Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 592510076 - Original comment: ** |
Wikispaces>TallKite **Imported revision 592510324 - 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>2016-09-18 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-18 04:01:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>592510324</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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Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.) | Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.) | ||
In the chart below, 13edo and 18edo use the narrower fifth. | |||
||= 5edo ||= pentatonic ||= ||= C/Db ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||= 5edo ||= pentatonic ||= ||= C/Db ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= 6edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||= 7edo ||= perfect ||= ||= C/C# ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||= 7edo ||= perfect ||= ||= C/C# ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= 8edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||= 8edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= 9edo ||= fourthward ||= ||= C/Db ||= C#/D ||= D# ||= ||= ||= ||= ||= ||= ||= || | ||= 9edo ||= fourthward ||= ||= C/Db ||= C#/D ||= D# ||= ||= ||= ||= ||= ||= ||= || | ||
||= 10edo ||= pentatonic ||= ||= C/Db ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||= || | ||= 10edo ||= pentatonic ||= ||= C/Db ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||= || | ||
||= 11edo ||= | ||= 11edo ||= fourthward ||= ||= C ||= D ||= C# ||= D# ||= ||= ||= ||= ||= ||= || | ||
||= 12edo ||= regular ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= || | ||= 12edo ||= regular ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= || | ||
||= 13edo ||= | ||= 13edo ||= fourthward ||= ||= C ||= D ||= * ||= C# ||= D# ||= ||= ||= ||= ||= || | ||
||= 14edo ||= perfect ||= ||= C/C# ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||= || | ||= 14edo ||= perfect ||= ||= C/C# ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||= || | ||
||= 15edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||= || | ||= 15edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||= || | ||
||= 16edo ||= fourthward ||= ||= C ||= C#/Db ||= D ||= D# ||= ||= ||= ||= ||= ||= || | ||= 16edo ||= fourthward ||= ||= C ||= C#/Db ||= D ||= D# ||= ||= ||= ||= ||= ||= || | ||
||= 17edo ||= regular ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= || | ||= 17edo ||= regular ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= || | ||
||= 18edo ||= | ||= 18edo ||= fourthward ||= ||= C/Db ||= * ||= C#/D ||= * ||= D# ||= ||= ||= ||= ||= || | ||
||= 19edo ||= regular ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= || | ||= 19edo ||= regular ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= || | ||
||= 20edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= || | ||= 20edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= || | ||
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||= ||= Keyspan of # || value of i ||= genspan of ^ ||= example ||= stepspan & | ||= ||= Keyspan of # || value of i ||= genspan of ^ ||= example ||= stepspan & | ||
quality of ^ || | quality of ^ || | ||
||= 11-tone ||= 2 ||= 1 ||= 2 ||= C^ = D ||= maj 2nd || | |||
||= 13b-tone ||= 3 ||= 1 ||= 2 ||= C^ = D ||= maj 2nd || | |||
||= 17-tone ||= 2 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||= 17-tone ||= 2 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
||= 22-tone ||= 3 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||= 22-tone ||= 3 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
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This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4122:&lt;img src=&quot;/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1002px; width: 800px;&quot; /&gt; --><img src="/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1002px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:4122 --><br /> | ||
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call <strong>kites</strong>, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br /> | The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call <strong>kites</strong>, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br /> | ||
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<br /> | <br /> | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4123:&lt;img src=&quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1035px; width: 800px;&quot; /&gt; --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:4123 --><br /> | ||
<br /> | <br /> | ||
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Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &quot;wide-fifth&quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &quot;narrow-fifth&quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)<br /> | Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &quot;wide-fifth&quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &quot;narrow-fifth&quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)<br /> | ||
<br /> | <br /> | ||
In the chart below, 13edo and 18edo use the narrower fifth.<br /> | |||
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</td> | </td> | ||
<td style="text-align: center;">C#/D<br /> | <td style="text-align: center;">C#/D<br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6edo<br /> | |||
</td> | |||
<td style="text-align: center;">fifthless<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,214: | Line 2,247: | ||
<td style="text-align: center;">11edo<br /> | <td style="text-align: center;">11edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">fourthward<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">D#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">13edo<br /> | <td style="text-align: center;">13edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">fourthward<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">D#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 2,410: | Line 2,443: | ||
<td style="text-align: center;">18edo<br /> | <td style="text-align: center;">18edo<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">fourthward<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C/Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C#/D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">D#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">stepspan &amp;<br /> | <td style="text-align: center;">stepspan &amp;<br /> | ||
quality of ^<br /> | quality of ^<br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">11-tone<br /> | |||
</td> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td style="text-align: center;">1<br /> | |||
</td> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td style="text-align: center;">C^ = D<br /> | |||
</td> | |||
<td style="text-align: center;">maj 2nd<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">13b-tone<br /> | |||
</td> | |||
<td style="text-align: center;">3<br /> | |||
</td> | |||
<td style="text-align: center;">1<br /> | |||
</td> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td style="text-align: center;">C^ = D<br /> | |||
</td> | |||
<td style="text-align: center;">maj 2nd<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||