TOP tuning: Difference between revisions

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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 (a [[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 (the tuning tempers away the syntonic comma).

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

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).

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 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 (a [[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 (the tuning tempers away the syntonic comma).
+For example, if M is |-4 4 -1&gt; then q = 81/80 (a [[syntonic comma]]). If T is &lt;1200 1900 2800| (a multiple of [[12edo]]) then &lt;T|M&gt; = -4800 + 7600 - 2800 = 0. Thus, while cents(q) = 21.506290, T(q) = 0 (i.e., the tuning tempers away the syntonic comma).
 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).