Cent: Difference between revisions

Revision as of 15:01, 26 January 2018

IMPORTED REVISION FROM WIKISPACES

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This revision was by author PiotrGrochowski and made on 2018-01-26 15:01:17 UTC.

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<span style="display: block; text-align: right;">[[xenharmonie/Cent|Deutsch]] - [[Centésimas|Español]]
</span>
=Definitions= 
A //cent// is an interval equal to exactly 1/100th of a [[12edo|12-EDO]] semitone. In other words, cents equally divide the half step (semitone) of 12-EDO into 100 equal parts. Cents are often used to express the size of intervals in different tuning systems.

The cent, which was first proposed in the late 19th century by [[http://en.wikipedia.org/wiki/Alexander_J._Ellis|Alexander Ellis]], is a logarithmic measure which may also be defined as the [[http://en.wikipedia.org/wiki/Logarithm|logarithm]] to the base 1200th root of 2. It may also be considered as exactly 1 step of 1200-EDO (dividing the octave into 1200 equal parts).

=Examples= 
The 12-EDO perfect fifth is exactly 700 cents, and the 12-EDO major third is exactly 400 cents. In contrast, the just perfect fifth, which corresponds to two notes in a frequency ratio of 3/2, is approximately 701.955 cents, and the just major third of 5/4 is ~386.314 cents. The 24-EDO neutral third is exactly 350 cents. The 22-EDO approximation to 3/2 is ~709.091 cents.

=How to calculate the size of an interval in cents= 
To find the size of a just interval in cents, you have to calculate the [[log2|binary logarithm]] (log<span style="font-size: 80%; vertical-align: sub;">2</span>) of its [[frequency ratio]], and multiply this by 1200.

Example (just perfect fifth): 1200 × log<span style="font-size: 80%; vertical-align: sub;">2</span>(3/2) = 1200 × ~0.584 = ~701.955 cents

If your pocket calculator has no //log2// key, but does have a //log// (log<span style="font-size: 80%; vertical-align: sub;">10</span>) or //ln// (log<span style="font-size: 80%; vertical-align: sub;">e</span>) key, you can key it this way:
[[media type="custom" key="28242337"]]
(This makes use of the property of logarithms that log<span style="font-size: 80%; vertical-align: sub;">2</span>(x) = log<span style="font-size: 80%; vertical-align: sub;">n</span>(x) / log<span style="font-size: 80%; vertical-align: sub;">n</span>(2). )
For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.

For EDO steps, which are already logarithmic, simply divide 1200 by the EDO size, then multiply by the number of steps. For example, 1 step of 31-EDO is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents.

=Other Units of Interval Measure= 
The cent is commonly used because of its ease in communicating information about intervals to a 12-EDO-savvy audience. However, some have suggested that the cent be deprecated, as other than societal convention there's no reason to give 12-EDO inherent importance over any other decent tuning. In contrast, others have suggested that cents are a useful unit of interval measure for purely mathematical reasons, even despite of 12-EDO's current status as the dominant tuning in Western society.

Whatever your stance, alternative measures of interval size can be found at [[Interval size measure]].

One prominent alternative interval measure is the [[millioctave]] (mO).

Additionally, a useful generalization for the cent measure is the **[[relative cent]],** which is one 100th of two neighboring [[pitch|pitches]] in any [[equal]] tuning.

=References= 
[[http://en.wikipedia.org/wiki/Cent_%28music%29|Wikipedia article on cents]]

Original HTML content:

<html><head><title>cent</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Cent">Deutsch</a> - <a class="wiki_link" href="/Cent%C3%A9simas">Español</a><br />
</span><br />
<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Definitions"></a><!-- ws:end:WikiTextHeadingRule:1 -->Definitions</h1>
 A <em>cent</em> is an interval equal to exactly 1/100th of a <a class="wiki_link" href="/12edo">12-EDO</a> semitone. In other words, cents equally divide the half step (semitone) of 12-EDO into 100 equal parts. Cents are often used to express the size of intervals in different tuning systems.<br />
<br />
The cent, which was first proposed in the late 19th century by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Alexander_J._Ellis" rel="nofollow">Alexander Ellis</a>, is a logarithmic measure which may also be defined as the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Logarithm" rel="nofollow">logarithm</a> to the base 1200th root of 2. It may also be considered as exactly 1 step of 1200-EDO (dividing the octave into 1200 equal parts).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:3 -->Examples</h1>
 The 12-EDO perfect fifth is exactly 700 cents, and the 12-EDO major third is exactly 400 cents. In contrast, the just perfect fifth, which corresponds to two notes in a frequency ratio of 3/2, is approximately 701.955 cents, and the just major third of 5/4 is ~386.314 cents. The 24-EDO neutral third is exactly 350 cents. The 22-EDO approximation to 3/2 is ~709.091 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="How to calculate the size of an interval in cents"></a><!-- ws:end:WikiTextHeadingRule:5 -->How to calculate the size of an interval in cents</h1>
 To find the size of a just interval in cents, you have to calculate the <a class="wiki_link" href="/log2">binary logarithm</a> (log<span style="font-size: 80%; vertical-align: sub;">2</span>) of its <a class="wiki_link" href="/frequency%20ratio">frequency ratio</a>, and multiply this by 1200.<br />
<br />
Example (just perfect fifth): 1200 × log<span style="font-size: 80%; vertical-align: sub;">2</span>(3/2) = 1200 × ~0.584 = ~701.955 cents<br />
<br />
If your pocket calculator has no <em>log2</em> key, but does have a <em>log</em> (log<span style="font-size: 80%; vertical-align: sub;">10</span>) or <em>ln</em> (log<span style="font-size: 80%; vertical-align: sub;">e</span>) key, you can key it this way:<br />
<!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/28242337?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;28242337&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><button>(frequency ratio)</button> <button>log</button> <button>÷</button> <button>2</button> <button>log</button> <button>=</button><!-- ws:end:WikiTextMediaRule:0 --><br />
(This makes use of the property of logarithms that log<span style="font-size: 80%; vertical-align: sub;">2</span>(x) = log<span style="font-size: 80%; vertical-align: sub;">n</span>(x) / log<span style="font-size: 80%; vertical-align: sub;">n</span>(2). )<br />
For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.<br />
<br />
For EDO steps, which are already logarithmic, simply divide 1200 by the EDO size, then multiply by the number of steps. For example, 1 step of 31-EDO is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc3"><a name="Other Units of Interval Measure"></a><!-- ws:end:WikiTextHeadingRule:7 -->Other Units of Interval Measure</h1>
 The cent is commonly used because of its ease in communicating information about intervals to a 12-EDO-savvy audience. However, some have suggested that the cent be deprecated, as other than societal convention there's no reason to give 12-EDO inherent importance over any other decent tuning. In contrast, others have suggested that cents are a useful unit of interval measure for purely mathematical reasons, even despite of 12-EDO's current status as the dominant tuning in Western society.<br />
<br />
Whatever your stance, alternative measures of interval size can be found at <a class="wiki_link" href="/Interval%20size%20measure">Interval size measure</a>.<br />
<br />
One prominent alternative interval measure is the <a class="wiki_link" href="/millioctave">millioctave</a> (mO).<br />
<br />
Additionally, a useful generalization for the cent measure is the <strong><a class="wiki_link" href="/relative%20cent">relative cent</a>,</strong> which is one 100th of two neighboring <a class="wiki_link" href="/pitch">pitches</a> in any <a class="wiki_link" href="/equal">equal</a> tuning.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc4"><a name="References"></a><!-- ws:end:WikiTextHeadingRule:9 -->References</h1>
 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cent_%28music%29" rel="nofollow">Wikipedia article on cents</a></body></html>

@@ 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:hstraub|hstraub]] and made on <tt>2016-11-10 04:42:57 UTC</tt>.<br>
+: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2018-01-26 15:01:17 UTC</tt>.<br>
-: The original revision id was <tt>599072716</tt>.<br>
+: The original revision id was <tt>625431819</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 24: / Line 24: @@
 [[media type="custom" key="28242337"]]
 (This makes use of the property of logarithms that log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;2&lt;/span&gt;(x) = log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;n&lt;/span&gt;(x) / log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;n&lt;/span&gt;(2). )
+For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.
 For EDO steps, which are already logarithmic, simply divide 1200 by the EDO size, then multiply by the number of steps. For example, 1 step of 31-EDO is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents.
@@ Line 57: / Line 58: @@
 &lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/28242337?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;28242337&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;button&gt;(frequency ratio)&lt;/button&gt; &lt;button&gt;log&lt;/button&gt; &lt;button&gt;÷&lt;/button&gt; &lt;button&gt;2&lt;/button&gt; &lt;button&gt;log&lt;/button&gt; &lt;button&gt;=&lt;/button&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
 (This makes use of the property of logarithms that log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;2&lt;/span&gt;(x) = log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;n&lt;/span&gt;(x) / log&lt;span style="font-size: 80%; vertical-align: sub;"&gt;n&lt;/span&gt;(2). )&lt;br /&gt;
+For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.&lt;br /&gt;
 &lt;br /&gt;
 For EDO steps, which are already logarithmic, simply divide 1200 by the EDO size, then multiply by the number of steps. For example, 1 step of 31-EDO is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents.&lt;br /&gt;

Cent: Difference between revisions

Revision as of 15:01, 26 January 2018

IMPORTED REVISION FROM WIKISPACES

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