Kite's ups and downs notation: Difference between revisions

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EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:

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.

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Ups and Downs Notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:36:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#A 22edo example"&gt;A 22edo example&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Other EDOs"&gt;Other EDOs&lt;/a&gt;&lt;/div&gt;

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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. When extended with highs and lows, it works with almost all rank 2 tunings (see the &lt;a class="wiki_link" href="/pergen"&gt;pergens&lt;/a&gt; page). 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;

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The up symbol means &amp;quot;sharpened by one edo-step&amp;quot; in any edo that uses them. The size in cents of the up changes greatly depending on the edo, from 120¢ in 10-edo to ~17¢ in 72-edo. The sharp symbol's cents size also depends on the edo, ranging from 240¢ in 5-edo to ~26¢ in 47edo.&lt;br /&gt;

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EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:&lt;br /&gt;

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This is in addition to the &lt;strong&gt;trivial&lt;/strong&gt; 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 &lt;strong&gt;kites&lt;/strong&gt;. 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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Ups and downs allow us to name any chord easily. The quality of an interval (major, minor, perfect, etc.) is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.&lt;br /&gt;

IV7(4) = &amp;quot;four-seven sus-four&amp;quot;&lt;br /&gt;

II7(v3) = &amp;quot;two-seven down-three&amp;quot;&lt;br /&gt;

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In chord names, the mid symbol &amp;quot;~&amp;quot; means &amp;quot;exactly midway between major and minor&amp;quot;, hence neutral. This only applies to even-sharpness edos. In sharp-2 edos (10, 17, 24, etc.), upminor equals downmajor, and &amp;quot;mid&amp;quot; replaces both terms. In sharp-4 edos (20, 27, 34, etc.), mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces both upmajor and downminor.&lt;br /&gt;

0-3-6-9 = D Fv A Cv = D.~7 = &amp;quot;D dot mid seven&amp;quot;, or D Fv A Bb = D~,b6 = &amp;quot;D mid flat-six&amp;quot;&lt;br /&gt;

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In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because &amp;quot;major 3rd&amp;quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.&lt;br /&gt;

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To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:&lt;br /&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;

If sharp is lower than flat, the chain of fifths is 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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&lt;span style="font-size: 150%;"&gt;This section is somewhat obsolete, see the &lt;a class="wiki_link" href="/pergen"&gt;pergens&lt;/a&gt; page instead.&lt;/span&gt;&lt;br /&gt;

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&lt;span style="font-size: 24px;"&gt;This section is somewhat obsolete, see the &lt;a class="wiki_link" href="/pergen"&gt;pergens&lt;/a&gt; page instead.&lt;/span&gt;&lt;br /&gt;

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