Otonality and utonality: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-03-15 09:42~~:46~~ UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-03-15 09:45:42 UTC</tt>.<br>

: The original revision id was <tt>~~608869997~~</tt>.<br>

: The original revision id was <tt>608870121</tt>.<br>

: The revision comment was: <tt>corrected an improper use of "octave reduction"</tt><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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=Precise definitions=

To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the ~~octave reduction~~ of this is {5/3, 5, 25}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).

To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the all-odd voicing of this is {5/3, 5/1, 25/1}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).

=Properties of types of chords=

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<br />

<h1 id="toc1"><a name="Precise definitions"></a>Precise definitions</h1>

To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the ~~octave reduction~~ of this is {5/3, 5, 25}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).<br />

To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the all-odd voicing of this is {5/3, 5/1, 25/1}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).<br />

<br />

<h1 id="toc2"><a name="Properties of types of chords"></a>Properties of types of chords</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-03-15 09:42:46 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-03-15 09:45:42 UTC</tt>.<br>
-: The original revision id was <tt>608869997</tt>.<br>
+: The original revision id was <tt>608870121</tt>.<br>
 : The revision comment was: <tt>corrected an improper use of "octave reduction"</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 15: / Line 15: @@
 =Precise definitions=
-To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the octave reduction of this is {5/3, 5, 25}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).
+To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the all-odd voicing of this is {5/3, 5/1, 25/1}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).
 =Properties of types of chords=
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Precise definitions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Precise definitions&lt;/h1&gt;
-  To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the octave reduction of this is {5/3, 5, 25}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).&lt;br /&gt;
+  To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by taking the odd parts of each member of the chord by removing factors of two from numerator and denominator, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the octave reduction by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator). For example, consider the chord {5/6, 5/3, 5/2, 25/16}; the all-odd voicing of this is {5/3, 5/1, 25/1}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 (inverse 6:8:9, with the same largest-odd-number) and 8:10:15 (inverse 8:12:15).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Properties of types of chords"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Properties of types of chords&lt;/h1&gt;