Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 584789887 - Original comment: ** |
Wikispaces>TallKite **Imported revision 584790759 - 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 08:03:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>584790759</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> | ||
| Line 915: | Line 915: | ||
|| 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 zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero. | 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. | ||
| Line 924: | Line 923: | ||
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. 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. | 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 other relevant frameworks of size 53 or less: | The other relevant frameworks of size 53 or less: | ||
| Line 943: | Line 943: | ||
||= 45-tone ||= 2 ||= 1 ||= -19 || <span style="display: block; text-align: center;">C^ = Dbbb | ||= 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 ||= | ||= 49-tone ||= 7 ||= -3 ||= +22 ||= C^ = G### ||= desc triple-dim 4th || | ||
||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd || | ||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd || | ||
||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd || | ||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd || | ||
The value of i equals the stepspan of the up | The value of i equals the stepspan of the up interval. A look at the scale fragments reveals why 29-tone has a negative i: | ||
17-tone: C Db C# D | 17-tone: C Db C# D | ||
| Line 953: | Line 953: | ||
29-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. | 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. The descending nature is not an issue, as the following charts show. | ||
The 29-tone genchain: | The 29-tone genchain: | ||
| Line 1,022: | Line 1,022: | ||
||= 28 ||= -12 ||= Cv ||= +17 ||= B^ || | ||= 28 ||= -12 ||= Cv ||= +17 ||= B^ || | ||
||= 29 ||= 0 ||= C ||= ||= || | ||= 29 ||= 0 ||= C ||= ||= || | ||
| Line 1,029: | Line 1,028: | ||
49-tone: C * Db * * * * C# * D | 49-tone: C * Db * * * * C# * D | ||
There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.</pre></div> | There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks. | ||
== == | |||
==__Generators other than a fifth__== | |||
Porcupine in 22-tone is generated by a 2nd = 3\22: | |||
Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb | |||
Scale fragment: D D# Eb E | |||
Keyboard: D * * E * * F * * G * * * A * * B * * C * * D | |||
Because K(#) = 1, ups and downs aren't needed. | |||
The 22-tone porcupine genchain: | |||
||= genspan from D ||= 22-tone keyspan from D || || | |||
||= -13 ||= 5 || Ex || | |||
||= -12 ||= 8 || Fx || | |||
||= -11 ||= 11 || Gx || | |||
||= -10 ||= 14 || A# || | |||
||= -9 ||= 17 || B# || | |||
||= -8 ||= 20 || C# || | |||
||= -7 ||= 1 || D# || | |||
||= -6 ||= 4 || E# || | |||
||= -5 ||= 7 || F# || | |||
||= -4 ||= 10 || G# || | |||
||= -3 ||= 13 || A || | |||
||= -2 ||= 16 || B || | |||
||= -1 ||= 19 || C || | |||
||= 0 ||= 0 || D || | |||
||= 1 ||= 3 || E || | |||
||= 2 ||= 6 || F || | |||
||= 3 ||= 9 || G || | |||
||= 4 ||= 12 || Ab || | |||
||= 5 ||= 15 || Bb || | |||
||= 6 ||= 18 || Cb || | |||
||= 7 ||= 21 || Db || | |||
||= 8 ||= 2 || Eb || | |||
||= 9 ||= 5 || Fb || | |||
||= 10 ||= 8 || Gb || | |||
||= 11 ||= 11 || Abb || | |||
||= 12 ||= 14 || Bbb || | |||
||= 13 ||= 17 || Cbb || | |||
||= 14 ||= 20 || Dbb || | |||
||= 15 ||= 1 || Ebb || | |||
||= 16 ||= 4 || Fbb || | |||
||= 17 ||= 7 || Gbb || | |||
||= ||= etc. || || | |||
Each of the 22 keys, with alternate tunings for the black keys: | |||
||= keyspan from D ||= genspan from D ||= note ||= genspan from D ||= note || | |||
||= 0 ||= 0 ||= D ||= ||= || | |||
||= 1 ||= -7 ||= D# ||= +15 ||= Ebb || | |||
||= 2 ||= -10 ||= Eb ||= +12 ||= Dx || | |||
||= 3 ||= -15 ||= E ||= ||= || | |||
||= 4 ||= +2 ||= D ||= ||= || | |||
||= 5 ||= -3 ||= Eb = D^ ||= +19 ||= D#vv = Ev3 || | |||
||= 6 ||= -8 ||= Eb^ = D^^ ||= +14 ||= D#v = Evv || | |||
||= 7 ||= -13 ||= Eb^^ = D^3 ||= +9 ||= D# = Ev || | |||
||= 8 ||= +4 ||= E ||= ||= || | |||
||= 9 ||= -1 ||= F ||= ||= || | |||
||= 10 ||= -6 ||= Gb = F^ ||= +16 ||= F#vv = Gv3 || | |||
||= 11 ||= -11 ||= Gb^ = F^^ ||= +11 ||= F#v = Gvv || | |||
||= 12 ||= -16 ||= Gb^^ = F^3 ||= +6 ||= F# = Gv || | |||
||= 13 ||= +1 ||= G ||= ||= || | |||
||= 14 ||= -4 ||= Ab = G^ ||= +18 ||= G#vv = Av3 || | |||
||= 15 ||= -9 ||= Ab^ = G^^ ||= +13 ||= G#v = Avv || | |||
||= 16 ||= -14 ||= Ab^^ = G^3 ||= +8 ||= G# = Av || | |||
||= 17 ||= +3 ||= A ||= ||= || | |||
||= 18 ||= -2 ||= Bb = A^ ||= +20 ||= A#vv = Bv3 || | |||
||= 19 ||= -7 ||= Bb^ = A^^ ||= +15 ||= A#v = Bvv || | |||
||= 20 ||= -12 ||= Bb^^ = A^3 ||= +10 ||= A# = Bv || | |||
||= 21 ||= +5 ||= B ||= ||= || | |||
||= 22 ||= 0 ||= C ||= ||= || | |||
P</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:3986:&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:3986 --><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 /> | ||
| Line 1,337: | Line 1,409: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:3987:&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:3987 --><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:3988:&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:3988 --></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> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><!-- ws:end:WikiTextHeadingRule:10 --> </h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><!-- ws:end:WikiTextHeadingRule:10 --> </h2> | ||
| Line 4,100: | Line 4,172: | ||
</td> | </td> | ||
</tr> | </tr> | ||
</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 zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.<br /> | |||
<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) mod N = 0, thus G(#vX) = i * N for some integer i<br /> | |||
7 - X * G(^) = i * N<br /> | |||
G(^) = - (i * N - 7) / X<br /> | |||
<br /> | |||
For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions, and<br /> | |||
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.<br /> | |||
<br /> | |||
The other relevant frameworks of size 53 or less:<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | <tr> | ||
<td> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;">Keyspan of #<br /> | ||
</td> | </td> | ||
<td> | <td>value of i<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">genspan of ^<br /> | |||
<td style="text-align: center;">genspan of ^<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">example<br /> | <td style="text-align: center;">example<br /> | ||
| Line 4,336: | Line 4,379: | ||
<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;">-3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+22<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">C^ = G | <td style="text-align: center;">C^ = G###<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">desc triple-dim 4th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,375: | Line 4,418: | ||
</table> | </table> | ||
The value of i equals the stepspan of the up | The value of i equals the stepspan of the up interval. A look at the scale fragments reveals why 29-tone has a negative i:<br /> | ||
<br /> | <br /> | ||
17-tone: C Db C# D<br /> | 17-tone: C Db C# D<br /> | ||
| Line 4,382: | Line 4,425: | ||
29-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 /> | 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. The descending nature is not an issue, as the following charts show.<br /> | ||
<br /> | <br /> | ||
The 29-tone genchain:<br /> | The 29-tone genchain:<br /> | ||
| Line 5,297: | Line 5,340: | ||
</table> | </table> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
| Line 5,304: | Line 5,346: | ||
49-tone: C * Db * * * * C# * D<br /> | 49-tone: C * Db * * * * C# * D<br /> | ||
<br /> | <br /> | ||
There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.</body></html></pre></div> | There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><!-- ws:end:WikiTextHeadingRule:38 --> </h2> | |||
<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Summary of EDO notation-Generators other than a fifth"></a><!-- ws:end:WikiTextHeadingRule:40 --><u>Generators other than a fifth</u></h2> | |||
<br /> | |||
Porcupine in 22-tone is generated by a 2nd = 3\22:<br /> | |||
Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb<br /> | |||
Scale fragment: D D# Eb E<br /> | |||
Keyboard: D * * E * * F * * G * * * A * * B * * C * * D<br /> | |||
Because K(#) = 1, ups and downs aren't needed.<br /> | |||
<br /> | |||
<br /> | |||
The 22-tone porcupine genchain:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td style="text-align: center;">genspan from D<br /> | |||
</td> | |||
<td style="text-align: center;">22-tone keyspan from D<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-13<br /> | |||
</td> | |||
<td style="text-align: center;">5<br /> | |||
</td> | |||
<td>Ex<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-12<br /> | |||
</td> | |||
<td style="text-align: center;">8<br /> | |||
</td> | |||
<td>Fx<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-11<br /> | |||
</td> | |||
<td style="text-align: center;">11<br /> | |||
</td> | |||
<td>Gx<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-10<br /> | |||
</td> | |||
<td style="text-align: center;">14<br /> | |||
</td> | |||
<td>A#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-9<br /> | |||
</td> | |||
<td style="text-align: center;">17<br /> | |||
</td> | |||
<td>B#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-8<br /> | |||
</td> | |||
<td style="text-align: center;">20<br /> | |||
</td> | |||
<td>C#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-7<br /> | |||
</td> | |||
<td style="text-align: center;">1<br /> | |||
</td> | |||
<td>D#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-6<br /> | |||
</td> | |||
<td style="text-align: center;">4<br /> | |||
</td> | |||
<td>E#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-5<br /> | |||
</td> | |||
<td style="text-align: center;">7<br /> | |||
</td> | |||
<td>F#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-4<br /> | |||
</td> | |||
<td style="text-align: center;">10<br /> | |||
</td> | |||
<td>G#<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-3<br /> | |||
</td> | |||
<td style="text-align: center;">13<br /> | |||
</td> | |||
<td>A<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-2<br /> | |||
</td> | |||
<td style="text-align: center;">16<br /> | |||
</td> | |||
<td>B<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-1<br /> | |||
</td> | |||
<td style="text-align: center;">19<br /> | |||
</td> | |||
<td>C<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">0<br /> | |||
</td> | |||
<td style="text-align: center;">0<br /> | |||
</td> | |||
<td>D<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1<br /> | |||
</td> | |||
<td style="text-align: center;">3<br /> | |||
</td> | |||
<td>E<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td style="text-align: center;">6<br /> | |||
</td> | |||
<td>F<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3<br /> | |||
</td> | |||
<td style="text-align: center;">9<br /> | |||
</td> | |||
<td>G<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4<br /> | |||
</td> | |||
<td style="text-align: center;">12<br /> | |||
</td> | |||
<td>Ab<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">5<br /> | |||
</td> | |||
<td style="text-align: center;">15<br /> | |||
</td> | |||
<td>Bb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6<br /> | |||
</td> | |||
<td style="text-align: center;">18<br /> | |||
</td> | |||
<td>Cb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">7<br /> | |||
</td> | |||
<td style="text-align: center;">21<br /> | |||
</td> | |||
<td>Db<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">8<br /> | |||
</td> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td>Eb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">9<br /> | |||
</td> | |||
<td style="text-align: center;">5<br /> | |||
</td> | |||
<td>Fb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">10<br /> | |||
</td> | |||
<td style="text-align: center;">8<br /> | |||
</td> | |||
<td>Gb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">11<br /> | |||
</td> | |||
<td style="text-align: center;">11<br /> | |||
</td> | |||
<td>Abb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">12<br /> | |||
</td> | |||
<td style="text-align: center;">14<br /> | |||
</td> | |||
<td>Bbb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">13<br /> | |||
</td> | |||
<td style="text-align: center;">17<br /> | |||
</td> | |||
<td>Cbb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">14<br /> | |||
</td> | |||
<td style="text-align: center;">20<br /> | |||
</td> | |||
<td>Dbb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">15<br /> | |||
</td> | |||
<td style="text-align: center;">1<br /> | |||
</td> | |||
<td>Ebb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">16<br /> | |||
</td> | |||
<td style="text-align: center;">4<br /> | |||
</td> | |||
<td>Fbb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">17<br /> | |||
</td> | |||
<td style="text-align: center;">7<br /> | |||
</td> | |||
<td>Gbb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;">etc.<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
Each of the 22 keys, with alternate tunings for the black keys:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td style="text-align: center;">keyspan from D<br /> | |||
</td> | |||
<td style="text-align: center;">genspan from D<br /> | |||
</td> | |||
<td style="text-align: center;">note<br /> | |||
</td> | |||
<td style="text-align: center;">genspan from D<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;">D<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;">-7<br /> | |||
</td> | |||
<td style="text-align: center;">D#<br /> | |||
</td> | |||
<td style="text-align: center;">+15<br /> | |||
</td> | |||
<td style="text-align: center;">Ebb<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td style="text-align: center;">-10<br /> | |||
</td> | |||
<td style="text-align: center;">Eb<br /> | |||
</td> | |||
<td style="text-align: center;">+12<br /> | |||
</td> | |||
<td style="text-align: center;">Dx<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3<br /> | |||
</td> | |||
<td style="text-align: center;">-15<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;">4<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;">5<br /> | |||
</td> | |||
<td style="text-align: center;">-3<br /> | |||
</td> | |||
<td style="text-align: center;">Eb = D^<br /> | |||
</td> | |||
<td style="text-align: center;">+19<br /> | |||
</td> | |||
<td style="text-align: center;">D#vv = Ev3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6<br /> | |||
</td> | |||
<td style="text-align: center;">-8<br /> | |||
</td> | |||
<td style="text-align: center;">Eb^ = D^^<br /> | |||
</td> | |||
<td style="text-align: center;">+14<br /> | |||
</td> | |||
<td style="text-align: center;">D#v = Evv<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">7<br /> | |||
</td> | |||
<td style="text-align: center;">-13<br /> | |||
</td> | |||
<td style="text-align: center;">Eb^^ = D^3<br /> | |||
</td> | |||
<td style="text-align: center;">+9<br /> | |||
</td> | |||
<td style="text-align: center;">D# = Ev<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">8<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;">9<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> | |||
<tr> | |||
<td style="text-align: center;">10<br /> | |||
</td> | |||
<td style="text-align: center;">-6<br /> | |||
</td> | |||
<td style="text-align: center;">Gb = F^<br /> | |||
</td> | |||
<td style="text-align: center;">+16<br /> | |||
</td> | |||
<td style="text-align: center;">F#vv = Gv3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">11<br /> | |||
</td> | |||
<td style="text-align: center;">-11<br /> | |||
</td> | |||
<td style="text-align: center;">Gb^ = F^^<br /> | |||
</td> | |||
<td style="text-align: center;">+11<br /> | |||
</td> | |||
<td style="text-align: center;">F#v = Gvv<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">12<br /> | |||
</td> | |||
<td style="text-align: center;">-16<br /> | |||
</td> | |||
<td style="text-align: center;">Gb^^ = F^3<br /> | |||
</td> | |||
<td style="text-align: center;">+6<br /> | |||
</td> | |||
<td style="text-align: center;">F# = Gv<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">13<br /> | |||
</td> | |||
<td style="text-align: center;">+1<br /> | |||
</td> | |||
<td style="text-align: center;">G<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">14<br /> | |||
</td> | |||
<td style="text-align: center;">-4<br /> | |||
</td> | |||
<td style="text-align: center;">Ab = G^<br /> | |||
</td> | |||
<td style="text-align: center;">+18<br /> | |||
</td> | |||
<td style="text-align: center;">G#vv = Av3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">15<br /> | |||
</td> | |||
<td style="text-align: center;">-9<br /> | |||
</td> | |||
<td style="text-align: center;">Ab^ = G^^<br /> | |||
</td> | |||
<td style="text-align: center;">+13<br /> | |||
</td> | |||
<td style="text-align: center;">G#v = Avv<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">16<br /> | |||
</td> | |||
<td style="text-align: center;">-14<br /> | |||
</td> | |||
<td style="text-align: center;">Ab^^ = G^3<br /> | |||
</td> | |||
<td style="text-align: center;">+8<br /> | |||
</td> | |||
<td style="text-align: center;">G# = Av<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">17<br /> | |||
</td> | |||
<td style="text-align: center;">+3<br /> | |||
</td> | |||
<td style="text-align: center;">A<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">18<br /> | |||
</td> | |||
<td style="text-align: center;">-2<br /> | |||
</td> | |||
<td style="text-align: center;">Bb = A^<br /> | |||
</td> | |||
<td style="text-align: center;">+20<br /> | |||
</td> | |||
<td style="text-align: center;">A#vv = Bv3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">19<br /> | |||
</td> | |||
<td style="text-align: center;">-7<br /> | |||
</td> | |||
<td style="text-align: center;">Bb^ = A^^<br /> | |||
</td> | |||
<td style="text-align: center;">+15<br /> | |||
</td> | |||
<td style="text-align: center;">A#v = Bvv<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">20<br /> | |||
</td> | |||
<td style="text-align: center;">-12<br /> | |||
</td> | |||
<td style="text-align: center;">Bb^^ = A^3<br /> | |||
</td> | |||
<td style="text-align: center;">+10<br /> | |||
</td> | |||
<td style="text-align: center;">A# = Bv<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">21<br /> | |||
</td> | |||
<td style="text-align: center;">+5<br /> | |||
</td> | |||
<td style="text-align: center;">B<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">22<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> | |||
</table> | |||
<br /> | |||
P</body></html></pre></div> | |||