UDP: Difference between revisions
Wikispaces>genewardsmith **Imported revision 275868640 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 275917892 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2011-11-15 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-15 21:54:59 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>275917892</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> | ||
| Line 18: | Line 18: | ||
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. In this case, 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.) | 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. In this case, 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.) | ||
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. If m shifts up, then | 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. 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. | ||
| Line 83: | Line 83: | ||
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] &gt;= 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. In this case, 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 /> | 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] &gt;= 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. In this case, 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 /> | <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. If m shifts up, then | 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. 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 /> | ||
<br /> | <br /> | ||