Whitewood family: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2011-01-02 12:58:13 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2011-01-02 14:17:49 UTC</tt>.<br>

: The original revision id was <tt>~~190489412~~</tt>.<br>

: The original revision id was <tt>190497118</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-EDO's, and whitewood is supported by 7n-EDO's, and the MOS's of both scales follow a similar pattern.

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.

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.

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.

=__5-limit__=

==Whitewood==

Commas: 2187/2048

[[POTE tuning|POTE generator]]: 374.469

Map: [<7 11 0|, <0 0 1|]

EDOs: 7, 35, 77

EDOs: 7, 21, 28, 35, 77

=____7-limit____=

==Whitewood==

Line 21:

Line 35:

Wedgie: <<7 -7 11 -11 -36||

EDOs: 7, 14, 21, 28, 49

==Other Whitewood==

Commas: 525/512, 2187/2048

[[POTE tuning|POTE generator]]: ???

Map: ???

Wedgie: ???

EDOs: 7, 35, 42

==Mujannab==

Line 32:

Line 56:

<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"><html><head><title>Apotome family</title></head><body>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.<br />

<br />

The 5-limit version of this temperament is called &quot;whitewood&quot; temperament, to serve in contrast with the &quot;blackwood&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-EDO's, and whitewood is supported by 7n-EDO's, and the MOS's of both scales follow a similar pattern.<br />

<br />

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 &quot;hit a wall&quot; if one moves by perfect fifth for long enough; the chain of fifths will eventually &quot;stop&quot; and make the next fifth a diminished fifth. This means that this scale is, in a sense, &quot;pantonal,&quot; since resolutions that work in one key will work in all other keys.<br />

<br />

Another interesting property is that it becomes possible to construct &quot;super linked&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 &quot;panmodal&quot; scales, in which you can construct &quot;modal&quot; sounding sonorities in one key that will work in all keys.<br />

<br />

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.<br />

<br />

<h1 id="toc0"><a name="x5-limit"></a><u>5-limit</u></h1>

<h2 id="toc1"><a name="x5-limit-Whitewood"></a>Whitewood</h2>

Commas: 2187/2048<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 374.469<br />

<br />

Map: [&lt;7 11 0|, &lt;0 0 1|]<br />

EDOs: 7, 35, 77<br />

EDOs: 7, 21, 28, 35, 77<br />

<br />

<h2 id="~~toc0~~"><a name="x-Whitewood"></a>Whitewood</h2>

<h1 id="toc2"><a name="x__7-limit__"></a>7-limit</h1>

<br />

<h2 id="toc3"><a name="x__7-limit__-Whitewood"></a>Whitewood</h2>

Commas: 36/35, 2187/2048<br />

<br />

Line 47:

Line 85:

EDOs: 7, 14, 21, 28, 49<br />

<br />

<h2 id="~~toc1~~"><a name="x-Mujannab"></a>Mujannab</h2>

<br />

<h2 id="toc4"><a name="x__7-limit__-Other Whitewood"></a>Other Whitewood</h2>

Commas: 525/512, 2187/2048<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ???<br />

<br />

Map: ???<br />

Wedgie: ???<br />

EDOs: 7, 35, 42<br />

<br />

<h2 id="toc5"><a name="x__7-limit__-Mujannab"></a>Mujannab</h2>

Commas: 54/49, 64/63<br />

<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2011-01-02 12:58:13 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-01-02 14:17:49 UTC</tt>.<br>
-: The original revision id was <tt>190489412</tt>.<br>
+: The original revision id was <tt>190497118</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-EDO's, and whitewood is supported by 7n-EDO's, and the MOS's of both scales follow a similar pattern.
+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.
+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.
+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.
+=__5-limit__=
+==Whitewood==
+Commas: 2187/2048
 [[POTE tuning|POTE generator]]: 374.469
 Map: [&lt;7 11 0|, &lt;0 0 1|]
-EDOs: 7, 35, 77
+EDOs: 7, 21, 28, 35, 77
+=____7-limit____=
 ==Whitewood==
@@ Line 21: / Line 35: @@
 Wedgie: &lt;&lt;7 -7 11 -11 -36||
 EDOs: 7, 14, 21, 28, 49
+==Other Whitewood==
+Commas: 525/512, 2187/2048
+[[POTE tuning|POTE generator]]: ???
+Map: ???
+Wedgie: ???
+EDOs: 7, 35, 42
 ==Mujannab==
@@ Line 32: / Line 56: @@
 <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;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;
+&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-EDO's, and whitewood is supported by 7n-EDO's, and the MOS's of both scales follow a similar pattern.&lt;br /&gt;
+&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;
+&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;
+&lt;br /&gt;
+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;
+&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;
+ Commas: 2187/2048&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 374.469&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;7 11 0|, &amp;lt;0 0 1|]&lt;br /&gt;
-EDOs: 7, 35, 77&lt;br /&gt;
+EDOs: 7, 21, 28, 35, 77&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Whitewood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Whitewood&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="x__7-limit__"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;7-limit&lt;/h1&gt;
+ &lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x__7-limit__-Whitewood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Whitewood&lt;/h2&gt;
   Commas: 36/35, 2187/2048&lt;br /&gt;
 &lt;br /&gt;
@@ Line 47: / Line 85: @@
 EDOs: 7, 14, 21, 28, 49&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Mujannab"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Mujannab&lt;/h2&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x__7-limit__-Other Whitewood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Other Whitewood&lt;/h2&gt;
+ Commas: 525/512, 2187/2048&lt;br /&gt;
+&lt;br /&gt;
+&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ???&lt;br /&gt;
+&lt;br /&gt;
+Map: ???&lt;br /&gt;
+Wedgie: ???&lt;br /&gt;
+EDOs: 7, 35, 42&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x__7-limit__-Mujannab"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Mujannab&lt;/h2&gt;
   Commas: 54/49, 64/63&lt;br /&gt;
 &lt;br /&gt;