UDP: Difference between revisions

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=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] &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. 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.

* 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.

 

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.

* Pajara[10] Standard Pentachordal Major, ssLsssLsss: 4|4(2) #8, 6|2(2) b3</pre></div>

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Given a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; S, a &lt;em&gt;modal shift&lt;/em&gt; by n may be defined as S'[i] = S[i+n]-S[n]. A modal shift is a &lt;em&gt;shift up&lt;/em&gt; if S'[i] &amp;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. 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.)&lt;br /&gt;

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 &lt;strong&gt;O&lt;/strong&gt; 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.&lt;br /&gt;

As an example&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;Meantone[7]'s ionian mode is 5|1(1), abbreviated 5|1 for short.&lt;/li&gt;&lt;li&gt;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.&lt;/li&gt;&lt;li&gt;Paul Erlich's standard pentachordal major is 4|4(2) #8, or alternatively 6|2(2) b3.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

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The UDP notation for any mode is U|D(P), where &amp;quot;u&amp;quot; specifies the number of chroma-aligned generators &amp;quot;up,&amp;quot; d specifies the number of chroma-aligned generators &amp;quot;down,&amp;quot; and p specifies the number of periods per equivalence interval. The chroma-aligned generator is the one such that more generators &amp;quot;up&amp;quot; also means more &amp;quot;major&amp;quot; scale degrees, or more generally, more &amp;quot;large&amp;quot; intervals that contain the root of the scale.&lt;br /&gt;

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  &lt;ul&gt;&lt;li&gt;Meantone[7] Ionian, LLsLLLs: 5|1&lt;/li&gt;&lt;li&gt;Meantone[7] Aeolian, LsLLsLL: 2|4&lt;/li&gt;&lt;li&gt;Mavila[7] Anti-Ionian, ssLsssL: 1|5&lt;/li&gt;&lt;li&gt;Mavila[7] Anti-Aeolian, Herman Miller's sLssLss mode: 4|2&lt;/li&gt;&lt;li&gt;Porcupine[7] Lssssss: 6|0&lt;/li&gt;&lt;li&gt;Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6: 6|0 b7&lt;/li&gt;&lt;li&gt;Porcupine[7] sssLsss: 3|3&lt;/li&gt;&lt;li&gt;Diminished[8] sLsLsLsL 0|4(4)&lt;/li&gt;&lt;li&gt;Diminished[8] LsLsLsLs 4|0(4)&lt;/li&gt;&lt;li&gt;Triforce[9] LLsLLsLLs: 6|0(3)&lt;/li&gt;&lt;li&gt;Meantone[5] minor pentatonic, LssLs: 3|1&lt;/li&gt;&lt;li&gt;Meantone[5] major pentatonic, ssLsL: 0|4&lt;/li&gt;&lt;li&gt;Sensi[11] LLsLLLsLLLs: 8|2&lt;/li&gt;&lt;li&gt;Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2)&lt;/li&gt;&lt;li&gt;Pajara[10] Standard Pentachordal Major, ssLsssLsss: 4|4(2) #8, 6|2(2) b3&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>