Blackwood temperament modal harmony (in 15edo)
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=Harmony in 15edo Blacksmith[10] //(in progress)//= [[toc]] Blacksmith[10] in 15edo refers to the 10-note symmetric 5L5s scale in 15edo, which has two modes: 2 1 2 1 2 1 2 1 2 1 and 1 2 1 2 1 2 1 2 1 2. It can be thought of as a 5-limit temperament tempering out 256/243 (the Pythagorean diatonic semitone), a 7-limit temperament tempering out 28/27 and 49/48, and an 11-limit temperament tempering out 28/27, 49/48, and 55/54 (though in 15edo 121/120 and 100/99 are both tempered out as well, making the tuning identical to Ferrier and the unnamed 5c&15 temperament). In 15edo it has a period of 240 cents (5 periods per octave) and a generator of 80 or 160 cents (though it is more commonly described as having a generator of 400 cents). ==Important features of Blacksmith[10] in 15edo== * 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. * 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; this is a property that is only held by other scales with 5 periods per octave. This is exceptionally efficient, and it's pretty sweet if you can find harmonies you like because you get a whole lot of them. * Another consequence of having 5 periods per octave and 10 notes is that the scale has only two modes, which makes it easy to navigate and memorize. ==Characteristic Consonant Triads==
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<html><head><title>Harmony in 15edo Blacksmith</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Harmony in 15edo Blacksmith[10] (in progress)"></a><!-- ws:end:WikiTextHeadingRule:0 -->Harmony in 15edo Blacksmith[10] <em>(in progress)</em></h1> <!-- ws:start:WikiTextTocRule:6:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><div style="margin-left: 1em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)">Harmony in 15edo Blacksmith[10] (in progress)</a></div> <!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --><div style="margin-left: 2em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)-Important features of Blacksmith[10] in 15edo">Important features of Blacksmith[10] in 15edo</a></div> <!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><div style="margin-left: 2em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)-Characteristic Consonant Triads">Characteristic Consonant Triads</a></div> <!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --></div> <!-- ws:end:WikiTextTocRule:10 -->Blacksmith[10] in 15edo refers to the 10-note symmetric 5L5s scale in 15edo, which has two modes: 2 1 2 1 2 1 2 1 2 1 and 1 2 1 2 1 2 1 2 1 2. It can be thought of as a 5-limit temperament tempering out 256/243 (the Pythagorean diatonic semitone), a 7-limit temperament tempering out 28/27 and 49/48, and an 11-limit temperament tempering out 28/27, 49/48, and 55/54 (though in 15edo 121/120 and 100/99 are both tempered out as well, making the tuning identical to Ferrier and the unnamed 5c&15 temperament). In 15edo it has a period of 240 cents (5 periods per octave) and a generator of 80 or 160 cents (though it is more commonly described as having a generator of 400 cents). <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Important features of Blacksmith[10] in 15edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Important features of Blacksmith[10] in 15edo</h2> <ul><li>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.</li><li>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; this is a property that is only held by other scales with 5 periods per octave. This is exceptionally efficient, and it's pretty sweet if you can find harmonies you like because you get a whole lot of them.</li><li>Another consequence of having 5 periods per octave and 10 notes is that the scale has only two modes, which makes it easy to navigate and memorize.</li></ul><!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Characteristic Consonant Triads"></a><!-- ws:end:WikiTextHeadingRule:4 -->Characteristic Consonant Triads</h2> </body></html>