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:<br>

: This revision was by author [[User:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2011-11-18 19:43:52 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-11-18 19:46:31 UTC</tt>.<br>

: The original revision id was <tt>~~277118666~~</tt>.<br>

: The original revision id was <tt>277119058</tt>.<br>

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

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The generator is chosen so that more generators "up" also equals more "major" scale degrees, so that the two are in harmony. This also means that the UDP generator has to point in the same direction on the lattice as the chroma, or is //chroma-aligned.//

=Definition=

Given a [[periodic scale]] S, a //~~modal shift~~// ~~by n may~~ be ~~defined as~~ S'[i] = ~~S[i+n]-S[n]. A modal shift is a~~ //~~shift up~~// ~~if S'[i] <= S[i] for all i. This definition applies to the case which especially concerns us, where~~ S is ~~a monotonically strictly increasing periodic scale defined by a MOS. Given a~~ MOS, ~~there will be both a top mode, which is the farthest mode up, and a bottom mode, which is the farthest down. So long as the shifting takes place between the bottom and top mode, depending on~~ the ~~choice of~~ generator ~~g, shifts up will occur either when n is positive (if m such that~~ S[m]=g ~~shifts up) or negative (if it shifts down~~.)

Given a [[periodic scale]] S with period S[p], let the //tonic// be S[0] = 0//.// If S is MOS, then let the generator S[m] = g such that g is the larger specific interval in its generic interval class.

~~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 any such S, there are two associated values u and d, called the number of generators located "up" and "down" from the tonic. Let u be the maximum integer such that S[m*u] = g*u, and 0 <= u < p. Likewise, let d be the maximum integer such that S[-m*d] = -g*d, and 0 <= d < p.

Let U = uP, and D = dP. The UDP notation for a given mode of an MOS is U|D(P). If P=1 we may omit it and just write U|D.

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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|| -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 || 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.

~~[[image:modes.jpg]]~~

=The Chroma-Aligned Generator=

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* Meantone[7]'s ionian mode is 5|1(1), abbreviated 5|1 for short.

* Melodic minor is 5|1(1) b3, abbreviated 5|1 b3 for short, but could also be 3|3(1) #7, abbreviated 3|3 #7 for short.

* Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.

* Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.</pre></div>

</pre></div>

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<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>Modal UDP Notation</title></head><body><h1 id="toc0"><a name="Modal UDP Notation"></a>Modal UDP Notation</h1>

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<br />

<h1 id="toc1"><a name="Definition"></a>Definition</h1>

Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> S, a <em>~~modal shift~~</em> ~~by n may~~ be ~~defined as~~ S'[i] = ~~S[i+n]-S[n]. A modal shift is a~~ <em>~~shift up~~</em> ~~if S'[i] &lt;= S[i] for all i. This definition applies to the case which especially concerns us, where~~ S is ~~a monotonically strictly increasing periodic scale defined by a MOS. Given a~~ MOS, ~~there will be both a top mode, which is the farthest mode up, and a bottom mode, which is the farthest down. So long as~~ the ~~shifting takes place between the bottom and top mode, depending on the choice of~~ generator ~~g, shifts up will occur either when n is positive (if m such that~~ S[m]=g ~~shifts up) or negative (if it shifts down~~.) <br />

Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> S with period S[p], let the <em>tonic</em> be S[0] = 0<em>.</em> If S is MOS, then let the generator S[m] = g such that g is the larger specific interval in its generic interval class.<br />

<br />

~~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~~ <~~strong~~>O</~~strong~~> ~~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.<br />

For any such S, there are two associated values u and d, called the number of generators located &quot;up&quot; and &quot;down&quot; from the tonic. Let u be the maximum integer such that S[m*u] = g*u, and 0 &lt;= u &lt; p. Likewise, let d be the maximum integer such that S[-m*d] = -g*d, and 0 &lt;= d &lt; p. <br />

<br />

Let U = uP, and D = dP. The UDP notation for a given mode of an MOS is U|D(P). If P=1 we may omit it and just write U|D.<br />

<br />

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:<br />

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</table>

~~<br />~~

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 <br />

~~<br />~~

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

~~<br />~~

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.<br />

~~<br />~~

<img src="/file/view/modes.jpg/277118632/modes.jpg" alt="modes.jpg" title="modes.jpg" /><br />

<br />

<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 19:43:52 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-11-18 19:46:31 UTC</tt>.<br>
-: The original revision id was <tt>277118666</tt>.<br>
+: The original revision id was <tt>277119058</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 15: / Line 15: @@
 The generator is chosen so that more generators "up" also equals more "major" scale degrees, so that the two are in harmony. This also means that the UDP generator has to point in the same direction on the lattice as the chroma, or is //chroma-aligned.//
 =Definition=
-Given a [[periodic scale]] S, a //modal shift// by n may be defined as S'[i] = S[i+n]-S[n]. A modal shift is a //shift up// if S'[i] &lt;= S[i] for all i. This definition applies to the case which especially concerns us, where S is a monotonically strictly increasing periodic scale defined by a MOS. Given a MOS, there will be both a top mode, which is the farthest mode up, and a bottom mode, which is the farthest down. So long as the shifting takes place between the bottom and top mode, depending on the choice of generator g, shifts up will occur either when n is positive (if m such that S[m]=g shifts up) or negative (if it shifts down.)
+Given a [[periodic scale]] S with period S[p], let the //tonic// be S[0] = 0//.// If S is MOS, then let the generator S[m] = g such that g is the larger specific interval in its generic interval class.
-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 any such S, there are two associated values u and d, called the number of generators located "up" and "down" from the tonic. Let u be the maximum integer such that S[m*u] = g*u, and 0 &lt;= u &lt; p. Likewise, let d be the maximum integer such that S[-m*d] = -g*d, and 0 &lt;= d &lt; p.
+Let U = uP, and D = dP. The UDP notation for a given mode of an MOS is U|D(P). If P=1 we may omit it and just write U|D.
 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:
@@ Line 24: / Line 26: @@
 || -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 || 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.
-[[image:modes.jpg]]
 =The Chroma-Aligned Generator=
@@ Line 82: / Line 76: @@
 * Meantone[7]'s ionian mode is 5|1(1), abbreviated 5|1 for short.
 * Melodic minor is 5|1(1) b3, abbreviated 5|1 b3 for short, but could also be 3|3(1) #7, abbreviated 3|3 #7 for short.
-* Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.
+* Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.</pre></div>
-</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Modal UDP Notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Modal UDP Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Modal UDP Notation&lt;/h1&gt;
@@ Line 93: / Line 86: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Definition&lt;/h1&gt;
-Given a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; S, a &lt;em&gt;modal shift&lt;/em&gt; by n may be defined as S'[i] = S[i+n]-S[n]. A modal shift is a &lt;em&gt;shift up&lt;/em&gt; if S'[i] &amp;lt;= S[i] for all i. This definition applies to the case which especially concerns us, where S is a monotonically strictly increasing periodic scale defined by a MOS. Given a MOS, there will be both a top mode, which is the farthest mode up, and a bottom mode, which is the farthest down. So long as the shifting takes place between the bottom and top mode, depending on the choice of generator g, shifts up will occur either when n is positive (if m such that S[m]=g shifts up) or negative (if it shifts down.) &lt;br /&gt;
+ Given a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; S with period S[p], let the &lt;em&gt;tonic&lt;/em&gt; be S[0] = 0&lt;em&gt;.&lt;/em&gt; If S is MOS, then let the generator S[m] = g such that g is the larger specific interval in its generic interval class.&lt;br /&gt;
 &lt;br /&gt;
-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;
+For any such S, there are two associated values u and d, called the number of generators located &amp;quot;up&amp;quot; and &amp;quot;down&amp;quot; from the tonic. Let u be the maximum integer such that S[m*u] = g*u, and 0 &amp;lt;= u &amp;lt; p. Likewise, let d be the maximum integer such that S[-m*d] = -g*d, and 0 &amp;lt;= d &amp;lt; p. &lt;br /&gt;
+&lt;br /&gt;
+Let U = uP, and D = dP. The UDP notation for a given mode of an 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 Lydian[i] with period 7 whose argument is the top row and value the bottom row of the following table:&lt;br /&gt;
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 &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:WikiTextLocalImageRule:148:&amp;lt;img src=&amp;quot;/file/view/modes.jpg/277118632/modes.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/modes.jpg/277118632/modes.jpg" alt="modes.jpg" title="modes.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:148 --&gt;&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;