Blackwood temperament modal harmony (in 15edo): Difference between revisions
Wikispaces>igliashon **Imported revision 416612362 - Original comment: ** |
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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- | : This revision was by author [[User:igliashon|igliashon]] and made on <tt>2013-03-21 21:52:11 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>416942398</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> | ||
<h4>Original Wikitext content:</h4> | <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"> | <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">[[toc]] | ||
[[toc]] | Blacksmith[10] in [[15edo]] refers to the 10-note symmetric 5L5s [[MOSscales|MOS]] scale in 15edo, which has two modes: 2 1 2 1 2 1 2 1 2 1 a | ||
Blacksmith[10] in [[15edo]] refers to the 10-note symmetric 5L5s [[MOSscales|MOS]] scale in 15edo, which has two modes: 2 1 2 1 2 1 2 1 2 1 | nd 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 tempe | ||
ring 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== | ==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. | * 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. | ||
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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]: | ||
[[image:Screen Shot 2013-03-20 at 6.01.58 PM.png | [[image:Screen Shot 2013-03-20 at 6.01.58 PM.png align="left"]] | ||
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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 " **,** " 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 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 is tempered out, 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> | 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"><html><head><title>Harmony in 15edo Blacksmith</title></head><body><!-- ws:start: | <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>Harmony in 15edo Blacksmith</title></head><body><!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 2em;"><a href="#x-Important features of Blacksmith[10] in 15edo">Important features of Blacksmith[10] in 15edo</a></div> | ||
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 2em;"><a href="#x-Interval Classes in Blacksmith[10]">Interval Classes in Blacksmith[10]</a></div> | |||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 2em;"><a href="#x-Chords of Blacksmith[10]">Chords of Blacksmith[10]</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 3em;"><a href="#x-Chords of Blacksmith[10]-Basic Functional Chords">Basic Functional Chords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 3em;"><a href="#x-Chords of Blacksmith[10]-Additional Functional Chords">Additional Functional Chords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#x-Diatonic Modal Harmony in Blacksmith[10]">Diatonic Modal Harmony in Blacksmith[10]</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --></div> | ||
< | <!-- ws:end:WikiTextTocRule:19 -->Blacksmith[10] in <a class="wiki_link" href="/15edo">15edo</a> refers to the 10-note symmetric 5L5s <a class="wiki_link" href="/MOSscales">MOS</a> scale in 15edo, which has two modes: 2 1 2 1 2 1 2 1 2 1 a<br /> | ||
nd 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 tempe<br /> | |||
ring 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&amp;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:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Important features of Blacksmith[10] in 15edo"></a><!-- ws:end:WikiTextHeadingRule:0 -->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>Blacksmith[10] has the most 5-odd-limit consonant triads it is possible to have in a 10-note 5-limit scale.</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, 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.</li><li>Blacksmith[10] is also a &quot;mode of limited transposition&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.</li><li>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).</li></ul><!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Interval Classes in Blacksmith[10]"></a><!-- ws:end:WikiTextHeadingRule:2 -->Interval Classes in Blacksmith[10]</h2> | |||
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*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.<br /> | *Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Chords of Blacksmith[10]"></a><!-- ws:end:WikiTextHeadingRule:4 -->Chords of Blacksmith[10]</h2> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Chords of Blacksmith[10]-Basic Functional Chords"></a><!-- ws:end:WikiTextHeadingRule:6 -->Basic Functional Chords</h3> | ||
All of the familiar triads and tetrads of the diatonic scale are found plentifully in Blacksmith[10], which is pretty obvious when you just look at the notes available in the major and minor modes:<br /> | All of the familiar triads and tetrads of the diatonic scale are found plentifully in Blacksmith[10], which is pretty obvious when you just look at the notes available in the major and minor modes:<br /> | ||
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</table> | </table> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Chords of Blacksmith[10]-Additional Functional Chords"></a><!-- ws:end:WikiTextHeadingRule:8 -->Additional Functional Chords</h3> | ||
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</table> | </table> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-Diatonic Modal Harmony in Blacksmith[10]"></a><!-- ws:end:WikiTextHeadingRule:10 -->Diatonic Modal Harmony in Blacksmith[10]</h2> | ||
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]:<br /> | 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]:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:662:&lt;img src=&quot;/file/view/Screen%20Shot%202013-03-20%20at%206.01.58%20PM.png/416606882/Screen%20Shot%202013-03-20%20at%206.01.58%20PM.png&quot; alt=&quot;&quot; title=&quot;&quot; align=&quot;left&quot; /&gt; --><img src="/file/view/Screen%20Shot%202013-03-20%20at%206.01.58%20PM.png/416606882/Screen%20Shot%202013-03-20%20at%206.01.58%20PM.png" alt="Screen Shot 2013-03-20 at 6.01.58 PM.png" title="Screen Shot 2013-03-20 at 6.01.58 PM.png" align="left" /><!-- ws:end:WikiTextLocalImageRule:662 --><br /> | ||
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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 &quot; <strong>,</strong> &quot; means that an interval is lowered by a syntonic comma (81/80), while the accidental &quot; <strong>'</strong> &quot; means an interval is raised by a syntonic comma. The &quot; <strong>#</strong> &quot; and &quot; <strong>b</strong> &quot; accidentals represent raising and lowering by the Pythagorean apotome (2187:2048). <br /> | 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 &quot; <strong>,</strong> &quot; means that an interval is lowered by a syntonic comma (81/80), while the accidental &quot; <strong>'</strong> &quot; means an interval is raised by a syntonic comma. The &quot; <strong>#</strong> &quot; and &quot; <strong>b</strong> &quot; accidentals represent raising and lowering by the Pythagorean apotome (2187:2048). <br /> | ||
<br /> | <br /> | ||
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 &quot; <strong>C</strong> &quot; is the same pitch-class as &quot; <strong>C,</strong> &quot;), 256/243 is tempered out, which makes &quot; <strong>C</strong> &quot; and &quot; <strong>B</strong> &quot; the same pitch-class. Thus &quot; <strong>C</strong> &quot; and &quot; <strong>B</strong> &quot; both correspond to the Blacksmith[10] note &quot; <strong>1</strong> &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 &quot; <strong>b</strong> &quot; accidental represents lowering a note by a 5-limit chromatic semitone (25/24). Sharps (&quot; <strong>#</strong> &quot;) are not pictured here, but would arise if another chain of 3/2s was added another 5/4 above the top-most chain.</body></html></pre></div> | 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 &quot; <strong>C</strong> &quot; is the same pitch-class as &quot; <strong>C,</strong> &quot;), 256/243 (the octave-normalized product of five 3/2s or 4/3s) is tempered out here instead, which makes &quot; <strong>C</strong> &quot; and &quot; <strong>B</strong> &quot; the same pitch-class. Thus &quot; <strong>C</strong> &quot; and &quot; <strong>B</strong> &quot; both correspond to the Blacksmith[10] note &quot; <strong>1</strong> &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 &quot; <strong>b</strong> &quot; accidental represents lowering a note by a 5-limit chromatic semitone (25/24). Sharps (&quot; <strong>#</strong> &quot;) are not pictured here, but would arise if another chain of 3/2s was added another 5/4 above the top-most chain.</body></html></pre></div> | ||