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Blackwood temperament modal harmony (in 15edo): Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2013-03-21 21:52:11 UTC</tt>.<br>
: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2013-03-21 23:11:27 UTC</tt>.<br>
: The original revision id was <tt>416942398</tt>.<br>
: The original revision id was <tt>416956404</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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|| Major-Diminished (Maj(b5)) || Major-Sus6 || 0-400-640 || E-Ab-Bb || 1-4-6 || I, III, V, VIII, IX || II, IV, VI, VIII, X ||
|| Major-Diminished (Maj(b5)) || Major-Sus6 || 0-400-640 || E-Ab-Bb || 1-4-6 || I, III, V, VIII, IX || II, IV, VI, VIII, X ||
==Diatonic Modal Harmony in Blacksmith[10]==  
==Diatonic Modal Harmony in Blacksmith[10]==  
When using the 10-note scale to organize 5-limit chord progressions that serve as accompaniment to a tonal melody (and is organized according to the principles of functional harmony), it is typical to approach the scale not as a single 10-note scale, but rather as a 7-note scale whose degrees are constantly shifting to keep pace with the chords. This is because, over a given 5-limit triad, three of the 10 notes will be relatively discordant compared to the other 7. It's also because any set of five 5-limit triads maximally connected by common tones will actually form a 7-note scale, which will have a tendency to make those 7 notes more tonally salient over those chords. This is perhaps better understood through diagrams comparing 5-limit JI with 15edo Blacksmith[10]:
When using the 10-note scale to organize 5-limit chord progressions that serve as accompaniment to a tonal melody (and is organized according to the principles of functional harmony), it is typical to approach the scale not as a single 10-note scale, but rather as a 7-note scale whose degrees are constantly shifting to keep pace with the chords. This is because, over a given 5-limit triad, three of the 10 notes will be relatively discordant compared to the other 7. It's also because any set of five 5-limit triads maximally connected by common tones will actually form a 7-note scale, which will have a tendency to make those 7 notes more tonally salient over those chords. This is perhaps better understood through diagrams comparing 5-limit JI with 15edo Blacksmith[10]:</pre></div>
 
[[image:Screen Shot 2013-03-20 at 6.01.58 PM.png align="left"]]
 
 
 
 
 
 
 
 
 
 
 
 
 
The top lattice represents three chains of 3/2s (along the horizontal axis) connected by chains of 5/4s (represented by /) or 6/5s (represented by \). The accidental " **,** " means that an interval is lowered by a syntonic comma (81/80), while the accidental " **'** " means an interval is raised by a syntonic comma. The " **#** " and " **b** " accidentals represent raising and lowering by the Pythagorean apotome (2187:2048).
 
The bottom lattice is what we get if we temper the top lattice in 15edo, and use decimal notation to notate it. Unlike in a meantone tuning, where 81/80 is tempered out (and thus " **C** " is the same pitch-class as " **C,** "), 256/243 (the octave-normalized product of five 3/2s or 4/3s) is tempered out here instead, which makes " **C** " and " **B** " the same pitch-class. Thus " **C** " and " **B** " both correspond to the Blacksmith[10] note " **1** ". (Note that this affects the apotome as well, causing it to widen from ~114¢ in JI to 240¢ in 15edo!). In the bottom lattice, the " **b** " accidental represents lowering a note by a 5-limit chromatic semitone (25/24). Sharps (" **#** ") are not pictured here, but would arise if another chain of 3/2s was added another 5/4 above the top-most chain. </pre></div>
<h4>Original HTML content:</h4>
<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;Harmony in 15edo Blacksmith&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x-Important features of Blacksmith[10] in 15edo"&gt;Important features of Blacksmith[10] in 15edo&lt;/a&gt;&lt;/div&gt;
<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;Harmony in 15edo Blacksmith&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x-Important features of Blacksmith[10] in 15edo"&gt;Important features of Blacksmith[10] in 15edo&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Diatonic Modal Harmony in Blacksmith[10]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Diatonic Modal Harmony in Blacksmith[10]&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Diatonic Modal Harmony in Blacksmith[10]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Diatonic Modal Harmony in Blacksmith[10]&lt;/h2&gt;
  When using the 10-note scale to organize 5-limit chord progressions that serve as accompaniment to a tonal melody (and is organized according to the principles of functional harmony), it is typical to approach the scale not as a single 10-note scale, but rather as a 7-note scale whose degrees are constantly shifting to keep pace with the chords. This is because, over a given 5-limit triad, three of the 10 notes will be relatively discordant compared to the other 7. It's also because any set of five 5-limit triads maximally connected by common tones will actually form a 7-note scale, which will have a tendency to make those 7 notes more tonally salient over those chords. This is perhaps better understood through diagrams comparing 5-limit JI with 15edo Blacksmith[10]:&lt;br /&gt;
  When using the 10-note scale to organize 5-limit chord progressions that serve as accompaniment to a tonal melody (and is organized according to the principles of functional harmony), it is typical to approach the scale not as a single 10-note scale, but rather as a 7-note scale whose degrees are constantly shifting to keep pace with the chords. This is because, over a given 5-limit triad, three of the 10 notes will be relatively discordant compared to the other 7. It's also because any set of five 5-limit triads maximally connected by common tones will actually form a 7-note scale, which will have a tendency to make those 7 notes more tonally salient over those chords. This is perhaps better understood through diagrams comparing 5-limit JI with 15edo Blacksmith[10]:&lt;/body&gt;&lt;/html&gt;</pre></div>
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The top lattice represents three chains of 3/2s (along the horizontal axis) connected by chains of 5/4s (represented by /) or 6/5s (represented by \). The accidental &amp;quot; &lt;strong&gt;,&lt;/strong&gt; &amp;quot; means that an interval is lowered by a syntonic comma (81/80), while the accidental &amp;quot; &lt;strong&gt;'&lt;/strong&gt; &amp;quot; means an interval is raised by a syntonic comma. The &amp;quot; &lt;strong&gt;#&lt;/strong&gt; &amp;quot; and &amp;quot; &lt;strong&gt;b&lt;/strong&gt; &amp;quot; accidentals represent raising and lowering by the Pythagorean apotome (2187:2048). &lt;br /&gt;
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The bottom lattice is what we get if we temper the top lattice in 15edo, and use decimal notation to notate it. Unlike in a meantone tuning, where 81/80 is tempered out (and thus &amp;quot; &lt;strong&gt;C&lt;/strong&gt; &amp;quot; is the same pitch-class as &amp;quot; &lt;strong&gt;C,&lt;/strong&gt; &amp;quot;), 256/243 (the octave-normalized product of five 3/2s or 4/3s) is tempered out here instead, which makes &amp;quot; &lt;strong&gt;C&lt;/strong&gt; &amp;quot; and &amp;quot; &lt;strong&gt;B&lt;/strong&gt; &amp;quot; the same pitch-class. Thus &amp;quot; &lt;strong&gt;C&lt;/strong&gt; &amp;quot; and &amp;quot; &lt;strong&gt;B&lt;/strong&gt; &amp;quot; both correspond to the Blacksmith[10] note &amp;quot; &lt;strong&gt;1&lt;/strong&gt; &amp;quot;. (Note that this affects the apotome as well, causing it to widen from ~114¢ in JI to 240¢ in 15edo!). In the bottom lattice, the &amp;quot; &lt;strong&gt;b&lt;/strong&gt; &amp;quot; accidental represents lowering a note by a 5-limit chromatic semitone (25/24). Sharps (&amp;quot; &lt;strong&gt;#&lt;/strong&gt; &amp;quot;) are not pictured here, but would arise if another chain of 3/2s was added another 5/4 above the top-most chain.&lt;/body&gt;&lt;/html&gt;</pre></div>
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