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 ~~size~~ of the ~~interval~~, 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).

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).

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)).

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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 size of the interval, 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).
+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).
 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)).