TOP tuning: Difference between revisions

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For example, if M is |-4 4 -1> then q = 81/80 ([[syntonic comma]]). If T is <1200 1900 2800| ([[12edo]]) then <T|M> = -4800 + 7600 - 2800 = 0. Thus, while cents(q) = 21.506290, T(q) = 0.

Given a tuning T and a rational number q in the domain of T, the ''signed error'' of T on q is defined as Err(q) = T(q) - cents(q). The ''absolute error'' Arr(q) = |Err(q)| is the absolute value of the signed error. The ''absolute proportional error'' is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(Ben(q)), where Ben(q) is the [[Benedetti_height|Benedetti height]], the product of the numerator and denominator of q. Similarly, the ''proportional error'' PE(q) = Err(q)/cents(Ben(q)).

Given a tuning T and a rational number q in the domain of T, the ''signed error'' of T on q is defined as Err(q) = T(q) - cents(q). The ''absolute error'' Arr(q) = |Err(q)| is the absolute value of the signed error. The ''absolute proportional error'' takes into account the [[Benedetti_height|Benedetti height]] or [[Tenney_Height|Tenney height]] of q. If q is expressed as a fraction n/d in lowest terms, then the Benedetti height is nd and the Tenney height is

The ''absolute proportional error'' is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(Ben(q)), where Ben(q) is the [[Benedetti_height|Benedetti height]], the product of the numerator and denominator of q. Similarly, the ''proportional error'' PE(q) = Err(q)/cents(Ben(q)).

While the above definition seems to use cents to define proportional error, any logarithm base will lead to the same result, so that the definition is not in fact based on cents.

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 For example, if M is |-4 4 -1&gt; then q = 81/80 ([[syntonic comma]]). If T is &lt;1200 1900 2800| ([[12edo]]) then &lt;T|M&gt; = -4800 + 7600 - 2800 = 0. Thus, while cents(q) = 21.506290, T(q) = 0.
-Given a tuning T and a rational number q in the domain of T, the ''signed error'' of T on q is defined as Err(q) = T(q) - cents(q). The ''absolute error'' Arr(q) = |Err(q)| is the absolute value of the signed error. The ''absolute proportional error'' is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(Ben(q)), where Ben(q) is the [[Benedetti_height|Benedetti height]], the product of the numerator and denominator of q. Similarly, the ''proportional error'' PE(q) = Err(q)/cents(Ben(q)).
+Given a tuning T and a rational number q in the domain of T, the ''signed error'' of T on q is defined as Err(q) = T(q) - cents(q). The ''absolute error'' Arr(q) = |Err(q)| is the absolute value of the signed error. The ''absolute proportional error'' takes into account the [[Benedetti_height|Benedetti height]] or [[Tenney_Height|Tenney height]] of q. If q is expressed as a fraction n/d in lowest terms, then the Benedetti height is nd and the Tenney height is
+The ''absolute proportional error'' is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(Ben(q)), where Ben(q) is the [[Benedetti_height|Benedetti height]], the product of the numerator and denominator of q. Similarly, the ''proportional error'' PE(q) = Err(q)/cents(Ben(q)).
 While the above definition seems to use cents to define proportional error, any logarithm base will lead to the same result, so that the definition is not in fact based on cents.