Cent: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 313602954 - Original comment: ** |
Wikispaces>spt3125 **Imported revision 509661674 - 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:spt3125|spt3125]] and made on <tt>2014-05-18 14:48:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509661674</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">= | <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">=Definitions= | ||
A //cent// is an interval equal to exactly 1/100th of a 12-EDO semitone. In other words, cents equally divide the 12-EDO | 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). | |||
The | =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= | =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. | |||
If | 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="25953772"]] | |||
(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 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= | =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. | 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]]. | 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]]</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>cent</title></head><body><!-- ws:start:WikiTextHeadingRule: | <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>cent</title></head><body><!-- 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 12-EDO semitone. In other words, cents equally divide the 12-EDO | 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 /> | <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 /> | <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 /> | <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 /> | <br /> | ||
If | 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/25953772?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;25953772&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><button>(number)</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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | 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 /> | 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 /> | <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 /> | 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 /> | <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 /> | <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></pre></div> | |||