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 [[MOSscales|MOS]] 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. * Blacksmith[10] has the most 5-odd-limit consonant triads it is possible to have in a 10-note 5-limit scale. * 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. * Blacksmith[10] is also a "mode of limited transposition" 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. * Another way to think about Blacksmith[10] is as a super-position of three modes of the diatonic scale: the major mode can be seen as a superposition of ionian, lydian, and mixolydian, while the minor mode can be seen as a superposition of dorian, phrygian, and aeolian (this will be expanded on below). This has radical implications for writing tonal and modal music in Blacksmith[10]. ==Interval Classes in Blacksmith[10]== || Step of 15edo || Cent Value || Interval Class || Guitar Notation || Decimal Notation || || 0 || 0 || Unison || E || 1 || || 1 || 80 || Minor 2nd, Augmented Unison* || E#, Gbb || 2b, 1# || || 2 || 160 || Major 2nd, Diminished 3rd || Gb, Ex || 2, 3b || || 3 || 240 || Perfect 3rd, Augmented 2nd, Diminished 4th || G || 3, 2#, 4bb || || 4 || 320 || Minor 4th, Augmented 3rd || G#, Abb || 4b, 3# || || 5 || 400 || Major 4th, Diminished 5th || Ab, Gx || 4, 5b || || 6 || 480 || Perfect 5th, Augmented 4th, Diminished 6th || A || 5, 4#, 6bb || || 7 || 560 || Minor 6th, Augmented 5th || A#, Bbb || 6b, 5# || || 8 || 640 || Major 6th, Diminished 7th || Bb, Ax || 6, 7b || || 9 || 720 || Perfect 7th, Augmented 6th, Diminished 8th || B || 7, 6#, 8bb || || 10 || 800 || Minor 8th, Augmented 7th || B#, Dbb || 8b, 7# || || 11 || 880 || Major 8th, Diminished 9th || Db, Bx || 8, 9b || || 12 || 960 || Perfect 9th, Augmented 8th, Diminished 10th || D || 9, 8#, 0bb || || 13 || 1040 || Minor 10th, Augmented 9th || D#, Ebb || 0b, 9# || || 14 || 1120 || Major 10th, Diminished Undecave || Eb, Dx || 0, 1b || || 15 || 1200 || Undecave (Octave) || E || 1 || *Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.
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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)-Interval Classes in Blacksmith[10]">Interval Classes in Blacksmith[10]</a></div>
<!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --></div>
<!-- ws:end:WikiTextTocRule:10 -->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 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>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 "mode of limited transposition" 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 super-position of three modes of the diatonic scale: the major mode can be seen as a superposition of ionian, lydian, and mixolydian, while the minor mode can be seen as a superposition of dorian, phrygian, and aeolian (this will be expanded on below). This has radical implications for writing tonal and modal music in Blacksmith[10].</li></ul><!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Interval Classes in Blacksmith[10]"></a><!-- ws:end:WikiTextHeadingRule:4 -->Interval Classes in Blacksmith[10]</h2>
<table class="wiki_table">
<tr>
<td>Step of 15edo<br />
</td>
<td>Cent Value<br />
</td>
<td>Interval Class<br />
</td>
<td>Guitar Notation<br />
</td>
<td>Decimal Notation<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>Unison<br />
</td>
<td>E<br />
</td>
<td>1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>80<br />
</td>
<td>Minor 2nd, Augmented Unison*<br />
</td>
<td>E#, Gbb<br />
</td>
<td>2b, 1#<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>160<br />
</td>
<td>Major 2nd, Diminished 3rd<br />
</td>
<td>Gb, Ex<br />
</td>
<td>2, 3b<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>240<br />
</td>
<td>Perfect 3rd, Augmented 2nd, Diminished 4th<br />
</td>
<td>G<br />
</td>
<td>3, 2#, 4bb<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>320<br />
</td>
<td>Minor 4th, Augmented 3rd<br />
</td>
<td>G#, Abb<br />
</td>
<td>4b, 3#<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>400<br />
</td>
<td>Major 4th, Diminished 5th<br />
</td>
<td>Ab, Gx<br />
</td>
<td>4, 5b<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>480<br />
</td>
<td>Perfect 5th, Augmented 4th, Diminished 6th<br />
</td>
<td>A<br />
</td>
<td>5, 4#, 6bb<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>560<br />
</td>
<td>Minor 6th, Augmented 5th<br />
</td>
<td>A#, Bbb<br />
</td>
<td>6b, 5#<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>640<br />
</td>
<td>Major 6th, Diminished 7th<br />
</td>
<td>Bb, Ax<br />
</td>
<td>6, 7b<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>720<br />
</td>
<td>Perfect 7th, Augmented 6th, Diminished 8th<br />
</td>
<td>B<br />
</td>
<td>7, 6#, 8bb<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>800<br />
</td>
<td>Minor 8th, Augmented 7th<br />
</td>
<td>B#, Dbb<br />
</td>
<td>8b, 7#<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>880<br />
</td>
<td>Major 8th, Diminished 9th<br />
</td>
<td>Db, Bx<br />
</td>
<td>8, 9b<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>960<br />
</td>
<td>Perfect 9th, Augmented 8th, Diminished 10th<br />
</td>
<td>D<br />
</td>
<td>9, 8#, 0bb<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>1040<br />
</td>
<td>Minor 10th, Augmented 9th<br />
</td>
<td>D#, Ebb<br />
</td>
<td>0b, 9#<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>1120<br />
</td>
<td>Major 10th, Diminished Undecave<br />
</td>
<td>Eb, Dx<br />
</td>
<td>0, 1b<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1200<br />
</td>
<td>Undecave (Octave)<br />
</td>
<td>E<br />
</td>
<td>1<br />
</td>
</tr>
</table>
*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.</body></html>