Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 584789155 - Original comment: ** |
Wikispaces>TallKite **Imported revision 584789637 - 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 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 07:00:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>584789637</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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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 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. The sharp is always genspan +7, and the flat is always genspan -7. By adding sharps/ | 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. | ||
For example, 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. | For example, 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. | ||
| Line 991: | Line 991: | ||
The 29-tone keyboard, with alternate tunings for the black keys: | The 29-tone keyboard, with alternate tunings for the black keys: | ||
||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note || | ||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note || | ||
||= 0 ||= 0 ||= C ||= ||= || | ||= 0 ||= 0 ||= C ||= ||= || | ||
||= 1 ||= -17 ||= Dbv = C#vv ||= +12 ||= C | ||= 1 ||= -17 ||= Dbv = C#vv ||= +12 ||= C^ || | ||
||= 2 ||= -5 ||= Db = C#v ||= +24 ||= C^^ = Dbb^3 || | ||= 2 ||= -5 ||= Db = C#v ||= +24 ||= C^^ = Dbb^3 || | ||
||= 3 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ || | ||= 3 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ || | ||
||= 4 ||= -10 ||= Dv | ||= 4 ||= -10 ||= Dv ||= +19 ||= C#^ = Db^^ || | ||
||= 5 ||= +2 ||= D ||= ||= || | ||= 5 ||= +2 ||= D ||= ||= || | ||
||= 6 ||= -15 ||= Ebv = D#vv ||= +14 ||= D | ||= 6 ||= -15 ||= Ebv = D#vv ||= +14 ||= D^ || | ||
||= 7 ||= -3 ||= Eb = D#v ||= +26 ||= D^^ = Ebb^3 || | ||= 7 ||= -3 ||= Eb = D#v ||= +26 ||= D^^ = Ebb^3 || | ||
||= 8 ||= -20 ||= Evv = Dxv3 ||= +9 ||= D# = Eb^ || | ||= 8 ||= -20 ||= Evv = Dxv3 ||= +9 ||= D# = Eb^ || | ||
||= 9 ||= -8 ||= Ev | ||= 9 ||= -8 ||= Ev ||= +21 ||= D#^ = Eb^^ || | ||
||= 10 ||= +4 ||= E ||= ||= || | ||= 10 ||= +4 ||= E ||= ||= || | ||
||= 11 ||= -13 ||= Fv | ||= 11 ||= -13 ||= Fv ||= +16 ||= E^ || | ||
||= 12 ||= -1 ||= F ||= ||= || | ||= 12 ||= -1 ||= F ||= ||= || | ||
||= | ||= 13 ||= -18 ||= Gbv = F#vv ||= +11 ||= F^ || | ||
||= | ||= 14 ||= -6 ||= Gb = F#v ||= +23 ||= F^^ = Gbb^3 || | ||
||= | ||= 15 ||= -23 ||= Gvv = Fxv3 ||= +6 ||= F# = Gb^ || | ||
||= | ||= 16 ||= -11 ||= Gv ||= +18 ||= F#^ = Gb^^ || | ||
||= | ||= 17 ||= +1 ||= G ||= ||= || | ||
||= | ||= 18 ||= -16 ||= Abv = G#vv ||= +13 ||= G^ || | ||
||= | ||= 19 ||= -4 ||= Ab = G#v ||= +25 ||= G^^ = Abb^3 || | ||
||= | ||= 20 ||= -21 ||= Avv = Gxv3 ||= +8 ||= G# = Ab^ || | ||
||= | ||= 21 ||= -9 ||= Av ||= +20 ||= G#^ = Ab^^ || | ||
||= | ||= 22 ||= +3 ||= A ||= ||= || | ||
||= | ||= 23 ||= -14 ||= Bbv = A#vv ||= +15 ||= A^ || | ||
||= | ||= 24 ||= -2 ||= Bb = A#v ||= +27 ||= A^^ = Bbb^3 || | ||
||= | ||= 25 ||= -19 ||= Bvv = Axv3 ||= +10 ||= A# = Bb^ || | ||
||= | ||= 26 ||= -7 ||= Bv ||= +22 ||= A#^ = Bb^^ || | ||
||= | ||= 27 ||= +5 ||= B ||= ||= || | ||
||= | ||= 28 ||= -12 ||= Cv ||= +17 ||= B^ || | ||
||= | ||= 29 ||= 0 ||= C ||= ||= || | ||
| Line 3,164: | Line 3,162: | ||
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 /> | 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. The sharp is always genspan +7, and the flat is always genspan -7. By adding sharps/ | 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. 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.<br /> | ||
<br /> | <br /> | ||
For example, 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.<br /> | For example, 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.<br /> | ||
| Line 4,922: | Line 4,920: | ||
<br /> | <br /> | ||
The 29-tone keyboard, with alternate tunings for the black keys:<br /> | The 29-tone keyboard, with alternate tunings for the black keys:<br /> | ||
| Line 4,960: | Line 4,956: | ||
<td style="text-align: center;">+12<br /> | <td style="text-align: center;">+12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,992: | Line 4,988: | ||
<td style="text-align: center;">-10<br /> | <td style="text-align: center;">-10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Dv | <td style="text-align: center;">Dv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+19<br /> | <td style="text-align: center;">+19<br /> | ||
| Line 5,020: | Line 5,016: | ||
<td style="text-align: center;">+14<br /> | <td style="text-align: center;">+14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">D | <td style="text-align: center;">D^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,052: | Line 5,048: | ||
<td style="text-align: center;">-8<br /> | <td style="text-align: center;">-8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Ev | <td style="text-align: center;">Ev<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+21<br /> | <td style="text-align: center;">+21<br /> | ||
| Line 5,076: | Line 5,072: | ||
<td style="text-align: center;">-13<br /> | <td style="text-align: center;">-13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Fv | <td style="text-align: center;">Fv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+16<br /> | <td style="text-align: center;">+16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">E | <td style="text-align: center;">E^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,096: | Line 5,092: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Gbv = F#vv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">F^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-6<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Gb = F#v<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+23<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">F^^ = Gbb^3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-23<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Gvv = Fxv3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+6<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">F# = Gb^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Gv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">F#^ = Gb^^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+1<br /> | <td style="text-align: center;">+1<br /> | ||
| Line 5,156: | Line 5,152: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Abv = G#vv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">G^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">19<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Ab = G#v<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+25<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">G^^ = Abb^3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">20<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-21<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Avv = Gxv3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">G# = Ab^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">21<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-9<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Av<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+20<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">G#^ = Ab^^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">22<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">A<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 5,216: | Line 5,212: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">23<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Bbv = A#vv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">A^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">24<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Bb = A#v<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+27<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">A^^ = Bbb^3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">25<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-19<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Bvv = Axv3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">A# = Bb^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">26<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-7<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Bv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+22<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">A#^ = Bb^^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">27<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">B<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 5,276: | Line 5,272: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">28<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">-12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Cv<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">B^<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">29<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||