UDP
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=Modal UDP Notation= 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. UDP notation is defined in such a way that it simultaneously describes the following properties of the mode in question: # How many scale degrees are of the "larger" or "major" variant, vs the "smaller" or "minor" variant. # How many generators up vs down from the tonic it requires to generate the mode. The generator is chosen so that more generators "up" also equals more "major" scale degrees, so that the two are in harmony. This generator can easily be chosen by picking the one which is the larger specific interval in its generic interval class. For example, the proper generator for meantone[7] is the perfect fifth, because it's larger than the other specific interval it shares a class with, the diminished fifth. This generator is called the //chroma-positive generator.// It should be noted that the chroma-positive generator will vary from MOS to MOS even within the same temperament. For example, the chroma-positive generator for meantone[7] is the ~3/2, but is the ~4/3 for meantone[5]. The UDP notation for any mode is then U|D(P), where P is the number of periods per equivalence interval, U is the number of chroma-positive generators going up from the tonic times P, and D is the number of chroma-positive generators going down from the tonic times P. If P = 1, then it can be omitted, so that the UDP notation is simply U|D. For example, meantone[7]'s Ionian mode is 5|1(1), which is 5|1 for short. Porcupine's Lssssss mode is 6|0, and sssLsss is 3|3. Mavila[7]'s ssLsssL anti-Ionian is 1|5, and Mavila[9]'s LLsLLLsLL "Olympian" mode is 4|4. We can add accidentals as well, so that meantone's harmonic minor is 2|4 #7, 2|4 being the UDP for Aeolian mode. =Mathematical Definition= 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). 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. 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 6|1. =Rationale= While the simplest interpretation of the modes is that they're only cyclic permutations of one another, a more advanced interpretation often utilized in musical styles where modal theory is prominent is to understand them as varying on a continuum from "brightest" to "darkest," meaning "most sharps" to "most flats" or "most major" to "most minor." This is the same as arranging the modes in descending order by the number of chroma-positive generators which go up from the tonic. Within the first interpretation, the diatonic scale's Ionian and Dorian modes would be adjacent, since they begin on adjacent notes within the scale. But within the second interpretation, Ionian and Lydian modes would be adjacent, since they occupy adjacent positions along the brightness-darkness continuum. Ionian and Mixolydian would similarly be adjacent. 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. This interpretation is what UDP notation generalizes. =Examples= * Meantone[7] Ionian, LLsLLLs: 5|1 * Meantone[7] Aeolian, LsLLsLL: 2|4 * Mavila[7] Anti-Ionian, ssLsssL: 1|5 * Mavila[7] Anti-Aeolian, Herman Miller's sLssLss mode: 4|2 * Porcupine[7] Lssssss: 6|0 * Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6: 6|0 b7 * Porcupine[7] sssLsss: 3|3 * Diminished[8] sLsLsLsL 0|4(4) * Diminished[8] LsLsLsLs 4|0(4) * Triforce[9] LLsLLsLLs: 6|0(3) * Meantone[5] minor pentatonic, LssLs: 3|1 * Meantone[5] major pentatonic, ssLsL: 0|4 * Sensi[11] LLsLLLsLLLs: 8|2 * Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2) * 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. * 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. =Diagram= [[image:xenharmonic/UDPTable.gif width="800" height="525"]]
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<html><head><title>Modal UDP Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Modal UDP Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->Modal UDP Notation</h1> 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 /> <br /> UDP notation is defined in such a way that it simultaneously describes the following properties of the mode in question:<br /> <ol><li>How many scale degrees are of the "larger" or "major" variant, vs the "smaller" or "minor" variant.</li><li>How many generators up vs down from the tonic it requires to generate the mode.</li></ol><br /> The generator is chosen so that more generators "up" also equals more "major" scale degrees, so that the two are in harmony. This generator can easily be chosen by picking the one which is the larger specific interval in its generic interval class. For example, the proper generator for meantone[7] is the perfect fifth, because it's larger than the other specific interval it shares a class with, the diminished fifth. This generator is called the <em>chroma-positive generator.</em> It should be noted that the chroma-positive generator will vary from MOS to MOS even within the same temperament. For example, the chroma-positive generator for meantone[7] is the ~3/2, but is the ~4/3 for meantone[5].<br /> <br /> The UDP notation for any mode is then U|D(P), where P is the number of periods per equivalence interval, U is the number of chroma-positive generators going up from the tonic times P, and D is the number of chroma-positive generators going down from the tonic times P. If P = 1, then it can be omitted, so that the UDP notation is simply U|D.<br /> <br /> For example, meantone[7]'s Ionian mode is 5|1(1), which is 5|1 for short. Porcupine's Lssssss mode is 6|0, and sssLsss is 3|3. Mavila[7]'s ssLsssL anti-Ionian is 1|5, and Mavila[9]'s LLsLLLsLL "Olympian" mode is 4|4. We can add accidentals as well, so that meantone's harmonic minor is 2|4 #7, 2|4 being the UDP for Aeolian mode.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Mathematical Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Mathematical Definition</h1> 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>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).<br /> <br /> 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 >= 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.<br /> <br /> 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 >= 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 6|1.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Rationale"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rationale</h1> While the simplest interpretation of the modes is that they're only cyclic permutations of one another, a more advanced interpretation often utilized in musical styles where modal theory is prominent is to understand them as varying on a continuum from "brightest" to "darkest," meaning "most sharps" to "most flats" or "most major" to "most minor." This is the same as arranging the modes in descending order by the number of chroma-positive generators which go up from the tonic.<br /> <br /> Within the first interpretation, the diatonic scale's Ionian and Dorian modes would be adjacent, since they begin on adjacent notes within the scale. But within the second interpretation, Ionian and Lydian modes would be adjacent, since they occupy adjacent positions along the brightness-darkness continuum. Ionian and Mixolydian would similarly be adjacent.<br /> <br /> 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.<br /> <br /> This interpretation is what UDP notation generalizes.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:6 -->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><li>Mavila/Mabila[9] Olympian, LLsLLLsLL 4|4<br /> <br /> <br /> </li><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:8:<h1> --><h1 id="toc4"><a name="Diagram"></a><!-- ws:end:WikiTextHeadingRule:8 -->Diagram</h1> <!-- ws:start:WikiTextLocalImageRule:60:<img src="http://xenharmonic.wikispaces.com/file/view/UDPTable.gif/277923180/800x525/UDPTable.gif" alt="" title="" style="height: 525px; width: 800px;" /> --><img src="http://xenharmonic.wikispaces.com/file/view/UDPTable.gif/277923180/800x525/UDPTable.gif" alt="UDPTable.gif" title="UDPTable.gif" style="height: 525px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:60 --></body></html>