Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 584785353 - Original comment: ** |
Wikispaces>TallKite **Imported revision 584785473 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 01: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 01:42:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>584785473</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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7-tone genchain Bb to E: C D E F G A Bb C | 7-tone genchain Bb to E: C D E F G A Bb 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 and | All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd is a descending interval. | ||
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. | |||
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. | 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. | ||
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||= 21 ||= +5 ||= B ||= ||= || | ||= 21 ||= +5 ||= B ||= ||= || | ||
||= 22 ||= 0 ||= C ||= ||= || | ||= 22 ||= 0 ||= C ||= ||= || | ||
"^3" means three ups. 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: | The genspan for the up symbol in 22-tone is calculated from the keyspans: | ||
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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) | 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: | |||
|| genchain of fifths || C || G || D || A || E || B || F# || C# || G# || D# || A# || E# || B# || | || genchain of fifths || C || G || D || A || E || B || F# || C# || G# || D# || A# || E# || B# || | ||
|| genspan from C || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || | || genspan from C || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || | ||
|| 12-tone keyspan from C || 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5 || 0 || | || 12-tone keyspan from C || 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5 || 0 || | ||
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 | 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(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation) | G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation) | ||
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For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd. | For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd. | ||
||= 5edo ||= pentatonic ||= | ||= 5edo ||= pentatonic ||= K(#) || || || | ||
||= 7edo ||= perfect ||= | ||= 7edo ||= perfect ||= 0 || || || | ||
||= 8edo ||= fifthless ||= || || || | ||= 8edo ||= fifthless ||= || || || | ||
||= 11edo ||= fifthless ||= || || || | ||= 11edo ||= fifthless ||= || || || | ||
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7-tone genchain Bb to E: C D E F G A Bb C<br /> | 7-tone genchain Bb to E: C D E F G A Bb C<br /> | ||
<br /> | <br /> | ||
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 and | All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd is a descending interval.<br /> | ||
<br /> | |||
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.<br /> | |||
<br /> | <br /> | ||
To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the generator chain, or <strong>genchain</strong>, one must travel to find the interval.<br /> | To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the generator chain, or <strong>genchain</strong>, one must travel to find the interval.<br /> | ||
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</table> | </table> | ||
<br /> | |||
&quot;^3&quot; means three ups. 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.<br /> | |||
<br /> | <br /> | ||
The genspan for the up symbol in 22-tone is calculated from the keyspans:<br /> | The genspan for the up symbol in 22-tone is calculated from the keyspans:<br /> | ||
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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)<br /> | 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)<br /> | ||
<br /> | <br /> | ||
&quot;#vX&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:<br /> | |||
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</table> | </table> | ||
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 | 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.<br /> | ||
<br /> | <br /> | ||
G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation)<br /> | G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation)<br /> | ||
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<td style="text-align: center;">pentatonic<br /> | <td style="text-align: center;">pentatonic<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">K(#)<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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<td style="text-align: center;">perfect<br /> | <td style="text-align: center;">perfect<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||