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. (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 T''p'' tuning:

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 T''p'' tuning:

* 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 first is called '''inharmonic T''p''''', because the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres. Inharmonic T''p'' depends on the basis used for the subgroup. In non-octave temperaments, inharmonic T''p'' 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 T''p'' 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 the TE tuning of the full prime-limit temperament.

* The second is called '''subgroup T''p''''', because it treats the temperament as a restriction of a full prime-limit temperament to a subgroup of the prime-limit. Subgroup T''p'' does not depend on the basis used for the subgroup, and as stated, extends naturally to the T''p'' tuning of the 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 (i.e. pairwise coprime) 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).

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+{{DISPLAYTITLE:T<sub>''p''</sub> tuning}}
 '''T<sub>''p''</sub> 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 T<sub>''p''</sub> tuning:
+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 T<sub>''p''</sub> tuning:
-* 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 first is called '''inharmonic T<sub>''p''</sub>''', because the basis entries are treated as if they were primes, reminiscent of some inharmonic timbres. Inharmonic T<sub>''p''</sub> depends on the basis used for the subgroup. In non-octave temperaments, inharmonic T<sub>''p''</sub> 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 T<sub>''p''</sub> 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 the TE tuning of the full prime-limit temperament.
+* The second is called '''subgroup T<sub>''p''</sub>''', because it treats the temperament as a restriction of a full prime-limit temperament to a subgroup of the prime-limit. Subgroup T<sub>''p''</sub> does not depend on the basis used for the subgroup, and as stated, extends naturally to the T<sub>''p''</sub> tuning of the 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 (i.e. pairwise coprime) 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).