UDP: Difference between revisions
Wikispaces>genewardsmith **Imported revision 277120038 - Original comment: ** |
Wikispaces>guest **Imported revision 277532598 - 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:guest|guest]] and made on <tt>2011-11-20 22:14:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>277532598</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.// | ||
=Definition= | =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, 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.) | ||
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. | 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. | ||
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|| -21 || -16 || -13 || -8 || -3 || 0 || 5 || 10 || 15 || 18 || 23 || 28 || 31 || 36 || 41 || 46 || 49 || 54 || 59 || 62 || 67 || | || -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 | 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 | Lydian(i) = V[(n-1 mod 7)+1] + 31 ceil(n/7) - 31 | ||
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=Rationale= | =Rationale= | ||
While the | While the simplest interpretation of the modes is that they're only 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 "brightest" to "darkest," meaning "most sharps" or "most major" to "most flats" or "most minor." This is the same as arranging the modes by the position of their roots along the 7-note diatonic generator chain, where more generators "up" is chosen to be the "more major" direction, and more generators "down" is chosen to be the "more minor" direction. | ||
Within the first interpretation, meantone's Ionian and Dorian modes would be adjacent, since they begin on adjacent notes within the diatonic scale. But within the second interpretation, meantone's Ionian and Lydian modes would be adjacent, since their roots occupy adjacent positions along the generator chain. | Within the first interpretation, meantone's Ionian and Dorian modes would be adjacent, since they begin on adjacent notes within the diatonic scale. But within the second interpretation, meantone's Ionian and Lydian modes would be adjacent, since their roots occupy adjacent positions along the generator chain. | ||
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* Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2) | * Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2) | ||
* Pajara[10] Standard Pentachordal Major, ssLsssLsss: 4|4(2) #8, 6|2(2) b3 | * Pajara[10] Standard Pentachordal Major, ssLsssLsss: 4|4(2) #8, 6|2(2) b3 | ||
* Mavila/Mabila[9] Olympian, LLsLLLsLL 4|4 | |||
* Meantone[7]'s ionian mode is 5|1(1), abbreviated 5|1 for short. | * 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. | * 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> | <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>Modal UDP Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Modal UDP Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->Modal UDP 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>Modal UDP Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Modal UDP Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->Modal UDP Notation</h1> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->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, 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 /> | ||
<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 /> | 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 /> | ||
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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 /> | 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 /> | <br /> | ||
Lydian(i) = V[(n-1 mod 7)+1] + 31 ceil(n/7) - 31<br /> | Lydian(i) = V[(n-1 mod 7)+1] + 31 ceil(n/7) - 31<br /> | ||
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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 /> | 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 /> | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:150:&lt;img src=&quot;/file/view/modes.jpg/277118632/modes.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/modes.jpg/277118632/modes.jpg" alt="modes.jpg" title="modes.jpg" /><!-- ws:end:WikiTextLocalImageRule:150 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The Chroma-Aligned Generator"></a><!-- ws:end:WikiTextHeadingRule:4 -->The Chroma-Aligned Generator</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The Chroma-Aligned Generator"></a><!-- ws:end:WikiTextHeadingRule:4 -->The Chroma-Aligned Generator</h1> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Rationale"></a><!-- ws:end:WikiTextHeadingRule:8 -->Rationale</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Rationale"></a><!-- ws:end:WikiTextHeadingRule:8 -->Rationale</h1> | ||
While the | While the simplest interpretation of the modes is that they're only 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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Within the first interpretation, meantone's Ionian and Dorian modes would be adjacent, since they begin on adjacent notes within the diatonic scale. But within the second interpretation, meantone's Ionian and Lydian modes would be adjacent, since their roots occupy adjacent positions along the generator chain.<br /> | Within the first interpretation, meantone's Ionian and Dorian modes would be adjacent, since they begin on adjacent notes within the diatonic scale. But within the second interpretation, meantone's Ionian and Lydian modes would be adjacent, since their roots occupy adjacent positions along the generator chain.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:10 -->Examples</h1> | <!-- 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><br /> | <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><li>Mavila/Mabila[9] Olympian, LLsLLLsLL 4|4</li></ul><br /> | ||
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<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></body></html></pre></div> | <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></body></html></pre></div> | ||