Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 584787447 - Original comment: ** |
Wikispaces>TallKite **Imported revision 584788485 - 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 06:00:25 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>584788485</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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16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C | 16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# 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 | 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. These correspond to open EDOs. Each node in the Stern-Brocot 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 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24). | For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot 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 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24). | ||
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5-tone: Bb * C G D * E = C D E G Bb C | 5-tone: Bb * C G D * E = C D E G Bb C | ||
12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C | 12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B 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, 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. | ||
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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. | ||
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. 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. | ||
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. | 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. | ||
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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) | ||
K(#) = X, K(b) = -X (X = keyspan of a sharp, i.e., how many keys wide | 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) | 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) | ||
| Line 926: | Line 924: | ||
G(^) = - (i * N - 7) / X | 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. Thus i = 1, G(^) = -5, and ^ = min 2nd. | For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. 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. | ||
||= ||= Keyspan of # ||= value of i ||= genspan of ^ ||= | The other relevant frameworks of size 53 or less: | ||
||= ||= Keyspan of # | |||
= K(A1) ||= value of i ||= genspan of ^ ||= example ||= stepspan & | |||
quality of ^ || | quality of ^ || | ||
||= 17-tone ||= 2 ||= 1 ||= -5 ||= C | ||= 17-tone ||= 2 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
||= 22-tone ||= 3 ||= 1 ||= -5 ||= C | ||= 22-tone ||= 3 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
||= 27-tone ||= 4 ||= 1 ||= -5 ||= C | ||= 27-tone ||= 4 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
||= 29-tone ||= 3 ||= | ||= 29-tone ||= 3 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd || | ||
||= 31-tone ||= 2 ||= 1 ||= -12 ||= C | ||= 31-tone ||= 2 ||= 1 ||= -12 ||= C^ = Dbb ||= dim 2nd || | ||
||= 32-tone ||= 5 ||= 1 ||= -5 ||= C | ||= 32-tone ||= 5 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
||= 37-tone ||= 6 ||= 1 ||= -5 ||= C | ||= 37-tone ||= 6 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th, desc double-dim 3rd || | ||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th, desc double-dim 3rd || | ||
||= 41-tone ||= 4 ||= | ||= 41-tone ||= 4 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd || | ||
||= 42-tone ||= 7 ||= 1 ||= -5 ||= C | ||= 42-tone ||= 7 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd || | ||
||= 43-tone ||= 3 ||= 1 ||= -12 ||= C | ||= 43-tone ||= 3 ||= 1 ||= -12 ||= C^ = Dbb ||= dim 2nd || | ||
||= 45-tone ||= 2 ||= 1 ||= -19 || <span style="display: block; text-align: center;">C | ||= 45-tone ||= 2 ||= 1 ||= -19 || <span style="display: block; text-align: center;">C^ = Dbbb | ||
</span> ||= double-dim 2nd || | </span> ||= double-dim 2nd || | ||
||= 49-tone ||= 7 ||= 0, 1, 2, -1 ||= 1, -6, -13, 8 ||= C-G, C-Gb, C-Gbb, C-G# ||= perf 5th, dim 5th, double-dim 5th, aug 5th || | ||= 49-tone ||= 7 ||= 0, 1, 2, -1 ||= 1, -6, -13, 8 ||= C-G, C-Gb, C-Gbb, C-G# ||= perf 5th, dim 5th, double-dim 5th, aug 5th || | ||
||= 50-tone ||= 3 ||= | ||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd || | ||
||= 53-tone ||= 5 ||= | ||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd || | ||
The value of i equals the stepspan of the up, except in the case of 49-tone. | The value of i equals the stepspan of the up, except in the case of 49-tone. A look at the scale fragments reveals why 29-tone has a negative i: | ||
17-tone: C Db C# D | |||
22-tone: C Db * C# D | |||
27-tone: C Db * * C# D | |||
29-tone: C * Db C# * D | |||
The 17-tone, 22-tone and 27-tone frameworks all have Db adjacent to C, so that C^ equals Db. For 29-tone, Db = C^^. To find a D-something that is adjacent to C, we must use Dbb, which is one key __below__ C. Thus Cv = Dbb, and C^ = B#, and ^ is a descending dim 2nd. In 41-tone, 50-tone and 53-tone, the up is also a descending 2nd. | |||
29-tone | |||
The 29-tone genchain: | The 29-tone genchain: | ||
| Line 988: | Line 989: | ||
||= 16 ||= 11 || || ||= Dx ||= E^ || Fb^^ || | ||= 16 ||= 11 || || ||= Dx ||= E^ || Fb^^ || | ||
||= 17 ||= 28 || || ||= Ax ||= B^ || Cb^^ || | ||= 17 ||= 28 || || ||= Ax ||= B^ || Cb^^ || | ||
||= ||= || || || etc. ||= || ||</pre></div> | ||= ||= || || || etc. ||= || || | ||
Some of the 29 keys, with alternate tunings for the black keys: | |||
||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note || | |||
||= 0 ||= 0 ||= C ||= ||= || | |||
||= 1 ||= -17 ||= Dbv = C#vv ||= +12 ||= C^ = Dbb^^ || | |||
||= 2 ||= -5 ||= Db = C#v ||= +24 ||= C^^ = Dbb^3 || | |||
||= 3 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ || | |||
||= 4 ||= -10 ||= Dv = Cxvv ||= +19 ||= C#^ = Db^^ || | |||
||= 5 ||= +2 ||= D ||= ||= || | |||
||= 6 ||= -15 ||= Ebv = D#vv ||= +14 ||= D^ = Ebb^^ || | |||
||= 7 ||= -3 ||= Eb = D#v ||= +26 ||= D^^ = Ebb^3 || | |||
||= 8 ||= -20 ||= Evv = Dxv3 ||= +9 ||= D# = Eb^ || | |||
||= 9 ||= -8 ||= Ev = Dxvv ||= +21 ||= D#^ = Eb^^ || | |||
||= 10 ||= +4 ||= E ||= ||= || | |||
||= 11 ||= -13 ||= Fv = E#vv ||= +16 ||= E^ = Fb^^ || | |||
||= 12 ||= -1 ||= F ||= ||= || | |||
The</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | ||
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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:3254:&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:3254 --><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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:3255:&lt;img src=&quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1035px; width: 800px;&quot; /&gt; --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:3255 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextLocalImageRule:3256:&lt;img src=&quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 957px; width: 800px;&quot; /&gt; --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:3256 --></h2> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><!-- ws:end:WikiTextHeadingRule:8 --> </h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><!-- ws:end:WikiTextHeadingRule:8 --> </h2> | ||
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16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C<br /> | 16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# 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 /> | 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. These correspond to open EDOs. Each node in the Stern-Brocot 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 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).<br /> | For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot 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 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).<br /> | ||
| Line 3,119: | Line 3,137: | ||
5-tone: Bb * C G D * E = C D E G Bb C<br /> | 5-tone: Bb * C G D * E = C D E G Bb C<br /> | ||
12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C<br /> | 12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B 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, 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 /> | ||
| Line 3,125: | Line 3,142: | ||
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. 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 /> | ||
<br /> | <br /> | ||
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 /> | 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 /> | ||
| Line 3,964: | Line 3,981: | ||
<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 /> | ||
K(#) = X, K(b) = -X (X = keyspan of a sharp, i.e., how many keys wide | K(#) = X, K(b) = -X (X = keyspan of a sharp, i.e., how many keys wide aug1 is. For 22-tone, X = 3)<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 /> | 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 /> | ||
| Line 4,101: | Line 4,118: | ||
G(^) = - (i * N - 7) / X<br /> | G(^) = - (i * N - 7) / X<br /> | ||
<br /> | <br /> | ||
For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd. The other relevant frameworks of 53 or less:<br /> | For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. 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.<br /> | ||
<br /> | |||
The other relevant frameworks of size 53 or less:<br /> | |||
<br /> | <br /> | ||
| Line 4,110: | Line 4,129: | ||
</td> | </td> | ||
<td style="text-align: center;">Keyspan of #<br /> | <td style="text-align: center;">Keyspan of #<br /> | ||
</td> | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="K(A1) ||"></a><!-- ws:end:WikiTextHeadingRule:38 --> K(A1)</td> | ||
<td | <td></h1> | ||
</td> | value of i</td> | ||
<td style="text-align: center;">genspan of ^<br /> | <td style="text-align: center;">genspan of ^<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">example<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">stepspan &amp;<br /> | <td style="text-align: center;">stepspan &amp;<br /> | ||
| Line 4,130: | Line 4,149: | ||
<td style="text-align: center;">-5<br /> | <td style="text-align: center;">-5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">min 2nd<br /> | <td style="text-align: center;">min 2nd<br /> | ||
| Line 4,144: | Line 4,163: | ||
<td style="text-align: center;">-5<br /> | <td style="text-align: center;">-5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">min 2nd<br /> | <td style="text-align: center;">min 2nd<br /> | ||
| Line 4,158: | Line 4,177: | ||
<td style="text-align: center;">-5<br /> | <td style="text-align: center;">-5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">min 2nd<br /> | <td style="text-align: center;">min 2nd<br /> | ||
| Line 4,168: | Line 4,187: | ||
<td style="text-align: center;">3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = B#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">desc dim 2nd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,186: | Line 4,205: | ||
<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^ = Dbb<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">dim 2nd<br /> | <td style="text-align: center;">dim 2nd<br /> | ||
| Line 4,200: | Line 4,219: | ||
<td style="text-align: center;">-5<br /> | <td style="text-align: center;">-5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">min 2nd<br /> | <td style="text-align: center;">min 2nd<br /> | ||
| Line 4,214: | Line 4,233: | ||
<td style="text-align: center;">-5<br /> | <td style="text-align: center;">-5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">min 2nd<br /> | <td style="text-align: center;">min 2nd<br /> | ||
| Line 4,238: | Line 4,257: | ||
<td style="text-align: center;">4<br /> | <td style="text-align: center;">4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = B#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">desc dim 2nd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,256: | Line 4,275: | ||
<td style="text-align: center;">-5<br /> | <td style="text-align: center;">-5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = Db<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">min 2nd<br /> | <td style="text-align: center;">min 2nd<br /> | ||
| Line 4,270: | Line 4,289: | ||
<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^ = Dbb<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">dim 2nd<br /> | <td style="text-align: center;">dim 2nd<br /> | ||
| Line 4,284: | Line 4,303: | ||
<td style="text-align: center;">-19<br /> | <td style="text-align: center;">-19<br /> | ||
</td> | </td> | ||
<td><span style="display: block; text-align: center;">C | <td><span style="display: block; text-align: center;">C^ = Dbbb<br /> | ||
</span><br /> | </span><br /> | ||
</td> | </td> | ||
| Line 4,309: | Line 4,328: | ||
<td style="text-align: center;">3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+19<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = Bx<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">desc double-dim 2nd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,323: | Line 4,342: | ||
<td style="text-align: center;">5<br /> | <td style="text-align: center;">5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C | <td style="text-align: center;">C^ = B#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">desc dim 2nd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
The value of i equals the stepspan of the up, except in the case of 49-tone.<br /> | The value of i equals the stepspan of the up, except in the case of 49-tone. A look at the scale fragments reveals why 29-tone has a negative i:<br /> | ||
<br /> | <br /> | ||
17-tone: C Db C# D<br /> | |||
22-tone: C Db * C# D<br /> | |||
27-tone: C Db * * C# D<br /> | |||
29-tone: C * Db C# * D<br /> | |||
<br /> | <br /> | ||
The 17-tone, 22-tone and 27-tone frameworks all have Db adjacent to C, so that C^ equals Db. For 29-tone, Db = C^^. To find a D-something that is adjacent to C, we must use Dbb, which is one key <u>below</u> C. Thus Cv = Dbb, and C^ = B#, and ^ is a descending dim 2nd. In 41-tone, 50-tone and 53-tone, the up is also a descending 2nd.<br /> | |||
C | |||
<br /> | <br /> | ||
The 29-tone genchain:<br /> | The 29-tone genchain:<br /> | ||
| Line 4,877: | Line 4,896: | ||
</table> | </table> | ||
</body></html></pre></div> | <br /> | ||
Some of the 29 keys, with alternate tunings for the black keys:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td style="text-align: center;">keyspan from C<br /> | |||
</td> | |||
<td style="text-align: center;">genspan from C<br /> | |||
</td> | |||
<td style="text-align: center;">note<br /> | |||
</td> | |||
<td style="text-align: center;">genspan from C<br /> | |||
</td> | |||
<td style="text-align: center;">note<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">0<br /> | |||
</td> | |||
<td style="text-align: center;">0<br /> | |||
</td> | |||
<td style="text-align: center;">C<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1<br /> | |||
</td> | |||
<td style="text-align: center;">-17<br /> | |||
</td> | |||
<td style="text-align: center;">Dbv = C#vv<br /> | |||
</td> | |||
<td style="text-align: center;">+12<br /> | |||
</td> | |||
<td style="text-align: center;">C^ = Dbb^^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td style="text-align: center;">-5<br /> | |||
</td> | |||
<td style="text-align: center;">Db = C#v<br /> | |||
</td> | |||
<td style="text-align: center;">+24<br /> | |||
</td> | |||
<td style="text-align: center;">C^^ = Dbb^3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3<br /> | |||
</td> | |||
<td style="text-align: center;">-22<br /> | |||
</td> | |||
<td style="text-align: center;">Dvv = Cxv3<br /> | |||
</td> | |||
<td style="text-align: center;">+7<br /> | |||
</td> | |||
<td style="text-align: center;">C# = Db^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4<br /> | |||
</td> | |||
<td style="text-align: center;">-10<br /> | |||
</td> | |||
<td style="text-align: center;">Dv = Cxvv<br /> | |||
</td> | |||
<td style="text-align: center;">+19<br /> | |||
</td> | |||
<td style="text-align: center;">C#^ = Db^^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">5<br /> | |||
</td> | |||
<td style="text-align: center;">+2<br /> | |||
</td> | |||
<td style="text-align: center;">D<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6<br /> | |||
</td> | |||
<td style="text-align: center;">-15<br /> | |||
</td> | |||
<td style="text-align: center;">Ebv = D#vv<br /> | |||
</td> | |||
<td style="text-align: center;">+14<br /> | |||
</td> | |||
<td style="text-align: center;">D^ = Ebb^^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">7<br /> | |||
</td> | |||
<td style="text-align: center;">-3<br /> | |||
</td> | |||
<td style="text-align: center;">Eb = D#v<br /> | |||
</td> | |||
<td style="text-align: center;">+26<br /> | |||
</td> | |||
<td style="text-align: center;">D^^ = Ebb^3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">8<br /> | |||
</td> | |||
<td style="text-align: center;">-20<br /> | |||
</td> | |||
<td style="text-align: center;">Evv = Dxv3<br /> | |||
</td> | |||
<td style="text-align: center;">+9<br /> | |||
</td> | |||
<td style="text-align: center;">D# = Eb^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">9<br /> | |||
</td> | |||
<td style="text-align: center;">-8<br /> | |||
</td> | |||
<td style="text-align: center;">Ev = Dxvv<br /> | |||
</td> | |||
<td style="text-align: center;">+21<br /> | |||
</td> | |||
<td style="text-align: center;">D#^ = Eb^^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">10<br /> | |||
</td> | |||
<td style="text-align: center;">+4<br /> | |||
</td> | |||
<td style="text-align: center;">E<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">11<br /> | |||
</td> | |||
<td style="text-align: center;">-13<br /> | |||
</td> | |||
<td style="text-align: center;">Fv = E#vv<br /> | |||
</td> | |||
<td style="text-align: center;">+16<br /> | |||
</td> | |||
<td style="text-align: center;">E^ = Fb^^<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">12<br /> | |||
</td> | |||
<td style="text-align: center;">-1<br /> | |||
</td> | |||
<td style="text-align: center;">F<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
The</body></html></pre></div> | |||