POTE tuning: Difference between revisions

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'''Destretched tunings''' are tuning [[optimization]] techniques with the [[tuning map]] scaled until a certain interval is just, that is, its stretch introduced in the optimization is removed. '''DTE tuning''' ('''destretched Tenney–Euclidean tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.

The most significant form of DTE tuning is pure-octave destretched, which is assumed unless specified otherwise. This has been called the '''POTE tuning''' ('''pure-octave Tenney–Euclidean tuning'''), although there are other ways to enforce a pure octave (→ [[Constrained tuning]]). POTE can be computed from [[TE tuning]] with all primes scaled until 2/1 is just.

== Motivation ==

POTE is the same as TE in the limit of very small intervals. This means it's most similar to TE for intervals smaller than an octave, and most divergent for intervals of several octaves. As a tuning for the full audible range, the logic is that smaller intervals are more common in chords and so more important to optimize for. There are other ways to do this. POTE is the simplest.

POTE can stand in for TE where a pure octave tuning is convenient for implementation constraints, like when a synthesizer has pure octave tuning tables. POTE is close to TE for melodic steps, so melodies can be translated between POTE and TE with minimal damage.

POTE has the conceptual advantage that it's a simple deformation of TE, itself a simple measure, and introduces no more free parameters. POTE can also be used to give a feel for how a tuning damages different odd primes and other simple intervals without requiring the mental arithmetic of juggling multiples of the damage of 2:1. (TE with a basis of 2:1, 3:2, 5:4, etc would also do this.)

POTE has practical advantages for tuning instruments constrained to pure octaves as part of a band targeting TE. You can set the absolute pitch reference for each instrument so that it agrees with the TE background for a target register. Guitars (or other fretted string instruments) can implement this within themselves by having the frets assuming pure octaves and the open strings following the TE stretch.

Psychoacoustics shows (Terhardt's website was a good reference for this) that many bands are tuned according to stretched octaves even when the instruments are producing harmonic timbres. This might be with each instrument having a stretched scale, or high-pitched instruments having a slightly sharp pitch reference. The magnitude of this stretch often swamps the optimal stretch for TE (which can be in either direction). So, if you aren't going to observe the TE stretch, you might as well simplify it out. There are other reasons for putting instruments deliberately out of tune, for example solo instruments can be tuned slightly sharp to make them stand out. This leads to an upward drift of pitch reference in European orchestras: pianos are tuned slightly sharp to make them sound bright, and then the orchestra sharpens up to follow them.

== Approximate Kees optimality ==

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 '''Destretched tunings''' are tuning [[optimization]] techniques with the [[tuning map]] scaled until a certain interval is just, that is, its stretch introduced in the optimization is removed. '''DTE tuning''' ('''destretched Tenney–Euclidean tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.
 The most significant form of DTE tuning is pure-octave destretched, which is assumed unless specified otherwise. This has been called the '''POTE tuning''' ('''pure-octave Tenney–Euclidean tuning'''), although there are other ways to enforce a pure octave (→ [[Constrained tuning]]). POTE can be computed from [[TE tuning]] with all primes scaled until 2/1 is just.
+== Motivation ==
+POTE is the same as TE in the limit of very small intervals.  This means it's most similar to TE for intervals smaller than an octave, and most divergent for intervals of several octaves.  As a tuning for the full audible range, the logic is that smaller intervals are more common in chords and so more important to optimize for.  There are other ways to do this.  POTE is the simplest.
+POTE can stand in for TE where a pure octave tuning is convenient for implementation constraints, like when a synthesizer has pure octave tuning tables.  POTE is close to TE for melodic steps, so melodies can be translated between POTE and TE with minimal damage.
+POTE has the conceptual advantage that it's a simple deformation of TE, itself a simple measure, and introduces no more free parameters.  POTE can also be used to give a feel for how a tuning damages different odd primes and other simple intervals without requiring the mental arithmetic of juggling multiples of the damage of 2:1.  (TE with a basis of 2:1, 3:2, 5:4, etc would also do this.)
+POTE has practical advantages for tuning instruments constrained to pure octaves as part of a band targeting TE.  You can set the absolute pitch reference for each instrument so that it agrees with the TE background for a target register.  Guitars (or other fretted string instruments) can implement this within themselves by having the frets assuming pure octaves and the open strings following the TE stretch.
+Psychoacoustics shows (Terhardt's website was a good reference for this) that many bands are tuned according to stretched octaves even when the instruments are producing harmonic timbres.  This might be with each instrument having a stretched scale, or high-pitched instruments having a slightly sharp pitch reference.  The magnitude of this stretch often swamps the optimal stretch for TE (which can be in either direction).  So, if you aren't going to observe the TE stretch, you might as well simplify it out.  There are other reasons for putting instruments deliberately out of tune, for example solo instruments can be tuned slightly sharp to make them stand out.  This leads to an upward drift of pitch reference in European orchestras: pianos are tuned slightly sharp to make them sound bright, and then the orchestra sharpens up to follow them.
 == Approximate Kees optimality ==