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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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 &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.
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>2010-06-06 17:30:51 UTC</tt>.<br>
: The original revision id was <tt>147268909</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>
<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 interval given by its logarhithmic size measure (like [[cent]]s or amount of edo-atoms) has an infinite list of //nearest just intervals//


==Examples==
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 600-cent interval (6 steps of [[12edo]], "Tritone") approximates following ratios:
|| **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
|| ... || ... || ... || ... ||
||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 ||
||= 7 / 5 ||= 5.129 ||= 582.51 ||= 17.49 ||
||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 ||
|| ... || ... || ... || ... ||


The 300-cent interval (3 steps of [[12edo]], "minor third") approximates following ratios:
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 &lt; q.
|| **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
|| ... || ... || ... || ... ||
||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 ||
||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 ||
||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 ||
|| ... || ... || ... || ... ||
</pre></div>
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Nearest just interval&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An interval given by its logarhithmic size measure (like &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s or amount of edo-atoms) has an infinite list of &lt;em&gt;nearest just intervals&lt;/em&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Examples&lt;/h2&gt;
The 600-cent interval (6 steps of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &amp;quot;Tritone&amp;quot;) approximates following ratios:&lt;br /&gt;


== Examples ==


&lt;table class="wiki_table"&gt;
=== Approximations for Ratios (of Pure Intervals) ===
    &lt;tr&gt;
The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... [[cent|cents]]):
        &lt;td&gt;&lt;strong&gt;freq. ratio&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;log(&lt;a class="wiki_link" href="/Tenney%20Height"&gt;Tenney Height&lt;/a&gt;)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;size&lt;/strong&gt; in cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;&amp;quot;error&amp;quot;&lt;/strong&gt; in cents&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3 / 2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.585&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;701.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;101.96&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;7 / 5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5.129&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;582.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;17.49&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;17 / 12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7.672&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;603.00&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
{| class="wikitable"
The 300-cent interval (3 steps of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &amp;quot;minor third&amp;quot;) approximates following ratios:&lt;br /&gt;
|-
| | '''Step\EDO'''
| | '''log([[Tenney_Height|Tenney Height]])'''
| | '''size''' in cents
| | '''"error"''' in cents
|-
| | ...
| | ...
| | ...
| | ...
|-
| style="text-align:center;" | 1 \ 1
| | 0.0
| style="text-align:center;" | 1200.0
| style="text-align:center;" | 498.04
|-
| style="text-align:center;" | 1 \ 2
| | 1.0
| style="text-align:center;" | 600.00
| style="text-align:center;" | -101.96
|-
| style="text-align:center;" | 2 \ 3
| | 2.585
| style="text-align:center;" | 800.00
| style="text-align:center;" | 98.045
|-
| style="text-align:center;" | 3 \ [[5edo|5]]
| | 3.907
| style="text-align:center;" | 720.00
| style="text-align:center;" | 18.045
|-
| style="text-align:center;" | 4 \ [[7edo|7]]
| | 4.807
| style="text-align:center;" | 685.7143
| style="text-align:center;" | -16.2407
|-
| style="text-align:center;" | 7 \ [[12edo|12]]
| | 6.392
| style="text-align:center;" | 700.00
| style="text-align:center;" | -1.955
|-
| style="text-align:center;" | 17 \ [[29edo|29]]
| | 8.945
| style="text-align:center;" | 703.4483
| style="text-align:center;" | 1.4933
|-
| style="text-align:center;" | 24 \ [[41edo|41]]
| | 9.943
| style="text-align:center;" | 702.43902
| style="text-align:center;" | 0.48402
|-
| style="text-align:center;" | 31 \ [[53edo|53]]
| | 10.682
| style="text-align:center;" | 701.88679
| style="text-align:center;" | -0.06821
|}


<ul><li>''for approximations of the harmonic seventh see [[7/4#Approximations|7_4]]''</li></ul>


&lt;table class="wiki_table"&gt;
=== Approximation for Logarihmic Measures ===
    &lt;tr&gt;
The 600-cent interval sqrt(2) (6 steps of [[12edo|12edo]], "Tritone") approximates following ratios:
        &lt;td&gt;&lt;strong&gt;freq. ratio&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;log(&lt;a class="wiki_link" href="/Tenney%20Height"&gt;Tenney Height&lt;/a&gt;)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;size&lt;/strong&gt; in cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;&amp;quot;error&amp;quot;&lt;/strong&gt; in cents&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6 / 5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4.907&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;315.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;15.64&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;19 / 16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8.248&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;297.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.49&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;25 / 21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9.036&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;301.84&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1.84&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;...&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
{| class="wikitable"
|-
| | '''freq. ratio'''
| | '''log2([[Tenney_Height|Tenney Height]])'''
| | '''size''' in cents
| | '''"error"''' in cents
|-
| | ...
| | ...
| | ...
| | ...
|-
| style="text-align:center;" | 1 / 1
| style="text-align:center;" | 0.0
| style="text-align:center;" | 0.0
| style="text-align:center;" | 600.0
|-
| style="text-align:center;" | [[3/2|3 / 2]]
| style="text-align:center;" | 2.585
| style="text-align:center;" | 701.96
| style="text-align:center;" | 101.96
|-
| style="text-align:center;" | [[4/3|4 / 3]]
| style="text-align:center;" | 3.585
| style="text-align:center;" | 498.04
| style="text-align:center;" | -101.96
|-
| style="text-align:center;" | [[7/5|7 / 5]]
| style="text-align:center;" | 5.129
| style="text-align:center;" | 582.51
| style="text-align:center;" | -17.49
|-
| style="text-align:center;" | [[17/12|17 / 12]]
| style="text-align:center;" | 7.672
| style="text-align:center;" | 603.000
| style="text-align:center;" | 3.000
|-
| style="text-align:center;" | 24 / 17
| |
| style="text-align:center;" | 597.000
| style="text-align:center;" | -3.000
|-
| style="text-align:center;" | 99 / 70
| |
| style="text-align:center;" | 600.0883
| style="text-align:center;" | 0.0883
|-
| style="text-align:center;" | 140 / 99
| |
| style="text-align:center;" | 599.9117
| style="text-align:center;" | -0.0883
|-
| | ...
| | ...
| | ...
| | ...
|}
 
The 300-cent interval 2^(1/4) (3 steps of [[12edo|12edo]], "minor third") approximates following ratios:
 
{| class="wikitable"
|-
| | '''freq. ratio'''
| | '''log([[Tenney_Height|Tenney Height]])'''
| | '''size''' in cents
| | '''"error"''' in cents
|-
| | ...
| | ...
| | ...
| | ...
|-
| style="text-align:center;" | 1 / 1
| style="text-align:center;" | 0.0
| style="text-align:center;" | 0.0
| style="text-align:center;" | 300.0
|-
| style="text-align:center;" | [[6/5|6 / 5]]
| style="text-align:center;" | 4.907
| style="text-align:center;" | 315.64
| style="text-align:center;" | 15.64
|-
| style="text-align:center;" | [[13/11|13 / 11]]
| style="text-align:center;" | 7.160
| style="text-align:center;" | 289.21
| style="text-align:center;" | -10.79
|-
| style="text-align:center;" | [[19/16|19 / 16]]
| style="text-align:center;" | 8.248
| style="text-align:center;" | 297.51
| style="text-align:center;" | -2.49
|-
| style="text-align:center;" | [[25/21|25 / 21]]
| style="text-align:center;" | 9.036
| style="text-align:center;" | 301.84
| style="text-align:center;" | 1.84
|-
| | ...
| | ...
| | ...
| | ...
|}
 
[[Category:Elementary math]]
[[Category:Approximation]]
[[Category:Just intonation]]
 
{{todo|add examples}}

Latest revision as of 04:38, 5 March 2023

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 such as 3/2 or ∜5 is often of interest.

The 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 best relative approximation. Here it is required that |qr - p| is less than |nr - m| for any n < q.

Examples

Approximations for Ratios (of Pure Intervals)

The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... cents):

Step\EDO log(Tenney Height) size in cents "error" in cents
... ... ... ...
1 \ 1 0.0 1200.0 498.04
1 \ 2 1.0 600.00 -101.96
2 \ 3 2.585 800.00 98.045
3 \ 5 3.907 720.00 18.045
4 \ 7 4.807 685.7143 -16.2407
7 \ 12 6.392 700.00 -1.955
17 \ 29 8.945 703.4483 1.4933
24 \ 41 9.943 702.43902 0.48402
31 \ 53 10.682 701.88679 -0.06821
  • for approximations of the harmonic seventh see 7_4

Approximation for Logarihmic Measures

The 600-cent interval sqrt(2) (6 steps of 12edo, "Tritone") approximates following ratios:

freq. ratio log2(Tenney Height) size in cents "error" in cents
... ... ... ...
1 / 1 0.0 0.0 600.0
3 / 2 2.585 701.96 101.96
4 / 3 3.585 498.04 -101.96
7 / 5 5.129 582.51 -17.49
17 / 12 7.672 603.000 3.000
24 / 17 597.000 -3.000
99 / 70 600.0883 0.0883
140 / 99 599.9117 -0.0883
... ... ... ...

The 300-cent interval 2^(1/4) (3 steps of 12edo, "minor third") approximates following ratios:

freq. ratio log(Tenney Height) size in cents "error" in cents
... ... ... ...
1 / 1 0.0 0.0 300.0
6 / 5 4.907 315.64 15.64
13 / 11 7.160 289.21 -10.79
19 / 16 8.248 297.51 -2.49
25 / 21 9.036 301.84 1.84
... ... ... ...
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