Tenney–Euclidean tuning: Difference between revisions

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== 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.

TE tuning combines the ideas of Tenney weighting of prime limits and optimizing the root mean square (RMS) of weighted error/damage. Prime limits are a way of choosing musically useful intervals without being too specific about which intervals you think are useful. Tenney weighting is a way of giving equal treatment to intervals of equal complexity without choosing a finite set of intervals. RMS optimization (least squares) implies optimizing for the average sensory dissonance of an interval, and so allows for good consonances to balance weak dissonances and well tuned intervals to balance poorly tuned intervals.

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.

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.

== Weaknesses ==

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 == 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.
+TE tuning combines the ideas of Tenney weighting of prime limits and optimizing the root mean square (RMS) of weighted error/damage.  Prime limits are a way of choosing musically useful intervals without being too specific about which intervals you think are useful.  Tenney weighting is a way of giving equal treatment to intervals of equal complexity without choosing a finite set of intervals.  RMS optimization (least squares) implies optimizing for the average sensory dissonance of an interval, and so allows for good consonances to balance weak dissonances and well tuned intervals to balance poorly tuned intervals.
+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.
 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.
 == Weaknesses ==