Glossary: Difference between revisions

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; [[just intonation]] (JI) : The set of intervals expressible as rational numbers.

; [[generator]]s : A set of intervals that generate a larger set through linear combination. e.g. the primes {2, 3} generate the ~~Pythogorean~~ scale.

; [[generator]]s : A set of intervals that generate a larger set through linear combination. e.g. the primes {2, 3} generate the Pythagorean scale.

; [[just intonation subgroup]] : The set of JI ratios obtainable by stacking (finitely many) copies of a finite set of generators up or down. For example, 7/6 and 49/32 are both in the 2.3.7 subgroup, the set of JI ratios obtained by stacking copies of 2/1, 3/1 and 7/1 up and down.

; [[just intonation subgroup]] : The set of JI ratios obtainable by stacking (finitely many) copies of a finite set of JI generators up or down. For example, 7/6 and 49/32 are both in the 2.3.7 subgroup, the set of JI ratios obtained by stacking copies of 2/1, 3/1 and 7/1 up and down.

; [[temperament]] : A mapping from a [[just intonation subgroup]] to a set of intervals with fewer generators. expressible as a [[mapping|mapping matrix]] whose columns are generators of the just intonation and whose rows are generators of the temperament. In particular, the row vectors are called [[val]]s.

; [[temperament]] : A mapping from a [[domain]] (such as a [[prime limit]] or [[just intonation subgroup]]) to a set of intervals with fewer generators. expressible as a [[mapping|mapping matrix]] whose columns are generators of the just intonation and whose rows are generators of the temperament. In particular, the row vectors are called [[val]]s or [[map]]s.

; [[comma]] : A rational number that maps to 1/1 in a given temperament.

; [[rank]] : The number of generators of a set of intervals, i.e. the set's ~~dimensionality~~. For example, 12edo is rank-1 because it can be generated by the semitone; the Pythagorean scale is rank-2 because it can be generated by the primes {2, 3}.

; [[rank]] : The number of generators of a set of intervals, i.e. the set's cardinality. For example, 12edo is rank-1 because it can be generated by the semitone; the Pythagorean scale is rank-2 because it can be generated by the primes {2, 3}.

; [[equal temperament]] : A rank-1 temperament. The temperament-agnostic term is [[equal-step tuning]].

; [[tuning system|tuning]] of a temperament : A concrete realization of a(n abstract) temperament, with concrete cent values (for each basis element of the temperament). A temperament's character does not depend on the exact choice of tuning, e.g. a semitone of 100.1 cents produces a scale similar to that produced by a semitone of 100.0 cents.

; [[tuning system|tuning]] of a temperament : A concrete realization of a(n abstract) temperament, with concrete cent values (for each [[basis element]] of the temperament). A temperament's character does not depend on the exact choice of tuning, e.g. a semitone of 100.1 cents produces a scale similar to that produced by a semitone of 100.0 cents.

; optimal tuning : A tuning of a temperament that optimizes some desired property (usually the accuracy of the temperament's approximations to just intonation). e.g. [[POTE tuning]].

@@ Line 21: / Line 21: @@
 ; [[just intonation]] (JI) : The set of intervals expressible as rational numbers.
-; [[generator]]s : A set of intervals that generate a larger set through linear combination. e.g. the primes {2, 3} generate the Pythogorean scale.
+; [[generator]]s : A set of intervals that generate a larger set through linear combination. e.g. the primes {2, 3} generate the Pythagorean scale.
-; [[just intonation subgroup]] : The set of JI ratios obtainable by stacking (finitely many) copies of a finite set of generators up or down. For example, 7/6 and 49/32 are both in the 2.3.7 subgroup, the set of JI ratios obtained by stacking copies of 2/1, 3/1 and 7/1 up and down.
+; [[just intonation subgroup]] : The set of JI ratios obtainable by stacking (finitely many) copies of a finite set of JI generators up or down. For example, 7/6 and 49/32 are both in the 2.3.7 subgroup, the set of JI ratios obtained by stacking copies of 2/1, 3/1 and 7/1 up and down.
-; [[temperament]] : A mapping from a [[just intonation subgroup]] to a set of intervals with fewer generators. expressible as a [[mapping|mapping matrix]] whose columns are generators of the just intonation and whose rows are generators of the temperament. In particular, the row vectors are called [[val]]s.
+; [[temperament]] : A mapping from a [[domain]] (such as a [[prime limit]] or [[just intonation subgroup]]) to a set of intervals with fewer generators. expressible as a [[mapping|mapping matrix]] whose columns are generators of the just intonation and whose rows are generators of the temperament. In particular, the row vectors are called [[val]]s or [[map]]s.
 ; [[comma]] : A rational number that maps to 1/1 in a given temperament.
-; [[rank]] : The number of generators of a set of intervals, i.e. the set's dimensionality. For example, 12edo is rank-1 because it can be generated by the semitone; the Pythagorean scale is rank-2 because it can be generated by the primes {2, 3}.
+; [[rank]] : The number of generators of a set of intervals, i.e. the set's cardinality. For example, 12edo is rank-1 because it can be generated by the semitone; the Pythagorean scale is rank-2 because it can be generated by the primes {2, 3}.
 ; [[equal temperament]] : A rank-1 temperament. The temperament-agnostic term is [[equal-step tuning]].
-; [[tuning system|tuning]] of a temperament : A concrete realization of a(n abstract) temperament, with concrete cent values (for each basis element of the temperament). A temperament's character does not depend on the exact choice of tuning, e.g. a semitone of 100.1 cents produces a scale similar to that produced by a semitone of 100.0 cents.
+; [[tuning system|tuning]] of a temperament : A concrete realization of a(n abstract) temperament, with concrete cent values (for each [[basis element]] of the temperament). A temperament's character does not depend on the exact choice of tuning, e.g. a semitone of 100.1 cents produces a scale similar to that produced by a semitone of 100.0 cents.
 ; optimal tuning : A tuning of a temperament that optimizes some desired property (usually the accuracy of the temperament's approximations to just intonation). e.g. [[POTE tuning]].