UDP: Difference between revisions
Wikispaces>guest **Imported revision 277532598 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 277836118 - 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>2011-11-21 15:34:42 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>277836118</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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=Definition= | =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 | 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. | ||
Then Ionian( | |||
=The Chroma-Aligned Generator= | =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. | 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 | 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. | ||
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. | |||
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. | 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. | 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. | ||
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. | 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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<br /> | <br /> | ||
<!-- 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> | ||
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 /> | |||
<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 &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 /> | |||
<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 &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 /> | |||
<br /> | <br /> | ||
<!-- 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> | ||
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 /> | ||
<br /> | <br /> | ||
Specifying a number of generators &quot;up&quot; or &quot;down&quot; is insufficient by itself to specify | 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 /> | ||
<br /> | <br /> | ||
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 /> | |||
<br /> | |||
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 /> | |||
<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 /> | 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 /> | <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 /> | 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 /> | <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 /> | 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="Rationale"></a><!-- ws:end:WikiTextHeadingRule:6 -->Rationale</h1> | ||
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 /> | 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 /> | ||
<br /> | <br /> | ||
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This interpretation is what UDP notation generalizes.<br /> | This interpretation is what UDP notation generalizes.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:8 -->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</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> | ||