UDP: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-18 18:26:59 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-18 18:55:16 UTC</tt>.

: The original revision id was <tt>~~277107336~~</tt>.

: The original revision id was <tt>277111706</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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If m shifts up, then U is such that mU shifts up to the top mode, and D is such that mD shifts down to the bottom mode; if m shifts down we reverse this so that -mD shifts up to the top mode and -mU to the bottom mode. If S is a periodic scale S such that the repetition interval **O** is some fraction 1/P of an octave, then the UDP notation for a given mode of a MOS is U|D(P). If P=1 we may omit it and just write U|D.

For example, consider the quasiperiodic function ~~with period 7~~ Lydian[i]:

For example, consider the quasiperiodic function Lydian[i] with period 7 whose argument is the top row and value the bottom row of the following table:

|| -5 || -4 || -3 || -2 || -1 || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 ||

|| -21 || -16 || -13 || -8 || -3 || 0 || 5 || 10 || 15 || 18 || 23 || 28 || 31 || 36 || 41 || 46 || 49 || || 54 || 59 || 62 ||

|| -21 || -16 || -13 || -8 || -3 || 0 || 5 || 10 || 15 || 18 || 23 || 28 || 31 || 36 || 41 || 46 || 49 || 54 || 59 || 62 || 67 ||

Extended to all the integers, this gives the Lydian mode of the diatonic scale, tuned to 31 equal. A formal closed-form definition can be given in terms of the list V[i] = [5, 10, 15, 18, 23, 28, 31] as

Lydian[i] = V[(n-1 mod 7)+1] + 31 ceil(n/7) - 31

Then Ionian[i] = Lydian[i+4]-Lydian[4], Mixolydian[i] = Lydian[i+8]-Lydian[8], Dorian[i]=Lydian[i+12]-Lydian[12], Aeolian[i] = Lydian[i+16]-Lydian[16], Phrygian[i] = Lydian[i+20]-Lydian[20], Locrian[i] = Lydian[i+24]-Lydian[24]. Lydian is the bottom mode, and Locrian is the top mode; if we attempt to go higher than Locrian by considering Locrian[i+4]-Locrian[4] = Lydian[i+28]-Lydian[28] we jump down the stack of modes to Lydian again. The "up" generator is Lydian[4] = 18, or in other words the meantone fifth. The Lydian mode takes the bottommost note of the chain of generators, choosing the "up" generator, and stacking notes, reducing modulo octaves, to obtain the scale. Starting with F, we go up the chain of fifths and get F-C-G-D-A-E-B, with F the tonic note, or 1/1 of the scale. Lydian, the bottom mode, has 6 modes above it and none below it, so that it is denoted 6|0. It is obtained by stacking six fifths (the "up" generator) above the tonic, and none below it, which is another way to get to 6|0. Ionian, with fives modes above and one below, is 5|1; this can also be seen as starting from C and going five fifths up to B, or one down to F.

=The Chroma-Aligned Generator=

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If m shifts up, then U is such that mU shifts up to the top mode, and D is such that mD shifts down to the bottom mode; if m shifts down we reverse this so that -mD shifts up to the top mode and -mU to the bottom mode. If S is a periodic scale S such that the repetition interval O is some fraction 1/P of an octave, then the UDP notation for a given mode of a MOS is U|D(P). If P=1 we may omit it and just write U|D.

For example, consider the quasiperiodic function ~~with period 7~~ Lydian[i]:

For example, consider the quasiperiodic function Lydian[i] with period 7 whose argument is the top row and value the bottom row of the following table:

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</td>

<td>49

~~</td>~~

~~<td> ~~

</td>

<td>54

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</td>

<td>62

</td>

<td>67

</td>

</tr>

</table>

Extended to all the integers, this gives the Lydian mode of the diatonic scale, tuned to 31 equal. A formal closed-form definition can be given in terms of the list V[i] = [5, 10, 15, 18, 23, 28, 31] as

Lydian[i] = V[(n-1 mod 7)+1] + 31 ceil(n/7) - 31

Then Ionian[i] = Lydian[i+4]-Lydian[4], Mixolydian[i] = Lydian[i+8]-Lydian[8], Dorian[i]=Lydian[i+12]-Lydian[12], Aeolian[i] = Lydian[i+16]-Lydian[16], Phrygian[i] = Lydian[i+20]-Lydian[20], Locrian[i] = Lydian[i+24]-Lydian[24]. Lydian is the bottom mode, and Locrian is the top mode; if we attempt to go higher than Locrian by considering Locrian[i+4]-Locrian[4] = Lydian[i+28]-Lydian[28] we jump down the stack of modes to Lydian again. The &quot;up&quot; generator is Lydian[4] = 18, or in other words the meantone fifth. The Lydian mode takes the bottommost note of the chain of generators, choosing the &quot;up&quot; generator, and stacking notes, reducing modulo octaves, to obtain the scale. Starting with F, we go up the chain of fifths and get F-C-G-D-A-E-B, with F the tonic note, or 1/1 of the scale. Lydian, the bottom mode, has 6 modes above it and none below it, so that it is denoted 6|0. It is obtained by stacking six fifths (the &quot;up&quot; generator) above the tonic, and none below it, which is another way to get to 6|0. Ionian, with fives modes above and one below, is 5|1; this can also be seen as starting from C and going five fifths up to B, or one down to F.

<h1 id="toc2"><a name="The Chroma-Aligned Generator"></a>The Chroma-Aligned Generator</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-18 18:26:59 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-18 18:55:16 UTC</tt>.<br>
-: The original revision id was <tt>277107336</tt>.<br>
+: The original revision id was <tt>277111706</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>
@@ Line 20: / Line 20: @@
 If m shifts up, then U is such that mU shifts up to the top mode, and D is such that mD shifts down to the bottom mode; if m shifts down we reverse this so that -mD shifts up to the top mode and -mU to the bottom mode. If S is a periodic scale S such that the repetition interval **O** is some fraction 1/P of an octave, then the UDP notation for a given mode of a MOS is U|D(P). If P=1 we may omit it and just write U|D.
-For example, consider the quasiperiodic function with period 7 Lydian[i]:
+For example, consider the quasiperiodic function Lydian[i] with period 7 whose argument is the top row and value the bottom row of the following table:
 || -5 || -4 || -3 || -2 || -1 || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 ||
-|| -21 || -16 || -13 || -8 || -3 || 0 || 5 || 10 || 15 || 18 || 23 || 28 || 31 || 36 || 41 || 46 || 49 || || 54 || 59 || 62 ||
+|| -21 || -16 || -13 || -8 || -3 || 0 || 5 || 10 || 15 || 18 || 23 || 28 || 31 || 36 || 41 || 46 || 49 || 54 || 59 || 62 || 67 ||
+Extended to all the integers, this gives the Lydian mode of the diatonic scale, tuned to 31 equal. A formal closed-form definition can be given in terms of the list V[i] = [5, 10, 15, 18, 23, 28, 31] as
+Lydian[i] = V[(n-1 mod 7)+1] + 31 ceil(n/7) - 31
+Then Ionian[i] = Lydian[i+4]-Lydian[4], Mixolydian[i] = Lydian[i+8]-Lydian[8], Dorian[i]=Lydian[i+12]-Lydian[12], Aeolian[i] = Lydian[i+16]-Lydian[16], Phrygian[i] = Lydian[i+20]-Lydian[20], Locrian[i] = Lydian[i+24]-Lydian[24]. Lydian is the bottom mode, and Locrian is the top mode; if we attempt to go higher than Locrian by considering Locrian[i+4]-Locrian[4] = Lydian[i+28]-Lydian[28] we jump down the stack of modes to Lydian again. The "up" generator is Lydian[4] = 18, or in other words the meantone fifth. The Lydian mode takes the bottommost note of the chain of generators, choosing the "up" generator, and stacking notes, reducing modulo octaves, to obtain the scale. Starting with F, we go up the chain of fifths and get F-C-G-D-A-E-B, with F the tonic note, or 1/1 of the scale. Lydian, the bottom mode, has 6 modes above it and none below it, so that it is denoted 6|0. It is obtained by stacking six fifths (the "up" generator) above the tonic, and none below it, which is another way to get to 6|0. Ionian, with fives modes above and one below, is 5|1; this can also be seen as starting from C and going five fifths up to B, or one down to F.
 =The Chroma-Aligned Generator=
@@ Line 89: / Line 95: @@
 If m shifts up, then U is such that mU shifts up to the top mode, and D is such that mD shifts down to the bottom mode; if m shifts down we reverse this so that -mD shifts up to the top mode and -mU to the bottom mode. If S is a periodic scale S such that the repetition interval &lt;strong&gt;O&lt;/strong&gt; is some fraction 1/P of an octave, then the UDP notation for a given mode of a MOS is U|D(P). If P=1 we may omit it and just write U|D.&lt;br /&gt;
 &lt;br /&gt;
-For example, consider the quasiperiodic function with period 7 Lydian[i]:&lt;br /&gt;
+For example, consider the quasiperiodic function Lydian[i] with period 7 whose argument is the top row and value the bottom row of the following table:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 172: / Line 178: @@
 &lt;/td&gt;
          &lt;td&gt;49&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;54&lt;br /&gt;
@@ Line 180: / Line 184: @@
 &lt;/td&gt;
          &lt;td&gt;62&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;67&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
 &lt;/table&gt;
+&lt;br /&gt;
+Extended to all the integers, this gives the Lydian mode of the diatonic scale, tuned to 31 equal. A formal closed-form definition can be given in terms of the list V[i] = [5, 10, 15, 18, 23, 28, 31] as &lt;br /&gt;
+&lt;br /&gt;
+Lydian[i] = V[(n-1 mod 7)+1] + 31 ceil(n/7) - 31&lt;br /&gt;
+&lt;br /&gt;
+Then Ionian[i] = Lydian[i+4]-Lydian[4], Mixolydian[i] = Lydian[i+8]-Lydian[8], Dorian[i]=Lydian[i+12]-Lydian[12], Aeolian[i] = Lydian[i+16]-Lydian[16], Phrygian[i] = Lydian[i+20]-Lydian[20], Locrian[i] = Lydian[i+24]-Lydian[24]. Lydian is the bottom mode, and Locrian is the top mode; if we attempt to go higher than Locrian by considering Locrian[i+4]-Locrian[4] = Lydian[i+28]-Lydian[28] we jump down the stack of modes to Lydian again. The &amp;quot;up&amp;quot; generator is Lydian[4] = 18, or in other words the meantone fifth. The Lydian mode takes the bottommost note of the chain of generators, choosing the &amp;quot;up&amp;quot; generator, and stacking notes, reducing modulo octaves, to obtain the scale. Starting with F, we go up the chain of fifths and get F-C-G-D-A-E-B, with F the tonic note, or 1/1 of the scale. Lydian, the bottom mode, has 6 modes above it and none below it, so that it is denoted 6|0. It is obtained by stacking six fifths (the &amp;quot;up&amp;quot; generator) above the tonic, and none below it, which is another way to get to 6|0. Ionian, with fives modes above and one below, is 5|1; this can also be seen as starting from C and going five fifths up to B, or one down to F.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The Chroma-Aligned Generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The Chroma-Aligned Generator&lt;/h1&gt;