TOP tuning: Difference between revisions

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__FORCETOC__

=Proportional error=

A ''tuning'' for a regular temperament is defined by a vector T in [[Vals_and_Tuning_Space#Vals and Monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [http://en.wikipedia.org/wiki/Bra-ket_notation bra vector], and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). For example, if M is |-4 4 -1>; then q = 81/80. If T is >;1200 1900 2800| then <;T|M>; = -4800 + 7600 - 2800 = 0. Thus, while cents(q) = 21.506290, T(q) = 0.

A ''tuning'' for a regular temperament is defined by a vector T in [[Vals_and_Tuning_Space#Vals and Monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [http://en.wikipedia.org/wiki/Bra-ket_notation bra vector], and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). For example, if M is |-4 4 -1> then q = 81/80. If T is >1200 1900 2800| 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 difference between the value in cents T assigns to q and the actual size in cents of q. 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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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''.

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 __FORCETOC__
 =Proportional error=
-A ''tuning'' for a regular temperament is defined by a vector T in [[Vals_and_Tuning_Space#Vals and Monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [http://en.wikipedia.org/wiki/Bra-ket_notation bra vector], and if M is a monzo then &lt;T|M&gt; is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). For example, if M is |-4 4 -1&gt;; then q = 81/80. If T is &gt;;1200 1900 2800| then &lt;;T|M&gt;; = -4800 + 7600 - 2800 = 0. Thus, while cents(q) = 21.506290, T(q) = 0.
+A ''tuning'' for a regular temperament is defined by a vector T in [[Vals_and_Tuning_Space#Vals and Monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [http://en.wikipedia.org/wiki/Bra-ket_notation bra vector], and if M is a monzo then &lt;T|M&gt; is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). For example, if M is |-4 4 -1&gt; then q = 81/80. If T is &gt;1200 1900 2800| 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 difference between the value in cents T assigns to q and the actual size in cents of q. 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.
- i
 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''.