Blackwood temperament modal harmony (in 15edo): Difference between revisions
Wikispaces>igliashon **Imported revision 415164532 - Original comment: ** |
Wikispaces>igliashon **Imported revision 416606494 - 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>2013-03-20 21:09:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>416606494</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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* 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. | * 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. | * 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 | * 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). | ||
==Interval Classes in Blacksmith[10]== | ==Interval Classes in Blacksmith[10]== | ||
|| Step of 15edo || Cent Value || Interval Class || Guitar Notation || Decimal Notation || Approximated Ratios || Pseudo-Diatonic Category || | || Step of 15edo || Cent Value || Interval Class || Guitar Notation || Decimal Notation || Approximated Ratios || Pseudo-Diatonic Category || | ||
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*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs. | *Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs. | ||
== | ==Chords of Blacksmith[10]== | ||
===Basic Functional Chords=== | ===Basic Functional Chords=== | ||
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: | 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: | ||
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|| Major Mode (cents) || 0 || 160 || 240 || 400 || 480 || 640 || 720 || 880 || 960 || 1120 || 1200 || | || Major Mode (cents) || 0 || 160 || 240 || 400 || 480 || 640 || 720 || 880 || 960 || 1120 || 1200 || | ||
|| Minor Mode (cents) || 0 || 80 || 240 || 320 || 480 || 560 || 720 || 800 || 960 || 1040 || 1200 || | || Minor Mode (cents) || 0 || 80 || 240 || 320 || 480 || 560 || 720 || 800 || 960 || 1040 || 1200 || | ||
Looking at this table, one can see approximations to all sorts of functional chords; if it's not immediately obvious, I'll spell it out in the following | Looking at this table, one can see approximations to all sorts of functional chords; if it's not immediately obvious, I'll spell it out in the following tables: | ||
|| Diatonic Chord Name || Decatonic Name | || Diatonic Chord Name || Decatonic Name | ||
(if different) || Tuning (cents) || Spelling 1 || Spelling 2 || Degrees of Major Mode Found On: || Degrees of Minor Mode Found On: || | (if different) || Tuning (cents) || Spelling 1 || Spelling 2 || Degrees of Major Mode Found On: || Degrees of Minor Mode Found On: || | ||
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|| Sus2 || Sus3 || 0-240-720 || E-G-B || 1-3-7 || All || All || | || Sus2 || Sus3 || 0-240-720 || E-G-B || 1-3-7 || All || All || | ||
|| Sus4 || Sus5 || 0-480-720 || E-A-B || 1-5-7 || All || All || | || Sus4 || Sus5 || 0-480-720 || E-A-B || 1-5-7 || All || All || | ||
|| Major 7th || Major 10th || 0-400-720-1120 || E-Ab-B-Eb || 1-4-7-0 || I, III, V, VII, IX || II, IV, VI, VIII, X || | || Major 7th (maj7) || Major 10th || 0-400-720-1120 || E-Ab-B-Eb || 1-4-7-0 || I, III, V, VII, IX || II, IV, VI, VIII, X || | ||
|| Minor 7th || Minor 10th || 0-320-720-1040 || E-G#-B-D# || 1-4b-7-0b || ii, iv, vi, viii, x || i, iii, v, vii, ix || | || Minor 7th (min7) || Minor 10th || 0-320-720-1040 || E-G#-B-D# || 1-4b-7-0b || ii, iv, vi, viii, x || i, iii, v, vii, ix || | ||
|| Dominant 7th || Major 9th || 0-400-720-960 || E-Ab-B-D || 1-4-7-9 || I, III, V, VII, IX || II, IV, VI, VIII, X || | || Dominant 7th (7) || Major 9th || 0-400-720-960 || E-Ab-B-D || 1-4-7-9 || I, III, V, VII, IX || II, IV, VI, VIII, X || | ||
|| Half-Diminished 7th || Diminished 10th || 0-320-560-1040 || E-G#-A#-D# || 1-4b-6b-0b || ii, iv, vi, viii, x || i, iii, v, vii, ix || | || Half-Diminished 7th (m7b5) || Diminished 10th || 0-320-560-1040 || E-G#-A#-D# || 1-4b-6b-0b || ii, iv, vi, viii, x || i, iii, v, vii, ix || | ||
|| Diminished 7th || Diminished 9th || 0-320-560-960 || E-G#-A#-D || 1-4b-6b-9 || ii, iv, vi, viii, x || i, iii, v, vii, ix || | |||
===Additional Functional Chords=== | |||
|| Diatonic Chord Name || Decatonic Name | |||
(if different) || Tuning (cents) || Spelling 1 || Spelling 2 || Degrees of Major Mode Found On: || Degrees of Minor Mode Found On: || | |||
|| | || Major 6th (M6) || Major 8th || 0-400-720-880 || E-Ab-B-Db || 1-4-7-8 || I, III, V, VII, IX || II, IV, VI, VIII, X || | ||
|| Minor-Major 6th (m6) || Minor 9th || 0-320-720-960 || E-G#-B-D || 1-4b-7-9 || ii, iv, vi, viii, x || i, iii, v, vii, ix || | |||
|| Diminished(b3) (Dim(b3)) || Sus3-Maj6 || 0-240-640 || E-G-Bb || 1-3-6 || I, III, V, VII, IX || II, IV, VI, VIII, X || | |||
|| Double-Diminished (Dim(b3)(b5)) || Sus3-Min6 || 0-240-560 || E-G-A# || 1-3-6b || ii, iv, vi, viii, x || i, iii, v, vii, ix || | |||
|| 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]== | |||
|| | 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> | ||
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<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:WikiTextHeadingRule:0:&lt;h1&gt; --><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> | <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:WikiTextHeadingRule:0:&lt;h1&gt; --><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> | ||
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<!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><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:15 --><!-- ws:start:WikiTextTocRule:16: --><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:16 --><!-- ws:start:WikiTextTocRule:17: --><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:16 --><!-- ws:start:WikiTextTocRule:17: --><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:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)- | <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)-Chords of Blacksmith[10]">Chords of Blacksmith[10]</a></div> | ||
<!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 3em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)- | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 3em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)-Chords of Blacksmith[10]-Basic Functional Chords">Basic Functional Chords</a></div> | ||
<!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 3em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)- | <!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 3em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)-Chords of Blacksmith[10]-Additional Functional Chords">Additional Functional Chords</a></div> | ||
<!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: | <!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 2em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)-Diatonic Modal Harmony in Blacksmith[10]">Diatonic Modal Harmony in Blacksmith[10]</a></div> | ||
<!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --></div> | <!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --></div> | ||
<!-- ws:end:WikiTextTocRule:22 -->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&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:end:WikiTextTocRule:22 -->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&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:2:&lt;h2&gt; --><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> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><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 &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 | <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:4:&lt;h2&gt; --><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> | ||
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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:6:&lt;h2&gt; --><h2 id="toc3"><a name="Harmony in 15edo Blacksmith[10] (in progress)- | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Chords of Blacksmith[10]"></a><!-- ws:end:WikiTextHeadingRule:6 -->Chords of Blacksmith[10]</h2> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Harmony in 15edo Blacksmith[10] (in progress)- | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Chords of Blacksmith[10]-Basic Functional Chords"></a><!-- ws:end:WikiTextHeadingRule:8 -->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> | ||
Looking at this table, one can see approximations to all sorts of functional chords; if it's not immediately obvious, I'll spell it out in the following | Looking at this table, one can see approximations to all sorts of functional chords; if it's not immediately obvious, I'll spell it out in the following tables:<br /> | ||
<br /> | |||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td>Major 7th<br /> | <td>Major 7th (maj7)<br /> | ||
</td> | </td> | ||
<td>Major 10th<br /> | <td>Major 10th<br /> | ||
| Line 592: | Line 560: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>Minor 7th<br /> | <td>Minor 7th (min7)<br /> | ||
</td> | </td> | ||
<td>Minor 10th<br /> | <td>Minor 10th<br /> | ||
| Line 608: | Line 576: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>Dominant 7th<br /> | <td>Dominant 7th (7)<br /> | ||
</td> | </td> | ||
<td>Major 9th<br /> | <td>Major 9th<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td>Half-Diminished 7th<br /> | <td>Half-Diminished 7th (m7b5)<br /> | ||
</td> | </td> | ||
<td>Diminished 10th<br /> | <td>Diminished 10th<br /> | ||
| Line 639: | Line 607: | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>Diminished 7th<br /> | ||
</td> | </td> | ||
<td | <td>Diminished 9th<br /> | ||
</td> | </td> | ||
<td | <td>0-320-560-960<br /> | ||
</td> | </td> | ||
<td | <td>E-G#-A#-D<br /> | ||
</td> | </td> | ||
<td | <td>1-4b-6b-9<br /> | ||
</td> | </td> | ||
<td | <td>ii, iv, vi, viii, x<br /> | ||
</td> | </td> | ||
<td | <td>i, iii, v, vii, ix<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | |||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Chords of Blacksmith[10]-Additional Functional Chords"></a><!-- ws:end:WikiTextHeadingRule:10 -->Additional Functional Chords</h3> | |||
<table class="wiki_table"> | |||
<tr> | <tr> | ||
<td | <td>Diatonic Chord Name<br /> | ||
</td> | </td> | ||
<td | <td>Decatonic Name<br /> | ||
(if different)<br /> | |||
</td> | </td> | ||
<td | <td>Tuning (cents)<br /> | ||
</td> | </td> | ||
<td | <td>Spelling 1<br /> | ||
</td> | </td> | ||
<td | <td>Spelling 2<br /> | ||
</td> | </td> | ||
<td | <td>Degrees of Major Mode Found On:<br /> | ||
</td> | </td> | ||
<td | <td>Degrees of Minor Mode Found On:<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>Major 6th (M6)<br /> | ||
</td> | </td> | ||
<td | <td>Major 8th<br /> | ||
</td> | </td> | ||
<td | <td>0-400-720-880<br /> | ||
</td> | </td> | ||
<td | <td>E-Ab-B-Db<br /> | ||
</td> | </td> | ||
<td | <td>1-4-7-8<br /> | ||
</td> | </td> | ||
<td | <td>I, III, V, VII, IX<br /> | ||
</td> | </td> | ||
<td | <td>II, IV, VI, VIII, X<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>Minor-Major 6th (m6)<br /> | ||
</td> | </td> | ||
<td | <td>Minor 9th<br /> | ||
</td> | </td> | ||
<td | <td>0-320-720-960<br /> | ||
</td> | </td> | ||
<td | <td>E-G#-B-D<br /> | ||
</td> | </td> | ||
<td | <td>1-4b-7-9<br /> | ||
</td> | </td> | ||
<td | <td>ii, iv, vi, viii, x<br /> | ||
</td> | </td> | ||
<td | <td>i, iii, v, vii, ix<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>Diminished(b3) (Dim(b3))<br /> | ||
</td> | </td> | ||
<td | <td>Sus3-Maj6<br /> | ||
</td> | </td> | ||
<td | <td>0-240-640<br /> | ||
</td> | </td> | ||
<td | <td>E-G-Bb<br /> | ||
</td> | </td> | ||
<td | <td>1-3-6<br /> | ||
</td> | </td> | ||
<td | <td>I, III, V, VII, IX<br /> | ||
</td> | </td> | ||
<td | <td>II, IV, VI, VIII, X<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>Double-Diminished (Dim(b3)(b5))<br /> | ||
</td> | </td> | ||
<td | <td>Sus3-Min6<br /> | ||
</td> | </td> | ||
<td | <td>0-240-560<br /> | ||
</td> | </td> | ||
<td | <td>E-G-A#<br /> | ||
</td> | </td> | ||
<td | <td>1-3-6b<br /> | ||
</td> | </td> | ||
<td | <td>ii, iv, vi, viii, x<br /> | ||
</td> | </td> | ||
<td | <td>i, iii, v, vii, ix<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>Major-Diminished (Maj(b5))<br /> | ||
</td> | </td> | ||
<td | <td>Major-Sus6<br /> | ||
</td> | </td> | ||
<td | <td>0-400-640<br /> | ||
</td> | </td> | ||
<td | <td>E-Ab-Bb<br /> | ||
</td> | </td> | ||
<td | <td>1-4-6<br /> | ||
</td> | </td> | ||
<td | <td>I, III, V, VIII, IX<br /> | ||
</td> | </td> | ||
<td | <td>II, IV, VI, VIII, X<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt; | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Diatonic Modal Harmony in Blacksmith[10]"></a><!-- ws:end:WikiTextHeadingRule:12 -->Diatonic Modal Harmony in Blacksmith[10]</h2> | ||
</body></html></pre></div> | 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]:</body></html></pre></div> | ||