Kite's ups and downs notation: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-02 16:58:38 UTC</tt>.<br>

: The original revision id was <tt>590981962</tt>.<br>

The 22-tone keyboard, with alternate tunings for the black keys ("^3" means triple-up):

||= 1 ||= -5 ||= Db = C^ ||= +17 ||= C#vv = Dv3 ||

||= 3 ||= -15 ||= Db^^ = C^3 ||= +7 ||= C# = Dv ||

||= 5 ||= -3 ||= Eb = D^ ||= +19 ||= D#vv = Ev3 ||

||= 7 ||= -13 ||= Eb^^ = D^3 ||= +9 ||= D# = Ev ||

||= 10 ||= -6 ||= Gb = F^ ||= +16 ||= F#vv = Gv3 ||

||= 12 ||= -16 ||= Gb^^ = F^3 ||= +6 ||= F# = Gv ||

||= 14 ||= -4 ||= Ab = G^ ||= +18 ||= G#vv = Av3 ||

||= 16 ||= -14 ||= Ab^^ = G^3 ||= +8 ||= G# = Av ||

||= 18 ||= -2 ||= Bb = A^ ||= +20 ||= A#vv = Bv3 ||

||= 20 ||= -12 ||= Bb^^ = A^3 ||= +10 ||= A# = Bv ||

K(#) = X, K(b) = -X (X = keyspan of a sharp, i.e., how many keys wide aug1 is. For 22-tone, X = 3)

K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)

"#vX" means one sharp plus X downs. Zero keyspans in the genchain only occur on every Nth step for a N-tone framework. E.g., 12-tone keyspans:

B#, genspan 12, has a zero keyspan, as does Dbb, genspan -12, and A###, genspan 24. Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.

G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)

G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i

7 - X * G(^) = i * N

G(^) = - (i * N - 7) / X

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 22-tone keyboard, with alternate tunings for the black keys (&amp;quot;^3&amp;quot; means triple-up):&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;C#vv = Dv3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;Db^^ = C^3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;D#vv = Ev3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;Eb^^ = D^3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;F#vv = Gv3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;Gb^^ = F^3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;G#vv = Av3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;Ab^^ = G^3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;A#vv = Bv3&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;Bb^^ = A^3&lt;br /&gt;

K(#) = X, K(b) = -X (X = keyspan of a sharp, i.e., how many keys wide aug1 is. For 22-tone, X = 3)&lt;br /&gt;

K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)&lt;br /&gt;

&amp;quot;#vX&amp;quot; means one sharp plus X downs. Zero keyspans in the genchain only occur on every Nth step for a N-tone framework. E.g., 12-tone keyspans:&lt;br /&gt;

B#, genspan 12, has a zero keyspan, as does Dbb, genspan -12, and A###, genspan 24. Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.&lt;br /&gt;

G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)&lt;br /&gt;

G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i&lt;br /&gt;

7 - X * G(^) = i * N&lt;br /&gt;

G(^) = - (i * N - 7) / X&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 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;