Greenwoodmic temperaments: Difference between revisions

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+ a brief intro to secund
 
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<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>
This is a collection of [[rank-2 temperament|rank-2]] '''greenwoodmic temperaments''', which [[tempering out|temper out]] [[405/392]], the [[greenwoodma]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-08 14:57:02 UTC</tt>.<br>
: The original revision id was <tt>299835366</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]]


These temper out the greenwoodma, |-3 4 1 -2&gt; = 405/392.
Temperaments discussed elsewhere are
* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
* ''[[August]]'' (+36/35) → [[Augmented family #August|Augmented family]]
* ''[[Superpelog]]'' (+49/48) → [[Semaphoresmic clan #Superpelog|Semaphoresmic clan]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]


=Greenwood=
Considered below are secund and semishallowtone.
Commas: 405/392, 1323/1280


POTE generator: ~8/7 = 241.490
== Secund ==
Secund tempers out the greendwoodma, the [[avicennma]], and the [[orwellisma]]. It may be described as the {{nowrap| 9 & 26 }} temperament, with a [[ploidacot]] signature of pentacot. It divides the [[3/2|perfect fifth]] into five [[~]][[16/15]] generators, two for [[7/6]] and three for [[9/7]].  


Map: [&lt;7 11 1 12|, &lt;0 0 2 1|]
[[Subgroup]]: 2.3.5.7
Wedgie: &lt;&lt;0 14 7 22 11 -23||
EDOs: 14c, 21, 35, 84bcd
Badness: 0.1218


==11-limit==
[[Comma list]]: 405/392, 525/512
Commas: 45/44, 99/98, 1323/1280


POTE generator: ~8/7 = 242.711
{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
: mapping generators: ~2, ~16/15


Map: [&lt;7 11 1 12 9|, &lt;0 0 2 1 2|]
[[Optimal tuning]]s:  
EDOs: 14c, 21, 35, 49bcde, 84bcde
* [[WE]]: ~2 = 1203.6786{{c}}, ~16/15 = 138.3805{{c}}
Badness: 0.0575
: [[error map]]: {{val| +3.679 -6.374 -5.561 +7.195 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 138.0648{{c}}
: error map: {{val| 0.000 -11.631 -14.702 -2.372 }}


==13-limit==
{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}
Commas: 27/26, 45/44, 99/98, 640/637


POTE generaor: ~8/7 = 238.607
[[Badness]] (Sintel): 2.27


Map: [&lt;7 11 1 12 9 26|, &lt;0 0 2 1 2 0|]
=== 11-limit ===
EDOs: 14c, 21, 35, 91bf
Subgroup: 2.3.5.7.11
Badness: 0.0540


=Secund=
Comma list: 45/44, 99/98, 385/384
Commas: 405/392, 525/512


POTE generator: ~16/15 = 137.958
Mapping: {{mapping| 1 1 3 2 3 | 0 5 -6 7 4 }}


Map: [&lt;1 1 3 2|, &lt;0 5 -6 7|]
Optimal tunings:  
Wedgie: &lt;&lt;5 -6 7 -21 -3 33||
* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
EDOs: 9, 17, 26, 61bc, 87bc
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}
Badness: 0.0898


==11-limit==
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}
Commas: 45/44, 99/98, 385/384


POTE generator: ~12/11 = 137.900
Badness (Sintel): 1.41


Map: [&lt;1 1 3 2 3|, &lt;0 5 -6 7 4|]
==== 13-limit ====
EDOs: 9, 17, 26, 61bc, 87bc
Subgroup: 2.3.5.7.11.13
Badness: 0.0426


==13-limit==
Comma list: 27/26, 45/44, 99/98, 385/384
Commas: 27/26, 45/44, 99/98, 385/384


POTE generator: ~12/11 = 136.937
Mapping: {{mapping| 1 1 3 2 3 2 | 0 5 -6 7 4 15 }}


Map: [&lt;1 1 3 2 3 2|, &lt;0 5 -6 7 4 15|]
Optimal tunings:  
EDOs: 9, 26f, 35
* WE: ~2 = 1202.4465{{c}}, ~12/11 = 137.2165{{c}}
Badness: 0.0497
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.0309{{c}}


</pre></div>
{{Optimal ET sequence|legend=0| 9, 35 }}
<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;Greenwoodmic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Greenwood"&gt;Greenwood&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Secund"&gt;Secund&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
Badness (Sintel): 2.05
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
 
These temper out the greenwoodma, |-3 4 1 -2&amp;gt; = 405/392.&lt;br /&gt;
==== Secundly ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Greenwood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Greenwood&lt;/h1&gt;
 
Commas: 405/392, 1323/1280&lt;br /&gt;
Comma list: 45/44, 65/64, 78/77, 99/98
&lt;br /&gt;
 
POTE generator: ~8/7 = 241.490&lt;br /&gt;
Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}
&lt;br /&gt;
 
Map: [&amp;lt;7 11 1 12|, &amp;lt;0 0 2 1|]&lt;br /&gt;
Optimal tunings:  
Wedgie: &amp;lt;&amp;lt;0 14 7 22 11 -23||&lt;br /&gt;
* WE: ~2 = 1203.0195{{c}}, ~13/12 = 138.2642{{c}}
EDOs: 14c, 21, 35, 84bcd&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 138.0317{{c}}
Badness: 0.1218&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Greenwood-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 45/44, 99/98, 1323/1280&lt;br /&gt;
Badness (Sintel): 1.08
&lt;br /&gt;
 
POTE generator: ~8/7 = 242.711&lt;br /&gt;
== Semishallowtone ==
&lt;br /&gt;
{{See also| Shallowtone }}
Map: [&amp;lt;7 11 1 12 9|, &amp;lt;0 0 2 1 2|]&lt;br /&gt;
 
EDOs: 14c, 21, 35, 49bcde, 84bcde&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Badness: 0.0575&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 405/392, 35721/32768
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Greenwood-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
 
Commas: 27/26, 45/44, 99/98, 640/637&lt;br /&gt;
{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
&lt;br /&gt;
: mapping generators: ~189/128, ~3
POTE generaor: ~8/7 = 238.607&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
Map: [&amp;lt;7 11 1 12 9 26|, &amp;lt;0 0 2 1 2 0|]&lt;br /&gt;
* [[WE]]: ~189/128 = 603.1653{{c}}, ~3/2 = 686.6365{{c}} (~64/63 = 83.4712{{c}})
EDOs: 14c, 21, 35, 91bf&lt;br /&gt;
: [[error map]]: {{val| +6.331 -8.988 -2.034 -0.248 }}
Badness: 0.0540&lt;br /&gt;
* [[CWE]]: ~189/128 = 600.0000{{c}}, ~3/2 = 682.6498{{c}} (~64/63 = 82.6498{{c}})
&lt;br /&gt;
: error map: {{val| 0.000 -19.305 -12.811 -16.775 }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Secund"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Secund&lt;/h1&gt;
 
Commas: 405/392, 525/512&lt;br /&gt;
{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}
&lt;br /&gt;
 
POTE generator: ~16/15 = 137.958&lt;br /&gt;
[[Badness]] (Sintel): 8.42
&lt;br /&gt;
 
Map: [&amp;lt;1 1 3 2|, &amp;lt;0 5 -6 7|]&lt;br /&gt;
=== 11-limit ===
Wedgie: &amp;lt;&amp;lt;5 -6 7 -21 -3 33||&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 9, 17, 26, 61bc, 87bc&lt;br /&gt;
 
Badness: 0.0898&lt;br /&gt;
Comma list: 45/44, 99/98, 35721/32768
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Secund-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}
Commas: 45/44, 99/98, 385/384&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
POTE generator: ~12/11 = 137.900&lt;br /&gt;
* WE: ~189/128 = 603.0536{{c}}, ~3/2 = 686.6175{{c}} (~64/63 = 83.5640{{c}})
&lt;br /&gt;
* CWE: ~189/128 = 600.0000{{c}}, ~3/2 = 682.7238{{c}} (~64/63 = 82.7238{{c}})
Map: [&amp;lt;1 1 3 2 3|, &amp;lt;0 5 -6 7 4|]&lt;br /&gt;
 
EDOs: 9, 17, 26, 61bc, 87bc&lt;br /&gt;
{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}
Badness: 0.0426&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 4.15
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Secund-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
 
Commas: 27/26, 45/44, 99/98, 385/384&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~12/11 = 136.937&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 27/26, 45/44, 99/98, 35721/32768
Map: [&amp;lt;1 1 3 2 3 2|, &amp;lt;0 5 -6 7 4 15|]&lt;br /&gt;
 
EDOs: 9, 26f, 35&lt;br /&gt;
Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}
Badness: 0.0497&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Optimal tunings:
* WE: ~91/64 = 601.8872{{c}}, ~3/2 = 684.4774{{c}} (~64/63 = 82.5902{{c}})
* CWE: ~91/64 = 600.0000{{c}}, ~3/2 = 682.1676{{c}} (~64/63 = 82.1676{{c}})
 
{{Optimal ET sequence|legend=0| 14c, 30b }}
 
Badness (Sintel): 4.38
 
[[Category:Temperament collections]]
[[Category:Greenwoodmic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 11:55, 4 July 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.

This is a collection of rank-2 greenwoodmic temperaments, which temper out 405/392, the greenwoodma.

Temperaments discussed elsewhere are

Considered below are secund and semishallowtone.

Secund

Secund tempers out the greendwoodma, the avicennma, and the orwellisma. It may be described as the 9 & 26 temperament, with a ploidacot signature of pentacot. It divides the perfect fifth into five ~16/15 generators, two for 7/6 and three for 9/7.

Subgroup: 2.3.5.7

Comma list: 405/392, 525/512

Mapping: [⟨1 1 3 2], ⟨0 5 -6 7]]

mapping generators: ~2, ~16/15

Optimal tunings:

  • WE: ~2 = 1203.6786 ¢, ~16/15 = 138.3805 ¢
error map: ⟨+3.679 -6.374 -5.561 +7.195]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 138.0648 ¢
error map: ⟨0.000 -11.631 -14.702 -2.372]

Optimal ET sequence: 9, 17, 26, 61bc, 87bcc

Badness (Sintel): 2.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 385/384

Mapping: [⟨1 1 3 2 3], ⟨0 5 -6 7 4]]

Optimal tunings:

  • WE: ~2 = 1202.9415 ¢, ~12/11 = 138.2377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 137.9970 ¢

Optimal ET sequence: 9, 17, 26, 61bc

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 385/384

Mapping: [⟨1 1 3 2 3 2], ⟨0 5 -6 7 4 15]]

Optimal tunings:

  • WE: ~2 = 1202.4465 ¢, ~12/11 = 137.2165 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 137.0309 ¢

Optimal ET sequence: 9, 35

Badness (Sintel): 2.05

Secundly

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 99/98

Mapping: [⟨1 1 3 2 3 3], ⟨0 5 -6 7 4 6]]

Optimal tunings:

  • WE: ~2 = 1203.0195 ¢, ~13/12 = 138.2642 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 138.0317 ¢

Optimal ET sequence: 9, 17, 26, 61bcf

Badness (Sintel): 1.08

Semishallowtone

See also: Shallowtone

Subgroup: 2.3.5.7

Comma list: 405/392, 35721/32768

Mapping: [⟨2 0 36 15], ⟨0 1 -10 -3]]

mapping generators: ~189/128, ~3

Optimal tunings:

  • WE: ~189/128 = 603.1653 ¢, ~3/2 = 686.6365 ¢ (~64/63 = 83.4712 ¢)
error map: ⟨+6.331 -8.988 -2.034 -0.248]
  • CWE: ~189/128 = 600.0000 ¢, ~3/2 = 682.6498 ¢ (~64/63 = 82.6498 ¢)
error map: ⟨0.000 -19.305 -12.811 -16.775]

Optimal ET sequence: 14c, 44bd, 58bcd

Badness (Sintel): 8.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 35721/32768

Mapping: [⟨2 0 36 15 32], ⟨0 1 -10 -3 -8]]

Optimal tunings:

  • WE: ~189/128 = 603.0536 ¢, ~3/2 = 686.6175 ¢ (~64/63 = 83.5640 ¢)
  • CWE: ~189/128 = 600.0000 ¢, ~3/2 = 682.7238 ¢ (~64/63 = 82.7238 ¢)

Optimal ET sequence: 14c, 44bd, 58bcde, 72bbccddee

Badness (Sintel): 4.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 35721/32768

Mapping: [⟨2 0 36 15 32 -2], ⟨0 1 -10 -3 -8 3]]

Optimal tunings:

  • WE: ~91/64 = 601.8872 ¢, ~3/2 = 684.4774 ¢ (~64/63 = 82.5902 ¢)
  • CWE: ~91/64 = 600.0000 ¢, ~3/2 = 682.1676 ¢ (~64/63 = 82.1676 ¢)

Optimal ET sequence: 14c, 30b

Badness (Sintel): 4.38

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