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'' ~~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

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 ''proportional error'' and ''absolute proportional error'' take 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 Benedetti height is nd and the Tenney height is log₂(nd).

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)).

The ''proportional error'' is defined as 0 when q equals 1 and otherwise PE(q) = Err(q)/cents(nd) = Err(q)/1200log₂(nd).

The ''absolute proportional error'' is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(nd) = Arr(q)/1200log₂(nd)

~~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.

Note that the same logarithmic measure - cents, expressed as 1200log₂ - is used in both numerator and denominator, so a logarithm with any other base would yield the same result. Thus, the definition is not in fact based on cents.

An equivalent way to write the above error metrics is as APE(q) = Arr(q)/(1200*Ten(q)) and PE(q) = Err(q)/(1200*Ten(q)), where Ten(q) is the [[Tenney_Height|Tenney height]], and the 1200 in the denominator is only necessary to cancel out the use of cents in the numerator. Due to the use of the log-weighting in the denominator, these metrics are often collectively referred to as ''Tenney-weighted error''.

These metrics are often collectively referred to as ''Tenney-weighted error''.

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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'' 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
+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 ''proportional error'' and ''absolute proportional error'' take 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 Benedetti height is nd and the Tenney height is log₂(nd).
-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)).
+The ''proportional error'' is defined as 0 when q equals 1 and otherwise PE(q) = Err(q)/cents(nd) = Err(q)/1200log₂(nd).
+The ''absolute proportional error'' is defined as 0 when q equals 1 and otherwise APE(q) = Arr(q)/cents(nd) = Arr(q)/1200log₂(nd)
-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.
+Note that the same logarithmic measure - cents, expressed as 1200log₂ - is used in both numerator and denominator, so a logarithm with any other base would yield the same result. Thus, the definition is not in fact based on cents.
-An equivalent way to write the above error metrics is as APE(q) = Arr(q)/(1200*Ten(q)) and PE(q) = Err(q)/(1200*Ten(q)), where Ten(q) is the [[Tenney_Height|Tenney height]], and the 1200 in the denominator is only necessary to cancel out the use of cents in the numerator. Due to the use of the log-weighting in the denominator, these metrics are often collectively referred to as ''Tenney-weighted error''.
+These metrics are often collectively referred to as ''Tenney-weighted error''.
 =TOP tuning=