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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | en = Cent |
| : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2016-11-10 04:42:57 UTC</tt>.<br>
| | | de = Cent |
| : The original revision id was <tt>599072716</tt>.<br>
| | | es = Centésimas |
| : The revision comment was: <tt></tt><br>
| | | ja = セント |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | | ko = 센트 |
| <h4>Original Wikitext content:</h4>
| | | ro = Centisunet |
| <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>
| | {{Wikipedia|Cent (music)}} |
| =Definitions= | | The '''cent''' (symbol: '''¢''') is a [[unit of interval size]] equal to exactly 1/100 (or 1%) of a [[12edo]] [[semitone (interval size measure)|semitone]]. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by {{w|Alexander John Ellis|Alexander J. Ellis}}, the cent may also be defined as the {{w|logarithm}} base 1200th root of 2 of a ratio. |
| 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).
| | Cents are often used to express the size of intervals in different tuning systems, sometimes to express the accuracy of the representation of a [[just intonation]] [[interval]] in a given system. |
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| =Examples= | | == 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 12edo perfect fifth is exactly 700 cents, and the 12edo 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 702 cents, and the just major third of [[5/4]] is about 386 cents. The [[24edo]] neutral third is exactly 350 cents. The [[22edo]] approximation to 3/2 is approximately 709 cents. |
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| =How to calculate the size of an interval in cents= | | == Conversion == |
| 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.
| | {{See also| Ratio #Conversion }} |
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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
| | === Ratio to cents === |
| | To find the size ''s'' of an interval in cents from its ratio ''r'', calculate the [[log2|binary logarithm]] (log<sub>2</sub>) of its [[frequency ratio]], and multiply it by 1200. |
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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:
| | <math>\displaystyle s = 1200 \cdot \log_2 (r)</math> |
| [[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 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 example, the size in cents of a just perfect fifth is 1200⋅log<sub>2</sub>(3/2) ≈ 1200 × 0.584 ≈ 701.955 cents. |
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| =Other Units of Interval Measure=
| | If your pocket calculator has no <code>log2</code> key, but does have a <code>log</code> (log<sub>10</sub>) or <code>ln</code> (log<sub>''e''</sub>) key, you can key it this way: |
| 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]].
| | <code>(frequency ratio)</code> <code>log</code> <code>÷</code> <code>2</code> <code>log</code> <code>=</code> |
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| One prominent alternative interval measure is the [[millioctave]] (mO).
| | This makes use of the property of logarithms that log<sub>2</sub>(''x'') = log<sub>''n''</sub>(''x'') / log<sub>''n''</sub>(2). |
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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.
| | For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed. |
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| =References= | | === Edosteps to cents === |
| [[http://en.wikipedia.org/wiki/Cent_%28music%29|Wikipedia article on cents]]</pre></div> | | For [[edo]]steps, which are already logarithmic, simply divide 1200 by the edo number, then multiply by the number of steps. |
| <h4>Original HTML 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;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 />
| | For example, 1 step of 31edo is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents. |
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| <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Definitions"></a><!-- ws:end:WikiTextHeadingRule:1 -->Definitions</h1>
| | === Monzo to cents === |
| 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 />
| | To find the size ''s'' of a just interval in cents from its [[monzo]] '''m''' = {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> ''m''<sub>3</sub> … }}, left-multiply '''m''' by the [[just tuning map]] in cents ''T''<sub>''J''</sub> = {{val| 1200.000 1901.955 2786.314 … }} |
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| 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 />
| | <math>\displaystyle s = T_J \cdot \vec m</math> |
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| <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:3 -->Examples</h1>
| | == Other interval size units == |
| 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 />
| | The cent is commonly used because of its ease in communicating information about intervals to a 12edo-savvy audience. However, some have suggested that the cent be deprecated, as other than societal convention there's no reason to give 12edo 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 12edo's current status as the dominant tuning in Western society. |
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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>
| | In the Xenharmonic Wiki there is broad agreement to stick to cents as a general interval measure. Under certain circumstances, alternative [[interval size measure]]s are provided in addition. |
| 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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| | == See also == |
| 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 />
| | * [[Relative cent]] – a useful generalization for the cent measure to ''any'' [[equal]] tuning |
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| | * [[Millioctave]] – one prominent alternative interval measure |
| 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 />
| | * [[Interval size measure]] – overview |
| <!-- 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 />
| | == External links == |
| <br />
| | * [http://tonalsoft.com/enc/c/cent.aspx cent, ¢, 1200-ed2] on [[Tonalsoft Encyclopedia]] |
| 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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| | [[Category:Interval size measures]] |
| <!-- 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>
| | [[Category:Elementary math]] |
| 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 />
| | [[Category:Terms]] |
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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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| One prominent alternative interval measure is the <a class="wiki_link" href="/millioctave">millioctave</a> (mO).<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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| <!-- 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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The cent (symbol: ¢) is a unit of interval size equal to exactly 1/100 (or 1%) of a 12edo semitone. In other words, cents divide the half step (semitone) of 12edo into 100 equal parts. First proposed in the late 19th century by Alexander J. Ellis, the cent may also be defined as the logarithm base 1200th root of 2 of a ratio.
Cents are often used to express the size of intervals in different tuning systems, sometimes to express the accuracy of the representation of a just intonation interval in a given system.
Examples
The 12edo perfect fifth is exactly 700 cents, and the 12edo 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 702 cents, and the just major third of 5/4 is about 386 cents. The 24edo neutral third is exactly 350 cents. The 22edo approximation to 3/2 is approximately 709 cents.
Conversion
Ratio to cents
To find the size s of an interval in cents from its ratio r, calculate the binary logarithm (log2) of its frequency ratio, and multiply it by 1200.
[math]\displaystyle{ \displaystyle s = 1200 \cdot \log_2 (r) }[/math]
For example, the size in cents of a just perfect fifth is 1200⋅log2(3/2) ≈ 1200 × 0.584 ≈ 701.955 cents.
If your pocket calculator has no log2 key, but does have a log (log10) or ln (loge) key, you can key it this way:
(frequency ratio) log ÷ 2 log =
This makes use of the property of logarithms that log2(x) = logn(x) / logn(2).
For scientific calculators, the order of buttons may be different, and a right parenthesis may be needed.
Edosteps to cents
For edosteps, which are already logarithmic, simply divide 1200 by the edo number, then multiply by the number of steps.
For example, 1 step of 31edo is 1200 ÷ 31 = ~38.710 cents; 5 steps of 31 is ~193.548 cents.
Monzo to cents
To find the size s of a just interval in cents from its monzo m = [m1 m2 m3 …⟩, left-multiply m by the just tuning map in cents TJ = ⟨1200.000 1901.955 2786.314 …]
[math]\displaystyle{ \displaystyle s = T_J \cdot \vec m }[/math]
Other interval size units
The cent is commonly used because of its ease in communicating information about intervals to a 12edo-savvy audience. However, some have suggested that the cent be deprecated, as other than societal convention there's no reason to give 12edo 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 12edo's current status as the dominant tuning in Western society.
In the Xenharmonic Wiki there is broad agreement to stick to cents as a general interval measure. Under certain circumstances, alternative interval size measures are provided in addition.
See also
External links
