Tenney–Euclidean tuning: Difference between revisions
Actual formulation of the problem. Display style |
Motivation |
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In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''. | In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''. | ||
== Motivation == | |||
TE shares with TOP tuning the insight that adding octaves to the optimization simplifies the calculation. Allowing for scale stretch balances intervals of different sizes. Tenney weighting weights each prime according to its complexity and if you consider an average over an infinite set of primes, it means similarly sized intervals get similar weight and arbitrarily complex intervals get an arbitrarily small weight. | |||
As an RMS measure, TE error is optimized by the least squares method, which is well known and simple and efficient to calculate. | |||
TE tuning is uniquely optimized for a given prime limit. There are no free parameters determining the weighting of different intervals or the balance of wide and narrow intervals: all follow from the definition of Tenney weighting of primes. | |||
== Definition == | == Definition == | ||