16edo: Difference between revisions
Wikispaces>k9assassin **Imported revision 599092924 - Original comment: ** |
Wikispaces>k9assassin **Imported revision 599093816 - 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:k9assassin|k9assassin]] and made on <tt>2016-11-10 09: | : This revision was by author [[User:k9assassin|k9assassin]] and made on <tt>2016-11-10 09:24:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>599093816</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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|| Degree || Cents (coarse, fine) | || Degree || Cents (coarse, fine) | ||
DMS ||= Approximate | DMS ||= Approximate | ||
Ratios* || Interval | Ratios* || Interval Names | ||
|| 0 || 0 ||= 1/1 || Unison || | Just || Interval Names | ||
Simplified || | |||
|| 0 || 0 ||= 1/1 || Unison || Unison || | |||
|| 1 || 75, 90 | || 1 || 75, 90 | ||
22º30' ||= 28/27, 27/26 || Subminor 2nd || | 22º30' ||= 28/27, 27/26 || Subminor 2nd || Minor Second || | ||
|| 2 || 150, 180 | || 2 || 150, 180 | ||
45° ||= 12/11, 35/32 || Neutral 2nd || | 45° ||= 12/11, 35/32 || Neutral 2nd || Major Second || | ||
|| 3 || 225, 270 | || 3 || 225, 270 | ||
67°30' ||= 8/7 || Supermajor 2nd, | 67°30' ||= 8/7 || Supermajor 2nd, | ||
Septimal Whole-Tone || | Septimal Whole-Tone || Perfect Second || | ||
|| 4 || 300, 360 | || 4 || 300, 360 | ||
90° ||= 19/16, 32/27 || Minor 3rd || | 90° ||= 19/16, 32/27 || Minor 3rd || Minor Third || | ||
|| 5 || 375, 450 | || 5 || 375, 450 | ||
112°30' ||= 5/4, 26/21 || Major 3rd || | 112°30' ||= 5/4, 26/21 || Major 3rd || Major Third || | ||
|| 6 || 450, 540 | || 6 || 450, 540 | ||
135° ||= 13/10, 35/27 || Sub-4th, | 135° ||= 13/10, 35/27 || Sub-4th, | ||
Supermajor 3rd || | Supermajor 3rd || Minor Fourth || | ||
|| 7 || 525, 630 | || 7 || 525, 630 | ||
157°30' ||= 27/20, 52/35, 256/189 || Wide 4th || | 157°30' ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth || | ||
|| 8 || 600, 720 | || 8 || 600, 720 | ||
180° ||= 7/5, 10/7 || Tritone || | 180° ||= 7/5, 10/7 || Tritone || Augmented fourth/ | ||
Diminished fifth || | |||
|| 9 || 675, 810 | || 9 || 675, 810 | ||
202°30' ||= 40/27, 35/26, 189/128 || Narrow 5th || | 202°30' ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth || | ||
|| 10 || 750, 900 | || 10 || 750, 900 | ||
225° ||= 20/13, 54/35 || Super-5th, | 225° ||= 20/13, 54/35 || Super-5th, | ||
Subminor 6th || | Subminor 6th || Major Fifth || | ||
|| 11 || 825, 990 | || 11 || 825, 990 | ||
247°30' ||= 8/5, 21/13 || Minor 6th || | 247°30' ||= 8/5, 21/13 || Minor 6th || Minor Sixth || | ||
|| 12 || 900, 1080 | || 12 || 900, 1080 | ||
270° ||= 27/16, 32/19 || Major 6th || | 270° ||= 27/16, 32/19 || Major 6th || Major Sixth || | ||
|| 13 || 975, 1170 | || 13 || 975, 1170 | ||
292°30' ||= 7/4 || Subminor 7th, | 292°30' ||= 7/4 || Subminor 7th, | ||
Septimal Minor 7th || | Septimal Minor 7th || Perfect Seventh || | ||
|| 14 || 1050, 1260 | || 14 || 1050, 1260 | ||
315° ||= 11/6, 64/35 || Neutral 7th || | 315° ||= 11/6, 64/35 || Neutral 7th || Minor Seventh || | ||
|| 15 || 1125, 1350 | || 15 || 1125, 1350 | ||
337°30' ||= 27/14, 52/27 || Supermajor 7th || | 337°30' ||= 27/14, 52/27 || Supermajor 7th || Major Seventh || | ||
|| 16 || 1200, 1440 | || 16 || 1200, 1440 | ||
360° ||= 2/1 || Octave || | 360° ||= 2/1 || Octave || Octave || | ||
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | ||
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==16 Tone Piano Layout Based on the Anti-diatonic Scale== | ==16 Tone Piano Layout Based on the Anti-diatonic Scale== | ||
[[image:16-EDO-PIano-Diagram.png width="748" height="293"]] | [[image:16-EDO-PIano-Diagram.png width="748" height="293"]] | ||
This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower. | This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower. | ||
=Rank two temperaments= | =Rank two temperaments= | ||
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<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Images"></a><!-- ws:end:WikiTextHeadingRule:3 -->Images</h1> | <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Images"></a><!-- ws:end:WikiTextHeadingRule:3 -->Images</h1> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals</h1> | <!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals</h1> | ||
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Ratios*<br /> | Ratios*<br /> | ||
</td> | </td> | ||
<td>Interval | <td>Interval Names<br /> | ||
Just<br /> | |||
</td> | |||
<td>Interval Names<br /> | |||
Simplified<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">1/1<br /> | <td style="text-align: center;">1/1<br /> | ||
</td> | |||
<td>Unison<br /> | |||
</td> | </td> | ||
<td>Unison<br /> | <td>Unison<br /> | ||
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</td> | </td> | ||
<td>Subminor 2nd<br /> | <td>Subminor 2nd<br /> | ||
</td> | |||
<td>Minor Second<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Neutral 2nd<br /> | <td>Neutral 2nd<br /> | ||
</td> | |||
<td>Major Second<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>Supermajor 2nd,<br /> | <td>Supermajor 2nd,<br /> | ||
Septimal Whole-Tone<br /> | Septimal Whole-Tone<br /> | ||
</td> | |||
<td>Perfect Second<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Minor 3rd<br /> | <td>Minor 3rd<br /> | ||
</td> | |||
<td>Minor Third<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Major 3rd<br /> | <td>Major 3rd<br /> | ||
</td> | |||
<td>Major Third<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>Sub-4th,<br /> | <td>Sub-4th,<br /> | ||
Supermajor 3rd<br /> | Supermajor 3rd<br /> | ||
</td> | |||
<td>Minor Fourth<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Wide 4th<br /> | <td>Wide 4th<br /> | ||
</td> | |||
<td>Major Fourth<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Tritone<br /> | <td>Tritone<br /> | ||
</td> | |||
<td>Augmented fourth/<br /> | |||
Diminished fifth<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Narrow 5th<br /> | <td>Narrow 5th<br /> | ||
</td> | |||
<td>Minor Fifth<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>Super-5th,<br /> | <td>Super-5th,<br /> | ||
Subminor 6th<br /> | Subminor 6th<br /> | ||
</td> | |||
<td>Major Fifth<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Minor 6th<br /> | <td>Minor 6th<br /> | ||
</td> | |||
<td>Minor Sixth<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Major 6th<br /> | <td>Major 6th<br /> | ||
</td> | |||
<td>Major Sixth<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>Subminor 7th,<br /> | <td>Subminor 7th,<br /> | ||
Septimal Minor 7th<br /> | Septimal Minor 7th<br /> | ||
</td> | |||
<td>Perfect Seventh<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Neutral 7th<br /> | <td>Neutral 7th<br /> | ||
</td> | |||
<td>Minor Seventh<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>Supermajor 7th<br /> | <td>Supermajor 7th<br /> | ||
</td> | |||
<td>Major Seventh<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">2/1<br /> | <td style="text-align: center;">2/1<br /> | ||
</td> | |||
<td>Octave<br /> | |||
</td> | </td> | ||
<td>Octave<br /> | <td>Octave<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc3"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:9 -->Hexadecaphonic Octave Theory</h1> | <!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc3"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:9 -->Hexadecaphonic Octave Theory</h1> | ||
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Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br /> | Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br /> | ||
<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:771:&lt;img src=&quot;http://www.ronsword.com/161928%20copy.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 179px; width: 316px;&quot; /&gt; --><img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" /><!-- ws:end:WikiTextRemoteImageRule:771 --><br /> | ||
<br /> | <br /> | ||
The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br /> | The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br /> | ||
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16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | 16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | ||
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<hr /> | <hr /> | ||
In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br /> | In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc5"><a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"></a><!-- ws:end:WikiTextHeadingRule:13 -->16 Tone Piano Layout Based on the Anti-diatonic Scale</h2> | <!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc5"><a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"></a><!-- ws:end:WikiTextHeadingRule:13 -->16 Tone Piano Layout Based on the Anti-diatonic Scale</h2> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:770:&lt;img src=&quot;/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 293px; width: 748px;&quot; /&gt; --><img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /><!-- ws:end:WikiTextLocalImageRule:770 --><br /> | ||
This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower. <br /> | This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc6"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:15 -->Rank two temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc6"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:15 -->Rank two temperaments</h1> | ||