Relative cent
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- This revision was by author genewardsmith and made on 2011-09-21 23:55:47 UTC.
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**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. 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]], 5300 relative cents and so forth. An existing example is the [[turkish cent]], which is the relative cent of [[106edo]]. The iota, the relative cent for [[17edo]], has been proposed by [[George Secor]] and [[Margo Schulter]] for use with 17edo, and [[Tútim Deft Wafil]] has advocated the [[purdal]], which divides the octave into 9900 parts. The [[millioctave]] is another such measure, as it can be viewed as the relative cent measure for [[10edo]]. 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. 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 ¢. ---- //...also the term [[centidegree]] was suggested, but this seems to be used already as a unit for temperature.//
Original HTML content:
<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 /> <br /> 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>, 5300 relative cents and so forth.<br /> <br /> An existing example is the <a class="wiki_link" href="/turkish%20cent">turkish cent</a>, which is the relative cent of <a class="wiki_link" href="/106edo">106edo</a>. The iota, the relative cent for <a class="wiki_link" href="/17edo">17edo</a>, has been proposed by <a class="wiki_link" href="/George%20Secor">George Secor</a> and <a class="wiki_link" href="/Margo%20Schulter">Margo Schulter</a> for use with 17edo, and <a class="wiki_link" href="/T%C3%BAtim%20Deft%20Wafil">Tútim Deft Wafil</a> has advocated the <a class="wiki_link" href="/purdal">purdal</a>, which divides the octave into 9900 parts. The <a class="wiki_link" href="/millioctave">millioctave</a> is another such measure, as it can be viewed as the relative cent measure for <a class="wiki_link" href="/10edo">10edo</a>. <br /> <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 1.995 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 /> <br /> 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 "approximations": <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 /> <br /> <hr /> <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>