UDP: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 275514280 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 275868640 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-15 19:20:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>275868640</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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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.// | 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.// | ||
=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 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 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 -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. | |||
As an example | As an example | ||
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<ol><li>How many scale degrees are of the &quot;larger&quot; or &quot;major&quot; variant, vs the &quot;smaller&quot; or &quot;minor&quot; variant.</li><li>How many generators up vs down it requires to generate the mode.</li></ol><br /> | <ol><li>How many scale degrees are of the &quot;larger&quot; or &quot;major&quot; variant, vs the &quot;smaller&quot; or &quot;minor&quot; variant.</li><li>How many generators up vs down it requires to generate the mode.</li></ol><br /> | ||
The generator is chosen so that more generators &quot;up&quot; also equals more &quot;major&quot; 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 <em>chroma-aligned.</em><br /> | The generator is chosen so that more generators &quot;up&quot; also equals more &quot;major&quot; 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 <em>chroma-aligned.</em><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Mathematical definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Mathematical 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] &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.)<br /> | |||
<br /> | |||
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 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 -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 <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 /> | |||
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As an example<br /> | As an example<br /> | ||
<ul><li>Meantone[7]'s ionian mode is 5|1(1), abbreviated 5|1 for short.</li><li>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.</li><li>Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.</li></ul><br /> | <ul><li>Meantone[7]'s ionian mode is 5|1(1), abbreviated 5|1 for short.</li><li>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.</li><li>Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.</li></ul><br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:4 -->Definition</h1> | ||
The UDP notation for any mode is U|D(P), where &quot;u&quot; specifies the number of chroma-aligned generators &quot;up,&quot; d specifies the number of chroma-aligned generators &quot;down,&quot; and p specifies the number of periods per equivalence interval. The chroma-aligned generator is the one such that more generators &quot;up&quot; also means more &quot;major&quot; scale degrees, or more generally, more &quot;large&quot; intervals that contain the root of the scale.<br /> | The UDP notation for any mode is U|D(P), where &quot;u&quot; specifies the number of chroma-aligned generators &quot;up,&quot; d specifies the number of chroma-aligned generators &quot;down,&quot; and p specifies the number of periods per equivalence interval. The chroma-aligned generator is the one such that more generators &quot;up&quot; also means more &quot;major&quot; scale degrees, or more generally, more &quot;large&quot; intervals that contain the root of the scale.<br /> | ||
<br /> | <br /> | ||
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Accidentals can follow the UDP string so as to specify a MODMOS. For example, in meantone[7], 5|1 b6 specifies harmonic major, and in porcupine[7], 6|0 b4 #7 specifies the 5-limit JI major scale.<br /> | Accidentals can follow the UDP string so as to specify a MODMOS. For example, in meantone[7], 5|1 b6 specifies harmonic major, and in porcupine[7], 6|0 b4 #7 specifies the 5-limit JI major scale.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="The Chroma-Aligned Generator"></a><!-- ws:end:WikiTextHeadingRule:6 -->The Chroma-Aligned Generator</h1> | ||
In general, any MOS scale is formed by stacking repeated instances of a generator on top of itself until one arrives at the desired MOS. The modes of this MOS will differ in how many generators you stack in the &quot;up&quot; direction, and how many you stack in the &quot;down&quot; direction.<br /> | In general, any MOS scale is formed by stacking repeated instances of a generator on top of itself until one arrives at the desired MOS. The modes of this MOS will differ in how many generators you stack in the &quot;up&quot; direction, and how many you stack in the &quot;down&quot; direction.<br /> | ||
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It should be noted that the chroma-aligned generator will change depending on which MOS of the temperament you're working within. For example, the chroma-aligned generator for mavila[7] is the 4/3, but for mavila[9] it's the 3/2.<br /> | It should be noted that the chroma-aligned generator will change depending on which MOS of the temperament you're working within. For example, the chroma-aligned generator for mavila[7] is the 4/3, but for mavila[9] it's the 3/2.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Rationale"></a><!-- ws:end:WikiTextHeadingRule:8 -->Rationale</h1> | ||
While the naive interpretation of the modes is that they're simply cyclic permutations of one another, a more advanced interpretation often taught in schools where modal theory is prominent (e.g. Jazz performance programs) is to understand them as varying on a continuum from &quot;brightest&quot; to &quot;darkest,&quot; meaning &quot;most sharps&quot; or &quot;most major&quot; to &quot;most flats&quot; or &quot;most minor.&quot; This is the same as arranging the modes by the position of their roots along the 7-note diatonic generator chain, where more generators &quot;up&quot; is chosen to be the &quot;more major&quot; direction, and more generators &quot;down&quot; is chosen to be the &quot;more minor&quot; direction.<br /> | While the naive interpretation of the modes is that they're simply cyclic permutations of one another, a more advanced interpretation often taught in schools where modal theory is prominent (e.g. Jazz performance programs) is to understand them as varying on a continuum from &quot;brightest&quot; to &quot;darkest,&quot; meaning &quot;most sharps&quot; or &quot;most major&quot; to &quot;most flats&quot; or &quot;most minor.&quot; This is the same as arranging the modes by the position of their roots along the 7-note diatonic generator chain, where more generators &quot;up&quot; is chosen to be the &quot;more major&quot; direction, and more generators &quot;down&quot; is chosen to be the &quot;more minor&quot; direction.<br /> | ||
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This interpretation is what UDP notation generalizes.<br /> | This interpretation is what UDP notation generalizes.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:10 -->Examples</h1> | ||
<ul><li>Meantone[7] Ionian, LLsLLLs: 5|1</li><li>Meantone[7] Aeolian, LsLLsLL: 2|4</li><li>Mavila[7] Anti-Ionian, ssLsssL: 1|5</li><li>Mavila[7] Anti-Aeolian, Herman Miller's sLssLss mode: 4|2</li><li>Porcupine[7] Lssssss: 6|0</li><li>Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6: 6|0 b7</li><li>Porcupine[7] sssLsss: 3|3</li><li>Diminished[8] sLsLsLsL 0|4(4)</li><li>Diminished[8] LsLsLsLs 4|0(4)</li><li>Triforce[9] LLsLLsLLs: 6|0(3)</li><li>Meantone[5] minor pentatonic, LssLs: 3|1</li><li>Meantone[5] major pentatonic, ssLsL: 0|4</li><li>Sensi[11] LLsLLLsLLLs: 8|2</li><li>Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2)</li><li>Pajara[10] Standard Pentachordal Major, ssLsssLsss: 4|4(2) #8, 6|2(2) b3</li></ul></body></html></pre></div> | <ul><li>Meantone[7] Ionian, LLsLLLs: 5|1</li><li>Meantone[7] Aeolian, LsLLsLL: 2|4</li><li>Mavila[7] Anti-Ionian, ssLsssL: 1|5</li><li>Mavila[7] Anti-Aeolian, Herman Miller's sLssLss mode: 4|2</li><li>Porcupine[7] Lssssss: 6|0</li><li>Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6: 6|0 b7</li><li>Porcupine[7] sssLsss: 3|3</li><li>Diminished[8] sLsLsLsL 0|4(4)</li><li>Diminished[8] LsLsLsLs 4|0(4)</li><li>Triforce[9] LLsLLsLLs: 6|0(3)</li><li>Meantone[5] minor pentatonic, LssLs: 3|1</li><li>Meantone[5] major pentatonic, ssLsL: 0|4</li><li>Sensi[11] LLsLLLsLLLs: 8|2</li><li>Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2)</li><li>Pajara[10] Standard Pentachordal Major, ssLsssLsss: 4|4(2) #8, 6|2(2) b3</li></ul></body></html></pre></div> | ||