Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-08-~~17 01~~:53:28 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-08-26 04:20:04 UTC</tt>.

: The original revision id was <tt>~~589462136~~</tt>.

: The original revision id was <tt>590192460</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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"fifth-less" EDOs, with fifths wider than 720¢

pentatonic EDOs, with a fifth = 720¢

"~~sweet~~" EDOs, ~~so-called because the~~ fifth hits the "sweet spot" between 720¢ and 686¢

"regular" EDOs, with a fifth that hits the "sweet spot" between 720¢ and 686¢

"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢

fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢

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Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.

This section will cover ~~sweet~~ EDOs and the other categories will be covered in later sections.

This section will cover regular EDOs and the other categories will be covered in later sections.

As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.

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"Perfect" EDOs (7, 14, 21, 28 and 35)

"Pentatonic" EDOs (5, 10, 15, 20, 25 and 30)

"~~Sweet~~" EDOs (all others)

"Regular" EDOs (all others)

The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The ~~sweet~~ EDOs may or may not.

The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The regular EDOs may or may not.

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

Line 734:

||= 10edo ||= pentatonic ||= ||= C/Db ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||

||= 11edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||

||= 12edo ||= ~~sweet~~ ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= ||

||= 12edo ||= regular ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= ||

||= 13edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||

||= 14edo ||= perfect ||= ||= C/C# ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||

||= 15edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||

||= 16edo ||= fourthward ||= ||= C ||= C#/Db ||= D ||= D# ||= ||= ||= ||= ||= ||= ||

||= 17edo ||= ~~sweet~~ ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= ||

||= 17edo ||= regular ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= ||

||= 18edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||

||= 19edo ||= ~~sweet~~ ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= ||

||= 19edo ||= regular ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= ||

||= 20edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||

||= 21edo ||= perfect ||= ||= C/C# ||= * ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||

||= 22edo ||= ~~sweet~~ ||= ||= C ||= Db ||= * ||= C# ||= D ||= ||= ||= ||= ||= ||

||= 22edo ||= regular ||= ||= C ||= Db ||= * ||= C# ||= D ||= ||= ||= ||= ||= ||

||= 23edo ||= fourthward ||= ||= C ||= C# ||= Db ||= D ||= D# ||= ||= ||= ||= ||= ||

||= 24edo ||= ~~sweet~~ ||= ||= C ||= * ||= C#/Db ||= * ||= D ||= ||= ||= ||= ||= ||

||= 24edo ||= regular ||= ||= C ||= * ||= C#/Db ||= * ||= D ||= ||= ||= ||= ||= ||

||= 25edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||

||= 26edo ||= ~~sweet~~ ||= ||= C ||= C# ||= * ||= Db ||= D ||= ||= ||= ||= ||= ||

||= 26edo ||= regular ||= ||= C ||= C# ||= * ||= Db ||= D ||= ||= ||= ||= ||= ||

||= 27edo ||= ~~sweet~~ ||= ||= C ||= Db ||= * ||= * ||= C# ||= D ||= ||= ||= ||= ||

||= 27edo ||= regular ||= ||= C ||= Db ||= * ||= * ||= C# ||= D ||= ||= ||= ||= ||

||= 28edo ||= perfect ||= ||= C/C# ||= * ||= * ||= * ||= Db/D ||= ||= ||= ||= ||= ||

||= 29edo ||= ~~sweet~~ ||= ||= C ||= * ||= Db ||= C# ||= * ||= D ||= ||= ||= ||= ||

||= 29edo ||= regular ||= ||= C ||= * ||= Db ||= C# ||= * ||= D ||= ||= ||= ||= ||

||= 30edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||

||= 31edo ||= ~~sweet~~ ||= ||= C ||= * ||= C# ||= Db ||= * ||= D ||= ||= ||= ||= ||

||= 31edo ||= regular ||= ||= C ||= * ||= C# ||= Db ||= * ||= D ||= ||= ||= ||= ||

||= 32edo ||= ~~sweet~~ ||= ||= C ||= Db ||= * ||= * ||= * ||= C# ||= D ||= ||= ||= ||

||= 32edo ||= regular ||= ||= C ||= Db ||= * ||= * ||= * ||= C# ||= D ||= ||= ||= ||

||= 33edo ||= ~~sweet~~ ||= ||= C ||= C# ||= * ||= * ||= Db ||= D ||= ||= ||= ||= ||

||= 33edo ||= regular ||= ||= C ||= C# ||= * ||= * ||= Db ||= D ||= ||= ||= ||= ||

||= 34edo ||= ~~sweet~~ ||= ||= C ||= * ||= Db ||= * ||= C# ||= * ||= D ||= ||= ||= ||

||= 34edo ||= regular ||= ||= C ||= * ||= Db ||= * ||= C# ||= * ||= D ||= ||= ||= ||

||= 35edo ||= perfect ||= ||= C/C# ||= * ||= * ||= * ||= * ||= Db/D ||= ||= ||= ||= ||

||= 36edo ||= ~~sweet~~ ||= ||= C ||= * ||= * ||= C#/Db ||= * ||= * ||= D ||= ||= ||= ||

||= 36edo ||= regular ||= ||= C ||= * ||= * ||= C#/Db ||= * ||= * ||= D ||= ||= ||= ||

||= 37edo ||= ~~sweet~~ ||= ||= C ||= Db ||= * ||= * ||= * ||= * ||= C# ||= D ||= ||= ||

||= 37edo ||= " ||= ||= C ||= Db ||= * ||= * ||= * ||= * ||= C# ||= D ||= ||= ||

||= 38edo ||= ~~sweet~~ ||= ||= C ||= * ||= C# ||= * ||= Db ||= * ||= D ||= ||= ||= ||

||= 38edo ||= " ||= ||= C ||= * ||= C# ||= * ||= Db ||= * ||= D ||= ||= ||= ||

||= 39edo ||= ~~sweet~~ ||= ||= C ||= * ||= Db ||= * ||= * ||= C# ||= * ||= D ||= ||= ||

||= 39edo ||= " ||= ||= C ||= * ||= Db ||= * ||= * ||= C# ||= * ||= D ||= ||= ||

||= 40edo ||= ~~sweet~~ ||= ||= C ||= C# ||= * ||= * ||= * ||= Db ||= D ||= ||= ||= ||

||= 40edo ||= " ||= ||= C ||= C# ||= * ||= * ||= * ||= Db ||= D ||= ||= ||= ||

||= 41edo ||= ~~sweet~~ ||= ||= C ||= * ||= * ||= Db ||= C# ||= * ||= * ||= D ||= ||= ||

||= 41edo ||= " ||= ||= C ||= * ||= * ||= Db ||= C# ||= * ||= * ||= D ||= ||= ||

||= 42edo ||= ~~sweet~~ ||= ||= C ||= Db ||= * ||= * ||= * ||= * ||= * ||= C# ||= D ||= ||

||= 42edo ||= " ||= ||= C ||= Db ||= * ||= * ||= * ||= * ||= * ||= C# ||= D ||= ||

||= 43edo ||= ~~sweet~~ ||= ||= C ||= * ||= * ||= C# ||= Db ||= * ||= * ||= D ||= ||= ||

||= 43edo ||= " ||= ||= C ||= * ||= * ||= C# ||= Db ||= * ||= * ||= D ||= ||= ||

||= 44ddo ||= ~~sweet~~ ||= ||= C ||= * ||= Db ||= * ||= * ||= * ||= C# ||= * ||= D ||= ||

||= 44ddo ||= " ||= ||= C ||= * ||= Db ||= * ||= * ||= * ||= C# ||= * ||= D ||= ||

||= 45edo ||= ~~sweet~~ ||= ||= C ||= * ||= C# ||= * ||= * ||= Db ||= * ||= D ||= ||= ||

||= 45edo ||= " ||= ||= C ||= * ||= C# ||= * ||= * ||= Db ||= * ||= D ||= ||= ||

||= 46edo ||= ~~sweet~~ ||= ||= C ||= * ||= * ||= Db ||= * ||= C# ||= * ||= * ||= D ||= ||

||= 46edo ||= " ||= ||= C ||= * ||= * ||= Db ||= * ||= C# ||= * ||= * ||= D ||= ||

||= 47edo ||= ~~sweet~~ ||= ||= C ||= C# ||= * ||= * ||= * ||= * ||= Db ||= D ||= ||= ||

||= 47edo ||= " ||= ||= C ||= C# ||= * ||= * ||= * ||= * ||= Db ||= D ||= ||= ||

||= 48edo ||= ~~sweet~~ ||= ||= C ||= * ||= * ||= * ||= C#/Db ||= * ||= * ||= * ||= D ||= ||

||= 48edo ||= " ||= ||= C ||= * ||= * ||= * ||= C#/Db ||= * ||= * ||= * ||= D ||= ||

||= 49edo ||= ~~sweet~~ ||= ||= C ||= * ||= Db ||= * ||= * ||= * ||= * ||= C# ||= * ||= D ||

||= 49edo ||= " ||= ||= C ||= * ||= Db ||= * ||= * ||= * ||= * ||= C# ||= * ||= D ||

||= 50edo ||= ~~sweet~~ ||= ||= C ||= * ||= * ||= C# ||= * ||= Db ||= * ||= * ||= D ||= ||

||= 50edo ||= " ||= ||= C ||= * ||= * ||= C# ||= * ||= Db ||= * ||= * ||= D ||= ||

||= 51edo ||= ~~sweet~~ ||= ||= C ||= * ||= * ||= Db ||= * ||= * ||= C# ||= * ||= * ||= D ||

||= 51edo ||= " ||= ||= C ||= * ||= * ||= Db ||= * ||= * ||= C# ||= * ||= * ||= D ||

||= 52edo ||= ~~sweet~~ ||= ||= C ||= * ||= C# ||= * ||= * ||= * ||= Db ||= * ||= D ||= ||

||= 52edo ||= " ||= ||= C ||= * ||= C# ||= * ||= * ||= * ||= Db ||= * ||= D ||= ||

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

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

=__**Summary of EDO notation**__=

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===__"~~Sweet~~" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)__===

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

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

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

Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.

Line 1,049:

12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C

If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, we need only consider ~~sweet~~ frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.

If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, we need only consider regular frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.

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.

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<div style="margin-left: 3em;"><a href="#Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)">Pentatonic EDOs (5, 10, 15, 20, 25 and 30)</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:">Alternative pentatonic notation for pentatonic EDOs:</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;~~Sweet~~&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)">&quot;~~Sweet~~&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Regular&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)">&quot;Regular&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)</a></div>

<div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div>

<div style="margin-left: 1em;"><a href="#Rank-2 Notation">Rank-2 Notation</a></div>

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&quot;fifth-less&quot; EDOs, with fifths wider than 720¢

pentatonic EDOs, with a fifth = 720¢

&quot;~~sweet~~&quot; EDOs, ~~so-called because the~~ fifth hits the &quot;sweet spot&quot; between 720¢ and 686¢

&quot;regular&quot; EDOs, with a fifth that hits the &quot;sweet spot&quot; between 720¢ and 686¢

&quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢

fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢

Line 1,447:

Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a &quot;natural&quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.

This section will cover ~~sweet~~ EDOs and the other categories will be covered in later sections.

This section will cover regular EDOs and the other categories will be covered in later sections.

As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.

Line 2,054:

&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)

&quot;Pentatonic&quot; EDOs (5, 10, 15, 20, 25 and 30)

&quot;~~Sweet~~&quot; EDOs (all others)

&quot;Regular&quot; EDOs (all others)

The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The ~~sweet~~ EDOs may or may not.

The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The regular EDOs may or may not.

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

Line 2,242:

<td style="text-align: center;">12edo

</td>