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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | #redirect [[Modal UDP notation]] |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-21 15:34:42 UTC</tt>.<br>
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| : The original revision id was <tt>277836118</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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;white-space: pre-wrap ! important" class="old-revision-html">=Modal UDP Notation=
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| Modal UDP notation is a way to uniquely specify a particular mode of any MOS. Its name is derived from up|down(period), or U|D(P), which is how the notation is defined. If accidentals are specified, it can also refer to the MODMOS's of those MOS's as well. | |
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| UDP notation is defined in such a way that it simultaneously describes the following properties of the mode in question:
| | [[Category:Acronyms]] |
| # How many scale degrees are of the "larger" or "major" variant, vs the "smaller" or "minor" variant.
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| # How many generators up vs down it requires to generate the mode.
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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.//
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| =Definition=
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| A [[periodic scale]] S associates an interval S(i) to every integer i, such that there is a period (strictly, a quasiperiod) Q>0 and an interval of repetition R such that S(i+Q) = S(i)+R. Q is chosen so as to be minimal; there is no smaller period. S is monotone if i<j implies that S(i)<S(j).
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| Given a monotone periodic scale S, suppose it is also a [[MOSScale|MOS]] or DE scale. Let the generator S(m) = g be such that g >= S(i+m)-S(i) for all i. If Q is the period of S, let u be the largest integer such that 0<=u<Q and S(m*u) = g*u, and d the largest integer such that 0<=d<Q and S(-m*d)=-g*d. If S(P*Q) = octave, so that P is the number of periods to an octave, let U = P*u and D = P*d. Then the UDP notation for the given mode is is U|D(P). If P=1 we may omit it and just write U|D.
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| For example, consider the quasiperiodic function Ionian(i) = V[(i+3 mod 7)+1] + 31ceil((n+4)/7)-49, where V = [5, 10, 15, 18, 23, 28, 31]. This has period 7, and Ionian(7) = 31, where the tuning is [[31edo]] so that 31 represents an octave. Going up from 0, it has values 0, 5, 10, 13, 18, 23, 28, 31, 36, 41 ... corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ..., and going down from 0, it gives 0, -3, -8, -13 ... corresponding to 0, -1, -2, -3 .... This gives the Ionian, or major, mode of the diatonic scale. Then Ionian(4)=18, the fifth, and 18 >= Ionian(i+4)-Ionian(i) for all i. We have Ionian(4)=18, Ionian(8)=36, Ionian(12)=54, Ionian(16)=72 and Ionian(20)=90. However, Ionian(4*6) = Ionian(24) is 106, which is less than 6*18 = 108. Hence the largest value for which Ionian(4*u) and 18*u are equal is u=5. Similarly, Ionian(-4)=-18, but Ionian(-8) is -34, not -36, and so d=1. Since Ionian(7)=31, which is the octave, P=1, so U=u=5, D=d=1, and the UDP notation for Ionian is 6|1(1), or simply ^|1.
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| =The Chroma-Aligned Generator=
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| 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 "up" direction, and how many you stack in the "down" direction.
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| Specifying a number of generators "up" or "down" without defining what is up and what is down is insufficient by itself to specify a mode, because even if we only consider generators which fit within the interval of repetition R, every MOS could have one of two generators: for example, Meantone[7]'s generator could be viewed as either the perfect fourth or the perfect fifth. Moreover, P, the number of periods in an octave, can be greater than one. In this case "u" and "d" denote total number of generators up and down per interval of repetition, and "U" and "D" per octave. For example, Paul Erlich's "Static Symmetrical Major" scale is 2 2 3 2 2 2 2 3 2 2; this is notated 4|4(2). UDP notation has the handy property that U+D+P = the total number of notes per octave. For example, 4+4+2 = 10, which tells us that Static Symmetrical Major is a decatonic scale, whereas 5+1+1 = 7, reflecting the fact that the diatonic scale has seven notes.
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| The **chroma-aligned generator** generator is the one which is the larger specific interval in its generic interval class in the MOS, or in other words the generator g such that g >= S(i+m)-S(i) for all i. The //chroma// of a MOS is c = m*R - L*g, where S(Q) = R and g is chroma-aligned. The chroma is positive, and may also be defined by c = L-s, where L is the larger of the two step sizes of the MOS, and s is the smaller.
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| The use of this generator means that more iterations "up" will equal more "sharpened" or "major-sized" intervals (which contain the root), so that U|D(P) doesn't just specify the number of generators up vs down, but simultaneously specifies the number of major vs minor scale degrees as well.
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| The generator can quickly be computed for some MOS scale pattern such as xLys by calculating g = (1/y mod (x+y)) + 1. This will tell you what scale degree in the scale is the desired generator.
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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.
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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.
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| =Rationale=
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| 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.
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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.
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| Ionian and Mixolydian would similarly be adjacent.
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| Movement to an adjacent mode "up" in this paradigm means a single interval will become sharpened, and moving "down" means that one will become flattened. For example, the movement "up" from Ionian to Lydian sharpens the 4th scale degree, and the movement "down" from Ionian to Mixolydian flattens the 7th.
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| This interpretation is what UDP notation generalizes.
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| =Examples=
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| * Meantone[7] Ionian, LLsLLLs: 5|1
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| * Meantone[7] Aeolian, LsLLsLL: 2|4
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| * Mavila[7] Anti-Ionian, ssLsssL: 1|5
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| * Mavila[7] Anti-Aeolian, Herman Miller's sLssLss mode: 4|2
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| * Porcupine[7] Lssssss: 6|0
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| * Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6: 6|0 b7
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| * Porcupine[7] sssLsss: 3|3
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| * Diminished[8] sLsLsLsL 0|4(4)
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| * Diminished[8] LsLsLsLs 4|0(4)
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| * Triforce[9] LLsLLsLLs: 6|0(3)
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| * Meantone[5] minor pentatonic, LssLs: 3|1
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| * Meantone[5] major pentatonic, ssLsL: 0|4
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| * Sensi[11] LLsLLLsLLLs: 8|2
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| * Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2)
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| * Pajara[10] Standard Pentachordal Major, ssLsssLsss: 4|4(2) #8, 6|2(2) b3
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| * Mavila/Mabila[9] Olympian, LLsLLLsLL 4|4
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| * Meantone[7]'s ionian mode is 5|1(1), abbreviated 5|1 for short.
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| * 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.
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| * Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.</pre></div>
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| <h4>Original HTML content:</h4>
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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><!-- 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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| Modal UDP notation is a way to uniquely specify a particular mode of any MOS. Its name is derived from up|down(period), or U|D(P), which is how the notation is defined. If accidentals are specified, it can also refer to the MODMOS's of those MOS's as well.<br />
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| <br />
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| UDP notation is defined in such a way that it simultaneously describes the following properties of the mode in question:<br />
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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 />
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| 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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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition</h1>
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| A <a class="wiki_link" href="/periodic%20scale">periodic scale</a> S associates an interval S(i) to every integer i, such that there is a period (strictly, a quasiperiod) Q&gt;0 and an interval of repetition R such that S(i+Q) = S(i)+R. Q is chosen so as to be minimal; there is no smaller period. S is monotone if i&lt;j implies that S(i)&lt;S(j). <br />
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| <br />
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| Given a monotone periodic scale S, suppose it is also a <a class="wiki_link" href="/MOSScale">MOS</a> or DE scale. Let the generator S(m) = g be such that g &gt;= S(i+m)-S(i) for all i. If Q is the period of S, let u be the largest integer such that 0&lt;=u&lt;Q and S(m*u) = g*u, and d the largest integer such that 0&lt;=d&lt;Q and S(-m*d)=-g*d. If S(P*Q) = octave, so that P is the number of periods to an octave, let U = P*u and D = P*d. Then the UDP notation for the given mode is is U|D(P). If P=1 we may omit it and just write U|D.<br />
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| <br />
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| For example, consider the quasiperiodic function Ionian(i) = V[(i+3 mod 7)+1] + 31ceil((n+4)/7)-49, where V = [5, 10, 15, 18, 23, 28, 31]. This has period 7, and Ionian(7) = 31, where the tuning is <a class="wiki_link" href="/31edo">31edo</a> so that 31 represents an octave. Going up from 0, it has values 0, 5, 10, 13, 18, 23, 28, 31, 36, 41 ... corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ..., and going down from 0, it gives 0, -3, -8, -13 ... corresponding to 0, -1, -2, -3 .... This gives the Ionian, or major, mode of the diatonic scale. Then Ionian(4)=18, the fifth, and 18 &gt;= Ionian(i+4)-Ionian(i) for all i. We have Ionian(4)=18, Ionian(8)=36, Ionian(12)=54, Ionian(16)=72 and Ionian(20)=90. However, Ionian(4*6) = Ionian(24) is 106, which is less than 6*18 = 108. Hence the largest value for which Ionian(4*u) and 18*u are equal is u=5. Similarly, Ionian(-4)=-18, but Ionian(-8) is -34, not -36, and so d=1. Since Ionian(7)=31, which is the octave, P=1, so U=u=5, D=d=1, and the UDP notation for Ionian is 6|1(1), or simply ^|1.<br />
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| <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>
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| 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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| <br />
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| Specifying a number of generators &quot;up&quot; or &quot;down&quot; without defining what is up and what is down is insufficient by itself to specify a mode, because even if we only consider generators which fit within the interval of repetition R, every MOS could have one of two generators: for example, Meantone[7]'s generator could be viewed as either the perfect fourth or the perfect fifth. Moreover, P, the number of periods in an octave, can be greater than one. In this case &quot;u&quot; and &quot;d&quot; denote total number of generators up and down per interval of repetition, and &quot;U&quot; and &quot;D&quot; per octave. For example, Paul Erlich's &quot;Static Symmetrical Major&quot; scale is 2 2 3 2 2 2 2 3 2 2; this is notated 4|4(2). UDP notation has the handy property that U+D+P = the total number of notes per octave. For example, 4+4+2 = 10, which tells us that Static Symmetrical Major is a decatonic scale, whereas 5+1+1 = 7, reflecting the fact that the diatonic scale has seven notes.<br />
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| <br />
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| The <strong>chroma-aligned generator</strong> generator is the one which is the larger specific interval in its generic interval class in the MOS, or in other words the generator g such that g &gt;= S(i+m)-S(i) for all i. The <em>chroma</em> of a MOS is c = m*R - L*g, where S(Q) = R and g is chroma-aligned. The chroma is positive, and may also be defined by c = L-s, where L is the larger of the two step sizes of the MOS, and s is the smaller.<br />
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| <br />
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| The use of this generator means that more iterations &quot;up&quot; will equal more &quot;sharpened&quot; or &quot;major-sized&quot; intervals (which contain the root), so that U|D(P) doesn't just specify the number of generators up vs down, but simultaneously specifies the number of major vs minor scale degrees as well.<br />
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| <br />
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| The generator can quickly be computed for some MOS scale pattern such as xLys by calculating g = (1/y mod (x+y)) + 1. This will tell you what scale degree in the scale is the desired generator.<br />
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| <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 />
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| <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 />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Rationale"></a><!-- ws:end:WikiTextHeadingRule:6 -->Rationale</h1>
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| 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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| <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 />
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| Ionian and Mixolydian would similarly be adjacent.<br />
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| <br />
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| Movement to an adjacent mode &quot;up&quot; in this paradigm means a single interval will become sharpened, and moving &quot;down&quot; means that one will become flattened. For example, the movement &quot;up&quot; from Ionian to Lydian sharpens the 4th scale degree, and the movement &quot;down&quot; from Ionian to Mixolydian flattens the 7th.<br />
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| <br />
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| This interpretation is what UDP notation generalizes.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:8 -->Examples</h1>
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| <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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| <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>
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