Otonality and utonality: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-21 14:37:40 UTC</tt>.

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Given a JI chord, how can we decide whether it is otonal or utonal? This might seem obvious at first, but it's actually surprisingly subtle. For example, the chord 10:12:15 is a 5-limit utonality (1/6:1/5:1/4), but it's also a 15-limit otonality, consisting of the 10th, 12th, and 15th harmonics of a fundamental. One reasonable definition is to say that a chord is otonal if its largest odd number is smaller than the largest odd number of its inverse, and utonal if the inverse has a smaller largest-odd-number. That way 4:5:6 is otonal because it's simpler than its inverse, 10:12:15, and 10:12:15 is utonal because it is more simply expressed as 1/6:1/5:1/4.

To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the octave reduction 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 (~~east~~ 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 octave reduction 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).

Some interesting properties of these chords follow from their definitions:

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Given a JI chord, how can we decide whether it is otonal or utonal? This might seem obvious at first, but it's actually surprisingly subtle. For example, the chord 10:12:15 is a 5-limit utonality (1/6:1/5:1/4), but it's also a 15-limit otonality, consisting of the 10th, 12th, and 15th harmonics of a fundamental. One reasonable definition is to say that a chord is otonal if its largest odd number is smaller than the largest odd number of its inverse, and utonal if the inverse has a smaller largest-odd-number. That way 4:5:6 is otonal because it's simpler than its inverse, 10:12:15, and 10:12:15 is utonal because it is more simply expressed as 1/6:1/5:1/4.

To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the octave reduction 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 (~~east~~ 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 octave reduction 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).

Some interesting properties of these chords follow from their definitions:

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2012-02-21 14:19:34 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-21 14:37:40 UTC</tt>.<br>
-: The original revision id was <tt>303748330</tt>.<br>
+: The original revision id was <tt>303756480</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>
@@ Line 10: / Line 10: @@
 Given a JI chord, how can we decide whether it is otonal or utonal? This might seem obvious at first, but it's actually surprisingly subtle. For example, the chord 10:12:15 is a 5-limit utonality (1/6:1/5:1/4), but it's also a 15-limit otonality, consisting of the 10th, 12th, and 15th harmonics of a fundamental. One reasonable definition is to say that a chord is otonal if its largest odd number is smaller than the largest odd number of its inverse, and utonal if the inverse has a smaller largest-odd-number. That way 4:5:6 is otonal because it's simpler than its inverse, 10:12:15, and 10:12:15 is utonal because it is more simply expressed as 1/6:1/5:1/4.
-To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the octave reduction 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 (east 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 octave reduction 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).
 Some interesting properties of these chords follow from their definitions:
@@ Line 34: / Line 34: @@
 Given a JI chord, how can we decide whether it is otonal or utonal? This might seem obvious at first, but it's actually surprisingly subtle. For example, the chord 10:12:15 is a 5-limit utonality (1/6:1/5:1/4), but it's also a 15-limit otonality, consisting of the 10th, 12th, and 15th harmonics of a fundamental. One reasonable definition is to say that a chord is otonal if its largest odd number is smaller than the largest odd number of its inverse, and utonal if the inverse has a smaller largest-odd-number. That way 4:5:6 is otonal because it's simpler than its inverse, 10:12:15, and 10:12:15 is utonal because it is more simply expressed as 1/6:1/5:1/4.&lt;br /&gt;
 &lt;br /&gt;
-To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the octave reduction 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 (east 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 octave reduction 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;
 &lt;br /&gt;
 Some interesting properties of these chords follow from their definitions:&lt;br /&gt;