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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''whitewood family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the Pythagorean apotome, [[2187/2048]]. Consequently the [[3/2|fifth]]s are always 4/7 of an [[octave]], a distinctly flat 685.714 [[cent]]s. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-08 14:03:40 UTC</tt>.<br>
: The original revision id was <tt>191845142</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">This family of temperaments tempers out the apotome, 2187/2048. Consequently the fifths are always 4/7 of an octave, a distinctly flat 685.714 cents. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.


The 5-limit version of this temperament is called "whitewood" temperament, to serve in contrast with the "blackwood" temperament which tempers out 256/243, the pythagorean limma. Whereas blackwood temperament can be thought of as a closed chain of 5 fifths and a major third generator, whitewood is a closed chain of 7 fifths and a major third generator. This means that blackwood is generally supported by 5n-EDOs, and whitewood is supported by 7n-EDOs, and the MOS of both scales follow a similar pattern.
== Whitewood ==
{{Main| Whitewood }}


Like blackwood, it shares a number of interesting properties which derive from the relatively small circle of fifths common to both: from any major or minor triad in the scale, one can always move away by ~3/2 or ~4/3 to reach another triad of the same type. This contrasts with the diatonic scale, in which one will eventually "hit a wall" if one moves by perfect fifth for long enough; the chain of fifths will eventually "stop" and make the next fifth a diminished fifth. This means that this scale is, in a sense, "pantonal," since resolutions that work in one key will work in all other keys.
Whitewood is the natural counterpart of [[blackwood]]: whereas blackwood can be thought of as a closed chain of five fifths and a [[5/4]] major third generator, whitewood is a closed chain of seven fifths and a 5/4 major third generator. This means that blackwood is generally supported by 5''n''-edos, and whitewood is supported by 7''n''-edos, and the [[mos]] of both scales follow a similar pattern.


Another interesting property is that it becomes possible to construct "super linked" 5-limit chords. In Whitewood[14] (or Blackwood[10]), if one stacks alternating major and minor thirds on top of one another, one will eventually come back to the root without ever hitting a wall, and hence the pattern can continue forever. Since all of the diatonic modes can be thought of as a stacked chain of 7 alternating thirds, placed in inversion, this means that Whitewood[14] and Blackwood[10] also make for excellent "panmodal" scales, in which you can construct "modal" sounding sonorities in one key that will work in all keys.
[[Subgroup]]: 2.3.5


Lastly, while blackwood fifths are sharp and thus necessitates the tuning as a whole to be sharp-leaning, whitewood fifths are flat and thus this tuning is generally flat-leaning.
[[Comma list]]: 2187/2048


=__5-limit__=
{{Mapping|legend=1| 7 11 0 | 0 0 1 }}
==Whitewood==
: mapping generators: ~9/8, ~5
Commas: 2187/2048


[[POTE tuning|POTE generator]]: 374.469
[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 172.1541{{c}}, ~5/4 = 376.0535{{c}} (~80/81 = 31.7453{{c}})
: [[error map]]: {{val| +5.079 -8.260 -0.102 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 378.3830{{c}} (~80/81 = 35.5258{{c}})
: error map: {{val| 0.000 -16.241 -7.931 }}


Map: [&lt;7 11 0|, &lt;0 0 1|]
{{Optimal ET sequence|legend=1| 7, 21, 28, 35, 77bbc }}
EDOs: 7, 21, 28, 35, 77


=__7-limit__=
[[Badness]] (Sintel): 3.63
==Whitewood==
Commas: 36/35, 2187/2048


[[POTE tuning|POTE generator]]: 392.700
Scales: [[7L 7s/13:7]] (140edo)


Map: [&lt;7 11 00 36|, &lt;0 0 1 -1|]
=== Overview to extensions ===
Wedgie: &lt;&lt;7 -7 11 -11 -36||
Temperaments discussed elsewhere include:  
EDOs: 7, 14, 21, 28, 49, 133
* ''[[Sept]]'' → [[Very low accuracy temperaments #Sept|Very low accuracy temperaments]]


==Redwood==
Considered below are septimal whitewood, redwood, greenwood, and jamesbond.
Commas: 525/512, 729/700


[[POTE tuning|POTE generator]]: 378.512
== Septimal whitewood ==
{{Main| Whitewood }}


Map: [&lt;7 11 0 52|, &lt;0 0 1 -2|]
[[Subgroup]]: 2.3.5.7
Wedgie: &lt;&lt;0 7 -14 11 -22 -52||
EDOs: 7, 35, 42


==Mujannab==
[[Comma list]]: 36/35, 2187/2048
Commas: 54/49, 64/63


[[POTE tuning|POTE generator]]: 395.187
{{Mapping|legend=1| 7 11 0 36 | 0 0 1 -1 }}


Map: [&lt;7 11 0 20|, &lt;0 0 1 0|]
[[Optimal tuning]]s:
Wedgie: &lt;&lt;0 7 0 11 0 -20||
* [[WE]]: ~9/8 = 171.5524{{c}}, ~5/4 = 392.9834{{c}} (~64/63 = 49.8786{{c}})
EDOs: 7, 21, 70, 91</pre></div>
: [[error map]]: {{val| +0.867 -14.879 +8.403 +12.343 }}
<h4>Original HTML content:</h4>
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 392.7412{{c}} (~64/63 = 49.8841{{c}})
<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;Apotome family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This family of temperaments tempers out the apotome, 2187/2048. Consequently the fifths are always 4/7 of an octave, a distinctly flat 685.714 cents. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.&lt;br /&gt;
: error map: {{val| 0.000 -16.241 +6.428 +9.861 }}
&lt;br /&gt;
 
The 5-limit version of this temperament is called &amp;quot;whitewood&amp;quot; temperament, to serve in contrast with the &amp;quot;blackwood&amp;quot; temperament which tempers out 256/243, the pythagorean limma. Whereas blackwood temperament can be thought of as a closed chain of 5 fifths and a major third generator, whitewood is a closed chain of 7 fifths and a major third generator. This means that blackwood is generally supported by 5n-EDOs, and whitewood is supported by 7n-EDOs, and the MOS of both scales follow a similar pattern.&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 14, 21, 28, 49b }}
&lt;br /&gt;
 
Like blackwood, it shares a number of interesting properties which derive from the relatively small circle of fifths common to both: from any major or minor triad in the scale, one can always move away by ~3/2 or ~4/3 to reach another triad of the same type. This contrasts with the diatonic scale, in which one will eventually &amp;quot;hit a wall&amp;quot; if one moves by perfect fifth for long enough; the chain of fifths will eventually &amp;quot;stop&amp;quot; and make the next fifth a diminished fifth. This means that this scale is, in a sense, &amp;quot;pantonal,&amp;quot; since resolutions that work in one key will work in all other keys.&lt;br /&gt;
[[Badness]] (Sintel): 2.88
&lt;br /&gt;
 
Another interesting property is that it becomes possible to construct &amp;quot;super linked&amp;quot; 5-limit chords. In Whitewood[14] (or Blackwood[10]), if one stacks alternating major and minor thirds on top of one another, one will eventually come back to the root without ever hitting a wall, and hence the pattern can continue forever. Since all of the diatonic modes can be thought of as a stacked chain of 7 alternating thirds, placed in inversion, this means that Whitewood[14] and Blackwood[10] also make for excellent &amp;quot;panmodal&amp;quot; scales, in which you can construct &amp;quot;modal&amp;quot; sounding sonorities in one key that will work in all keys.&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Lastly, while blackwood fifths are sharp and thus necessitates the tuning as a whole to be sharp-leaning, whitewood fifths are flat and thus this tuning is generally flat-leaning.&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 36/35, 45/44, 2079/2048
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x5-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;u&gt;5-limit&lt;/u&gt;&lt;/h1&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x5-limit-Whitewood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Whitewood&lt;/h2&gt;
Mapping: {{mapping| 7 11 0 36 8 | 0 0 1 -1 1 }}
Commas: 2187/2048&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 374.469&lt;br /&gt;
* WE: ~11/10 = 171.4451{{c}}, ~5/4 = 390.0053{{c}} (~64/63 = 47.1151{{c}})
&lt;br /&gt;
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 389.9864{{c}} (~64/63 = 47.1293{{c}})
Map: [&amp;lt;7 11 0|, &amp;lt;0 0 1|]&lt;br /&gt;
 
EDOs: 7, 21, 28, 35, 77&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="x7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;u&gt;7-limit&lt;/u&gt;&lt;/h1&gt;
Badness (Sintel): 2.01
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x7-limit-Whitewood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Whitewood&lt;/h2&gt;
 
Commas: 36/35, 2187/2048&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 392.700&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 27/26, 36/35, 45/44, 512/507
Map: [&amp;lt;7 11 00 36|, &amp;lt;0 0 1 -1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;7 -7 11 -11 -36||&lt;br /&gt;
Mapping: {{mapping| 7 11 0 36 8 26 | 0 0 1 -1 1 0 }}
EDOs: 7, 14, 21, 28, 49, 133&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x7-limit-Redwood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Redwood&lt;/h2&gt;
* WE: ~11/10 = 171.3236{{c}}, ~5/4 = 390.4957{{c}} (~64/63 = 47.8484{{c}})
Commas: 525/512, 729/700&lt;br /&gt;
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 390.6336{{c}} (~64/63 = 47.7765{{c}})
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 378.512&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28 }}
&lt;br /&gt;
 
Map: [&amp;lt;7 11 0 52|, &amp;lt;0 0 1 -2|]&lt;br /&gt;
Badness (Sintel): 1.65
Wedgie: &amp;lt;&amp;lt;0 7 -14 11 -22 -52||&lt;br /&gt;
 
EDOs: 7, 35, 42&lt;br /&gt;
== Redwood ==
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x7-limit-Mujannab"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Mujannab&lt;/h2&gt;
 
Commas: 54/49, 64/63&lt;br /&gt;
[[Comma list]]: 525/512, 729/700
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 395.187&lt;br /&gt;
{{Mapping|legend=1| 7 11 0 52 | 0 0 1 -2 }}
&lt;br /&gt;
 
Map: [&amp;lt;7 11 0 20|, &amp;lt;0 0 1 0|]&lt;br /&gt;
[[Optimal tuning]]s:
Wedgie: &amp;lt;&amp;lt;0 7 0 11 0 -20||&lt;br /&gt;
* [[WE]]: ~9/8 = 172.0521{{c}}, ~5/4 = 379.5277{{c}} (~36/35 = 35.4234{{c}})
EDOs: 7, 21, 70, 91&lt;/body&gt;&lt;/html&gt;</pre></div>
: [[error map]]: {{val| +4.365 -9.382 +1.944 +1.370 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 377.7903{{c}} (~36/35 = 34.9331{{c}})
: error map: {{val| 0.000 -16.241 -8.523 -10.121 }}
 
{{Optimal ET sequence|legend=1| 7, 28d, 35 }}
 
[[Badness]] (Sintel): 4.18
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 385/384, 729/700
 
Mapping: {{mapping| 7 11 0 52 8 | 0 0 1 -2 1 }}
 
Optimal tunings:
* WE: ~11/10 = 171.9390{{c}}, ~5/4 = 377.8321{{c}} (~36/35 = 33.9542{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 376.7162{{c}} (~36/35 = 33.8590{{c}})
 
{{Optimal ET sequence|legend=0| 7, 28d, 35 }}
 
Badness (Sintel): 2.59
 
== Greenwood ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 405/392, 1323/1280
 
{{Mapping|legend=1| 7 11 1 12 | 0 0 2 1 }}
 
: mapping generators: ~9/8, ~15/7
 
[[Optimal tuning]]s:  
* [[WE]]: ~9/8 = 172.1073{{c}}, ~15/14 = 101.7681{{c}} (~21/20 = 70.3391{{c}})
: [[error map]]: {{val| +4.751 -8.775 -1.169 +2.980 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~15/14 = 103.3802{{c}} (~21/20 = 68.0484{{c}})
: error map: {{val| 0.000 -16.241 -8.125 -8.303 }}
 
{{Optimal ET sequence|legend=1| 7c, 14c, 21, 35, 84bbccd }}
 
[[Badness]] (Sintel): 3.08
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 99/98, 1323/1280
 
Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}
 
Optimal tunings:  
* WE: ~11/10 = 172.0795{{c}}, ~15/14 = 100.5259{{c}} (~21/20 = 71.5536{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 102.1866{{c}} (~21/20 = 69.2419{{c}})
 
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35, 49bcde }}
 
Badness (Sintel): 1.90
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 45/44, 99/98, 640/637
 
Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}
 
Optimal tunings:
* WE: ~11/10 = 171.6777{{c}}, ~15/14 = 104.4016{{c}} (~21/20 = 67.2761{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 104.8518{{c}} (~21/20 = 66.5768{{c}})
 
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}
 
Badness (Sintel): 2.23
 
== Jamesbond ==
This temperament uses exactly the same 5-limit as 7et, but the harmonic 7 is mapped to an independent generator. It is so named because its "[[wedgie]]" (a kind of mathematical object representing the temperament) starts with {{multival| 0 0 7 … }} (in fact, it is {{multival| 0 0 7 0 11 16 }})
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 25/24, 81/80
 
{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }}
: mapping generators: ~10/9, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~10/9 = 172.790{{c}}, ~7/4 = 949.343{{c}}
: [[error map]]: {{val| +9.533 -1.261 -21.668 -0.418 }}
* [[CWE]]: ~10/9 = 171.429{{c}}, ~7/4 = 948.499{{c}}
: error map: {{val| -0.000 -16.241 -43.457 -20.327 }}
 
{{Optimal ET sequence|legend=1| 7(d), 14c }}
 
[[Badness]] (Sintel): 1.06
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 33/32, 45/44
 
Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}
 
Optimal tunings:  
* WE: ~10/9 = 172.830{{c}}, ~7/4 = 948.784{{c}}
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 946.554{{c}}
 
{{Optimal ET sequence|legend=0| 7(d), 14c }}
 
Badness (Sintel): 0.778
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 27/26, 33/32, 40/39
 
Mapping: {{mapping| 7 11 16 0 24 26 | 0 0 0 1 0 0 }}
 
Optimal tunings:  
* WE: ~10/9 = 172.390{{c}}, ~7/4 = 954.559{{c}}
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 952.367{{c}}
 
{{Optimal ET sequence|legend=0| 7(d), 14c }}
 
Badness (Sintel): 0.951
 
==== Austinpowers ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 33/32, 45/44, 65/63
 
Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }}
 
Optimal tunings:
* WE: ~10/9 = 172.873{{c}}, ~7/4 = 960.581{{c}}
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 958.793{{c}}
 
{{Optimal ET sequence|legend=0| 7(df), 14cf }}
 
Badness (Sintel): 0.933
 
[[Category:Whitewood family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Whitewood_family"