Relative cent: Difference between revisions

'''Relative cent''' ('''rct''', '''r¢''', '''¢_EDO''') is a logarithmic [[interval size measure]] 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]] (¢).

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]], 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_Dennsuul_Wafiil|Tútim Dennsuul]] has advocated the [[purdal]], which divides the octave into 9900 parts (being relative cents of [[99edo]]). 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 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|natural seventh]] with the ratio [[7/4]]: the absolute distance of 4\5 in 5edo is 8.826 ¢ or 3.677 r¢ flat of 7/4. But the approximations of its multiple edos [[10edo]] (7.355 r¢), [[15edo]] (11.032 r¢) ... become progressively worse (in a relative sense). So in [[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 ¢ flat. See [[Pepper ambiguity]] for a mathematical approach to quantify the approximations for sets of intervals.

 

* [[Relative cent/Problem]] -- a WIP addressing the shortcomings of the relative cent

* [[centidegree]] -- was suggested as an alias, but this seems to be used already as a unit for temperature

[[Category:Interval measure]]

[[Category:Practical help]]

[[Category:Relative measure]]

[[Category:Term]]

[[Category:Unit]]