UDP: Difference between revisions
Wikispaces>genewardsmith **Imported revision 277933300 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278343148 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-11- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:55:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278343148</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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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). | 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 | 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 5|1(1), or simply 5|1. | 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 5|1(1), or simply 5|1. | ||
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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 /> | 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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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 | 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&lt;=u&lt;Q and S(m*u) = g*u, and d the largest integer such that 0≤=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 5|1(1), or simply 5|1.<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 5|1(1), or simply 5|1.<br /> | ||