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Encyclopedia of Microtonal Music Theory

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[From John deLaubenfels, private communication with Joe Monzo]

"Pain" is the word I use to describe the feeling from less-than-ideal tuning; it is therefore the thing to be minimized for the closest possible approach to ideal tuning in a given piece.

I model three sources for pain when doing adaptive-tuning work:

  1. Actual tuning of intervals, at a given moment, that is not ideal, which for me, means "not JI".
  2. A cetain note's tuning changing over time, especially if the note is continuously sounding as its tuning changes.
  3. The overall tuning center drifting away from where it began (similar to #2, but "strategic" where it is "tactical").

I hypothesize that pain is proportional to the square of deviation from ideal; thus a major third tuned 14 cents sharp, as in 12-tET, is modeled as four times as painful as one tuned 7 cents sharp (or flat).

My tuning model uses an analog to a physical model of nodes connected by springs. In this model, deviation from JI is equivalent to deflection of a spring from rest position, and pain is equivalent to energy stored by a single spring, or by the summation over the entire matrix of springs. Such a model is very amenable to relaxation to minimum energy, thus minimum "pain", thus best possible tuning.

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