Xenharmonic Wiki

Gravity family: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 251255716 - Original comment: **
 
(42 intermediate revisions by 13 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 '''gravity family''' tempers out [[graviton]], the 5-limit comma 129140163/128000000 = {{monzo| -13 17 -6 }}. The graviton equals (81/80)<sup>4</sup>/(25/24), so that four 81/80 commas come to a classic chromatic semitone. The generator of gravity temperament is a grave fifth of [[~]][[40/27]], and hence the name. It is part of the [[syntonic&ndash;chromatic equivalence continuum]], whereby (81/80)<sup>''k''</sup> = 25/24.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-06 14:07:45 UTC</tt>.<br>
: The original revision id was <tt>251255716</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">[[toc|flat]]


=Gravity=
== Gravity ==
The gravity family tempers out [[graviton]], the 5-limit comma 129140163/128000000 = |-13 17 -6&gt;. The graviton equals (81/80)^4/(25/24), so that four 81/80 commas come to a chromatic semitone. The generator of gravity temperament is a grave fifth of 40/27, and hence the name. It is part of the Mavila -&gt; Dicot -&gt; Porcupine -&gt; Tetracot -&gt; Amity continuum, whereby (81/80)^n = 25/24.
{{main|Gravity}}


Comma: 129140163/128000000
[[Subgroup]]: 2.3.5


POTE generator: ~40/27 = 683.156
[[Comma list]]: 129140163/128000000


Map: [&lt;1 5 12|, &lt;0 -6 -17|]
{{Mapping|legend=1| 1 5 12 | 0 -6 -17 }}
EDOs: [[7edo|7]], [[58edo|58]], [[65edo|65]], [[137edo|137]], [[202edo|202]], [[267edo|267]], [[469edo|469]]
Badness: 0.0932


=Harry=
: mapping generators: ~2, ~40/27
Commas: 2401/2400, 19683/19600


POTE generator: ~21/20 = 83.156
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.844


Map: [&lt;2 4 7 7|, &lt;0 -6 -17 -10|]
{{Optimal ET sequence|legend=1| 7, 37cc, 44c, 51c, 58, 65, 137, 202, 267, 469 }}
Wedgie: &lt;&lt;12 34 20 26 -2 -49||
EDOs: 58, 72, 130, 202, 534
Badness: 0.0341


==11-limit==
[[Badness]]: 0.093184
Commas: 243/242, 441/440, 4000/3993


POTE generator: ~21/20 = 83.167
=== Overview to extensions ===
Full 7-limit extensions of gravity include marvo (65d &amp; 72), zarvo (65 &amp; 72), gravid (58 &amp; 65), and harry (58 &amp; 72), all considered below. A notable subgroup extension is larry.


Map: [&lt;2 4 7 7 9|, &lt;0 -6 -17 -10 -15|]
There is also an unnamed 58 & 65d extension by tempering [[176/175]] to extend larry to include prime 7 and tempering [[847/845]] to extend it to the 13-limit, with an [[S-expression]]-based comma list of {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]], ([[847/845|S11/S13]],) [[144/143|S12]]}.
EDOs: 58, 72, 130, 202
Badness: 0.0159


==13-limit==
=== 2.3.5.11 subgroup (larry) ===
Commas: 243/242, 351/350, 441/440, 676/675
Subgroup: 2.3.5.11


POTE generator: ~21/20 = 83.116
Comma list: 243/242, 4000/3993


Map: [&lt;2 4 7 7 9 11|, &lt;0 -6 -17 -10 -15 -26|]
Sval mapping: {{mapping| 1 5 12 12 | 0 -6 -17 -15 }}
EDOs: 58, 72, 130
Badness: 0.0130


</pre></div>
Gencom mapping: {{mapping| 1 5 12 0 12 | 0 -6 -17 0 -15 }}
<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;Gravity family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Gravity"&gt;Gravity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Harry"&gt;Harry&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
: gencom: [2 40/27; 243/242 4000/3993]
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Gravity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Gravity&lt;/h1&gt;
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834
The gravity family tempers out &lt;a class="wiki_link" href="/graviton"&gt;graviton&lt;/a&gt;, the 5-limit comma 129140163/128000000 = |-13 17 -6&amp;gt;. The graviton equals (81/80)^4/(25/24), so that four 81/80 commas come to a chromatic semitone. The generator of gravity temperament is a grave fifth of 40/27, and hence the name. It is part of the Mavila -&amp;gt; Dicot -&amp;gt; Porcupine -&amp;gt; Tetracot -&amp;gt; Amity continuum, whereby (81/80)^n = 25/24.&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}
Comma: 129140163/128000000&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.0125
POTE generator: ~40/27 = 683.156&lt;br /&gt;
 
&lt;br /&gt;
== Marvo ==
Map: [&amp;lt;1 5 12|, &amp;lt;0 -6 -17|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58&lt;/a&gt;, &lt;a class="wiki_link" href="/65edo"&gt;65&lt;/a&gt;, &lt;a class="wiki_link" href="/137edo"&gt;137&lt;/a&gt;, &lt;a class="wiki_link" href="/202edo"&gt;202&lt;/a&gt;, &lt;a class="wiki_link" href="/267edo"&gt;267&lt;/a&gt;, &lt;a class="wiki_link" href="/469edo"&gt;469&lt;/a&gt;&lt;br /&gt;
 
Badness: 0.0932&lt;br /&gt;
[[Comma list]]: 225/224, 78125000/78121827
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Harry"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harry&lt;/h1&gt;
{{Mapping|legend=1| 1 5 12 29 | 0 -6 -17 -46 }}
Commas: 2401/2400, 19683/19600&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.694
POTE generator: ~21/20 = 83.156&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
Map: [&amp;lt;2 4 7 7|, &amp;lt;0 -6 -17 -10|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;12 34 20 26 -2 -49||&lt;br /&gt;
[[Badness]]: 0.097627
EDOs: 58, 72, 130, 202, 534&lt;br /&gt;
 
Badness: 0.0341&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Harry-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
 
Commas: 243/242, 441/440, 4000/3993&lt;br /&gt;
Comma list: 225/224, 243/242, 4000/3993
&lt;br /&gt;
 
POTE generator: ~21/20 = 83.167&lt;br /&gt;
Mapping: {{mapping| 1 5 12 29 12 | 0 -6 -17 -46 -15 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 4 7 7 9|, &amp;lt;0 -6 -17 -10 -15|]&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.699
EDOs: 58, 72, 130, 202&lt;br /&gt;
 
Badness: 0.0159&lt;br /&gt;
{{Optimal ET sequence|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Harry-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Badness: 0.031685
Commas: 243/242, 351/350, 441/440, 676/675&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
POTE generator: ~21/20 = 83.116&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;2 4 7 7 9 11|, &amp;lt;0 -6 -17 -10 -15 -26|]&lt;br /&gt;
Comma list: 225/224, 243/242, 351/350, 1625/1617
EDOs: 58, 72, 130&lt;br /&gt;
 
Badness: 0.0130&lt;/body&gt;&lt;/html&gt;</pre></div>
Mapping: {{mapping| 1 5 12 29 12 39 | 0 -6 -17 -46 -15 -62 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.730
 
{{Optimal ET sequence|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
 
Badness: 0.026882
 
== Zarvo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 33075/32768
 
{{Mapping|legend=1| 1 5 12 -12 | 0 -6 -17 26 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.702
 
{{Optimal ET sequence|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}
 
[[Badness]]: 0.096840
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 385/384, 4000/3993
 
Mapping: {{mapping| 1 5 12 -12 12 | 0 -6 -17 26 -15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.691
 
{{Optimal ET sequence|legend=1| 65, 72, 353cd }}
 
Badness: 0.034773
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 243/242, 325/324, 385/384
 
Mapping: {{mapping| 1 5 12 -12 12 -2 | 0 -6 -17 26 -15 10 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.667
 
{{Optimal ET sequence|legend=1| 65f, 72 }}
 
Badness: 0.027584
 
== Gravid ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 1605632/1594323
 
{{Mapping|legend=1| 1 5 12 25 | 0 -6 -17 -39 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 517.140
 
{{Optimal ET sequence|legend=1| 58, 123, 181c }}
 
[[Badness]]: 0.131153
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 896/891
 
Mapping: {{mapping| 1 5 12 25 12 | 0 -6 -17 -39 -15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 517.155
 
{{Optimal ET sequence|legend=1| 58, 123, 181ce }}
 
Badness: 0.047283
 
== Harry ==
{{Main| Harry }}
 
Harry adds the [[breedsma]], 2401/2400, and the [[cataharry comma]], 19683/19600, to the set of commas, and may be described as the 58 &amp; 72 temperament. The [[period]] is half an [[octave]], and the generator ~21/20, with generator tunings of [[130edo|9\130]] or [[202edo|14\202]] being good choices. [[Mos]] of size 14, 16, 30, 44 or 58 are among the scale choices.
 
It becomes much more interesting as we move to the 11-limit, where we can add [[243/242]], [[441/440]] and [[540/539]] to the set of commas. 9\130 and especially 14\202 still make for good tuning choices.
 
Similar comments apply to the 13-limit, where we can add [[351/350]], [[364/363]], and [[729/728]] to the commas. 130edo is again a good tuning choice, but even better might be tuning the harmonic 7 justly, which can be done via a generator of 83.1174 [[cent]]s. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 19683/19600
 
{{Mapping|legend=1| 2 4 7 7 | 0 -6 -17 -10 }}
 
: mapping generators: ~567/400, ~21/20
 
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~27/20 = 516.844 (~21/20 = 83.156)
 
{{Optimal ET sequence|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}
 
[[Badness]]: 0.034077
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 4000/3993
 
Mapping: {{mapping| 2 4 7 7 9 | 0 -6 -17 -10 -15 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~27/20 = 516.833 (~21/20 = 83.167)
 
{{Optimal ET sequence|legend=1| 14c, 58, 72, 130, 202 }}
 
Badness: 0.015867
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 364/363, 441/440
 
Mapping: {{mapping| 2 4 7 7 9 11 | 0 -6 -17 -10 -15 -26 }}
 
Optimal tuning (POTE): ~55/39 = 1\2, ~27/20 = 516.884 (~21/20 = 83.116)
 
{{Optimal ET sequence|legend=1| 14cf, 58, 72, 130, 332f, 462ef }}
 
Badness: 0.013046
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 221/220, 243/242, 289/288, 351/350, 441/440
 
Mapping: {{mapping| 2 4 7 7 9 11 9 | 0 -6 -17 -10 -15 -26 -6 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~27/20 = 516.832 (~21/20 = 83.168)
 
{{Optimal ET sequence|legend=1| 14cf, 58, 72, 130, 202g }}
 
Badness: 0.012657
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Gravity family| ]] <!-- main article -->
[[Category:Gravity| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Gravity_family"