Otonality and utonality
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<span style="text-align: right;
display: block;">[[OtonalityとUtonality|日本語]]</span>
[[toc|flat]]
=Introduction=
* For the basic concepts, see the Wikipedia article [[http://en.wikipedia.org/wiki/Otonality_and_Utonality|Otonality and Utonality]].
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.
=Precise definitions=
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).
=Properties of types of chords=
==Otonal==
* If we represent an otonal chord as a set of integers in the form A<span style="vertical-align: sub;">1</span>:A<span style="vertical-align: sub;">2</span>: ... :A<span style="vertical-align: sub;">n, we may add any additional integers without affecting the chord's otonality.</span>
* <span style="vertical-align: sub;">All chords with [[Linear chord|isoratios]] that can be reduced to 1:1, 1:1:1, 1:1:1:1 etc., are otonal.</span>
==Utonal==
* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [[http://tech.groups.yahoo.com/group/tuning-math/message/20310|[proof]]]
==Ambitonal==
* [[Dyadic chord|Essentially tempered]] chords can be ambitonal, even though they do not have unique representations in the harmonic series.
=Ambitonal chord theorem=
A chord can be represented as a set of integers whose gcd is 1. (If octave equivalence is assumed we take the largest odd factors of all of these integers.) The inverse of this chord is the set of integers LCM(original chord)/x for each integer x in the original chord.
Assume a chord is ambitonal. Then its largest integer, max(chord), is equal to the largest integer of its inverse, which is LCM(chord)/min(chord). Therefore min(chord)*max(chord) = LCM(chord). Conversely, if a set of integers has gcd 1 and also satisfies this, then it is an ambitonal chord.
Thus, for any given odd number N (which ought to be composite to get non-trivial results), all ambitonal chords with LCM N can easily be found by considering subsets of the factors of N. If a subset has gcd 1 and also satisfies min(subset)*max(subset) = N, then it is an ambitonal chord.
=Scales=
These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, Euler genera, combination product sets, or tonality diamonds are necessarily ambitonal, whereas dwarf scales are always either otonal or ambitonal.
=Essentially tempered chords=
This kind of reduction can also be used to analyze [[Dyadic chord|essentially tempered chords]]. Consider for example the [[sinbadmic tetrad]], which is the 1001/1000-tempering of 1-11/10-13/10-10/7. The reduction of the JI version of this chord is {25, 35, 77, 91}; discarding the lowest number, 25, and reducing again gives {5, 11, 13}. This tells us the chord can be analyzed as an otonbal 1-11/10-13/10 chord plus a 10/7 addition requiring essential tempering.ed Original HTML content:
<html><head><title>Otonality and utonality</title></head><body><span style="text-align: right;
display: block;"><a class="wiki_link" href="/Otonality%E3%81%A8Utonality">日本語</a></span><br />
<!-- ws:start:WikiTextTocRule:18:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Precise definitions">Precise definitions</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Properties of types of chords">Properties of types of chords</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Ambitonal chord theorem">Ambitonal chord theorem</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Essentially tempered chords">Essentially tempered chords</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: -->
<!-- ws:end:WikiTextTocRule:28 --><br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1>
<ul><li>For the basic concepts, see the Wikipedia article <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Otonality_and_Utonality" rel="nofollow">Otonality and Utonality</a>.</li></ul><br />
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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Precise definitions"></a><!-- ws:end:WikiTextHeadingRule:2 -->Precise definitions</h1>
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).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Properties of types of chords"></a><!-- ws:end:WikiTextHeadingRule:4 -->Properties of types of chords</h1>
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Properties of types of chords-Otonal"></a><!-- ws:end:WikiTextHeadingRule:6 -->Otonal</h2>
<ul><li>If we represent an otonal chord as a set of integers in the form A<span style="vertical-align: sub;">1</span>:A<span style="vertical-align: sub;">2</span>: ... :A<span style="vertical-align: sub;">n, we may add any additional integers without affecting the chord's otonality.</span></li><li><span style="vertical-align: sub;">All chords with <a class="wiki_link" href="/Linear%20chord">isoratios</a> that can be reduced to 1:1, 1:1:1, 1:1:1:1 etc., are otonal.</span></li></ul><br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Properties of types of chords-Utonal"></a><!-- ws:end:WikiTextHeadingRule:8 -->Utonal</h2>
<ul><li>The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. <a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/20310" rel="nofollow">[proof</a>]</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Properties of types of chords-Ambitonal"></a><!-- ws:end:WikiTextHeadingRule:10 -->Ambitonal</h2>
<ul><li><a class="wiki_link" href="/Dyadic%20chord">Essentially tempered</a> chords can be ambitonal, even though they do not have unique representations in the harmonic series.</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Ambitonal chord theorem"></a><!-- ws:end:WikiTextHeadingRule:12 -->Ambitonal chord theorem</h1>
A chord can be represented as a set of integers whose gcd is 1. (If octave equivalence is assumed we take the largest odd factors of all of these integers.) The inverse of this chord is the set of integers LCM(original chord)/x for each integer x in the original chord.<br />
<br />
Assume a chord is ambitonal. Then its largest integer, max(chord), is equal to the largest integer of its inverse, which is LCM(chord)/min(chord). Therefore min(chord)*max(chord) = LCM(chord). Conversely, if a set of integers has gcd 1 and also satisfies this, then it is an ambitonal chord.<br />
<br />
Thus, for any given odd number N (which ought to be composite to get non-trivial results), all ambitonal chords with LCM N can easily be found by considering subsets of the factors of N. If a subset has gcd 1 and also satisfies min(subset)*max(subset) = N, then it is an ambitonal chord.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->Scales</h1>
These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, Euler genera, combination product sets, or tonality diamonds are necessarily ambitonal, whereas dwarf scales are always either otonal or ambitonal.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Essentially tempered chords"></a><!-- ws:end:WikiTextHeadingRule:16 -->Essentially tempered chords</h1>
This kind of reduction can also be used to analyze <a class="wiki_link" href="/Dyadic%20chord">essentially tempered chords</a>. Consider for example the <a class="wiki_link" href="/sinbadmic%20tetrad">sinbadmic tetrad</a>, which is the 1001/1000-tempering of 1-11/10-13/10-10/7. The reduction of the JI version of this chord is {25, 35, 77, 91}; discarding the lowest number, 25, and reducing again gives {5, 11, 13}. This tells us the chord can be analyzed as an otonbal 1-11/10-13/10 chord plus a 10/7 addition requiring essential tempering.ed</body></html>