Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 584781639 - Original comment: ** |
Wikispaces>TallKite **Imported revision 584785353 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 01:28:25 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>584785353</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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==__Rank-2 Notation__== | ==__Rank-2 Notation__== | ||
Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.) | Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.) | ||
Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain: | Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain: | ||
...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ... | ...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ... | ||
Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on | Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on either side of the 4\7 kite: | ||
12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C | 12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C | ||
| Line 813: | Line 813: | ||
16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C | 16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C | ||
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. | For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot EDO chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24). | ||
All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. | All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs: | ||
5-tone genchain C to E: C D E G A C | |||
5-tone genchain F to A: C D F G A C | |||
7-tone genchain C to F#: C D E F# G A B 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 pentatonic 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. | |||
For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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. | |||
The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D. | |||
Here's part of the 22-tone genchain. There are more than 22 notes because the genchain is theoretically infinite: | Here's part of the 22-tone genchain. There are more than 22 notes because the genchain is theoretically infinite: | ||
| Line 857: | Line 863: | ||
||= keyspan ||= genspan ||= note ||= genspan ||= note || | ||= keyspan ||= genspan ||= note ||= genspan ||= note || | ||
||= 0 ||= 0 ||= C ||= ||= || | ||= 0 ||= 0 ||= C ||= ||= || | ||
||= 1 ||= -5 ||= Db = C^ ||= +17 || | ||= 1 ||= -5 ||= Db = C^ ||= +17 ||= C#vv = Dv3 || | ||
||= 2 ||= -10 || | ||= 2 ||= -10 ||= Db^ = C^^ ||= +12 ||= C#v = Dvv || | ||
||= 3 ||= -15 || | ||= 3 ||= -15 ||= Db^^ = C^3 ||= +7 ||= C# = Dv || | ||
||= 4 ||= +2 ||= D ||= ||= || | ||= 4 ||= +2 ||= D ||= ||= || | ||
||= 5 ||= -3 ||= Eb = D^ ||= +19 || | ||= 5 ||= -3 ||= Eb = D^ ||= +19 ||= D#vv = Ev3 || | ||
||= 6 ||= -8 || | ||= 6 ||= -8 ||= Eb^ = D^^ ||= +14 ||= D#v = Evv || | ||
||= 7 ||= -13 || | ||= 7 ||= -13 ||= Eb^^ = D^3 ||= +9 ||= D# = Ev || | ||
||= 8 ||= +4 ||= E ||= ||= || | ||= 8 ||= +4 ||= E ||= ||= || | ||
||= 9 ||= -1 ||= F ||= ||= || | ||= 9 ||= -1 ||= F ||= ||= || | ||
||= 10 ||= -6 ||= Gb = F^ ||= +16 || | ||= 10 ||= -6 ||= Gb = F^ ||= +16 ||= F#vv = Gv3 || | ||
||= 11 ||= -11 || | ||= 11 ||= -11 ||= Gb^ = F^^ ||= +11 ||= F#v = Gvv || | ||
||= 12 ||= -16 || | ||= 12 ||= -16 ||= Gb^^ = F^3 ||= +6 ||= F# = Gv || | ||
||= 13 ||= +1 ||= G ||= ||= || | ||= 13 ||= +1 ||= G ||= ||= || | ||
||= 14 ||= -4 ||= Ab = G^ ||= +18 || | ||= 14 ||= -4 ||= Ab = G^ ||= +18 ||= G#vv = Av3 || | ||
||= 15 ||= -9 || | ||= 15 ||= -9 ||= Ab^ = G^^ ||= +13 ||= G#v = Avv || | ||
||= 16 ||= -14 || | ||= 16 ||= -14 ||= Ab^^ = G^3 ||= +8 ||= G# = Av || | ||
||= 17 ||= +3 ||= A ||= ||= || | ||= 17 ||= +3 ||= A ||= ||= || | ||
||= 18 ||= -2 ||= Bb = A^ ||= +20 || | ||= 18 ||= -2 ||= Bb = A^ ||= +20 ||= A#vv = Bv3 || | ||
||= 19 ||= -7 || | ||= 19 ||= -7 ||= Bb^ = A^^ ||= +15 ||= A#v = Bvv || | ||
||= 20 ||= -12 || | ||= 20 ||= -12 ||= Bb^^ = A^3 ||= +10 ||= A# = Bv || | ||
||= 21 ||= +5 ||= B ||= ||= || | ||= 21 ||= +5 ||= B ||= ||= || | ||
||= 22 ||= 0 ||= C ||= ||= || | ||= 22 ||= 0 ||= C ||= ||= || | ||
The genspan for the up symbol in 22-tone is calculated from the keyspans: | |||
The genspan is calculated from the keyspans: | |||
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1) | K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1) | ||
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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) | ||
"v3" means three downs. "#vX" means one sharp plus X downs. | "v3" means three downs. "#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: | ||
|| 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# || | ||
|| 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5 || 0 || | || genspan from C || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || | ||
Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be either zero, N, -N, 2N, -2N, etc. | || 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 either zero, N, -N, 2N, -2N, etc. | |||
G(#) = 7 (by definition, the sharp's genspan = 7, assuming heptatonic notation) | G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation) | ||
G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^) | 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 | G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i | ||
| Line 1,028: | Line 1,033: | ||
fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br /> | fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:2478:&lt;img src=&quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1035px; width: 800px;&quot; /&gt; --><img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:2478 --><br /> | ||
<br /> | <br /> | ||
This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | ||
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<br /> | <br /> | ||
Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | ||
<br /> | <br /> | ||
Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:<br /> | Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:<br /> | ||
...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...<br /> | ...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...<br /> | ||
<br /> | <br /> | ||
Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on | Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on either side of the 4\7 kite:<br /> | ||
<br /> | <br /> | ||
12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C<br /> | 12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C<br /> | ||
| Line 3,046: | Line 3,051: | ||
16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C<br /> | 16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C<br /> | ||
<br /> | <br /> | ||
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. | For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot EDO chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).<br /> | ||
<br /> | |||
All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:<br /> | |||
<br /> | |||
5-tone genchain C to E: C D E G A C<br /> | |||
5-tone genchain F to A: C D F G A C<br /> | |||
<br /> | <br /> | ||
7-tone genchain C to F#: C D E F# G A B 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, ups and downs aren't needed to notate rank-2. | 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 pentatonic 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 /> | ||
<br /> | <br /> | ||
For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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 a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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 /> | ||
<br /> | <br /> | ||
The usual genchain | The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.<br /> | ||
<br /> | <br /> | ||
Here's part of the 22-tone genchain. There are more than 22 notes because the genchain is theoretically infinite:<br /> | Here's part of the 22-tone genchain. There are more than 22 notes because the genchain is theoretically infinite:<br /> | ||
| Line 3,426: | Line 3,437: | ||
<td style="text-align: center;">+17<br /> | <td style="text-align: center;">+17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">C#vv = Dv3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,434: | Line 3,445: | ||
<td style="text-align: center;">-10<br /> | <td style="text-align: center;">-10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Db^ = C^^<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+12<br /> | <td style="text-align: center;">+12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">C#v = Dvv<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,447: | Line 3,457: | ||
<td style="text-align: center;">-15<br /> | <td style="text-align: center;">-15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Db^^ = C^3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+7<br /> | <td style="text-align: center;">+7<br /> | ||
| Line 3,475: | Line 3,485: | ||
<td style="text-align: center;">+19<br /> | <td style="text-align: center;">+19<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">D#vv = Ev3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,483: | Line 3,493: | ||
<td style="text-align: center;">-8<br /> | <td style="text-align: center;">-8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Eb^ = D^^<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+14<br /> | <td style="text-align: center;">+14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">D#v = Evv<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,495: | Line 3,505: | ||
<td style="text-align: center;">-13<br /> | <td style="text-align: center;">-13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Eb^^ = D^3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+9<br /> | <td style="text-align: center;">+9<br /> | ||
| Line 3,535: | Line 3,545: | ||
<td style="text-align: center;">+16<br /> | <td style="text-align: center;">+16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">F#vv = Gv3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,543: | Line 3,553: | ||
<td style="text-align: center;">-11<br /> | <td style="text-align: center;">-11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Gb^ = F^^<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+11<br /> | <td style="text-align: center;">+11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">F#v = Gvv<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,556: | Line 3,565: | ||
<td style="text-align: center;">-16<br /> | <td style="text-align: center;">-16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Gb^^ = F^3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+6<br /> | <td style="text-align: center;">+6<br /> | ||
| Line 3,584: | Line 3,593: | ||
<td style="text-align: center;">+18<br /> | <td style="text-align: center;">+18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">G#vv = Av3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,592: | Line 3,601: | ||
<td style="text-align: center;">-9<br /> | <td style="text-align: center;">-9<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Ab^ = G^^<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+13<br /> | <td style="text-align: center;">+13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">G#v = Avv<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,604: | Line 3,613: | ||
<td style="text-align: center;">-14<br /> | <td style="text-align: center;">-14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Ab^^ = G^3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+8<br /> | <td style="text-align: center;">+8<br /> | ||
| Line 3,632: | Line 3,641: | ||
<td style="text-align: center;">+20<br /> | <td style="text-align: center;">+20<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">A#vv = Bv3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,641: | Line 3,649: | ||
<td style="text-align: center;">-7<br /> | <td style="text-align: center;">-7<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Bb^ = A^^<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+15<br /> | <td style="text-align: center;">+15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">A#v = Bvv<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 3,653: | Line 3,661: | ||
<td style="text-align: center;">-12<br /> | <td style="text-align: center;">-12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Bb^^ = A^3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+10<br /> | <td style="text-align: center;">+10<br /> | ||
| Line 3,687: | Line 3,695: | ||
<br /> | <br /> | ||
The genspan for the up symbol in 22-tone is calculated from the keyspans:<br /> | |||
The genspan is calculated from the keyspans:<br /> | |||
<br /> | <br /> | ||
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br /> | K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br /> | ||
| Line 3,695: | Line 3,701: | ||
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;v3&quot; means three downs. &quot;#vX&quot; means one sharp plus X downs. | &quot;v3&quot; means three downs. &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 /> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<td>genchain of fifths<br /> | |||
</td> | |||
<td>C<br /> | <td>C<br /> | ||
</td> | </td> | ||
| Line 3,728: | Line 3,736: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>genspan from C<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td>1<br /> | |||
</td> | |||
<td>2<br /> | |||
</td> | |||
<td>3<br /> | |||
</td> | |||
<td>4<br /> | |||
</td> | |||
<td>5<br /> | |||
</td> | |||
<td>6<br /> | |||
</td> | |||
<td>7<br /> | |||
</td> | |||
<td>8<br /> | |||
</td> | |||
<td>9<br /> | |||
</td> | |||
<td>10<br /> | |||
</td> | |||
<td>11<br /> | |||
</td> | |||
<td>12<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12-tone keyspan from C<br /> | |||
</td> | |||
<td>0<br /> | <td>0<br /> | ||
</td> | </td> | ||
| Line 3,757: | Line 3,797: | ||
</table> | </table> | ||
Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be either zero, N, -N, 2N, -2N, etc.<br /> | 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 either zero, N, -N, 2N, -2N, etc.<br /> | ||
<br /> | <br /> | ||
G(#) = 7 (by definition, the sharp's genspan = 7, assuming heptatonic notation)<br /> | G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation)<br /> | ||
G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)<br /> | G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)<br /> | ||
G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i<br /> | G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i<br /> | ||