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 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 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. 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 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. As an example * 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. =Definition= The UDP notation for any mode is U|D(P), where "u" specifies the number of chroma-aligned generators "up," d specifies the number of chroma-aligned generators "down," and p specifies the number of periods per equivalence interval. The chroma-aligned generator is the one such that more generators "up" also means more "major" scale degrees, or more generally, more "large" intervals that contain the root of the scale. When the period is one, the (p) can be left off by convention for the short form of UDP notation, such that meantone's Ionian mode can simply be stated 5|1 instead of 5|1(1). When the period is greater than one, "u" and "d" should be taken to represent the total number of generators up and down per **equivalence interval**, not per period. 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). This has the handy property of u+d+p = the total number of notes in the scale. For example, 4+4+2 = 10, which tells us that Static Symmetrical Major is a decatonic scale. 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. =The Chroma-Aligned Generator= 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. Specifying a number of generators "up" or "down" is insufficient by itself to specify any mode in particular, because even if we only consider generators which fit within the period, 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. UDP notation is hence defined with respect to the **chroma-aligned generator**, which is the generator that points in the same direction as the MOS's chroma, equal to L-s, on the lattice. This generator is, intuitively speaking, the one which is the larger specific interval in its generic interval class in the MOS. 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. 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. 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. =Rationale= 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 "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. 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
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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 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 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 /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><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] >= 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. <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. 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 /> 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 <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 /> <br /> <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 /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:4 -->Definition</h1> The UDP notation for any mode is U|D(P), where "u" specifies the number of chroma-aligned generators "up," d specifies the number of chroma-aligned generators "down," and p specifies the number of periods per equivalence interval. The chroma-aligned generator is the one such that more generators "up" also means more "major" scale degrees, or more generally, more "large" intervals that contain the root of the scale.<br /> <br /> When the period is one, the (p) can be left off by convention for the short form of UDP notation, such that meantone's Ionian mode can simply be stated 5|1 instead of 5|1(1).<br /> <br /> When the period is greater than one, "u" and "d" should be taken to represent the total number of generators up and down per <strong>equivalence interval</strong>, not per period. 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). This has the handy property of u+d+p = the total number of notes in the scale. For example, 4+4+2 = 10, which tells us that Static Symmetrical Major is a decatonic scale.<br /> <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 /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><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 "up" direction, and how many you stack in the "down" direction.<br /> <br /> Specifying a number of generators "up" or "down" is insufficient by itself to specify any mode in particular, because even if we only consider generators which fit within the period, 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.<br /> <br /> UDP notation is hence defined with respect to the <strong>chroma-aligned generator</strong>, which is the generator that points in the same direction as the MOS's chroma, equal to L-s, on the lattice. This generator is, intuitively speaking, the one which is the larger specific interval in its generic interval class in the MOS. 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.<br /> <br /> 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 /> <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 /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><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 "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.<br /> <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 /> 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:10:<h1> --><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>