Kite's ups and downs notation: Difference between revisions

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=__**Summary of EDO notation**__=  

(12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)

All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.

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Ups and Downs is a notation system developed by &lt;a class="wiki_link" href="/KiteGiedraitis"&gt;Kite&lt;/a&gt; that works with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &amp;quot;^&amp;quot; and the down symbol &amp;quot;v&amp;quot;.&lt;br /&gt;

This is in addition to the trivial EDOs, 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.&lt;br /&gt;

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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 &amp;quot;generation&amp;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 &lt;strong&gt;kites&lt;/strong&gt;, 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 &lt;strong&gt;spinal&lt;/strong&gt; node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Chord names in other EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;u&gt;Chord names in other EDOs&lt;/u&gt;&lt;/h1&gt;

&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Summary of EDO notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;u&gt;&lt;strong&gt;Summary of EDO notation&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;

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(12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)&lt;br /&gt;

All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;

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  (7, 14, 21, 28 and 35)&lt;br /&gt;

All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.&lt;br /&gt;

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  (9, 11, 13b, 16, 18b and 23)&lt;br /&gt;

All superflat EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;

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  (5, 10, 15, 20, 25 and 30)&lt;br /&gt;

All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;

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  (8, 11b, 13 and 18)&lt;br /&gt;

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Ups and downs can be avoided entirely by using some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth.&lt;br /&gt;

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Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about &lt;strong&gt;frameworks&lt;/strong&gt; like 12-tone. The generator chain is called a &lt;strong&gt;genchain&lt;/strong&gt;. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (chroma 1 or -1):&lt;br /&gt;

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Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &amp;quot;genweb&amp;quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.&lt;br /&gt;

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An example of a rank-2 tuning with a non-fifth generator is porcupine. Porcupine in 22-tone is generated by the natural generator, the 2nd = 3\22.&lt;br /&gt;

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Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:&lt;br /&gt;