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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | The '''relative cent''' (symbol: '''r¢''', '''rct''', '''¢<sub>EDO</sub>''') is a [[unit of interval size]] based on a given [[equal]]-stepped tonal system (especially [[EDO]] systems). In being the size of 1 percent of the distance between adjacent pitches, it is the generalization of the [[common cent]] (¢). |
| 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:xenwolf|xenwolf]] and made on <tt>2011-06-29 07:53:58 UTC</tt>.<br>
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| : The original revision id was <tt>239300435</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> | |
| <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">**Relative cent** (**rct**, **r¢**) is a logarithmic [[interval size measure]] based on a given [[equal]]-stepped tonal system. Its size is 1 percent of the distance between adjacent pitches.
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| Given any N [[EDO]], the size of an interval in //relative cents// is N/12 times its size in [[Cent|cents]]; or equivalently, 100 N times its logarithm base 2. Hence in [[7edo]], the octave is 700 relative cents, in [[53edo]], 53000 relative cents and so forth.
| | The term ''centidegree'' was suggested as an alias, but this seems to be used already as a unit for temperature. |
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| Measuring the error of an approximation of an interval in an edo in terms of relative cents gives the relative error, which so long as the corresponding val is used is additive. For instance, the fifth of 12edo is 1.995 cents flat, or -1.955 cents sharp, which is therefore also its error in relative cents. The fifth of [[41edo]] is 1.654 relative cents sharp. Thus for 53=41+12, the fifth is -1.955 + 1.654 = -0.301 relative cents sharp, and hence (-0.301)*(12/53) = -0.068 cents sharp, which is to say 0.068 cents flat.
| | Given any ''N'' [[EDO]], the size of an interval in ''relative cents'' is ''N''/12 times its size in [[cent]]s; or equivalently, 100''N'' times its logarithm base 2. Hence in [[7edo|7EDO]], the octave is 700 relative cents, in [[53edo|53EDO]], 5300 relative cents and so forth. |
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| If you want to quantify the approximation of a given [[JI]] interval in an equal-stepped tonal systems, you can consider the absolute distance of 50 relative cent as the worst possible and 0 relative cent as the best possible. For example, [[5edo]] has a relatively good approximated [[natural seventh]] with the ratio [[7_4|7/4]]: the absolute distance of the 4th pith in 5edo is 8.826 ¢ or 3.677 r¢. But the approximations of its multiple edos [[10edo]] (7.355 r¢), [[15edo]] (11,03 r¢) ... gets relatively worse. So it's obvious that there will be multiple edos with a real bad "approximations": [[65edo]] has the 7/4 just between adjacent pithes (47,81 r¢), but its absolute approximation of this interval in cents is still the same as for 5edo: 8.826 ¢.
| | An existing example is the [[turkish cent]], which is the relative cent of [[106edo|106EDO]]. The iota, the relative cent for [[17edo|17EDO]], has been proposed by [[George Secor]] and [[Margo Schulter]] for use with 17EDO, and [[Tútim Dennsuul Wafiil|Tútim Dennsuul]] has advocated the [[purdal]], which divides the octave into 9900 parts (being relative cents of [[99edo|99EDO]]). The [[millioctave]] is another such measure, as it can be viewed as the relative cent measure for [[10edo|10EDO]]. The śat, for use with Armodue, divides the octave into 1600 parts (Armodue being a theory of [[16edo|16EDO]]). |
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| | == Relative error == |
| //...also the term [[centidegree]] was suggested, but this seems to be used already as a unit for temperature.//</pre></div>
| | {{Main| Relative interval error }} |
| <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>Relative cent</title></head><body><strong>Relative cent</strong> (<strong>rct</strong>, <strong>r¢</strong>) is a logarithmic <a class="wiki_link" href="/interval%20size%20measure">interval size measure</a> based on a given <a class="wiki_link" href="/equal">equal</a>-stepped tonal system. Its size is 1 percent of the distance between adjacent pitches.<br />
| | Measuring the error of an approximation of an interval in an EDO in terms of relative cents gives the '''relative error''', which so long as the corresponding val is used is additive. For instance, the fifth of 12EDO is 1.955{{c}} flat, or −1.955 cents sharp, which is therefore also its error in relative cents. The fifth of [[41edo|41EDO]] is 1.654 relative cents sharp. Thus for {{nowrap|53 {{=}} 41 + 12}}, the fifth is {{nowrap|−1.955 + 1.654 {{=}} −0.301}} relative cents sharp, and hence {{nowrap|(−0.301) × (12/53) {{=}} −0.068{{c}}}} sharp, which is to say 0.068{{c}} flat. |
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| Given any N <a class="wiki_link" href="/EDO">EDO</a>, the size of an interval in <em>relative cents</em> is N/12 times its size in <a class="wiki_link" href="/Cent">cents</a>; or equivalently, 100 N times its logarithm base 2. Hence in <a class="wiki_link" href="/7edo">7edo</a>, the octave is 700 relative cents, in <a class="wiki_link" href="/53edo">53edo</a>, 53000 relative cents and so forth.<br />
| | == Application for quantifying approximation == |
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| | If you want to quantify the approximation of a given [[JI]] interval in a given [[Equal|equal-stepped]] tonal system, you can consider the absolute distance of 50 r¢ as the worst possible and 0 r¢ as the best possible. For example, [[5edo|5EDO]] has a relatively good approximated [[natural seventh]] with the ratio [[7/4]]: the absolute distance of 4\5 in 5EDO is 8.82{{c}} or 3.677 r¢ flat of 7/4. But the approximations of its multiple EDOs [[10edo|10EDO]] (7.355 r¢), [[15edo|15EDO]] (11.032 r¢) … become progressively worse (in a relative sense). So in [[65edo|65EDO]], there is the 7/4 situated halfway between two adjacent pitches (off by at least 47.807 r¢), but its absolute distance from this interval in cents is still the same as for 5EDO: 8.826{{c}} flat. See [[Pepper ambiguity]] for a mathematical approach to quantify the approximations for sets of intervals. |
| Measuring the error of an approximation of an interval in an edo in terms of relative cents gives the relative error, which so long as the corresponding val is used is additive. For instance, the fifth of 12edo is 1.995 cents flat, or -1.955 cents sharp, which is therefore also its error in relative cents. The fifth of <a class="wiki_link" href="/41edo">41edo</a> is 1.654 relative cents sharp. Thus for 53=41+12, the fifth is -1.955 + 1.654 = -0.301 relative cents sharp, and hence (-0.301)*(12/53) = -0.068 cents sharp, which is to say 0.068 cents flat.<br /> | | |
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| | == See also == |
| If you want to quantify the approximation of a given <a class="wiki_link" href="/JI">JI</a> interval in an equal-stepped tonal systems, you can consider the absolute distance of 50 relative cent as the worst possible and 0 relative cent as the best possible. For example, <a class="wiki_link" href="/5edo">5edo</a> has a relatively good approximated <a class="wiki_link" href="/natural%20seventh">natural seventh</a> with the ratio <a class="wiki_link" href="/7_4">7/4</a>: the absolute distance of the 4th pith in 5edo is 8.826 ¢ or 3.677 r¢. But the approximations of its multiple edos <a class="wiki_link" href="/10edo">10edo</a> (7.355 r¢), <a class="wiki_link" href="/15edo">15edo</a> (11,03 r¢) ... gets relatively worse. So it's obvious that there will be multiple edos with a real bad &quot;approximations&quot;: <a class="wiki_link" href="/65edo">65edo</a> has the 7/4 just between adjacent pithes (47,81 r¢), but its absolute approximation of this interval in cents is still the same as for 5edo: 8.826 ¢.<br /> | | * [[Relative error]] − a measure for mapping quality |
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| | [[Category:Approximation]] |
| <em>...also the term <a class="wiki_link" href="/centidegree">centidegree</a> was suggested, but this seems to be used already as a unit for temperature.</em></body></html></pre></div>
| | [[Category:Interval size measures]] |
| | [[Category:Elementary math]] |
| | [[Category:Relative measures]] |
| | [[Category:Terms]] |