Kite's ups and downs notation: Difference between revisions

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Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works very well 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 "^" and the down symbol "v". There's also the optional mid symbol "~" which undoes ups and downs (see the Cancelling section).

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

Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!

Let's review the 22edo interval names:

== ==  

== ==  

When applied to notes, the mid symbol "~"means "neither up nor down". But in chord names it means "exactly midway between major and minor", hence neutral. This only applies to certain "neutral EDOs" in which the sharp equals an even number of EDOsteps. For example, in every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.), a sharp is two EDOsteps, upminor equals downmajor, and "mid" replaces both terms. In 20edo, 27edo, 34edo, 41edo, etc., a sharp is four EDOsteps, and mid replaces both double-upminor and double-downmajor.

Applying "dot up" or "dot down" to a chord raises or lowers the 3rd, and also the 6th or the 7th, if present. Thus "C dot down nine" is the usual C9 chord with the 3rd and 7th lowered: C.v9 = C Ev G Bbv D. A "dot mid" chord has a neutral 3rd and a neutral 6th/7th. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th.

__**Various triads:**__

0-3-6-9 = D Fv A Cv = D.~7 = "D dot mid seven", or D Fv A Bb = D~(b6) = "D mid flat-six"

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 "major 3rd" 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.

Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.

Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:

||= 53edo ||= sweet ||=  ||= C ||= * ||= * ||= * ||= Db ||= C# ||= * ||= * ||= * ||= D ||

===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__===  

Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:

Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)

== ==  

The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.

||= 22 ||= 0 ||= D ||=  ||=  ||</pre></div>

<h4>Original HTML content:</h4>

  &lt;br /&gt;

Ups and Downs is a notation system developed by &lt;a class="wiki_link" href="/KiteGiedraitis"&gt;Kite&lt;/a&gt; that works very well 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;. There's also the optional mid symbol &amp;quot;~&amp;quot; which undoes ups and downs (see the Cancelling section).&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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

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.&lt;br /&gt;

&lt;br /&gt;

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. The colored regions of the tree are what I call &lt;strong&gt;kites&lt;/strong&gt;, and this version of the Stern-Brocot tree I call the &lt;strong&gt;Tree of 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;

&lt;br /&gt;

Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Let's review the 22edo interval names:&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt; &lt;/h2&gt;

  &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt; &lt;/h2&gt;

  &lt;br /&gt;

When applied to notes, the mid symbol &amp;quot;~&amp;quot;means &amp;quot;neither up nor down&amp;quot;. But in chord names it means &amp;quot;exactly midway between major and minor&amp;quot;, hence neutral. This only applies to certain &amp;quot;neutral EDOs&amp;quot; in which the sharp equals an even number of EDOsteps. For example, in every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.), a sharp is two EDOsteps, upminor equals downmajor, and &amp;quot;mid&amp;quot; replaces both terms. In 20edo, 27edo, 34edo, 41edo, etc., a sharp is four EDOsteps, and mid replaces both double-upminor and double-downmajor.&lt;br /&gt;

Applying &amp;quot;dot up&amp;quot; or &amp;quot;dot down&amp;quot; to a chord raises or lowers the 3rd, and also the 6th or the 7th, if present. Thus &amp;quot;C dot down nine&amp;quot; is the usual C9 chord with the 3rd and 7th lowered: C.v9 = C Ev G Bbv D. A &amp;quot;dot mid&amp;quot; chord has a neutral 3rd and a neutral 6th/7th. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;u&gt;&lt;strong&gt;Various triads:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;

C Dbv Ev G Bbv = C.v7(vb9) = &amp;quot;C dot down seven downflat-nine&amp;quot;, or C.vb9(v9)&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

14edo: D * E * F * G * A * B * C * D, zero keys per #/b.&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;

&lt;br /&gt;

  &lt;br /&gt;

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;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

&lt;strong&gt;&lt;u&gt;8edo&lt;/u&gt;:&lt;/strong&gt; (generator = 1\8 = perfect 2nd = 150¢)&lt;br /&gt;

E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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

&lt;br /&gt;

&lt;br /&gt;

  &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;

P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)&lt;br /&gt;

There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for rank-2 fifth-generated tunings in these two frameworks.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.&lt;br /&gt;