TOP tuning: Difference between revisions
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'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]]. | |||
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 unison) with minimal APE. This minimal proportional error is called 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 unison) with minimal APE. This minimal proportional error is called the TOP error. | ||
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Most importantly, the above result holds without any hitches for prime-limit subgroups as the limit tends to infinity, and in particular for infinite-limit generalized patent vals, where the TOP tuning minimizes the error on all rationals regardless of whether those rationals are mapped "consistently" given the mapping on the primes, or "inconsistently" given their direct rounding to the nearest edostep. | Most importantly, the above result holds without any hitches for prime-limit subgroups as the limit tends to infinity, and in particular for infinite-limit generalized patent vals, where the TOP tuning minimizes the error on all rationals regardless of whether those rationals are mapped "consistently" given the mapping on the primes, or "inconsistently" given their direct rounding to the nearest edostep. | ||
[[Category:Acronyms]] | [[Category:Acronyms]] | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category: | [[Category:Regular temperament tuning]] | ||