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Wikispaces>Sarzadoce **Imported revision 303550258 - Original comment: ** |
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<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: | : This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2012-02-21 00:48:55 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>303550258</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">For the basic concepts, see the Wikipedia article [[http://en.wikipedia.org/wiki/Otonality_and_Utonality|Otonality and Utonality]]. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">* 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-numer. 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. | 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-numer. 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. | ||
A chord that satisfies neither of these, meaning the chord itself and its inverse have equal largest-odd-numbers, is neither obviously otonal nor obviously utonal, so it's called "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). | A chord that satisfies neither of these, meaning the chord itself and its inverse have equal largest-odd-numbers, is neither obviously otonal nor obviously utonal, so it's called "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: | |||
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= | =Ambitonal chord theorem= | ||
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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.</pre></div> | 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.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Otonality and utonality</title></head><body>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>.< | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Otonality and utonality</title></head><body><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 /> | ||
<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-numer. 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 /> | 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-numer. 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 /> | <br /> | ||
A chord that satisfies neither of these, meaning the chord itself and its inverse have equal largest-odd-numbers, is neither obviously otonal nor obviously utonal, so it's called &quot;ambitonal&quot;. 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 /> | A chord that satisfies neither of these, meaning the chord itself and its inverse have equal largest-odd-numbers, is neither obviously otonal nor obviously utonal, so it's called &quot;ambitonal&quot;. 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 /> | <br /> | ||
Some interesting properties of these chords follow from their definitions:<br /> | |||
<br /> | |||
Otonal:<br /> | |||
<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 /> | |||
Utonal:<br /> | |||
<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 /> | |||
Ambitonal:<br /> | |||
<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:0:&lt;h1&gt; --><h1 id="toc0"><a name="Ambitonal chord theorem"></a><!-- ws:end:WikiTextHeadingRule:0 -->Ambitonal chord theorem</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Ambitonal chord theorem"></a><!-- ws:end:WikiTextHeadingRule:0 -->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 /> | 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 /> | ||
Revision as of 00:48, 21 February 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Sarzadoce and made on 2012-02-21 00:48:55 UTC.
- The original revision id was 303550258.
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
* 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-numer. 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. A chord that satisfies neither of these, meaning the chord itself and its inverse have equal largest-odd-numbers, is neither obviously otonal nor obviously utonal, so it's called "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: 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.
Original HTML content:
<html><head><title>Otonality and utonality</title></head><body><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-numer. 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 /> A chord that satisfies neither of these, meaning the chord itself and its inverse have equal largest-odd-numbers, is neither obviously otonal nor obviously utonal, so it's called "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 /> Some interesting properties of these chords follow from their definitions:<br /> <br /> Otonal:<br /> <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 /> Utonal:<br /> <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 /> Ambitonal:<br /> <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:0:<h1> --><h1 id="toc0"><a name="Ambitonal chord theorem"></a><!-- ws:end:WikiTextHeadingRule:0 -->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.</body></html>