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 || Approximated Ratios || Pseudo-Diatonic Category ||
|| 0 || 0 || Unison || E || 1 || 1/1 || Unison ||
|| 1 || 80 || Minor 2nd, Augmented Unison* || E#, Gbb || 2b, 1# || 16/15, 21/20, 22/21, 25/24 || Minor 2nd ||
|| 2 || 160 || Major 2nd, Diminished 3rd || Gb, Ex || 2, 3b || 10/9, 11/10, 12/11, 15/14 || Flat Major 2nd ||
|| 3 || 240 || Perfect 3rd, Augmented 2nd, Diminished 4th || G || 3, 2#, 4bb || 7/6, 8/7, 9/8 || Major 2nd/Subminor 3rd ||
|| 4 || 320 || Minor 4th, Augmented 3rd || G#, Abb || 4b, 3# || 6/5, 11/9 || Minor 3rd ||
|| 5 || 400 || Major 4th, Diminished 5th || Ab, Gx || 4, 5b || 5/4, 14/11, || Major 3rd ||
|| 6 || 480 || Perfect 5th, Augmented 4th, Diminished 6th || A || 5, 4#, 6bb || 4/3, 21/16, 9/7 || Perfect Fourth ||
|| 7 || 560 || Minor 6th, Augmented 5th || A#, Bbb || 6b, 5# || 7/5, 11/8, || Augmented Fourth ||
|| 8 || 640 || Major 6th, Diminished 7th || Bb, Ax || 6, 7b || 10/7, 16/11 || Diminished 5th ||
|| 9 || 720 || Perfect 7th, Augmented 6th, Diminished 8th || B || 7, 6#, 8bb || 3/2, 32/21, 14/9 || Perfect Fifth ||
|| 10 || 800 || Minor 8th, Augmented 7th || B#, Dbb || 8b, 7# || 8/5, 11/7, || Minor 6th ||
|| 11 || 880 || Major 8th, Diminished 9th || Db, Bx || 8, 9b || 5/3, 18/11 || Major 6th ||
|| 12 || 960 || Perfect 9th, Augmented 8th, Diminished 10th || D || 9, 8#, 0bb || 12/7, 7/4, 16/9 || Minor 7th/Supermajor 6th ||
|| 13 || 1040 || Minor 10th, Augmented 9th || D#, Ebb || 0b, 9# || 9/5, 20/11, 11/6, 28/15 || Sharp Minor 7th ||
|| 14 || 1120 || Major 10th, Diminished Undecave || Eb, Dx || 0, 1b || 15/8, 40/21, 48/25 || Major 7th ||
|| 15 || 1200 || Undecave ("Octave") || E || 1 || 2/1 || Octave ||
*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.
==Modes and Chords of Blacksmith[10]==
===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:
|| || 1st || 2nd || 3rd || 4th || 5th || 6th || 7th || 8th || 9th || 10th || 11-ave ||
|| 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 ||
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 table:
|| 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 Triad || Same || 0-400-720 || E-Ab-B || 1-4-7 || I, III, V, VII, IX || II, IV, VI, VIII, X ||
|| Minor Triad || Same || 0-320-720 || E-G#-B || 1-4b-7 || ii, iv, vi, viii, x || i, iii, v, vii, ix ||
|| Diminished || Same || 0-320-560 || E-G#-A# || 1-4b-6b || ii, iv, vi, viii, x || i, iii, v, vii, ix ||
|| 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 ||
|| 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 ||
|| 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 ||
|| 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 ||
|| 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 ||
===Complete List of Unaltered Triads //(in progress)//===
||< Decimal Notation ||< Guitar Notation ||< Cents ||< Steps of 15edo ||< Approximated
Harmonics #1 ||< Approximated
Harmonics #2 ||< Triad Name ||< Comments ||
||< 1-2b-3 ||< E-E#-G ||< 0-80-240 ||< 0-1-3 ||< 21:22:24 ||< ||< Minor Cluster ||< ||
||< 1-2-3 ||< E-Gb-G ||< 0-160-240 ||< 0-2-3 ||< 1/(21:22:24) ||< ||< Major Cluster ||< ||
||< 1-2b-4b ||< E-E#-G# ||< 0-80-320 ||< 0-1-4 ||< 20:21:24 ||< ||< ||< ||
||< 1-3-4b ||< E-G-G# ||< 0-240-320 ||< 0-3-4 ||< ||< ||< ||< ||
||< 1-2-4 ||< E-Gb-Ab ||< 0-160-400 ||< 0-2-5 ||< 11:12:14 ||< 1/(8:9:10) ||< ||< ||
||< 1-3-4 ||< E-G-Ab ||< 0-240-400 ||< 0-3-5 ||< 8:9:10 ||< 1/(11:12:14) ||< ||< ||
||< 1-2b-5 ||< E-E#-A ||< 0-80-480 ||< 0-1-6 ||< ||< ||< ||< ||
||< 1-2-5 ||< E-Gb-A ||< 0-160-480 ||< 0-2-6 ||< 9:10:12 ||< ||< ||< ||
||< 1-3-5 ||< E-G-A ||< 0-240-480 ||< 0-3-6 ||< 6:7:8 ||< 1/(6:7:8) ||< ||< ||
||< 1-4b-5 ||< E-G#-A ||< 0-320-480 ||< 0-4-6 ||< 9:11:12 ||< ||< ||< ||
||< 1-4-5 ||< E-Ab-A ||< 0-400-480 ||< 0-5-6 ||< 16:20:21 ||< ||< ||< ||
||< 1-2b-6b ||< E-E#-A# ||< 0-80-560 ||< 0-1-7 ||< 15:16:21 ||< ||< ||< ||
||< 1-3-6b ||< E-G-A# ||< 0-240-560 ||< 0-3-7 ||< 1/(5:6:7) ||< ||< ||< ||
||< 1-4b-6b ||< E-G#-A# ||< 0-320-560 ||< 0-4-7 ||< 5:6:7 ||< ||< Diminished ||< ||
||< 1-5-6b ||< E-A-A# ||< 0-480-560 ||< 0-6-7 ||< ||< ||< ||< ||
||< 1-2-6 ||< E-Gb-Bb ||< 0-160-640 ||< 0-2-8 ||< 11:12:16 ||< ||< ||< ||
||< 1-3-6 ||< E-G-Bb ||< 0-240-640 ||< 0-3-8 ||< 7:8:10 ||< 1/(11:14:16) ||< ||< ||
||< 1-4-6 ||< E-Ab-Bb ||< 0-400-640 ||< 0-5-8 ||< 1/(7:8:10) ||< 11:14:16 ||< Major-Diminished ||< ||
||< 1-5-6 ||< E-A-Bb ||< 0-480-640 ||< 0-6-8 ||< ||< ||< ||< ||
||< 1-2b-7 ||< E-E#-B ||< 0-80-720 ||< 0-1-9 ||< 21:22:32 ||< 1/(14:20:21) ||< ||< ||
||< 1-2-7 ||< E-Gb-B ||< 0-160-720 ||< 0-2-9 ||< ||< ||< ||< ||
||< 1-3-7 ||< E-G-B ||< 0-240-720 ||< 0-3-9 ||< 6:7:9 ||< 8:9:12 ||< Sus3 ||< ||
||< 1-4b-7 ||< E-G#-B ||< 0-320-720 ||< 0-4-9 ||< 1/(4:5:6) ||< 10:12:15 ||< Minor Triad ||< ||
||< 1-4-7 ||< E-Ab-B ||< 0-400-720 ||< 0-5-9 ||< 4:5:6 ||< ||< Major Triad ||< ||
|| 1-5-7 || E-A-B || 0-480-720 || 0-6-9 || 6:8:9 || 1/(6:7:9) || Sus5 || ||
|| 1-6b-7 || E-A#-B || 0-560-720 || 0-7-9 || 8:11:12 || || || ||
|| 1-6-7 || E-Bb-B || 0-640-720 || 0-8-9 || 14:20:21 || 1/(21:22:32) || || ||
|| 1-2b-8b || E-E#-B# || 0-80-800 || 0-1-10 || 15:16:24 || || || ||
|| 1-3-8b || E-G-B# || 0-240-800 || 0-3-10 || 7:8:11 || || || ||
|| 1-4b-8b || E-G#-B# || 0-320-800 || 0-4-10 || 5:6:8 || || Major Triad, 1st inversion || ||
|| 1-5-8b || E-A-B# || 0-480-800 || 0-6-10 || 1/(5:6:8) || 15:20:24 || Minor Triad, 2nd inversion || ||
|| 1-6b-8b || E-A#-B# || 0-560-800 || 0-7-10 || 5:7:8 || || || ||
|| 1-7-8b || E-B-B# || 0-720-800 || 0-9-10 || || || || ||
=== === Original HTML content:
<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:14:<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:14 --><!-- ws:start:WikiTextTocRule:15: --><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: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:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#Harmony in 15edo Blacksmith[10] (in progress)-Modes and Chords of Blacksmith[10]">Modes and 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)-Modes and 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)-Modes and Chords of Blacksmith[10]-Complete List of Unaltered Triads (in progress)">Complete List of Unaltered Triads (in progress)</a></div>
<!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 3em;"><a href="#toc6"> </a></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&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>
<td>Approximated Ratios<br />
</td>
<td>Pseudo-Diatonic Category<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>Unison<br />
</td>
<td>E<br />
</td>
<td>1<br />
</td>
<td>1/1<br />
</td>
<td>Unison<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>
<td>16/15, 21/20, 22/21, 25/24<br />
</td>
<td>Minor 2nd<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>
<td>10/9, 11/10, 12/11, 15/14<br />
</td>
<td>Flat Major 2nd<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>
<td>7/6, 8/7, 9/8<br />
</td>
<td>Major 2nd/Subminor 3rd<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>
<td>6/5, 11/9<br />
</td>
<td>Minor 3rd<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>
<td>5/4, 14/11,<br />
</td>
<td>Major 3rd<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>
<td>4/3, 21/16, 9/7<br />
</td>
<td>Perfect Fourth<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>
<td>7/5, 11/8,<br />
</td>
<td>Augmented Fourth<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>
<td>10/7, 16/11<br />
</td>
<td>Diminished 5th<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>
<td>3/2, 32/21, 14/9<br />
</td>
<td>Perfect Fifth<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>
<td>8/5, 11/7,<br />
</td>
<td>Minor 6th<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>
<td>5/3, 18/11<br />
</td>
<td>Major 6th<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>
<td>12/7, 7/4, 16/9<br />
</td>
<td>Minor 7th/Supermajor 6th<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>
<td>9/5, 20/11, 11/6, 28/15<br />
</td>
<td>Sharp Minor 7th<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>
<td>15/8, 40/21, 48/25<br />
</td>
<td>Major 7th<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>
<td>2/1<br />
</td>
<td>Octave<br />
</td>
</tr>
</table>
*Augmented and diminished intervals do not occur in the 10-note MOS scale, but can occur in chromatically-altered MODMOSs.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Modes and Chords of Blacksmith[10]"></a><!-- ws:end:WikiTextHeadingRule:6 -->Modes and Chords of Blacksmith[10]</h2>
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Modes and 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 />
<table class="wiki_table">
<tr>
<td><br />
</td>
<td>1st<br />
</td>
<td>2nd<br />
</td>
<td>3rd<br />
</td>
<td>4th<br />
</td>
<td>5th<br />
</td>
<td>6th<br />
</td>
<td>7th<br />
</td>
<td>8th<br />
</td>
<td>9th<br />
</td>
<td>10th<br />
</td>
<td>11-ave<br />
</td>
</tr>
<tr>
<td>Major Mode (cents)<br />
</td>
<td>0<br />
</td>
<td>160<br />
</td>
<td>240<br />
</td>
<td>400<br />
</td>
<td>480<br />
</td>
<td>640<br />
</td>
<td>720<br />
</td>
<td>880<br />
</td>
<td>960<br />
</td>
<td>1120<br />
</td>
<td>1200<br />
</td>
</tr>
<tr>
<td>Minor Mode (cents)<br />
</td>
<td>0<br />
</td>
<td>80<br />
</td>
<td>240<br />
</td>
<td>320<br />
</td>
<td>480<br />
</td>
<td>560<br />
</td>
<td>720<br />
</td>
<td>800<br />
</td>
<td>960<br />
</td>
<td>1040<br />
</td>
<td>1200<br />
</td>
</tr>
</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 table:<br />
<br />
<table class="wiki_table">
<tr>
<td>Diatonic Chord Name<br />
</td>
<td>Decatonic Name<br />
(if different)<br />
</td>
<td>Tuning (cents)<br />
</td>
<td>Spelling 1<br />
</td>
<td>Spelling 2<br />
</td>
<td>Degrees of Major Mode Found On:<br />
</td>
<td>Degrees of Minor Mode Found On:<br />
</td>
</tr>
<tr>
<td>Major Triad<br />
</td>
<td>Same<br />
</td>
<td>0-400-720<br />
</td>
<td>E-Ab-B<br />
</td>
<td>1-4-7<br />
</td>
<td>I, III, V, VII, IX<br />
</td>
<td>II, IV, VI, VIII, X<br />
</td>
</tr>
<tr>
<td>Minor Triad<br />
</td>
<td>Same<br />
</td>
<td>0-320-720<br />
</td>
<td>E-G#-B<br />
</td>
<td>1-4b-7<br />
</td>
<td>ii, iv, vi, viii, x<br />
</td>
<td>i, iii, v, vii, ix<br />
</td>
</tr>
<tr>
<td>Diminished<br />
</td>
<td>Same<br />
</td>
<td>0-320-560<br />
</td>
<td>E-G#-A#<br />
</td>
<td>1-4b-6b<br />
</td>
<td>ii, iv, vi, viii, x<br />
</td>
<td>i, iii, v, vii, ix<br />
</td>
</tr>
<tr>
<td>Sus2<br />
</td>
<td>Sus3<br />
</td>
<td>0-240-720<br />
</td>
<td>E-G-B<br />
</td>
<td>1-3-7<br />
</td>
<td>All<br />
</td>
<td>All<br />
</td>
</tr>
<tr>
<td>Sus4<br />
</td>
<td>Sus5<br />
</td>
<td>0-480-720<br />
</td>
<td>E-A-B<br />
</td>
<td>1-5-7<br />
</td>
<td>All<br />
</td>
<td>All<br />
</td>
</tr>
<tr>
<td>Major 7th<br />
</td>
<td>Major 10th<br />
</td>
<td>0-400-720-1120<br />
</td>
<td>E-Ab-B-Eb<br />
</td>
<td>1-4-7-0<br />
</td>
<td>I, III, V, VII, IX<br />
</td>
<td>II, IV, VI, VIII, X<br />
</td>
</tr>
<tr>
<td>Minor 7th<br />
</td>
<td>Minor 10th<br />
</td>
<td>0-320-720-1040<br />
</td>
<td>E-G#-B-D#<br />
</td>
<td>1-4b-7-0b<br />
</td>
<td>ii, iv, vi, viii, x<br />
</td>
<td>i, iii, v, vii, ix<br />
</td>
</tr>
<tr>
<td>Dominant 7th<br />
</td>
<td>Major 9th<br />
</td>
<td>0-400-720-960<br />
</td>
<td>E-Ab-B-D<br />
</td>
<td>1-4-7-9<br />
</td>
<td>I, III, V, VII, IX<br />
</td>
<td>II, IV, VI, VIII, X<br />
</td>
</tr>
<tr>
<td>Half-Diminished 7th<br />
</td>
<td>Diminished 10th<br />
</td>
<td>0-320-560-1040<br />
</td>
<td>E-G#-A#-D#<br />
</td>
<td>1-4b-6b-0b<br />
</td>
<td>ii, iv, vi, viii, x<br />
</td>
<td>i, iii, v, vii, ix<br />
</td>
</tr>
</table>
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="Harmony in 15edo Blacksmith[10] (in progress)-Modes and Chords of Blacksmith[10]-Complete List of Unaltered Triads (in progress)"></a><!-- ws:end:WikiTextHeadingRule:10 -->Complete List of Unaltered Triads <em>(in progress)</em></h3>
<table class="wiki_table">
<tr>
<td style="text-align: left;">Decimal Notation<br />
</td>
<td style="text-align: left;">Guitar Notation<br />
</td>
<td style="text-align: left;">Cents<br />
</td>
<td style="text-align: left;">Steps of 15edo<br />
</td>
<td style="text-align: left;">Approximated<br />
Harmonics #1<br />
</td>
<td style="text-align: left;">Approximated<br />
Harmonics #2<br />
</td>
<td style="text-align: left;">Triad Name<br />
</td>
<td style="text-align: left;">Comments<br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2b-3<br />
</td>
<td style="text-align: left;">E-E#-G<br />
</td>
<td style="text-align: left;">0-80-240<br />
</td>
<td style="text-align: left;">0-1-3<br />
</td>
<td style="text-align: left;">21:22:24<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;">Minor Cluster<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2-3<br />
</td>
<td style="text-align: left;">E-Gb-G<br />
</td>
<td style="text-align: left;">0-160-240<br />
</td>
<td style="text-align: left;">0-2-3<br />
</td>
<td style="text-align: left;">1/(21:22:24)<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;">Major Cluster<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2b-4b<br />
</td>
<td style="text-align: left;">E-E#-G#<br />
</td>
<td style="text-align: left;">0-80-320<br />
</td>
<td style="text-align: left;">0-1-4<br />
</td>
<td style="text-align: left;">20:21:24<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-3-4b<br />
</td>
<td style="text-align: left;">E-G-G#<br />
</td>
<td style="text-align: left;">0-240-320<br />
</td>
<td style="text-align: left;">0-3-4<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2-4<br />
</td>
<td style="text-align: left;">E-Gb-Ab<br />
</td>
<td style="text-align: left;">0-160-400<br />
</td>
<td style="text-align: left;">0-2-5<br />
</td>
<td style="text-align: left;">11:12:14<br />
</td>
<td style="text-align: left;">1/(8:9:10)<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-3-4<br />
</td>
<td style="text-align: left;">E-G-Ab<br />
</td>
<td style="text-align: left;">0-240-400<br />
</td>
<td style="text-align: left;">0-3-5<br />
</td>
<td style="text-align: left;">8:9:10<br />
</td>
<td style="text-align: left;">1/(11:12:14)<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2b-5<br />
</td>
<td style="text-align: left;">E-E#-A<br />
</td>
<td style="text-align: left;">0-80-480<br />
</td>
<td style="text-align: left;">0-1-6<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2-5<br />
</td>
<td style="text-align: left;">E-Gb-A<br />
</td>
<td style="text-align: left;">0-160-480<br />
</td>
<td style="text-align: left;">0-2-6<br />
</td>
<td style="text-align: left;">9:10:12<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-3-5<br />
</td>
<td style="text-align: left;">E-G-A<br />
</td>
<td style="text-align: left;">0-240-480<br />
</td>
<td style="text-align: left;">0-3-6<br />
</td>
<td style="text-align: left;">6:7:8<br />
</td>
<td style="text-align: left;">1/(6:7:8)<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-4b-5<br />
</td>
<td style="text-align: left;">E-G#-A<br />
</td>
<td style="text-align: left;">0-320-480<br />
</td>
<td style="text-align: left;">0-4-6<br />
</td>
<td style="text-align: left;">9:11:12<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-4-5<br />
</td>
<td style="text-align: left;">E-Ab-A<br />
</td>
<td style="text-align: left;">0-400-480<br />
</td>
<td style="text-align: left;">0-5-6<br />
</td>
<td style="text-align: left;">16:20:21<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2b-6b<br />
</td>
<td style="text-align: left;">E-E#-A#<br />
</td>
<td style="text-align: left;">0-80-560<br />
</td>
<td style="text-align: left;">0-1-7<br />
</td>
<td style="text-align: left;">15:16:21<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-3-6b<br />
</td>
<td style="text-align: left;">E-G-A#<br />
</td>
<td style="text-align: left;">0-240-560<br />
</td>
<td style="text-align: left;">0-3-7<br />
</td>
<td style="text-align: left;">1/(5:6:7)<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-4b-6b<br />
</td>
<td style="text-align: left;">E-G#-A#<br />
</td>
<td style="text-align: left;">0-320-560<br />
</td>
<td style="text-align: left;">0-4-7<br />
</td>
<td style="text-align: left;">5:6:7<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;">Diminished<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-5-6b<br />
</td>
<td style="text-align: left;">E-A-A#<br />
</td>
<td style="text-align: left;">0-480-560<br />
</td>
<td style="text-align: left;">0-6-7<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2-6<br />
</td>
<td style="text-align: left;">E-Gb-Bb<br />
</td>
<td style="text-align: left;">0-160-640<br />
</td>
<td style="text-align: left;">0-2-8<br />
</td>
<td style="text-align: left;">11:12:16<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-3-6<br />
</td>
<td style="text-align: left;">E-G-Bb<br />
</td>
<td style="text-align: left;">0-240-640<br />
</td>
<td style="text-align: left;">0-3-8<br />
</td>
<td style="text-align: left;">7:8:10<br />
</td>
<td style="text-align: left;">1/(11:14:16)<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-4-6<br />
</td>
<td style="text-align: left;">E-Ab-Bb<br />
</td>
<td style="text-align: left;">0-400-640<br />
</td>
<td style="text-align: left;">0-5-8<br />
</td>
<td style="text-align: left;">1/(7:8:10)<br />
</td>
<td style="text-align: left;">11:14:16<br />
</td>
<td style="text-align: left;">Major-Diminished<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-5-6<br />
</td>
<td style="text-align: left;">E-A-Bb<br />
</td>
<td style="text-align: left;">0-480-640<br />
</td>
<td style="text-align: left;">0-6-8<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2b-7<br />
</td>
<td style="text-align: left;">E-E#-B<br />
</td>
<td style="text-align: left;">0-80-720<br />
</td>
<td style="text-align: left;">0-1-9<br />
</td>
<td style="text-align: left;">21:22:32<br />
</td>
<td style="text-align: left;">1/(14:20:21)<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-2-7<br />
</td>
<td style="text-align: left;">E-Gb-B<br />
</td>
<td style="text-align: left;">0-160-720<br />
</td>
<td style="text-align: left;">0-2-9<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-3-7<br />
</td>
<td style="text-align: left;">E-G-B<br />
</td>
<td style="text-align: left;">0-240-720<br />
</td>
<td style="text-align: left;">0-3-9<br />
</td>
<td style="text-align: left;">6:7:9<br />
</td>
<td style="text-align: left;">8:9:12<br />
</td>
<td style="text-align: left;">Sus3<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-4b-7<br />
</td>
<td style="text-align: left;">E-G#-B<br />
</td>
<td style="text-align: left;">0-320-720<br />
</td>
<td style="text-align: left;">0-4-9<br />
</td>
<td style="text-align: left;">1/(4:5:6)<br />
</td>
<td style="text-align: left;">10:12:15<br />
</td>
<td style="text-align: left;">Minor Triad<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: left;">1-4-7<br />
</td>
<td style="text-align: left;">E-Ab-B<br />
</td>
<td style="text-align: left;">0-400-720<br />
</td>
<td style="text-align: left;">0-5-9<br />
</td>
<td style="text-align: left;">4:5:6<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;">Major Triad<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td>1-5-7<br />
</td>
<td>E-A-B<br />
</td>
<td>0-480-720<br />
</td>
<td>0-6-9<br />
</td>
<td>6:8:9<br />
</td>
<td>1/(6:7:9)<br />
</td>
<td>Sus5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-6b-7<br />
</td>
<td>E-A#-B<br />
</td>
<td>0-560-720<br />
</td>
<td>0-7-9<br />
</td>
<td>8:11:12<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-6-7<br />
</td>
<td>E-Bb-B<br />
</td>
<td>0-640-720<br />
</td>
<td>0-8-9<br />
</td>
<td>14:20:21<br />
</td>
<td>1/(21:22:32)<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-2b-8b<br />
</td>
<td>E-E#-B#<br />
</td>
<td>0-80-800<br />
</td>
<td>0-1-10<br />
</td>
<td>15:16:24<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-3-8b<br />
</td>
<td>E-G-B#<br />
</td>
<td>0-240-800<br />
</td>
<td>0-3-10<br />
</td>
<td>7:8:11<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-4b-8b<br />
</td>
<td>E-G#-B#<br />
</td>
<td>0-320-800<br />
</td>
<td>0-4-10<br />
</td>
<td>5:6:8<br />
</td>
<td><br />
</td>
<td>Major Triad, 1st inversion<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-5-8b<br />
</td>
<td>E-A-B#<br />
</td>
<td>0-480-800<br />
</td>
<td>0-6-10<br />
</td>
<td>1/(5:6:8)<br />
</td>
<td>15:20:24<br />
</td>
<td>Minor Triad, 2nd inversion<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-6b-8b<br />
</td>
<td>E-A#-B#<br />
</td>
<td>0-560-800<br />
</td>
<td>0-7-10<br />
</td>
<td>5:7:8<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1-7-8b<br />
</td>
<td>E-B-B#<br />
</td>
<td>0-720-800<br />
</td>
<td>0-9-10<br />
</td>
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</td>
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</td>
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</td>
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</td>
</tr>
</table>
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