Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 590296076 - Original comment: ** |
Wikispaces>TallKite **Imported revision 590981962 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-02 16:58:38 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>590981962</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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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. | 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. | ||
[[image:The Tree | [[image:The Scale Tree.png width="800" height="1002"]] | ||
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 "generation" 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. The colored regions of the tree are what I call **kites**, and | 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 "generation" 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 **kites**, 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 **spinal** node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others. | ||
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs. | Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs. | ||
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To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a **keyspan** (always +1) but also a **genspan**, which indicates how many steps forward or backwards along the generator chain, or **genchain**, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain. | To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a **keyspan** (always +1) but also a **genspan**, which indicates how many steps forward or backwards along the generator chain, or **genchain**, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain. | ||
Every node on the | Every node on the scale tree, other than the spinal nodes, heads up a kite and is on the side of two other kites. These two kites can be used to find the rank-2 interval with keyspan of 1. For example, the 13\22 node is on the side of the 10\17 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the __right__ (fifthward) side of the 10\17 kite, we know that 17 __fifths__ add up to 1\22. Because it's on the __left__ (fourthward) side of the 3\5 kite, 5 __fourths__ add up to 1\22. Between the two, choose the interval with smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occupy the same key on the keyboard, just as C# may not equal Db in 12-tone. | ||
The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D. | The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D. | ||
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In 22-tone, positive genspans, which lie on the fifthward half of the genchain, create sharps and downs. Negative genspans, from the fourthward half of the genchain, create flats and ups. | In 22-tone, positive genspans, which lie on the fifthward half of the genchain, create sharps and downs. Negative genspans, from the fourthward half of the genchain, create flats and ups. | ||
The genspan for the up symbol in 22-tone can be found from the | The genspan for the up symbol in 22-tone can be found from the scale tree. Or it can be derived more rigorously if calculated from the keyspans: | ||
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1) | K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1) | ||
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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:4066:&lt;img src=&quot;/file/view/The%20Tree | <!-- ws:start:WikiTextLocalImageRule:4066:&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:4066 --><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. The colored regions of the tree are what I call <strong>kites</strong>, and | 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 /> | ||
<br /> | <br /> | ||
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.<br /> | Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.<br /> | ||
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To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the generator chain, or <strong>genchain</strong>, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain.<br /> | To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the generator chain, or <strong>genchain</strong>, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain.<br /> | ||
<br /> | <br /> | ||
Every node on the | Every node on the scale tree, other than the spinal nodes, heads up a kite and is on the side of two other kites. These two kites can be used to find the rank-2 interval with keyspan of 1. For example, the 13\22 node is on the side of the 10\17 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> (fifthward) side of the 10\17 kite, we know that 17 <u>fifths</u> add up to 1\22. Because it's on the <u>left</u> (fourthward) side of the 3\5 kite, 5 <u>fourths</u> add up to 1\22. Between the two, choose the interval with smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occupy the same key on the keyboard, just as C# may not equal Db in 12-tone.<br /> | ||
<br /> | <br /> | ||
The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.<br /> | The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.<br /> | ||
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In 22-tone, positive genspans, which lie on the fifthward half of the genchain, create sharps and downs. Negative genspans, from the fourthward half of the genchain, create flats and ups.<br /> | In 22-tone, positive genspans, which lie on the fifthward half of the genchain, create sharps and downs. Negative genspans, from the fourthward half of the genchain, create flats and ups.<br /> | ||
<br /> | <br /> | ||
The genspan for the up symbol in 22-tone can be found from the | The genspan for the up symbol in 22-tone can be found from the scale tree. Or it can be derived more rigorously if calculated from the keyspans:<br /> | ||
<br /> | <br /> | ||
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br /> | K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br /> | ||