TOP tuning: Difference between revisions

Line 7:

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 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.

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''.

@@ Line 7: / Line 7: @@
 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 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.
+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''.