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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | <span style="display: block; text-align: right;">[[:de:Cent Deutsch]] - [[Centésimas|Español]]</span> |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2018-01-26 15:01:17 UTC</tt>.<br>
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| : The original revision id was <tt>625431819</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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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.<br>
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| <h4>Original Wikitext content:</h4>
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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"><span style="display: block; text-align: right;">[[xenharmonie/Cent|Deutsch]] - [[Centésimas|Español]]
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| </span>
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| =Definitions=
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| 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.
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| 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).
| | =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. |
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| =Examples= | | 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). |
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| | =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. | | 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. |
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| =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. | | 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|frequency ratio]], and multiply this by 1200. |
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| | Example (just perfect fifth): 1200 × log<span style="font-size: 80%; vertical-align: sub;">2</span>(3/2) = 1200 × ~0.584 = ~701.955 cents |
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| | 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: |
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| | (frequency ratio) log ÷ 2 log = |
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| Example (just perfect fifth): 1200 × log<span style="font-size: 80%; vertical-align: sub;">2</span>(3/2) = 1200 × ~0.584 = ~701.955 cents
| | (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). ) |
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| 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:
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| [[media type="custom" key="28242337"]]
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| (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). )
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| For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed. | | For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed. |
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| 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. | | 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. |
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| =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. |
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| 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|Interval size measure]]. |
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| One prominent alternative interval measure is the [[millioctave]] (mO). | | One prominent alternative interval measure is the [[millioctave|millioctave]] (mO). |
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| 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. | | Additionally, a useful generalization for the cent measure is the '''[[Relative_cent|relative cent]],''' which is one 100th of two neighboring [[pitch|pitches]] in any [[Equal|equal]] tuning. |
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| =References= | | =References= |
| [[http://en.wikipedia.org/wiki/Cent_%28music%29|Wikipedia article on cents]]</pre></div>
| | [http://en.wikipedia.org/wiki/Cent_%28music%29 Wikipedia article on cents] [[Category:absolute_measure]] |
| <h4>Original HTML content:</h4>
| | [[Category:interval]] |
| <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><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 />
| | [[Category:interval_measure]] |
| </span><br />
| | [[Category:interval_size_measure]] |
| <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Definitions"></a><!-- ws:end:WikiTextHeadingRule:1 -->Definitions</h1>
| | [[Category:logarithmic_measure]] |
| 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 />
| | [[Category:measure]] |
| <br />
| | [[Category:size]] |
| 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 />
| | [[Category:unit]] |
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| <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:3 -->Examples</h1>
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| 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 />
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| <!-- 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>
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| 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 />
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| 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 />
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| 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 />
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| <!-- 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 />
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| (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 />
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| For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.<br />
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| <br />
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| 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 />
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| <br />
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| <!-- 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>
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| 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 />
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| <br />
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| 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 />
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| <br />
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| One prominent alternative interval measure is the <a class="wiki_link" href="/millioctave">millioctave</a> (mO).<br />
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| <br />
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| 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 />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc4"><a name="References"></a><!-- ws:end:WikiTextHeadingRule:9 -->References</h1>
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| <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>
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