16edo: Difference between revisions
Wikispaces>TallKite **Imported revision 599359142 - Original comment: ** |
Wikispaces>TallKite **Imported revision 599359330 - 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:TallKite|TallKite]] and made on <tt>2016-11-14 02: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-14 02:53:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>599359330</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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=Intervals= | =Intervals= | ||
|| Degree || Cents | || Degree || Cents (coarse, | ||
(coarse, fine) | fine) DMS ||= Approximate | ||
DMS ||= Approximate | |||
Ratios* || Interval Names | Ratios* || Interval Names | ||
Just || Interval Names | Just || Interval Names | ||
Simplified || Interval | Simplified || Interval Names with | ||
major wider than minor || | |||
|| 0 || 0 ||= 1/1 || Unison || Unison || || | || 0 || 0 ||= 1/1 || Unison || Unison || || | ||
|| 1 || 75, 90 | || 1 || 75, 90 | ||
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16edo can be notated with conventional notation, including the staff, note names, chord names, relative notation, etc. in two ways. The first preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always higher pitched than the latter: | 16edo can be notated with conventional notation, including the staff, note names, chord names, relative notation, etc. in two ways. The first preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always higher pitched than the latter: | ||
||= D ||= | ||= ---D--- ||= -------- ||= ---E--- ||= --------- ||= --------- ||= ---F--- ||= --------- ||= ---G--- ||= --------- ||= ---A--- ||= --------- ||= ---B--- ||= --------- ||= --------- ||= ---C--- ||= --------- ||= ---D--- || | ||
|| D || D#/Eb || E || E# || Fb || F || F#/Gb || G || G#/Ab || A || A#/Bb || B || B# || Cb || C || C#/Db || D || | ||= D ||= D#/Eb ||= E ||= E# ||= Fb ||= F ||= F#/Gb ||= G ||= G#/Ab ||= A ||= A#/Bb ||= B ||= B# ||= Cb ||= C ||= C#/Db ||= D || | ||
||= P1 ||= A1/d2 ||= m2 ||= M2 ||= m3 ||= M3 ||= A3/d4 ||= P4 ||= A4/d5 ||= P5 ||= A5/d6 ||= m6 ||= M6 ||= m7 ||= M7 ||= A7/d8 ||= P8 || | ||= P1 ||= A1/d2 ||= m2 ||= M2 ||= m3 ||= M3 ||= A3/d4 ||= P4 ||= A4/d5 ||= P5 ||= A5/d6 ||= m6 ||= M6 ||= m7 ||= M7 ||= A7/d8 ||= P8 || | ||
The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. | The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. | ||
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|| D || Db/E# || E || Eb || F# || F || Fb/G# || G || Gb/A# || A || Ab/B# || B || Bb || C# || C || Cb/D# || D || | || D || Db/E# || E || Eb || F# || F || Fb/G# || G || Gb/A# || A || Ab/B# || B || Bb || C# || C || Cb/D# || D || | ||
||= P1 ||= d1/A2 ||= M2 ||= m2 ||= M3 ||= m3 ||= d3/A4 ||= P4 ||= d4/A5 ||= P5 ||= d5/A6 ||= M6 ||= m6 ||= M7 ||= m7 ||= d7/A8 ||= P8 || | ||= P1 ||= d1/A2 ||= M2 ||= m2 ||= M3 ||= m3 ||= d3/A4 ||= P4 ||= d4/A5 ||= P5 ||= d5/A6 ||= M6 ||= m6 ||= M7 ||= m7 ||= d7/A8 ||= P8 || | ||
Sharp is lower in pitch than flat, major is narrower than minor, etc. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly". | |||
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. | 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. | ||
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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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<!-- 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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<td>Degree<br /> | <td>Degree<br /> | ||
</td> | </td> | ||
<td>Cents | <td>Cents (coarse, <br /> | ||
(coarse, | fine) DMS<br /> | ||
DMS<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Approximate<br /> | <td style="text-align: center;">Approximate<br /> | ||
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Simplified<br /> | Simplified<br /> | ||
</td> | </td> | ||
<td>Interval | <td>Interval Names with<br /> | ||
major wider than minor<br /> | |||
</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
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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 /> | ||
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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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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<td style="text-align: center;">D<br /> | <td style="text-align: center;">---D---<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">E<br /> | <td style="text-align: center;">---E---<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">F<br /> | <td style="text-align: center;">---F---<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">G<br /> | <td style="text-align: center;">---G---<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">A<br /> | <td style="text-align: center;">---A---<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">B<br /> | <td style="text-align: center;">---B---<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">C<br /> | <td style="text-align: center;">---C---<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">< | <td style="text-align: center;"><hr /> | ||
</td> | </td> | ||
<td style="text-align: center;">D<br /> | <td style="text-align: center;">---D---<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>D<br /> | <td style="text-align: center;">D<br /> | ||
</td> | </td> | ||
<td>D#/Eb<br /> | <td style="text-align: center;">D#/Eb<br /> | ||
</td> | </td> | ||
<td>E<br /> | <td style="text-align: center;">E<br /> | ||
</td> | </td> | ||
<td>E#<br /> | <td style="text-align: center;">E#<br /> | ||
</td> | </td> | ||
<td>Fb<br /> | <td style="text-align: center;">Fb<br /> | ||
</td> | </td> | ||
<td>F<br /> | <td style="text-align: center;">F<br /> | ||
</td> | </td> | ||
<td>F#/Gb<br /> | <td style="text-align: center;">F#/Gb<br /> | ||
</td> | </td> | ||
<td>G<br /> | <td style="text-align: center;">G<br /> | ||
</td> | </td> | ||
<td>G#/Ab<br /> | <td style="text-align: center;">G#/Ab<br /> | ||
</td> | </td> | ||
<td>A<br /> | <td style="text-align: center;">A<br /> | ||
</td> | </td> | ||
<td>A#/Bb<br /> | <td style="text-align: center;">A#/Bb<br /> | ||
</td> | </td> | ||
<td>B<br /> | <td style="text-align: center;">B<br /> | ||
</td> | </td> | ||
<td>B#<br /> | <td style="text-align: center;">B#<br /> | ||
</td> | </td> | ||
<td>Cb<br /> | <td style="text-align: center;">Cb<br /> | ||
</td> | </td> | ||
<td>C<br /> | <td style="text-align: center;">C<br /> | ||
</td> | </td> | ||
<td>C#/Db<br /> | <td style="text-align: center;">C#/Db<br /> | ||
</td> | </td> | ||
<td>D<br /> | <td style="text-align: center;">D<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</table> | </table> | ||
Sharp is lower in pitch than flat, major is narrower than minor, etc. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo &quot;on the fly&quot;.<br /> | |||
<br /> | <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 /> | 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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<!-- 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> | ||
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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 /> | ||