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

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

This approach allows one to apply the usual principles of diatonic tonality and modality, with the caveat that each familiar mode of the diatonic scale will come in two flavors, depending which of the 7 notes one wants to build consonant triads on. The two flavors will share six notes in common, but one of the seven will differ by 81/80 (i.e. one step of 15edo). Interesting relationships do arise if one maintains the tonic but switches through its different modes (i.e. 1 mixolydian to 1 ionian to 1 lydian), and an "extra" mode appears, because of the fact that 256/243 is tempered out (a way to understand 256/243 vanishing in musical terms is that V/V/V/V = IV, unlike in meantone where V/V/V/V = iii). All together, there are 14 modes of the 7-note diatonic to be found in Blacksmith[10] if we keep the same tonic, and 20 if we allow alterations of the tonic.

<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:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Introduction&lt;/h1&gt;

&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Introduction-Important features of Blacksmith[10] in 15edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Important features of Blacksmith[10] in 15edo&lt;/h2&gt;

  &lt;ul&gt;&lt;li&gt;As an 11-limit temperament, Blacksmith is extremely simple and efficient, and while it does fairly high damage to many ratios of 3 and 9, it does a very acceptable job of approximating most ratios of 5, 7, and 11. 9/8, 7/6, 11/9, 4/3, and their octave inversions are the most heavily-damaged, but 6/5, 12/11, and their octave inversions are tuned with good to tolerable accuracy.&lt;/li&gt;&lt;li&gt;Blacksmith[10] has the most 5-odd-limit consonant triads it is possible to have in a 10-note 5-limit scale.&lt;/li&gt;&lt;li&gt;Because it is a 10-note scale with a period of 1/5 of an octave, any arbitrary harmony will occur either 5 or 10 times within the 10-note scale, and for otonal harmonies consisting of three or more notes, the utonal counterpart of the harmony will also occur either 5 or 10 times within the scale; this is a property that is only held by other scales with 5 periods per octave.&lt;/li&gt;&lt;li&gt;Blacksmith[10] is also a &amp;quot;mode of limited transposition&amp;quot; like the Diminished and Augmented scales in 12edo: since the scale is built by applying the generator only a single time within each period, the scale has only two modes.&lt;/li&gt;&lt;li&gt;Another way to think about Blacksmith[10] is as a superposition of seven 7-note 5-limit Fokker blocks, representing untempered variations of the diatonic modes, built on a single tonic (more on this below).&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Introduction-Interval Classes in Blacksmith[10]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Interval Classes in Blacksmith[10]&lt;/h2&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="MODMOSes of Blacksmith[10]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;MODMOSes of Blacksmith[10]&lt;/h1&gt;

&lt;/body&gt;&lt;/html&gt;</pre></div>