Tp tuning: Difference between revisions

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'''T''p'' tuning''' is a generalization of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning.

'''T''p'' tuning''' is a generalization of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning. (In this article ''p'' denotes a parameter, ''p'' ≥ 1; it does not denote a prime.)

For a subgroup temperament over a general JI subgroup, and for a given choice of ''p'' (most commonly ''p'' = 2), there are two notions of optimization:

* The first is called '''inharmonic TE''', because the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres. Inharmonic TE depends on the basis used for the subgroup. In non-octave temperaments, inharmonic TE could be used when optimizing a specific voicing of a tempered JI chord. For example in 3/2.7/4.5/2 semiwolf temperament which tempers out 245/243, the 3/2.7/4.5/2 inharmonic TE optimizes the 4:6:7:10 chord.

* The second is called '''subgroup TE''', because it treats the temperament as a restriction of a full prime-limit temperament to a subgroup of the prime-limit. Subgroup TE does not depend on the basis used for the subgroup, and as stated, extends naturally to a full prime-limit temperament.

The two notions agree exactly when the temperament is defined on a JI subgroup with a basis consisting of rationally independent members. That is, the subgroup has a basis where no two elements share a prime factor (examples: 2.3.5 and 2.9.5; nonexample: 2.9.5.21).

== Definition ==

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-'''T''p'' tuning''' is a generalization of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning.
+'''T''p'' tuning''' is a generalization of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning. (In this article ''p'' denotes a parameter, ''p'' ≥ 1; it does not denote a prime.)
+For a subgroup temperament over a general JI subgroup, and for a given choice of ''p'' (most commonly ''p'' = 2), there are two notions of optimization:
+* The first is called '''inharmonic TE''', because the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres. Inharmonic TE depends on the basis used for the subgroup. In non-octave temperaments, inharmonic TE could be used when optimizing a specific voicing of a tempered JI chord. For example in 3/2.7/4.5/2 semiwolf temperament which tempers out 245/243, the 3/2.7/4.5/2 inharmonic TE optimizes the 4:6:7:10 chord.
+* The second is called '''subgroup TE''', because it treats the temperament as a restriction of a full prime-limit temperament to a subgroup of the prime-limit. Subgroup TE does not depend on the basis used for the subgroup, and as stated, extends naturally to a full prime-limit temperament.
+The two notions agree exactly when the temperament is defined on a JI subgroup with a basis consisting of rationally independent members. That is, the subgroup has a basis where no two elements share a prime factor (examples: 2.3.5 and 2.9.5; nonexample: 2.9.5.21).
 == Definition ==