Nearest just interval - Revision history

Fredg999 category edits: Categories

2023-03-05T04:38:34Z

Fredg999: Misc. edits, categories

2021-12-21T06:31:21Z

Misc. edits, categories

← Older revision		Revision as of 06:31, 21 December 2021
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	\| \| ...		\| \| ...
	\|}		\|}
	[[Category:~~approximation~~]]		[[Category:Approximation]]
	[[Category:~~hack~~]]		[[Category:Just intonation]]
	[[Category:~~just~~]]		[[Category:Math]]
	[[Category:~~math~~]]		[[Category:Theory]]
	~~[[Category:theory]]~~
	~~[[Category:~~todo~~:add_examples]]~~		{{todo\|add examples}}

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>xenwolf: Imported revision 391770730 - Original comment: added some links to intervals

2012-12-12T17:25:20Z

**Imported revision 391770730 - Original comment: added some links to intervals**

← Older revision		Revision as of 17:25, 12 December 2012
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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:~~genewardsmith~~\|~~genewardsmith~~]] and made on <tt>~~2011~~-11-~~22 21~~:06:00 UTC</tt>.<br>		: This revision was by author [[User:xenwolf\|xenwolf]] and made on <tt>2012-12-12 17:25:20 UTC</tt>.<br>
	: The original revision id was <tt>~~278345590~~</tt>.<br>		: The original revision id was <tt>391770730</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt>added some links to intervals</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>
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	\|\| ... \|\| ... \|\| ... \|\| ... \|\|		\|\| ... \|\| ... \|\| ... \|\| ... \|\|
	\|\|= 1 / 1 \|\|= 0.0 \|\|= 0.0 \|\|= 600.0 \|\|		\|\|= 1 / 1 \|\|= 0.0 \|\|= 0.0 \|\|= 600.0 \|\|
	\|\|= 3 / 2 \|\|= 2.585 \|\|= 701.96 \|\|= 101.96 \|\|		\|\|= [[3_2\|3 / 2]] \|\|= 2.585 \|\|= 701.96 \|\|= 101.96 \|\|
	\|\|= 4 / 3 \|\|= 3.585 \|\|= 498.04 \|\|= -101.96 \|\|		\|\|= [[4_3\|4 / 3]] \|\|= 3.585 \|\|= 498.04 \|\|= -101.96 \|\|
	\|\|= 7 / 5 \|\|= 5.129 \|\|= 582.51 \|\|= -17.49 \|\|		\|\|= [[7_5\|7 / 5]] \|\|= 5.129 \|\|= 582.51 \|\|= -17.49 \|\|
	\|\|= 17 / 12 \|\|= 7.672 \|\|= 603.000 \|\|= 3.000 \|\|		\|\|= [[17_12\|17 / 12]] \|\|= 7.672 \|\|= 603.000 \|\|= 3.000 \|\|
	\|\|= 24 / 17 \|\| \|\|= 597.000 \|\|= -3.000 \|\|		\|\|= 24 / 17 \|\| \|\|= 597.000 \|\|= -3.000 \|\|
	\|\|= 99 / 70 \|\| \|\|= 600.0883 \|\|= 0.0883 \|\|		\|\|= 99 / 70 \|\| \|\|= 600.0883 \|\|= 0.0883 \|\|
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	\|\| ... \|\| ... \|\| ... \|\| ... \|\|		\|\| ... \|\| ... \|\| ... \|\| ... \|\|
	\|\|= 1 / 1 \|\|= 0.0 \|\|= 0.0 \|\|= 300.0 \|\|		\|\|= 1 / 1 \|\|= 0.0 \|\|= 0.0 \|\|= 300.0 \|\|
	\|\|= 6 / 5 \|\|= 4.907 \|\|= 315.64 \|\|= 15.64 \|\|		\|\|= [[6_5\|6 / 5]] \|\|= 4.907 \|\|= 315.64 \|\|= 15.64 \|\|
	\|\|= 13 / 11 \|\|= 7.160 \|\|= 289.21 \|\|= -10.79 \|\|		\|\|= [[13_11\|13 / 11]] \|\|= 7.160 \|\|= 289.21 \|\|= -10.79 \|\|
	\|\|= 19 / 16 \|\|= 8.248 \|\|= 297.51 \|\|= -2.49 \|\|		\|\|= [[19_16\|19 / 16]] \|\|= 8.248 \|\|= 297.51 \|\|= -2.49 \|\|
	\|\|= 25 / 21 \|\|= 9.036 \|\|= 301.84 \|\|= 1.84 \|\|		\|\|= [[25_21\|25 / 21]] \|\|= 9.036 \|\|= 301.84 \|\|= 1.84 \|\|
	\|\| ... \|\| ... \|\| ... \|\| ... \|\|		\|\| ... \|\| ... \|\| ... \|\| ... \|\|

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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">3 / 2<br />		<td style="text-align: center;"><a class="wiki_link" href="/3_2">3 / 2</a><br />
	</td>		</td>
	<td style="text-align: center;">2.585<br />		<td style="text-align: center;">2.585<br />
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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">4 / 3<br />		<td style="text-align: center;"><a class="wiki_link" href="/4_3">4 / 3</a><br />
	</td>		</td>
	<td style="text-align: center;">3.585<br />		<td style="text-align: center;">3.585<br />
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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">7 / 5<br />		<td style="text-align: center;"><a class="wiki_link" href="/7_5">7 / 5</a><br />
	</td>		</td>
	<td style="text-align: center;">5.129<br />		<td style="text-align: center;">5.129<br />
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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">17 / 12<br />		<td style="text-align: center;"><a class="wiki_link" href="/17_12">17 / 12</a><br />
	</td>		</td>
	<td style="text-align: center;">7.672<br />		<td style="text-align: center;">7.672<br />
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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">6 / 5<br />		<td style="text-align: center;"><a class="wiki_link" href="/6_5">6 / 5</a><br />
	</td>		</td>
	<td style="text-align: center;">4.907<br />		<td style="text-align: center;">4.907<br />
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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">13 / 11<br />		<td style="text-align: center;"><a class="wiki_link" href="/13_11">13 / 11</a><br />
	</td>		</td>
	<td style="text-align: center;">7.160<br />		<td style="text-align: center;">7.160<br />
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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">19 / 16<br />		<td style="text-align: center;"><a class="wiki_link" href="/19_16">19 / 16</a><br />
	</td>		</td>
	<td style="text-align: center;">8.248<br />		<td style="text-align: center;">8.248<br />
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	</tr>		</tr>
	<tr>		<tr>
	<td style="text-align: center;">25 / 21<br />		<td style="text-align: center;"><a class="wiki_link" href="/25_21">25 / 21</a><br />
	</td>		</td>
	<td style="text-align: center;">9.036<br />		<td style="text-align: center;">9.036<br />

Wikispaces>genewardsmith: Imported revision 278345590 - Original comment:

2011-11-22T21:06:00Z

**Imported revision 278345590 - Original comment: **

Wikispaces>genewardsmith: Imported revision 278345318 - Original comment:

2011-11-22T21:04:40Z

**Imported revision 278345318 - Original comment: **

← Older revision		Revision as of 21:04, 22 November 2011
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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:~~xenwolf~~\|~~xenwolf~~]] and made on <tt>2011-06-~~16 02~~:43:39 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-11-22 21:04:40 UTC</tt>.<br>
	: The original revision id was <tt>~~237020004~~</tt>.<br>		: The original revision id was <tt>278345318</tt>.<br>
	: The revision comment was: <tt>~~jump into section~~</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">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from ~~sqrt(2)~~ = 600 cents, but \|4/3 - ~~sqrt(2)~~\| = .08088 whereas \|3/2 - ~~sqrt(2)~~\| = 0.08479, which is larger.		<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">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but \|4/3 - √2\| = .08088 whereas \|3/2 - √2\| = 0.08479, which is larger.

	Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.		Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.
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	</pre></div>		</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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from ~~sqrt(2)~~ = 600 cents, but \|4/3 - ~~sqrt(2)~~\| = .08088 whereas \|3/2 - ~~sqrt(2)~~\| = 0.08479, which is larger.<br />		<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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but \|4/3 - √2\| = .08088 whereas \|3/2 - √2\| = 0.08479, which is larger.<br />
	<br />		<br />
	Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.<br />		Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.<br />

Wikispaces>xenwolf: Imported revision 237020004 - Original comment: jump into section

2011-06-16T02:43:39Z

**Imported revision 237020004 - Original comment: jump into section**

Wikispaces>xenwolf: Imported revision 237019882 - Original comment:

2011-06-16T02:42:08Z

**Imported revision 237019882 - Original comment: **

← Older revision		Revision as of 02:42, 16 June 2011
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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:xenwolf\|xenwolf]] and made on <tt>2011-06-16 02:40:58 UTC</tt>.<br>		: This revision was by author [[User:xenwolf\|xenwolf]] and made on <tt>2011-06-16 02:42:08 UTC</tt>.<br>
	: The original revision id was <tt>~~237019800~~</tt>.<br>		: The original revision id was <tt>237019882</tt>.<br>
	: The revision comment was: <tt>~~page reordered - the reverse approximations is the less interesting (in my opinion)~~</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>
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	\|\|= 24 \ [[41edo\|41]] \|\| 9.943 \|\|= 702.43902 \|\|= 0.48402 \|\|		\|\|= 24 \ [[41edo\|41]] \|\| 9.943 \|\|= 702.43902 \|\|= 0.48402 \|\|
	\|\|= 31 \ [[53edo\|53]] \|\| 10.682 \|\|= 701.88679 \|\|= -0.06821 \|\|		\|\|= 31 \ [[53edo\|53]] \|\| 10.682 \|\|= 701.88679 \|\|= -0.06821 \|\|

			* //for approximations of the harmonic seventh see [[7_4]]//

	=== Approximation for Logarihmic Measures ===		=== Approximation for Logarihmic Measures ===
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	<br />		<br />
			<ul><li><em>for approximations of the harmonic seventh see <a class="wiki_link" href="/7_4">7_4</a></em></li></ul><br />
	<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Examples-Approximation for Logarihmic Measures"></a><!-- ws:end:WikiTextHeadingRule:4 --> Approximation for Logarihmic Measures </h3>		<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Examples-Approximation for Logarihmic Measures"></a><!-- ws:end:WikiTextHeadingRule:4 --> Approximation for Logarihmic Measures </h3>
	The 600-cent interval sqrt(2) (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;Tritone&quot;) approximates following ratios:<br />		The 600-cent interval sqrt(2) (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;Tritone&quot;) approximates following ratios:<br />

Wikispaces>xenwolf: Imported revision 237019800 - Original comment: page reordered - the reverse approximations is the less interesting (in my opinion)

2011-06-16T02:40:58Z

**Imported revision 237019800 - Original comment: page reordered - the reverse approximations is the less interesting (in my opinion)**

Show changes

Wikispaces>genewardsmith: Imported revision 236965526 - Original comment:

2011-06-15T20:32:29Z

**Imported revision 236965526 - Original comment: **

← Older revision		Revision as of 20:32, 15 June 2011
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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:~~xenwolf~~\|~~xenwolf~~]] and made on <tt>2011-06-15 17:25:23 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-06-15 20:32:29 UTC</tt>.<br>
	: The original revision id was <tt>~~236928382~~</tt>.<br>		: The original revision id was <tt>236965526</tt>.<br>
	: The revision comment was: <tt>~~(the first cent link on this page)~~</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>
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	\|\|= 7 \ [[12edo\|12]] \|\| 6.392 \|\|= 700.00 \|\|= -1.955 \|\|		\|\|= 7 \ [[12edo\|12]] \|\| 6.392 \|\|= 700.00 \|\|= -1.955 \|\|
	\|\|= 17 \ [[29edo\|29]] \|\| 8.945 \|\|= 703.4483 \|\|= 1.4933 \|\|		\|\|= 17 \ [[29edo\|29]] \|\| 8.945 \|\|= 703.4483 \|\|= 1.4933 \|\|
	\|\|= 24 \ [[41edo\|41]] \|\| 9.943 \|\|= 702.43902 \|\|= 0.48402 \|\|</pre></div>		\|\|= 24 \ [[41edo\|41]] \|\| 9.943 \|\|= 702.43902 \|\|= 0.48402 \|\|
			\|\|= 31 \ [[53edo\|53]] \|\| 10.682 \|\|= 701.88679 \|\|= -0.06821 \|\|</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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but \|4/3 - sqrt(2)\| = .08088 whereas \|3/2 - sqrt(2)\| = 0.08479, which is larger.<br />		<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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but \|4/3 - sqrt(2)\| = .08088 whereas \|3/2 - sqrt(2)\| = 0.08479, which is larger.<br />
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	</td>		</td>
	<td style="text-align: center;">0.48402<br />		<td style="text-align: center;">0.48402<br />
			</td>
			</tr>
			<tr>
			<td style="text-align: center;">31 \ <a class="wiki_link" href="/53edo">53</a><br />
			</td>
			<td>10.682<br />
			</td>
			<td style="text-align: center;">701.88679<br />
			</td>
			<td style="text-align: center;">-0.06821<br />
	</td>		</td>
	</tr>		</tr>

← Older revision		Revision as of 04:38, 5 March 2023
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	\| \| ...		\| \| ...
	\|}		\|}

			[[Category:Elementary math]]
	[[Category:Approximation]]		[[Category:Approximation]]
	[[Category:Just intonation]]		[[Category:Just intonation]]
	~~[[Category:Math]]~~
	~~[[Category:Theory]]~~

	{{todo\|add examples}}		{{todo\|add examples}}

← Older revision		Revision as of 21:06, 22 November 2011
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	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>
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	: The original revision id was <tt>~~278345318~~</tt>.<br>		: The original revision id was <tt>278345590</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>
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	<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">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but \|4/3 - √2\| = .08088 whereas \|3/2 - √2\| = 0.08479, which is larger.		<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">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but \|4/3 - √2\| = .08088 whereas \|3/2 - √2\| = 0.08479, which is larger.

	Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number ~~of interest~~ such as 3/2 or ~~5^(1/4)~~ is often of interest.		Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest.

	The [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents\|semiconvergents]] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [[http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations\|best relative approximation]]. Here it is required that \|qr - p\| is less than \|nr - m\| for any n < q.		The [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents\|semiconvergents]] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [[http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations\|best relative approximation]]. Here it is required that \|qr - p\| is less than \|nr - m\| for any n < q.
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	<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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but \|4/3 - √2\| = .08088 whereas \|3/2 - √2\| = 0.08479, which is larger.<br />		<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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but \|4/3 - √2\| = .08088 whereas \|3/2 - √2\| = 0.08479, which is larger.<br />
	<br />		<br />
	Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number ~~of interest~~ such as 3/2 or ~~5^(1/4)~~ is often of interest.<br />		Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest.<br />
	<br />		<br />
	The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations" rel="nofollow">best relative approximation</a>. Here it is required that \|qr - p\| is less than \|nr - m\| for any n &lt; q.<br />		The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations" rel="nofollow">best relative approximation</a>. Here it is required that \|qr - p\| is less than \|nr - m\| for any n &lt; q.<br />

← Older revision		Revision as of 02:43, 16 June 2011
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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:xenwolf\|xenwolf]] and made on <tt>2011-06-16 02:42:08 UTC</tt>.<br>		: This revision was by author [[User:xenwolf\|xenwolf]] and made on <tt>2011-06-16 02:43:39 UTC</tt>.<br>
	: The original revision id was <tt>~~237019882~~</tt>.<br>		: The original revision id was <tt>237020004</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt>jump into section</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>
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	\|\|= 31 \ [[53edo\|53]] \|\| 10.682 \|\|= 701.88679 \|\|= -0.06821 \|\|		\|\|= 31 \ [[53edo\|53]] \|\| 10.682 \|\|= 701.88679 \|\|= -0.06821 \|\|

	* //for approximations of the harmonic seventh see [[7_4]]//		* //for approximations of the harmonic seventh see [[7_4#Approximations]]//

	=== Approximation for Logarihmic Measures ===		=== Approximation for Logarihmic Measures ===
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	<br />		<br />
	<ul><li><em>for approximations of the harmonic seventh see <a class="wiki_link" href="/7_4">7_4</a></em></li></ul><br />		<ul><li><em>for approximations of the harmonic seventh see <a class="wiki_link" href="/7_4#Approximations">7_4</a></em></li></ul><br />
	<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Examples-Approximation for Logarihmic Measures"></a><!-- ws:end:WikiTextHeadingRule:4 --> Approximation for Logarihmic Measures </h3>		<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Examples-Approximation for Logarihmic Measures"></a><!-- ws:end:WikiTextHeadingRule:4 --> Approximation for Logarihmic Measures </h3>
	The 600-cent interval sqrt(2) (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;Tritone&quot;) approximates following ratios:<br />		The 600-cent interval sqrt(2) (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;Tritone&quot;) approximates following ratios:<br />

Nearest just interval - Revision history

Fredg999 category edits: Categories

Fredg999: Misc. edits, categories

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>xenwolf: **Imported revision 391770730 - Original comment: added some links to intervals**

Wikispaces>genewardsmith: **Imported revision 278345590 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 278345318 - Original comment: **

Wikispaces>xenwolf: **Imported revision 237020004 - Original comment: jump into section**

Wikispaces>xenwolf: **Imported revision 237019882 - Original comment: **

Wikispaces>xenwolf: **Imported revision 237019800 - Original comment: page reordered - the reverse approximations is the less interesting (in my opinion)**