Xenharmonic Wiki

Gamelismic family: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 165583609 - Original comment: **
m Recategorize
 
(55 intermediate revisions by 11 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>2010-09-26 22:52:32 UTC</tt>.<br>
: The original revision id was <tt>165583609</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 [[Harmonic Limit|7-limit]] [[Regular Temperaments|temperament]] tempering out 1029/1024.  


===Vital statistics===
See [[Gamelismic clan]] for the rank-2 temperament without the last generator of gamelismic, and its various extensions.  
Comma c = 1029/1024
7-limit minimax tuning: 3, 5, and 7 1/4c flat
9-limit minimax tuning: 3 1/7c flat, 5 and 7 2/7c flat
TOP-RMS: &lt;1200.486 1901.833 2786.314 3367.676|
Minkowski lattice basis: 8/7 length 0.5192, 5/4 length = log2(5)
Angle(8/7, 5/4) = 90 degrees
Map to lattice: [&lt;0 3 0 -1|, &lt;0 0 1 0|]
EDOs: 118, 159


==11-limit temperaments==
== Gamelismic ==
{{Main| Gamelismic and portent }}


===Portent===
[[Subgroup]]: 2.3.5.7
Portent has a [[Normal lists|normal comma list]] [1029/1024, 385/384] and also tempers out 441/440.  


Commas c = 1029/1024, e = 385/384
[[Comma list]]: 1029/1024


Minimax:  3 1/7c flat, 5 and 7 2/7c flat, 11  e-3/7c flat
{{Mapping|legend=1| 1 1 0 3 | 0 3 0 -1 | 0 0 1 0 }}
[|1 0 0 0 0&gt;, |10/7 6/7 0 -3/7 0&gt;,
: mapping generators: ~2, ~8/7, ~5
|39/14 4/7 1/2 -2/7 -1/2&gt;,
|20/7 -2/7 0 1/7 0&gt;,
|39/14 4/7 -1/2 -2/7 1/2&gt;]
Eigemonzos: 2, 11/10, 9/7


Minkowski lattice basis: 8/7 length 0.46467, 12/11 length 1.931
[[Mapping to lattice]]: [{{val| 0 3 0 -1 }}, {{val| 0 0 1 0 }}]
Angle(8/7, 12/11) = 86.657 degrees
 
Map to lattice: [&lt;0 3 1 -1 3|, &lt;0 0 1 0 -1|]
[[Minkowski lattice basis]]:
EDOs 72, 118, 159, 190</pre></div>
: 8/7 length = 0.5192, 5/4 length = log<sub>2</sub>5
<h4>Original HTML content:</h4>
: Angle (8/7, 5/4) = 90 degrees
<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 &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;7-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;temperament&lt;/a&gt; tempering out 1029/1024. &lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&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;
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}, ~5/4 = 385.3412{{c}}
Comma c = 1029/1024&lt;br /&gt;
: [[error map]]: {{val| +0.486 -0.123 -0.001 -1.151 }}
7-limit minimax tuning: 3, 5, and 7 1/4c flat&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.7474{{c}}, ~5/4 = 385.5205{{c}}
9-limit minimax tuning: 3 1/7c flat, 5 and 7 2/7c flat&lt;br /&gt;
: error map: {{val| 0.000 -0.713 -0.793 -2.573 }}
TOP-RMS: &amp;lt;1200.486 1901.833 2786.314 3367.676|&lt;br /&gt;
 
Minkowski lattice basis: 8/7 length 0.5192, 5/4 length = log2(5)&lt;br /&gt;
[[Minimax tuning]]: ''c'' = 1029/1024
Angle(8/7, 5/4) = 90 degrees&lt;br /&gt;
* [[7-odd-limit]]: 3, 5, and 7 (1/4)''c'' flat
Map to lattice: [&amp;lt;0 3 0 -1|, &amp;lt;0 0 1 0|] &lt;br /&gt;
: {{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 }}
EDOs: 118, 159&lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3.5/3
&lt;br /&gt;
* [[9-odd-limit]]: 3 1/7-comma flat, 5 and 7 2/7-comma flat
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-11-limit temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit temperaments&lt;/h2&gt;
: {{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 }}
&lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11-limit temperaments-Portent"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Portent&lt;/h3&gt;
 
Portent has a &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; [1029/1024, 385/384] and also tempers out 441/440. &lt;br /&gt;
{{Optimal ET sequence|legend=1| 5, 10, 15, 26, 31, 41, 72, 118, 190 }}
&lt;br /&gt;
 
Commas c = 1029/1024, e = 385/384&lt;br /&gt;
[[Badness]] (Sintel): 0.777
&lt;br /&gt;
 
Minimax: 3 1/7c flat, 5 and 7 2/7c flat, 11 e-3/7c flat&lt;br /&gt;
[[Projection pair]]: 3 1024/343 to 2.5.7
[|1 0 0 0 0&amp;gt;, |10/7 6/7 0 -3/7 0&amp;gt;, &lt;br /&gt;
 
|39/14 4/7 1/2 -2/7 -1/2&amp;gt;,&lt;br /&gt;
Scales: [[portent26]]
|20/7 -2/7 0 1/7 0&amp;gt;, &lt;br /&gt;
 
|39/14 4/7 -1/2 -2/7 1/2&amp;gt;]&lt;br /&gt;
== Portent ==
Eigemonzos: 2, 11/10, 9/7&lt;br /&gt;
{{Main| Gamelismic and portent }}
&lt;br /&gt;
 
Minkowski lattice basis: 8/7 length 0.46467, 12/11 length 1.931&lt;br /&gt;
Portent tempers out [[385/384]] and [[441/440]] and is the main extension of gamelismic. Notice the identity 1029/1024 = (385/384)⋅(441/440).  
Angle(8/7, 12/11) = 86.657 degrees&lt;br /&gt;
 
Map to lattice: [&amp;lt;0 3 1 -1 3|, &amp;lt;0 0 1 0 -1|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
EDOs 72, 118, 159, 190&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[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:Gamelismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]
Retrieved from "https://en.xen.wiki/w/Gamelismic_family"