Xenharmonic Wiki

Whitewood family: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>guest
**Imported revision 197667110 - Original comment: **
m Recategorize
 
(43 intermediate revisions by 12 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 '''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:guest|guest]] and made on <tt>2011-02-01 00:06:24 UTC</tt>.<br>
: The original revision id was <tt>197667110</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 }}


The 14-note MOS of whitewood, like the 10-note MOS of blackwood, 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 in the scale, at least keys that share the same chord quality.
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 necessitate 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;
 
The 14-note MOS of whitewood, like the 10-note MOS of blackwood, 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 in the scale, at least keys that share the same chord quality.&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 necessitate 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"