Kite's ups and downs notation: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-07-22 21:56:16 UTC</tt>.<br>

: The original revision id was <tt>587606199</tt>.<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 "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**. 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 (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.

15\22 = ^m6

There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.

0-3-9 = m (or possibly sus2)

0-4-9 = ^m

0-5-9 = vM

0-6-9 = M (or possibly sus4)

0-5-9-12 = 7(v3)

16edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.

0-3-9 = sus2

0-4-9 = m

0-5-9 = M (in practice, no symbol, as in "C" for the C chord)

0-5-10 = aug (the conventional aug chord)

0-6-9 = (A3) (aug 3rd, perfect 5th)

0-7-9 = sus4

0-4-8-12 = dim7 (the conventional dim tetrad)

chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7

0-4-10 = m

0-5-10 = ^m or vM (probably choose vM over ^m whenever possible)

0-6-10 = M

0-7-10 = sus4

0-4-10-14 = m7

0-5-10-15 = vM7

0-6-10-16 = M7

0-4-11 = (d3) (dim 3rd, perfect 5th)

0-4-10 = dim(d3)

0-5-11 = m

0-5-10 = dim (conventional dim chord)

0-6-11 = M

0-7-11 = (A3) (aug 3rd, perfect 5th)

0-6-12 = aug (conventional aug chord)

0-7-12 = aug(A3)

0-8-11 = sus4

Quality can also be omitted in the chord names if we use the mid symbol "~":

0-3-12 = sus2

0-4-12 = vv or sus^2

0-5-12 = v (a down chord, e.g. C.v = "C dot down")

0-6-12 = ~ (a mid chord, e.g. D.~ = "D dot mid")

0-7-12 = ^ (an up chord, e.g. E.^ = "E dot up")

0-7-14 = ^m or vM or ~

0-9-18 = ^m or vM or ~

Every node on the Tree of Kites, 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__ side of the 10\17 kite, we know that 17 __fifths__ add up to 1\22. Because it's on the __left__ 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 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 occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.

Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.

The genspan for the up symbol in 22-tone is calculated from the keyspans:

For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions, and

produces an interval with a keyspan of 1. Thus i = 1, G(^) = -5, and ^ = min 2nd. In order to provide alternate names for each note, the ^ should always be a 2nd. However as we'll see, this isn't always possible.

The other relevant frameworks of size 53 or less:

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 these two frameworks.

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;. 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 (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.&lt;br /&gt;

15\22 = ^m6&lt;br /&gt;

There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.&lt;br /&gt;

0-3-9 = m (or possibly sus2)&lt;br /&gt;

0-4-9 = ^m&lt;br /&gt;

0-5-9 = vM&lt;br /&gt;

0-6-9 = M (or possibly sus4)&lt;br /&gt;

0-5-9-12 = 7(v3)&lt;br /&gt;

16edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.&lt;br /&gt;

0-3-9 = sus2&lt;br /&gt;

0-4-9 = m&lt;br /&gt;

0-5-9 = M (in practice, no symbol, as in &amp;quot;C&amp;quot; for the C chord)&lt;br /&gt;

0-5-10 = aug (the conventional aug chord)&lt;br /&gt;

0-6-9 = (A3) (aug 3rd, perfect 5th)&lt;br /&gt;

0-7-9 = sus4&lt;br /&gt;

0-4-8-12 = dim7 (the conventional dim tetrad)&lt;br /&gt;

chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7&lt;br /&gt;

0-4-10 = m&lt;br /&gt;

0-5-10 = ^m or vM (probably choose vM over ^m whenever possible)&lt;br /&gt;

0-6-10 = M&lt;br /&gt;

0-7-10 = sus4&lt;br /&gt;

0-4-10-14 = m7&lt;br /&gt;

0-5-10-15 = vM7&lt;br /&gt;

0-6-10-16 = M7&lt;br /&gt;

0-4-11 = (d3) (dim 3rd, perfect 5th)&lt;br /&gt;

0-4-10 = dim(d3)&lt;br /&gt;

0-5-11 = m&lt;br /&gt;

0-5-10 = dim (conventional dim chord)&lt;br /&gt;

0-6-11 = M&lt;br /&gt;

0-7-11 = (A3) (aug 3rd, perfect 5th)&lt;br /&gt;

0-6-12 = aug (conventional aug chord)&lt;br /&gt;

0-7-12 = aug(A3)&lt;br /&gt;

0-8-11 = sus4&lt;br /&gt;

Quality can also be omitted in the chord names if we use the mid symbol &amp;quot;~&amp;quot;:&lt;br /&gt;

0-3-12 = sus2&lt;br /&gt;

0-4-12 = vv or sus^2&lt;br /&gt;

0-5-12 = v (a down chord, e.g. C.v = &amp;quot;C dot down&amp;quot;)&lt;br /&gt;

0-6-12 = ~ (a mid chord, e.g. D.~ = &amp;quot;D dot mid&amp;quot;)&lt;br /&gt;

0-7-12 = ^ (an up chord, e.g. E.^ = &amp;quot;E dot up&amp;quot;)&lt;br /&gt;

0-7-14 = ^m or vM or ~&lt;br /&gt;

0-9-18 = ^m or vM or ~&lt;br /&gt;

Every node on the Tree of Kites, 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 &lt;u&gt;right&lt;/u&gt; side of the 10\17 kite, we know that 17 &lt;u&gt;fifths&lt;/u&gt; add up to 1\22. Because it's on the &lt;u&gt;left&lt;/u&gt; side of the 3\5 kite, 5 &lt;u&gt;fourths&lt;/u&gt; 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 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 occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.&lt;br /&gt;

Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.&lt;br /&gt;

The genspan for the up symbol in 22-tone is calculated from the keyspans:&lt;br /&gt;

For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions, and&lt;br /&gt;

produces an interval with a keyspan of 1. Thus i = 1, G(^) = -5, and ^ = min 2nd. In order to provide alternate names for each note, the ^ should always be a 2nd. However as we'll see, this isn't always possible.&lt;br /&gt;

The other relevant frameworks of size 53 or less:&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 these two frameworks.&lt;br /&gt;