TOP tuning: Difference between revisions

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=TOP tuning=

For any tuning T, we may define the absolute proportional error APE(T) of T as the [http://mathworld.wolfram.com/Supremum.html supremum] (maximum) of the absolute proportional errors of all q belonging to the domain of T~~; that is, for which T provides a value~~. A '''TOP tuning''' for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal APE. This minimal proportional error is ~~a measure of the error of the temperament, which we might call~~ the TOP error.

For any tuning T, we may define the absolute proportional error APE(T) of T as the [http://mathworld.wolfram.com/Supremum.html supremum] (maximum) of the absolute proportional errors of all q belonging to the domain of T. A '''TOP tuning''' for a regular temperament is a tuning supporting the temperament (i.e., one which sends commas of the temperament to 0) with minimal APE. This minimal proportional error is called the TOP error.

The concept of a TOP tuning was first suggested by [[Paul_Erlich|Paul Erlich]]~~, who gave it its name, which~~ stands for both Tenney OPtimal and Tempered Octaves Please~~, the~~ latter ~~due~~ to the fact that ~~usually~~ the ~~octaves are tempered~~.

The concept of a TOP tuning was first suggested and named by [[Paul_Erlich|Paul Erlich]]. The acronym stands for both Tenney OPtimal and Tempered Octaves Please. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1.

=Maximal error semigroups=

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 =TOP tuning=
-For any tuning T, we may define the absolute proportional error APE(T) of T as the [http://mathworld.wolfram.com/Supremum.html supremum] (maximum) of the absolute proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A '''TOP tuning''' for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal APE. This minimal proportional error is a measure of the error of the temperament, which we might call the TOP error.
+For any tuning T, we may define the absolute proportional error APE(T) of T as the [http://mathworld.wolfram.com/Supremum.html supremum] (maximum) of the absolute proportional errors of all q belonging to the domain of T. A '''TOP tuning''' for a regular temperament is a tuning supporting the temperament (i.e., one which sends commas of the temperament to 0) with minimal APE. This minimal proportional error is called the TOP error.
-The concept of a TOP tuning was first suggested by [[Paul_Erlich|Paul Erlich]], who gave it its name, which stands for both Tenney OPtimal and Tempered Octaves Please, the latter due to the fact that usually the octaves are tempered.
+The concept of a TOP tuning was first suggested and named by [[Paul_Erlich|Paul Erlich]]. The acronym stands for both Tenney OPtimal and Tempered Octaves Please. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1.
 =Maximal error semigroups=