31edo: Difference between revisions
Wikispaces>TallKite **Imported revision 599964184 - Original comment: ** |
Wikispaces>TallKite **Imported revision 599965604 - 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>2016-11-21 07: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-21 07:49:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>599965604</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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A single step of 31-edo is about 38.71¢. Intervals around this size are called //dieses// (singular '//diesis//'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the [[11-limit]], the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in [[SpiralProgressions]]. | A single step of 31-edo is about 38.71¢. Intervals around this size are called //dieses// (singular '//diesis//'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the [[11-limit]], the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in [[SpiralProgressions]]. | ||
===2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor | ===2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second or downminor 2nd=== | ||
The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two "microtones". In meantone, it is the //chromatic semitone//, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates [[Starling temperaments#Valentine%20temperament|valentine temperament]] - aka [[Armodue theory#Semi-equalized%20Armodue|semi-equalized Armodue]]. | The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two "microtones". In meantone, it is the //chromatic semitone//, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates [[Starling temperaments#Valentine%20temperament|valentine temperament]] - aka [[Armodue theory#Semi-equalized%20Armodue|semi-equalized Armodue]]. | ||
====MOS Scales generated by 2\31:==== | ====MOS Scales generated by 2\31:==== | ||
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|| 16-tone || [[15L 1s]] || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 1 || | || 16-tone || [[15L 1s]] || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 2 || || 1 || | ||
===3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor | ===3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor Second or minor 2nd=== | ||
The larger and clunkier of the 31edo semitones. In meantone, it is the //diatonic semitone// which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a //secor// and generates [[Gamelismic clan|miracle temperament]]. | The larger and clunkier of the 31edo semitones. In meantone, it is the //diatonic semitone// which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a //secor// and generates [[Gamelismic clan|miracle temperament]]. | ||
====MOS Scales generated by 3\31:==== | ====MOS Scales generated by 3\31:==== | ||
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===15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished | ===15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished Fifth or downdim 5th=== | ||
In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates [[tritonic]] temperament. | In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates [[tritonic]] temperament. | ||
====MOS Scales generated by 15\31:==== | ====MOS Scales generated by 15\31:==== | ||
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===16\31 | ===16\31 Large Tritone or dim 5th=== | ||
Etc. | |||
31edo can be notated with [[ups and downs notation]] like so: | 31edo can be notated with [[ups and downs notation]] like so: | ||
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C | Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C | ||
Relative notation: P1 ^P1 vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc. | |||
Absolute notation: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc. | |||
All 31edo chords can be named using ups and downs: | All 31edo chords can be named using ups and downs: | ||
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A single step of 31-edo is about 38.71¢. Intervals around this size are called <em>dieses</em> (singular '<em>diesis</em>'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the <a class="wiki_link" href="/11-limit">11-limit</a>, the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in <a class="wiki_link" href="/SpiralProgressions">SpiralProgressions</a>.<br /> | A single step of 31-edo is about 38.71¢. Intervals around this size are called <em>dieses</em> (singular '<em>diesis</em>'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the <a class="wiki_link" href="/11-limit">11-limit</a>, the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in <a class="wiki_link" href="/SpiralProgressions">SpiralProgressions</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:9:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor | <!-- ws:start:WikiTextHeadingRule:9:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second or downminor 2nd"></a><!-- ws:end:WikiTextHeadingRule:9 -->2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second or downminor 2nd</h3> | ||
The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two &quot;microtones&quot;. In meantone, it is the <em>chromatic semitone</em>, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament">valentine temperament</a> - aka <a class="wiki_link" href="/Armodue%20theory#Semi-equalized%20Armodue">semi-equalized Armodue</a>.<br /> | The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two &quot;microtones&quot;. In meantone, it is the <em>chromatic semitone</em>, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament">valentine temperament</a> - aka <a class="wiki_link" href="/Armodue%20theory#Semi-equalized%20Armodue">semi-equalized Armodue</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule:11:&lt;h4&gt; --><h4 id="toc5"><a name="Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor | <!-- ws:start:WikiTextHeadingRule:11:&lt;h4&gt; --><h4 id="toc5"><a name="Intervals-Selected just intervals by error-2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second or downminor 2nd-MOS Scales generated by 2\31:"></a><!-- ws:end:WikiTextHeadingRule:11 -->MOS Scales generated by 2\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule:13:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor | <!-- ws:start:WikiTextHeadingRule:13:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor Second or minor 2nd"></a><!-- ws:end:WikiTextHeadingRule:13 -->3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor Second or minor 2nd</h3> | ||
The larger and clunkier of the 31edo semitones. In meantone, it is the <em>diatonic semitone</em> which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a <em>secor</em> and generates <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>.<br /> | The larger and clunkier of the 31edo semitones. In meantone, it is the <em>diatonic semitone</em> which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a <em>secor</em> and generates <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>.<br /> | ||
<!-- ws:start:WikiTextHeadingRule:15:&lt;h4&gt; --><h4 id="toc7"><a name="Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor | <!-- ws:start:WikiTextHeadingRule:15:&lt;h4&gt; --><h4 id="toc7"><a name="Intervals-Selected just intervals by error-3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor Second or minor 2nd-MOS Scales generated by 3\31:"></a><!-- ws:end:WikiTextHeadingRule:15 -->MOS Scales generated by 3\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule:61:&lt;h3&gt; --><h3 id="toc30"><a name="Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished | <!-- ws:start:WikiTextHeadingRule:61:&lt;h3&gt; --><h3 id="toc30"><a name="Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished Fifth or downdim 5th"></a><!-- ws:end:WikiTextHeadingRule:61 -->15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished Fifth or downdim 5th</h3> | ||
In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates <a class="wiki_link" href="/tritonic">tritonic</a> temperament.<br /> | In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates <a class="wiki_link" href="/tritonic">tritonic</a> temperament.<br /> | ||
<!-- ws:start:WikiTextHeadingRule:63:&lt;h4&gt; --><h4 id="toc31"><a name="Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished | <!-- ws:start:WikiTextHeadingRule:63:&lt;h4&gt; --><h4 id="toc31"><a name="Intervals-Selected just intervals by error-15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished Fifth or downdim 5th-MOS Scales generated by 15\31:"></a><!-- ws:end:WikiTextHeadingRule:63 -->MOS Scales generated by 15\31:</h4> | ||
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<!-- ws:start:WikiTextHeadingRule:65:&lt;h3&gt; --><h3 id="toc32"><a name="Intervals-Selected just intervals by error-16\31 | <!-- ws:start:WikiTextHeadingRule:65:&lt;h3&gt; --><h3 id="toc32"><a name="Intervals-Selected just intervals by error-16\31 Large Tritone or dim 5th"></a><!-- ws:end:WikiTextHeadingRule:65 -->16\31 Large Tritone or dim 5th</h3> | ||
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Etc.<br /> | |||
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31edo can be notated with <a class="wiki_link" href="/ups%20and%20downs%20notation">ups and downs notation</a> like so:<br /> | 31edo can be notated with <a class="wiki_link" href="/ups%20and%20downs%20notation">ups and downs notation</a> like so:<br /> | ||
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C<br /> | Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C<br /> | ||
Relative notation: P1 ^P1 vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc.<br /> | |||
Absolute notation: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.<br /> | |||
<br /> | <br /> | ||
All 31edo chords can be named using ups and downs:<br /> | All 31edo chords can be named using ups and downs:<br /> | ||