0⟩, [13/8 -1/2 1/8 0 0 1/8⟩, [13/4 -3 5/4 0 0 1/4⟩, [7/2 0 1/2 0 0 -1/2⟩, [25/8 -9/2 5/8 0 0 13/8⟩, [13/4 -3 1/4 0 0 5/4⟩]

unchanged-interval (eigenmonzo) basis: 2.11.13/7

Optimal ET sequence: 41, 53, 58, 94, 111, 152f, 415dff*

* optimal patent val: 205

Badness (Smith): 0.822 × 10^-3

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 352/351, 400/399, 456/455, 495/494

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 = 1200.0000, ~3/2 = 702.4062, ~5/4 = 385.5254

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

* optimal patent val: 205

Badness (Smith): 0.661 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 561/560

Mapping: [⟨1 0 0 10 -18 -13 32], ⟨0 1 0 -6 15 12 -22], ⟨0 0 1 1 -1 -1 3]]

Minimax tuning:

17-odd-limit

[[1 0 0 0 0 0 0⟩, [13/12 0 0 1/12 1/6 -1/12 0⟩, [-7/4 0 0 5/4 3/2 -5/4 0⟩, [7/4 0 0 3/4 1/2 -3/4 0⟩, [0 0 0 0 1 0 0⟩, [7/4 0 0 -1/4 1/2 1/4 0⟩, [35/12 0 0 23/12 5/6 -23/12 0⟩]

unchanged-interval (eigenmonzo) basis: 2.11.13/7

Optimal ET sequence: 58, 94, 111, 152f, 205, 263df

Badness (Smith): 1.19 × 10^-3

Akea

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187

Mapping: [⟨1 0 0 10 -3], ⟨0 1 0 -6 7], ⟨0 0 1 1 -2]]

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

Minimax tuning:

11-odd-limit

[[1 0 0 0 0⟩, [5/3 0 1/6 -1/6 0⟩, [26/9 0 13/18 -7/18 -1/3⟩, [26/9 0 -5/18 11/18 -1/3⟩, [26/9 0 -5/18 -7/18 2/3⟩]

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

Optimal ET sequence: 34, 41, 53, 87, 140, 181, 321

Badness (Smith): 0.998 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384

Mapping: [⟨1 0 0 10 -3 2], ⟨0 1 0 -6 7 4], ⟨0 0 1 1 -2 -2]]

Lattice basis:

3/2 length = 0.5354, 27/20 length = 1.0463

Angle (3/2, 27/20) = 80.5628 degrees

Mapping to lattice: [⟨0 1 3 -3 1 -2], ⟨0 0 -1 -1 2 2]]

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

Minimax tuning:

13- and 15-odd-limit

[[1 0 0 0 0 0⟩, [5/3 0 1/6 -1/6 0 0⟩, [26/9 0 13/18 -7/18 -1/3 0⟩, [26/9 0 -5/18 11/18 -1/3 0⟩, [26/9 0 -5/18 -7/18 2/3 0⟩, [26/9 0 -7/9 1/9 2/3 0⟩]

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

Optimal ET sequence: 34, 41, 46, 53, 87, 140, 321, 461e

Badness (Smith): 0.822 × 10^-3

Scales: akea46_13

Lono

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103

Mapping: [⟨1 0 0 10 6], ⟨0 1 0 -6 -6], ⟨0 0 1 1 3]]

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

Optimal ET sequence: 46, 53, 58, 99, 111, 268cd

Badness (Smith): 1.18 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845

Mapping: [⟨1 0 0 10 6 11], ⟨0 1 0 -6 -6 -9], ⟨0 0 1 1 3 3]]

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

Optimal ET sequence: 46, 53, 58, 99, 104c, 111, 268cd

Badness (Smith): 0.908 × 10^-3

Kapo

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5120/5103

Mapping: [⟨1 0 0 10 7], ⟨0 1 1 -5 -2], ⟨0 0 2 2 -1]]

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

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

Minimax tuning:

11-odd-limit:

[[1 0 0 0 0⟩, [8/5 2/5 0 -1/15 -2/15⟩, [14/5 6/5 0 7/15 -16/15⟩, [16/5 -6/5 0 13/15 -4/15⟩, [16/5 -6/5 0 -2/15 11/15⟩]

unchanged-interval (eigenmonzo) basis: 2.9/7.11/9

Optimal ET sequence: 41, 87, 111, 152, 239, 391

Badness (Smith): 0.994 × 10^-3

Namaka

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 5120/5103

Mapping: [⟨1 0 0 10 -6], ⟨0 2 0 -12 9], ⟨0 0 1 1 1]]

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

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

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198

Badness (Smith): 1.74 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845

Mapping: [⟨1 0 0 10 -6 -1], ⟨0 2 0 -12 9 3], ⟨0 0 1 1 1 1]]

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

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198

Badness (Smith): 0.781 × 10^-3

Notes

↑ Yahoo! Tuning Group | Laka 17-limit minimax planar temperament

[1] Yahoo! Tuning Group | Laka 17-limit minimax planar temperament

[1]

Hemifamity family: Difference between revisions

Latest revision as of 00:41, 24 June 2025

Contents

Hemifamity

Overview to extensions

11- and 13-limit extensions

Subgroup extensions

Counterpyth

Pele

13-limit

17-limit

Laka

13-limit

2.3.5.7.11.13.19 subgroup

17-limit

Akea

13-limit

Lono

13-limit

Kapo

Namaka

13-limit

Notes

Navigation menu

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Technical data page}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+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.
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-16 01:04:26 UTC</tt>.<br>
-: The original revision id was <tt>193574288</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">===Vital statistics===
-[[Comma]] c = 5120/5103
--limit minimax: 3 and 7 1/7c sharp, 5 just
+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).
-[|1 0 0 0&gt;, |10/7 1/7 1/7 -1/7&gt;,
-|0 0 1 0&gt;, |10/7 -6/7 1/7 6/7&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 7/6
--limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
+== Hemifamity ==
-[|1 0 0 0&gt;, |5/4 1/4 1/8 -1/8&gt;,
+[[Subgroup]]: 2.3.5.7
-|0 0 1 0&gt;, |5/2 -3/2 1/4 3/4&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 9/7
-Lattice basis: 3/2 length 0.5670, 10/9 length 1.8063
+[[Comma list]]: [[5120/5103]]
-Angle(3/2, 10/9) = 82.112 degrees
-Map to lattice: [&lt;0 1 2 -4|, &lt;0 0 1 1|]
-Map: [&lt;1 0 0 10|, &lt;0 1 0 -6|, &lt;0 0 1 1|]
+{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
-Generators: 2, 3, 5
-[[edo|EDOs]]: [[99edo|99]], [[140edo|140]], [[239edo|239]], [[292edo|292]], [[490edo|490]], [[531edo|531]]
-Badness: 0.000153
-Music
+: mapping generators: ~2, ~3, ~5
-By Gene Ward Smith
-http://www.archive.org/details/Choraled
-By Chris Vaisvil
-http://tinyurl.com/2d6k6xo
-==Pele==
+[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]
-[[Comma|Commas]]: 441/440, 896/891
+Lattice basis:
+: 3/2 length = 0.5670, 10/9 length = 1.8063
+: Angle (3/2, 10/9) = 82.112 degrees
+[[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|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|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]] (Smith): 0.153 × 10<sup>-3</sup>
+[[Projection pair]]s: 7 5120/729
+; Music
+* [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]
+* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]
+=== Overview to extensions ===
+==== 11- and 13-limit extensions ====
+Strong extensions of hemifamity are [[#Pele|pele]], [[#Laka|laka]], [[#Akea|akea]], and [[#Lono|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] at the down diminished fifth (C–vGb); laka, up augmented third (C–^E#); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F#). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third (C–Eb), tempering out [[352/351]], [[847/845]], and [[2080/2079]].
+==== Subgroup extensions ====
+A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.
+=== Counterpyth ===
+{{Main| Counterpyth }}
+Developed analogous to [[parapyth]], counterpyth is an extension of hemifamity with an even milder fifth, as it finds [[19/15]] at the major third (C–E) and [[19/10]] at the major seventh (C–B). Notice the factorization {{nowrap| 5120/5103 {{=}} ([[400/399]])⋅([[1216/1215]]) }}. Other important ratios are [[21/19]] at the diminished third (C–Ebb) and [[19/14]] at the augmented third (C–E#).
+It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.
+Subgroup: 2.3.5.7.19
+Comma list: 400/399, 1216/1215
+Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.6411, ~5/4 = 385.4452
+{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
+Badness (Smith): 0.212 × 10<sup>-3</sup>
+== Pele ==
+{{Main| Pele }}
+{{See also| Pentacircle clan }}
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 441/440, 896/891
+{{Mapping|legend=1| 1 0 0 10 17 | 0 1 0 -6 -10 | 0 0 1 1 1 }}
+[[Mapping to lattice]]: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]
+Lattice basis:
+: 3/2 length = 0.3812, 56/55 length = 1.5893
+: Angle(3/2, 56/55) = 90.4578 degrees
+[[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|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
+{{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}
+[[Badness]] (Smith): 0.648 × 10<sup>-3</sup>
+[[Projection pair]]s: 7 5120/729 11 655360/59049
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 196/195, 352/351, 364/363
+Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.4398, ~5/4 = 386.8933
+Minimax tuning:
+* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
+* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9
+{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}
+Badness (Smith): 0.703 × 10<sup>-3</sup>
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 196/195, 256/255, 352/351, 364/363
+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 = 1200.0000, ~3/2 = 703.5544, ~5/4 = 387.9654
+{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}
+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
+[[Comma list]]: 540/539, 5120/5103
+{{Mapping|legend=1| 1 0 0 10 -18 | 0 1 0 -6 15 | 0 0 1 1 -1 }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.5133, ~5/4 = 385.5563
 [[Minimax tuning]]
-[|1 0 0 0 0&gt;, |17/10 0 1/10 0 -1/10&gt;,
+* [[11-odd-limit]]
-|17/5 -2 6/5 0 -1/5&gt;,
+: [{{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 }}]
-|16/5 -2 3/5 0 2/5&gt;, |17/5 -2 1/5 0 4/5&gt;]
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
-[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9
+{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
-Lattice basis: 3/2 length 0.3812 56/55 length 1.5893
+[[Badness]] (Smith): 0.825 × 10<sup>-3</sup>
-Angle(3/2, 56/55) = 90.4578 degrees
-Map to lattice: [&lt;0 1 4 -2 -6|, &lt;0 0 -1 -1 -1|]
-Map: [&lt;1 0 0 10 17|, &lt;0 1 0 -6 -10|, &lt;0 0 1 1 1|]
+[[Projection pair]]s: 5120/729 11 14348907/1310720
-[[Generator|Generators]]: 2, 3, 5
-[[edo|Edos]]: [[87edo|87]], [[128edo|128]], [[145edo|145]], [[232edo|232]]
-Badness: 0.000648
-===13-limit===
+=== 13-limit ===
-Commas: 196/195, 352/351, 364/363
+Subgroup: 2.3.5.7.11.13
--limit minimax
+Comma list: 352/351, 540/539, 729/728
-Eigenmonzos: 2, 10/9, 13/10
--limit minimax
+Mapping: {{mapping| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}
-Eigenmonzos: 2, 15/13, 6/5
-Map: [&lt;1 0 0 10 17 22|, &lt;0 1 0 -6 -10 -13|, &lt;0 0 1 1 1 1|]
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4078, ~5/4 = 385.5405
-EDOs: 29, 41, 46, 58, 87, 145, 232
-Badness: 0.000703
-===Laka===
+Minimax tuning:
-[[Comma|Commas]]: 5120/5103, 540/539
+* 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 }}]
+: unchanged-interval (eigenmonzo) basis: 2.11.13/7
-[[Minimax tuning]]
+{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }}*
-[|1 0 0 0 0&gt;, |4/3 0 2/21 -1/21 1/21&gt;,
-|0 0 1 0 0&gt;, |2 0 3/7 2/7 -2/7&gt;,
+<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
-|2 0 3/7 -5/7 5/7&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 14/11
+Badness (Smith): 0.822 × 10<sup>-3</sup>
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
+Comma list: 352/351, 400/399, 456/455, 495/494
+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 = 1200.0000, ~3/2 = 702.4062, ~5/4 = 385.5254
+{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }}*
+<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
+Badness (Smith): 0.661 × 10<sup>-3</sup>
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 352/351, 442/441, 540/539, 561/560
+Mapping: {{mapping| 1 0 0 10 -18 -13 32 | 0 1 0 -6 15 12 -22 | 0 0 1 1 -1 -1 3 }}
+Minimax tuning:
+* 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 }}]
+: unchanged-interval (eigenmonzo) basis: 2.11.13/7
+{{Optimal ET sequence|legend=0| 58, 94, 111, 152f, 205, 263df }}
+Badness (Smith): 1.19 × 10<sup>-3</sup>
+== Akea ==
+[[File:Lattice Akea.png|thumb|Lattice for 13-limit akea.]]
+[[File:Lattice Akea-commatic.png|thumb|Ditto, but rearranged to basis {~2, ~3, ~81/80}.]]
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 385/384, 2200/2187
+{{Mapping|legend=1| 1 0 0 10 -3 | 0 1 0 -6 7 | 0 0 1 1 -2 }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
-limit
+{{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}
-[[Comma|Commas]]: 352/351, 540/539, 847/845
-and 15 limit minimax tuning
+[[Badness]] (Smith): 0.998 × 10<sup>-3</sup>
-[|1 0 0 0 0 0&gt;, |13/8 -1/2 1/8 0 0 1/8&gt;,
-|13/4 -3 5/4 0 0 1/4&gt;, |7/2 0 1/2 0 0 -1/2&gt;,
-|25/8 -9/2 5/8 0 0 13/8&gt;,
-|13/4 -3 1/4 0 0 5/4&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 14/13
-limit
+=== 13-limit ===
-[[Comma|Commas]]: 352/351, 540/539, 847/845, 442/441
+Subgroup: 2.3.5.7.11.13
--limit minimax
+Comma list: 325/324, 352/351, 385/384
-[|1 0 0 0 0 0 0&gt;,
-|13/12 0 0 1/12 1/6 -1/12 0&gt;,
-|-7/4 0 0 5/4 3/2 -5/4 0&gt;,
-|7/4 0 0 3/4 1/2 -3/4 0&gt;,
-|0 0 0 0 1 0 0&gt;,
-|7/4 0 0 -1/4 1/2 1/4 0&gt;,
-|35/12 0 0 23/12 5/6 -23/12 0&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 14/13
-Map: [&lt;1 0 0 10 -18 -13 32|,
+Mapping: {{mapping| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }}
-&lt;0 1 0 -6 15 12 -22|, &lt;0 0 1 1 -1 -1 3|]
-[[Generator|Generators]]: 2, 3, 5
-[[edo|Edos]]: [[94edo|94]], [[111edo|111]], [[152edo|152]], [[205edo|205]], [[345edo|345]]
-==Akea==
+Lattice basis:
-[[Comma|Commas]]: 385/384, 2200/2187
+: 3/2 length = 0.5354, 27/20 length = 1.0463
+: Angle (3/2, 27/20) = 80.5628 degrees
-[[Minimax tuning]]
+Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]
-[|1 0 0 0 0&gt;, |5/3 0 1/6 -1/6 0&gt;,
-|26/9 0 13/18 -7/18 -1/3&gt;,
-|26/9 0 -5/18 11/18 -1/3&gt;,
-|26/9 0 -5/18 -7/18 2/3&gt;]
-[[Eigenmonzo|Eigenmonzos]]: 2, 14/11, 7/5
-Map: [&lt;1 0 0 10 -3|, &lt;0 1 0 -6 7|, &lt;0 0 1 1 -2|]
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.9018, ~5/4 = 385.4158
-[[Generator|Generators]]: 2, 3, 5
-EDOs: 34, 41, 53, 87, 140, 181, 321
-Badness: 0.000998
-===13-limit===
+Minimax tuning:
-Commas: 325/324, 352/351, 385/384
+* 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 }}]
+: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5
-and 15 limit minimax
+{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}
-[|1 0 0 0 0 0&gt;, |5/3 0 1/6 -1/6 0 0&gt;,
-|26/9 0 13/18 -7/18 -1/3 0&gt;,
-|26/9 0 -5/18 11/18 -1/3 0&gt;,
-|26/9 0 -5/18 -7/18 2/3 0&gt;,
-|26/9 0 -7/9 1/9 2/3 0&gt;]
-Eigenmonzos: 2, 14/11, 7/5
-Lattice basis: 3/2 length 0.5354 27/20 length 1.0463
+Badness (Smith): 0.822 × 10<sup>-3</sup>
-Angle(3/2, 27/20) = 80.5628 degrees
-Map to lattice: [&lt;0 1 3 -3 1 -2|, &lt;0 0 -1 -1 2 2|]
-Map: [&lt;1 0 0 10 -3 2|, &lt;0 1 0 -6 7 4|, &lt;0 0 1 1 -2 -2|]
-Generators: 2, 3, 5
-[[edo|EDOs]]: [[94edo|94]], [[140edo|140]], [[181edo|181]], [[408edo|408]]
-Badness: 0.000822
 Scales: [[akea46_13]]
-==Kapo==
+== Lono ==
-[[Comma|Commas]]: 5120/5103, 3025/3024
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 176/175, 5120/5103
+{{Mapping|legend=1| 1 0 0 10 6 | 0 1 0 -6 -6 | 0 0 1 1 3 }}
+[[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]] (Smith): 1.18 × 10<sup>-3</sup>
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 176/175, 351/350, 847/845
+Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.8670, ~5/4 = 388.6277
+{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}
+Badness (Smith): 0.908 × 10<sup>-3</sup>
+== Kapo ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 3025/3024, 5120/5103
+{{Mapping|legend=1| 1 0 0 10 7 | 0 1 1 -5 -2 | 0 0 2 2 -1 }}
+: mapping generators: ~2, ~3, ~128/99
+[[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|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9
+{{Optimal ET sequence|legend=1| 41, 87, 111, 152, 239, 391 }}
+[[Badness]] (Smith): 0.994 × 10<sup>-3</sup>
+== Namaka ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 3388/3375, 5120/5103
-[[Minimax tuning]]
+{{Mapping|legend=1| 1 0 0 10 -6 | 0 2 0 -12 9 | 0 0 1 1 1 }}
-[|1 0 0 0 0&gt;, |8/5 2/5 0 -1/15 -2/15&gt;,
-|14/5 6/5 0 7/15 -16/15&gt;,
+: mapping generators: ~2, ~400/231, ~5
-|16/5 -6/5 0 13/15 -4/15&gt;,
-|16/5 -6/5 0 -2/15 11/15&gt;]
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~400/231 = 951.4956, ~5/4 = 386.7868
-[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 9/7
+{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
+[[Badness]] (Smith): 1.74 × 10<sup>-3</sup>
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 352/351, 676/675, 847/845
+Mapping: {{mapping| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}
+Optimal tuning (CTE): ~2 = 1200.0000, ~26/15 = 951.4871, ~5/4 = 386.6606
+{{Optimal ET sequence|legend=0| 29, 53, 58, 87, 111, 140, 198 }}
-Map: [&lt;1 0 0 10 7|, &lt;0 1 1 -5 -2|, &lt;0 0 2 2 -1|]
+Badness (Smith): 0.781 × 10<sup>-3</sup>
-[[Generator|Generators]]: 2, 3, 128/99
-[[edo|EDOs]]: [[152edo|152]], [[198edo|198]], [[239edo|239]], [[350edo|350]], [[391edo|391]], [[478edo|478]], [[589edo|589]]
-EDOs: 41, 87, 111, 152, 239, 391
-Badness: 0.000994
-==Lono==
+== Notes ==
-Commas: 176/175, 5120/5103
-Map: [&lt;1 0 0 10 6|, &lt;0 1 0 -6 -6|, &lt;0 0 1 1 3|]
+[[Category:Temperament families]]
-EDOs: 7, 12, 39, 46, 53, 58, 99, 111
+[[Category:Pages with mostly numerical content]]
-Badness: 0.00118
+[[Category:Hemifamity family| ]] <!-- main article -->
-</pre></div>
+[[Category:Hemifamity| ]] <!-- key article -->
-<h4>Original HTML content:</h4>
+[[Category:Rank 3]]
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Hemifamity family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h3&gt;
+[[Category:Listen]]
-&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 5120/5103&lt;br /&gt;
-&lt;br /&gt;
--limit minimax: 3 and 7 1/7c sharp, 5 just&lt;br /&gt;
-[|1 0 0 0&amp;gt;, |10/7 1/7 1/7 -1/7&amp;gt;, &lt;br /&gt;
-|0 0 1 0&amp;gt;, |10/7 -6/7 1/7 6/7&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 7/6&lt;br /&gt;
-&lt;br /&gt;
--limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp&lt;br /&gt;
-[|1 0 0 0&amp;gt;, |5/4 1/4 1/8 -1/8&amp;gt;, &lt;br /&gt;
-|0 0 1 0&amp;gt;, |5/2 -3/2 1/4 3/4&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 9/7&lt;br /&gt;
-&lt;br /&gt;
-Lattice basis: 3/2 length 0.5670, 10/9 length 1.8063&lt;br /&gt;
-Angle(3/2, 10/9) = 82.112 degrees&lt;br /&gt;
-Map to lattice: [&amp;lt;0 1 2 -4|, &amp;lt;0 0 1 1|]&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10|, &amp;lt;0 1 0 -6|, &amp;lt;0 0 1 1|]&lt;br /&gt;
-Generators: 2, 3, 5&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/239edo"&gt;239&lt;/a&gt;, &lt;a class="wiki_link" href="/292edo"&gt;292&lt;/a&gt;, &lt;a class="wiki_link" href="/490edo"&gt;490&lt;/a&gt;, &lt;a class="wiki_link" href="/531edo"&gt;531&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000153&lt;br /&gt;
-&lt;br /&gt;
-Music&lt;br /&gt;
-By Gene Ward Smith&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:216:http://www.archive.org/details/Choraled --&gt;&lt;a class="wiki_link_ext" href="http://www.archive.org/details/Choraled" rel="nofollow"&gt;http://www.archive.org/details/Choraled&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:216 --&gt;&lt;br /&gt;
-By Chris Vaisvil&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:217:http://tinyurl.com/2d6k6xo --&gt;&lt;a class="wiki_link_ext" href="http://tinyurl.com/2d6k6xo" rel="nofollow"&gt;http://tinyurl.com/2d6k6xo&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:217 --&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Pele"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Pele&lt;/h2&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 441/440, 896/891&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |17/10 0 1/10 0 -1/10&amp;gt;, &lt;br /&gt;
-|17/5 -2 6/5 0 -1/5&amp;gt;,&lt;br /&gt;
-|16/5 -2 3/5 0 2/5&amp;gt;, |17/5 -2 1/5 0 4/5&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 11/9&lt;br /&gt;
-&lt;br /&gt;
-Lattice basis: 3/2 length 0.3812 56/55 length 1.5893&lt;br /&gt;
-Angle(3/2, 56/55) = 90.4578 degrees&lt;br /&gt;
-Map to lattice: [&amp;lt;0 1 4 -2 -6|, &amp;lt;0 0 -1 -1 -1|]&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10 17|, &amp;lt;0 1 0 -6 -10|, &amp;lt;0 0 1 1 1|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/128edo"&gt;128&lt;/a&gt;, &lt;a class="wiki_link" href="/145edo"&gt;145&lt;/a&gt;, &lt;a class="wiki_link" href="/232edo"&gt;232&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000648&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Pele-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h3&gt;
-Commas: 196/195, 352/351, 364/363&lt;br /&gt;
-&lt;br /&gt;
--limit minimax&lt;br /&gt;
-Eigenmonzos: 2, 10/9, 13/10&lt;br /&gt;
-&lt;br /&gt;
--limit minimax&lt;br /&gt;
-Eigenmonzos: 2, 15/13, 6/5&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10 17 22|, &amp;lt;0 1 0 -6 -10 -13|, &amp;lt;0 0 1 1 1 1|]&lt;br /&gt;
-EDOs: 29, 41, 46, 58, 87, 145, 232&lt;br /&gt;
-Badness: 0.000703&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Pele-Laka"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Laka&lt;/h3&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 5120/5103, 540/539&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |4/3 0 2/21 -1/21 1/21&amp;gt;, &lt;br /&gt;
-|0 0 1 0 0&amp;gt;, |2 0 3/7 2/7 -2/7&amp;gt;, &lt;br /&gt;
-|2 0 3/7 -5/7 5/7&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 14/11&lt;br /&gt;
-&lt;br /&gt;
-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 352/351, 540/539, 847/845&lt;br /&gt;
-&lt;br /&gt;
-and 15 limit minimax tuning&lt;br /&gt;
-[|1 0 0 0 0 0&amp;gt;, |13/8 -1/2 1/8 0 0 1/8&amp;gt;, &lt;br /&gt;
-|13/4 -3 5/4 0 0 1/4&amp;gt;, |7/2 0 1/2 0 0 -1/2&amp;gt;, &lt;br /&gt;
-|25/8 -9/2 5/8 0 0 13/8&amp;gt;, &lt;br /&gt;
-|13/4 -3 1/4 0 0 5/4&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/8, 14/13&lt;br /&gt;
-&lt;br /&gt;
-limit&lt;br /&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 352/351, 540/539, 847/845, 442/441&lt;br /&gt;
-&lt;br /&gt;
--limit minimax&lt;br /&gt;
-[|1 0 0 0 0 0 0&amp;gt;, &lt;br /&gt;
-|13/12 0 0 1/12 1/6 -1/12 0&amp;gt;, &lt;br /&gt;
-|-7/4 0 0 5/4 3/2 -5/4 0&amp;gt;, &lt;br /&gt;
-|7/4 0 0 3/4 1/2 -3/4 0&amp;gt;, &lt;br /&gt;
-|0 0 0 0 1 0 0&amp;gt;, &lt;br /&gt;
-|7/4 0 0 -1/4 1/2 1/4 0&amp;gt;, &lt;br /&gt;
-|35/12 0 0 23/12 5/6 -23/12 0&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/8, 14/13&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10 -18 -13 32|, &lt;br /&gt;
-&amp;lt;0 1 0 -6 15 12 -22|, &amp;lt;0 0 1 1 -1 -1 3|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/111edo"&gt;111&lt;/a&gt;, &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/205edo"&gt;205&lt;/a&gt;, &lt;a class="wiki_link" href="/345edo"&gt;345&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Akea"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Akea&lt;/h2&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 385/384, 2200/2187&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |5/3 0 1/6 -1/6 0&amp;gt;, &lt;br /&gt;
-|26/9 0 13/18 -7/18 -1/3&amp;gt;,&lt;br /&gt;
-|26/9 0 -5/18 11/18 -1/3&amp;gt;, &lt;br /&gt;
-|26/9 0 -5/18 -7/18 2/3&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 14/11, 7/5&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10 -3|, &amp;lt;0 1 0 -6 7|, &amp;lt;0 0 1 1 -2|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
-EDOs: 34, 41, 53, 87, 140, 181, 321&lt;br /&gt;
-Badness: 0.000998&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Akea-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h3&gt;
-Commas: 325/324, 352/351, 385/384&lt;br /&gt;
-&lt;br /&gt;
-and 15 limit minimax&lt;br /&gt;
-[|1 0 0 0 0 0&amp;gt;, |5/3 0 1/6 -1/6 0 0&amp;gt;, &lt;br /&gt;
-|26/9 0 13/18 -7/18 -1/3 0&amp;gt;, &lt;br /&gt;
-|26/9 0 -5/18 11/18 -1/3 0&amp;gt;, &lt;br /&gt;
-|26/9 0 -5/18 -7/18 2/3 0&amp;gt;, &lt;br /&gt;
-|26/9 0 -7/9 1/9 2/3 0&amp;gt;]&lt;br /&gt;
-Eigenmonzos: 2, 14/11, 7/5&lt;br /&gt;
-&lt;br /&gt;
-Lattice basis: 3/2 length 0.5354 27/20 length 1.0463&lt;br /&gt;
-Angle(3/2, 27/20) = 80.5628 degrees&lt;br /&gt;
-Map to lattice: [&amp;lt;0 1 3 -3 1 -2|, &amp;lt;0 0 -1 -1 2 2|]&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10 -3 2|, &amp;lt;0 1 0 -6 7 4|, &amp;lt;0 0 1 1 -2 -2|]&lt;br /&gt;
-Generators: 2, 3, 5&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/181edo"&gt;181&lt;/a&gt;, &lt;a class="wiki_link" href="/408edo"&gt;408&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000822&lt;br /&gt;
-Scales: &lt;a class="wiki_link" href="/akea46_13"&gt;akea46_13&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Kapo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Kapo&lt;/h2&gt;
-&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 5120/5103, 3025/3024&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
-[|1 0 0 0 0&amp;gt;, |8/5 2/5 0 -1/15 -2/15&amp;gt;, &lt;br /&gt;
-|14/5 6/5 0 7/15 -16/15&amp;gt;, &lt;br /&gt;
-|16/5 -6/5 0 13/15 -4/15&amp;gt;, &lt;br /&gt;
-|16/5 -6/5 0 -2/15 11/15&amp;gt;]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9, 9/7&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10 7|, &amp;lt;0 1 1 -5 -2|, &amp;lt;0 0 2 2 -1|]&lt;br /&gt;
-&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 128/99&lt;br /&gt;
-&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/198edo"&gt;198&lt;/a&gt;, &lt;a class="wiki_link" href="/239edo"&gt;239&lt;/a&gt;, &lt;a class="wiki_link" href="/350edo"&gt;350&lt;/a&gt;, &lt;a class="wiki_link" href="/391edo"&gt;391&lt;/a&gt;, &lt;a class="wiki_link" href="/478edo"&gt;478&lt;/a&gt;, &lt;a class="wiki_link" href="/589edo"&gt;589&lt;/a&gt;&lt;br /&gt;
-EDOs: 41, 87, 111, 152, 239, 391&lt;br /&gt;
-Badness: 0.000994&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-Lono"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Lono&lt;/h2&gt;
-Commas: 176/175, 5120/5103&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;1 0 0 10 6|, &amp;lt;0 1 0 -6 -6|, &amp;lt;0 0 1 1 3|]&lt;br /&gt;
-EDOs: 7, 12, 39, 46, 53, 58, 99, 111&lt;br /&gt;
-Badness: 0.00118&lt;/body&gt;&lt;/html&gt;</pre></div>

Hemifamity family: Difference between revisions

Latest revision as of 00:41, 24 June 2025

Hemifamity

Overview to extensions

11- and 13-limit extensions

Subgroup extensions

Counterpyth

Pele

13-limit

17-limit

Laka

13-limit

2.3.5.7.11.13.19 subgroup

17-limit

Akea

13-limit

Lono

13-limit

Kapo

Namaka

13-limit

Notes

Navigation menu

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