Gamelismic family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 149668185 - Original comment: **
 
Squib (talk | contribs)
363 edits
Tags: Mobile edit Mobile web edit Advanced mobile edit
 
(62 intermediate revisions by 10 users not shown)
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 '''gamelismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[gamelisma]], 1029/1024. The head of this family, gamelismic, tempers out 1029/1024 alone in the full 7-limit, so it has the same [[2.3.7 subgroup|2.3.7-subgroup]] structure as [[slendric]] but giving [[prime harmonic|prime]] [[5/1|5]] an independent generator.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-20 02:12:22 UTC</tt>.<br>
: The original revision id was <tt>149668185</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">//Gamelismic temperament// is the rank three 7-limit temperament tempering out 1029/1024. It has a [[Minkowski reduced generator basis]] given by the tempered versions of [8/7, 2, 5] (in that order.) The vals [&lt;0 3 0 -1|, &lt;1 1 0 3|, &lt;0 0 1 0|] map 7-limit just intonation to the three basis elements, which may be tuned to [[118edo]] as [23/118, 118/118, 274/118].


==11-limit temperaments==
See [[Gamelismic clan]] for the rank-2 temperament without the last generator of gamelismic, and its various extensions.


===Prodigy===
== Gamelismic ==
Prodigy has normal comma list [1029/1024, 385/384] and also tempers out 441/440. It has a Minkowski generator basis [2, 8/7, 12/11], which may be mapped from the 11-limit by [&lt;1 1 2 3 3|, &lt;0 3 1 -1 3|, &lt;0 0 1 0 -1|]. Once again [[118edo]] is a good tuning choice, and [[159edo]] is another one. </pre></div>
{{Main| Gamelismic and portent }}
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Gamelismic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;Gamelismic temperament&lt;/em&gt; is the rank three 7-limit temperament tempering out 1029/1024. It has a &lt;a class="wiki_link" href="/Minkowski%20reduced%20generator%20basis"&gt;Minkowski reduced generator basis&lt;/a&gt; given by the tempered versions of [8/7, 2, 5] (in that order.) The vals [&amp;lt;0 3 0 -1|, &amp;lt;1 1 0 3|, &amp;lt;0 0 1 0|] map 7-limit just intonation to the three basis elements, which may be tuned to &lt;a class="wiki_link" href="/118edo"&gt;118edo&lt;/a&gt; as [23/118, 118/118, 274/118].&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-11-limit temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;11-limit temperaments&lt;/h2&gt;
[[Comma list]]: 1029/1024
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-11-limit temperaments-Prodigy"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Prodigy&lt;/h3&gt;
{{Mapping|legend=1| 1 1 0 3 | 0 3 0 -1 | 0 0 1 0 }}
Prodigy has normal comma list [1029/1024, 385/384] and also tempers out 441/440. It has a Minkowski generator basis [2, 8/7, 12/11], which may be mapped from the 11-limit by [&amp;lt;1 1 2 3 3|, &amp;lt;0 3 1 -1 3|,  &amp;lt;0 0 1 0 -1|]. Once again &lt;a class="wiki_link" href="/118edo"&gt;118edo&lt;/a&gt; is a good tuning choice, and &lt;a class="wiki_link" href="/159edo"&gt;159edo&lt;/a&gt; is another one.&lt;/body&gt;&lt;/html&gt;</pre></div>
: mapping generators: ~2, ~8/7, ~5
 
[[Mapping to lattice]]: [{{val| 0 3 0 -1 }}, {{val| 0 0 1 0 }}]
 
[[Minkowski lattice basis]]:
: 8/7 length = 0.5192, 5/4 length = log<sub>2</sub>5
: Angle (8/7, 5/4) = 90 degrees
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}, ~5/4 = 385.3412{{c}}
: [[error map]]: {{val| +0.486 -0.123 -0.001 -1.151 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.7474{{c}}, ~5/4 = 385.5205{{c}}
: error map: {{val| 0.000 -0.713 -0.793 -2.573 }}
 
[[Minimax tuning]]: ''c'' = 1029/1024
* [[7-odd-limit]]: 3, 5, and 7 (1/4)''c'' flat
: {{Monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 5/2 -1/4 1 -3/4 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3.5/3
* [[9-odd-limit]]: 3 1/7-comma flat, 5 and 7 2/7-comma flat
: {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 10/7 -1/7 1 -3/7 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7
 
{{Optimal ET sequence|legend=1| 5, 10, 15, 26, 31, 41, 72, 118, 190 }}
 
[[Badness]] (Sintel): 0.777
 
[[Projection pair]]: 3 1024/343 to 2.5.7
 
Scales: [[portent26]]
 
== Portent ==
{{Main| Gamelismic and portent }}
 
Portent tempers out [[385/384]] and [[441/440]] and is the main extension of gamelismic. Notice the identity 1029/1024 = (385/384)⋅(441/440).
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 385/384, 441/440
 
{{Mapping|legend=1| 1 1 0 3 5 | 0 3 0 -1 4 | 0 0 1 0 -1 }}
 
[[Mapping to lattice]]: [{{val| 0 3 1 -1 3 }}, {{val| 0 0 1 0 -1 }}]
 
Minkowski lattice basis:
: 8/7 length = 0.46467, 12/11 length = 1.931
: Angle (8/7, 12/11) = 86.657 degrees
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4902{{c}}, ~8/7 = 233.7839{{c}}, ~5/4 = 385.3191{{c}}
: [[error map]]: {{val| +0.490 -0.113 -0.014 -1.139 -0.031 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.7616{{c}}, ~5/4 = 385.3149{{c}}
: error map: {{val| 0.000 -0.670 -0.999 -2.587 -1.586 }}
 
[[Minimax tuning]]: ''c''<sub>1</sub> = 1029/1024, ''c''<sub>2</sub> = 385/384
* [[11-odd-limit]]: 3 (1/7)''c''<sub>1</sub> flat, 5 and 7 (2/7)''c''<sub>1</sub> flat, 11 (''c''<sub>2</sub> - (3/7)''c''<sub>1</sub>) flat
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 39/14 4/7 1/2 -2/7 -1/2 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 39/14 4/7 -1/2 -2/7 1/2 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5
 
{{Optimal ET sequence|legend=1| 15, 26, 31, 41, 72, 118, 159, 190 }}
 
[[Badness]] (Sintel): 0.281
 
[[Projection pair]]s: 3 1024/343 11 131072/12005 to 2.5.7
 
Scales: [[portent26]]
 
=== Portending ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 385/384
 
Mapping: {{mapping| 1 1 0 3 5 6 | 0 3 0 -1 4 12 | 0 0 1 0 -1 -2 }}
 
Optimal tunings:  
* WE: ~2 = 1200.4540{{c}}, ~8/7 = 234.0013{{c}}, ~5/4 = 384.8733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9748{{c}}, ~5/4 = 384.8812{{c}}
 
{{Optimal ET sequence|legend=0| 15, 26, 31f, 41, 46, 72, 87, 159 }}
 
Badness (Sintel): 0.587
 
Complexity spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13
 
=== Portentous ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 441/440, 625/624
 
Mapping: {{mapping| 1 1 0 3 5 -5 | 0 3 0 -1 4 -3 | 0 0 1 0 -1 4 }}
 
Optimal tunings:
* WE: ~2 = 1200.4888{{c}}, ~8/7 = 233.7795{{c}}, ~5/4 = 385.1398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.7575{{c}}, ~5/4 = 385.1447{{c}}
 
{{Optimal ET sequence|legend=0| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 452cdef, 611cddef }}
 
Badness (Sintel): 0.618
 
=== Ominous ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 385/384, 441/440
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 0 -1 4 -10 | 0 0 1 0 -1 2 }}
 
Optimal tunings:
* WE: ~2 = 1200.7019{{c}}, ~8/7 = 233.5453{{c}}, ~5/4 = 385.6079{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.4510{{c}}, ~5/4 = 385.6739{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}
 
Badness (Sintel): 0.702
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 351/350, 385/384, 441/440
 
Mapping: {{mapping| 1 1 0 3 5 1 1 | 0 3 0 -1 4 -10 -8 | 0 0 1 0 -1 2 2 }}
 
Mapping to lattice: [{{val| 0 1 1 0 0 -1 0 }}, {{val| 0 -1 -1 0 -1 2 1 }}]
 
Lattice basis:
: 8/7 length = 0.3859, 6/5 length = 1.1303
: Angle (8/7, 6/5) = 98.6015
 
Optimal tunings:
* WE: ~2 = 1200.6745{{c}}, ~8/7 = 233.5625{{c}}, ~5/4 = 385.5056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.4679{{c}}, ~5/4 = 385.5892{{c}}
 
Minimax tuning:
* 17-odd-limit
: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 7/4 9/10 0 0 -3/10 -3/20 0 }}, {{monzo| 5/2 7/5 0 0 -4/5 1/10 0 }}, {{monzo| 11/4 -3/10 0 0 1/10 1/20 0 }}, {{monzo| 7/2 -1/5 0 0 2/5 -3/10 0 }}, {{monzo| 7/2 -1/5 0 0 -3/5 7/10 0 }}, {{monzo| 4 2/5 0 0 -4/5 3/5 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
 
{{Optimal ET sequence|legend=0| 15f, 20c, 26, 31, 46, 72, 103, 149, 221ef }}
 
Badness (Sintel): 0.582
 
=== Momentous ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 385/384
 
Mapping: {{mapping| 1 1 0 3 5 7 | 0 3 0 -4 1 -5 | 0 0 1 0 -1 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.0652{{c}}, ~8/7 = 234.1856{{c}}, ~5/4 = 386.6199{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1748{{c}}, ~5/4 = 386.5951{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 21e, 31, 41, 46, 72f, 77, 87, 118, 164, 205d }}
 
Badness (Sintel): 0.778
 
=== Foreboding ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 275/273
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 0 -1 4 2 | 0 0 1 0 -1 1 }}
 
Optimal tunings:  
* WE: ~2 = 1200.2251{{c}}, ~8/7 = 233.4102{{c}}, ~5/4 = 382.4142{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.4017{{c}}, ~5/4 = 382.4261{{c}}
 
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e, 26, 31, 41, 72f }}
 
Badness (Sintel): 0.816
 
=== Portannic ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 441/440, 10985/10976
 
Mapping: {{mapping| 1 1 2 3 3 4 | 0 3 0 -1 4 -1 | 0 0 3 0 -3 -1 }}
: mapping generators: ~2, ~8/7, ~14/13
 
Optimal tunings:  
* WE: ~2 = 1200.5451{{c}}, ~8/7 = 233.7495{{c}}, ~14/13 = 128.4023{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6930{{c}}, ~14/13 = 128.3684{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}
 
Badness (Sintel): 1.67
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 385/384, 441/440, 8624/8619
 
Mapping: {{mapping| 1 1 2 3 3 4 4 | 0 3 0 -1 4 -1 1 | 0 0 3 0 -3 -1 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.4416{{c}}, ~8/7 = 233.7663{{c}}, ~14/13 = 128.4622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.7173{{c}}, ~14/13 = 128.4269{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}
 
Badness (Sintel): 1.24
 
== Gamel ==
This esoteric alternative extension tempers out [[540/539]], and sometimes comes up in temperament searches. In practice however, it is almost always desirable to further temper it to [[miracle]], in which case is is identical to portent anyway. This enables many more [[essentially tempered chord]]s while introducing virtually no additional error.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 1029/1024
 
{{Mapping|legend=1| 1 1 0 3 -1 | 0 3 0 -1 11 | 0 0 1 0 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6462{{c}}, ~8/7 = 233.4166{{c}}, ~5/4 = 384.4189{{c}}
: [[error map]]: {{val| +0.646 -1.059 -0.602 -0.304 +1.329 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.3289{{c}}, ~5/4 = 384.5627{{c}}
: error map: {{val| 0.000 -1.968 -1.751 -2.155 -0.137 }}
 
{{Optimal ET sequence|legend=1| 5e, 10, 21e, 26e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
 
[[Badness]] (Sintel): 1.02
 
[[Category:Temperament families]]
[[Category:Gamelismic family| ]] <!-- main article -->
[[Category:Rank 3]]

Latest revision as of 19:54, 2 March 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 gamelismic family of rank-3 temperaments tempers out the gamelisma, 1029/1024. The head of this family, gamelismic, tempers out 1029/1024 alone in the full 7-limit, so it has the same 2.3.7-subgroup structure as slendric but giving prime 5 an independent generator.

See Gamelismic clan for the rank-2 temperament without the last generator of gamelismic, and its various extensions.

Gamelismic

Main article: Gamelismic and portent

Subgroup: 2.3.5.7

Comma list: 1029/1024

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

mapping generators: ~2, ~8/7, ~5

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

Minkowski lattice basis:

8/7 length = 0.5192, 5/4 length = log25
Angle (8/7, 5/4) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1200.4859 ¢, ~8/7 = 233.7822 ¢, ~5/4 = 385.3412 ¢
error map: ⟨+0.486 -0.123 -0.001 -1.151]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7474 ¢, ~5/4 = 385.5205 ¢
error map: ⟨0.000 -0.713 -0.793 -2.573]

Minimax tuning: c = 1029/1024

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [5/2 -1/4 1 -3/4⟩, [5/2 -1/4 0 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.7/3.5/3
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [10/7 -1/7 1 -3/7⟩, [20/7 -2/7 0 1/7⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.9/7

Optimal ET sequence: 5, 10, 15, 26, 31, 41, 72, 118, 190

Badness (Sintel): 0.777

Projection pair: 3 1024/343 to 2.5.7

Scales: portent26

Portent

Main article: Gamelismic and portent

Portent tempers out 385/384 and 441/440 and is the main extension of gamelismic. Notice the identity 1029/1024 = (385/384)⋅(441/440).

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440

Mapping: [⟨1 1 0 3 5], ⟨0 3 0 -1 4], ⟨0 0 1 0 -1]]

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

Minkowski lattice basis:

8/7 length = 0.46467, 12/11 length = 1.931
Angle (8/7, 12/11) = 86.657 degrees

Optimal tunings:

  • WE: ~2 = 1200.4902 ¢, ~8/7 = 233.7839 ¢, ~5/4 = 385.3191 ¢
error map: ⟨+0.490 -0.113 -0.014 -1.139 -0.031]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7616 ¢, ~5/4 = 385.3149 ¢
error map: ⟨0.000 -0.670 -0.999 -2.587 -1.586]

Minimax tuning: c1 = 1029/1024, c2 = 385/384

  • 11-odd-limit: 3 (1/7)c1 flat, 5 and 7 (2/7)c1 flat, 11 (c2 - (3/7)c1) flat
[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [39/14 4/7 1/2 -2/7 -1/2⟩, [20/7 -2/7 0 1/7 0⟩, [39/14 4/7 -1/2 -2/7 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7.11/5

Optimal ET sequence: 15, 26, 31, 41, 72, 118, 159, 190

Badness (Sintel): 0.281

Projection pairs: 3 1024/343 11 131072/12005 to 2.5.7

Scales: portent26

Portending

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384

Mapping: [⟨1 1 0 3 5 6], ⟨0 3 0 -1 4 12], ⟨0 0 1 0 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1200.4540 ¢, ~8/7 = 234.0013 ¢, ~5/4 = 384.8733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.9748 ¢, ~5/4 = 384.8812 ¢

Optimal ET sequence: 15, 26, 31f, 41, 46, 72, 87, 159

Badness (Sintel): 0.587

Complexity spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13

Portentous

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624

Mapping: [⟨1 1 0 3 5 -5], ⟨0 3 0 -1 4 -3], ⟨0 0 1 0 -1 4]]

Optimal tunings:

  • WE: ~2 = 1200.4888 ¢, ~8/7 = 233.7795 ¢, ~5/4 = 385.1398 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7575 ¢, ~5/4 = 385.1447 ¢

Optimal ET sequence: 15, 31, 56, 72, 87, 103, 159, 190, 262df, 452cdef, 611cddef

Badness (Sintel): 0.618

Ominous

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 0 -1 4 -10], ⟨0 0 1 0 -1 2]]

Optimal tunings:

  • WE: ~2 = 1200.7019 ¢, ~8/7 = 233.5453 ¢, ~5/4 = 385.6079 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4510 ¢, ~5/4 = 385.6739 ¢

Optimal ET sequence: 15f, 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff

Badness (Sintel): 0.702

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440

Mapping: [⟨1 1 0 3 5 1 1], ⟨0 3 0 -1 4 -10 -8], ⟨0 0 1 0 -1 2 2]]

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

Lattice basis:

8/7 length = 0.3859, 6/5 length = 1.1303
Angle (8/7, 6/5) = 98.6015

Optimal tunings:

  • WE: ~2 = 1200.6745 ¢, ~8/7 = 233.5625 ¢, ~5/4 = 385.5056 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4679 ¢, ~5/4 = 385.5892 ¢

Minimax tuning:

  • 17-odd-limit
[[1 0 0 0 0 0 0⟩, [7/4 9/10 0 0 -3/10 -3/20 0⟩, [5/2 7/5 0 0 -4/5 1/10 0⟩, [11/4 -3/10 0 0 1/10 1/20 0⟩, [7/2 -1/5 0 0 2/5 -3/10 0⟩, [7/2 -1/5 0 0 -3/5 7/10 0⟩, [4 2/5 0 0 -4/5 3/5 0⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9.13/9

Optimal ET sequence: 15f, 20c, 26, 31, 46, 72, 103, 149, 221ef

Badness (Sintel): 0.582

Momentous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384

Mapping: [⟨1 1 0 3 5 7], ⟨0 3 0 -4 1 -5], ⟨0 0 1 0 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0652 ¢, ~8/7 = 234.1856 ¢, ~5/4 = 386.6199 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.1748 ¢, ~5/4 = 386.5951 ¢

Optimal ET sequence: 15f, 21e, 31, 41, 46, 72f, 77, 87, 118, 164, 205d

Badness (Sintel): 0.778

Foreboding

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 0 -1 4 2], ⟨0 0 1 0 -1 1]]

Optimal tunings:

  • WE: ~2 = 1200.2251 ¢, ~8/7 = 233.4102 ¢, ~5/4 = 382.4142 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4017 ¢, ~5/4 = 382.4261 ¢

Optimal ET sequence: 5, 10, 15, 25e, 26, 31, 41, 72f

Badness (Sintel): 0.816

Portannic

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 10985/10976

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

mapping generators: ~2, ~8/7, ~14/13

Optimal tunings:

  • WE: ~2 = 1200.5451 ¢, ~8/7 = 233.7495 ¢, ~14/13 = 128.4023 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6930 ¢, ~14/13 = 128.3684 ¢

Optimal ET sequence: 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff

Badness (Sintel): 1.67

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 385/384, 441/440, 8624/8619

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

Optimal tunings:

  • WE: ~2 = 1200.4416 ¢, ~8/7 = 233.7663 ¢, ~14/13 = 128.4622 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7173 ¢, ~14/13 = 128.4269 ¢

Optimal ET sequence: 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def

Badness (Sintel): 1.24

Gamel

This esoteric alternative extension tempers out 540/539, and sometimes comes up in temperament searches. In practice however, it is almost always desirable to further temper it to miracle, in which case is is identical to portent anyway. This enables many more essentially tempered chords while introducing virtually no additional error.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1029/1024

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

Optimal tunings:

  • WE: ~2 = 1200.6462 ¢, ~8/7 = 233.4166 ¢, ~5/4 = 384.4189 ¢
error map: ⟨+0.646 -1.059 -0.602 -0.304 +1.329]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.3289 ¢, ~5/4 = 384.5627 ¢
error map: ⟨0.000 -1.968 -1.751 -2.155 -0.137]

Optimal ET sequence: 5e, 10, 21e, 26e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde

Badness (Sintel): 1.02

Retrieved from "https://en.xen.wiki/index.php?title=Gamelismic_family&oldid=225041"