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-~~15 21~~:54:59 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-16 00:00:24 UTC</tt>.

: The original revision id was <tt>~~275917892~~</tt>.

: The original revision id was <tt>275947548</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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=Mathematical 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. ~~In this case~~, 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, 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 MOS, there will be both a top mode, which is the farthest mode up, and a bottom mode, which is the farthest down.~~ If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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.

If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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.

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<h1 id="toc1"><a name="Mathematical definition"></a>Mathematical definition</h1>

Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> 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] &gt;= 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. ~~In this case~~, 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 <a class="wiki_link" href="/periodic%20scale">periodic scale</a> 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] &gt;= 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 MOS, there will be both a top mode, which is the farthest mode up, and a bottom mode, which is the farthest down.~~ If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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.

If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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.

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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>2011-11-15 21:54:59 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-16 00:00:24 UTC</tt>.<br>
-: The original revision id was <tt>275917892</tt>.<br>
+: The original revision id was <tt>275947548</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 16: / Line 16: @@
 =Mathematical 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] &gt;= 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. In this case, 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, 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] &gt;= 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 MOS, there will be both a top mode, which is the farthest mode up, and a bottom mode, which is the farthest down. If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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.
+If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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.
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Mathematical definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Mathematical 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;gt;= 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. In this case, 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, 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;gt;= 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. &lt;br /&gt;
+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;
 &lt;br /&gt;
-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. If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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;
+If m shifts up, then D is such that mD shifts up to the top mode, and U is such that mU shifts down to the bottom mode; if m shifts down we reverse this so that -mU shifts up to the top mode and -mD 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;
 &lt;br /&gt;