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).

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 k is the rational number which M represents, then we may also write this quantity as T(k).

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).

For example, if M is |-4 4 -1> then k = 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(k) = 21.506290, T(k) = 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.

Given a tuning T and any 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 terms ''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 ''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)~~. Note that the same logarithmic measure - cents, expressed as 1200log₂ - is being 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, which are used simply for convenience.

The terms ''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 ''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 the absolute value of the ''proportional error''. Note that the same logarithmic measure - cents, expressed as 1200log₂ - is being 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, which are used simply for convenience.

These quantities 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).
+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 k is the rational number which M represents, then we may also write this quantity as T(k).
-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).
+For example, if M is |-4 4 -1&gt; then k = 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(k) = 21.506290, T(k) = 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.
+Given a tuning T and any 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 terms ''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 ''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). Note that the same logarithmic measure - cents, expressed as 1200log₂ - is being 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, which are used simply for convenience.
+The terms ''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 ''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 the absolute value of the ''proportional error''. Note that the same logarithmic measure - cents, expressed as 1200log₂ - is being 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, which are used simply for convenience.
 These quantities are often collectively referred to as ''Tenney-weighted error''.