User:Zhenlige/Logical tempering: Difference between revisions

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'''Logical tempering'''{{idio}} means using a [[regular temperament]] (called the '''logical temperament''') as a ~~category system~~. For example, ~~diatonic interval degrees are the result~~ of ~~logical tempering~~ [[3-limit]] ~~into~~ [[~~7edo|7et~~]]~~, and~~ [[~~User:Zhenlige/Chromatic notation~~|~~chromatic degrees~~]] are the ~~result of~~ [[~~12edo|12et~~]].

'''Logical tempering'''{{idio}} means using a [[regular temperament]] (called the '''logical temperament'''{{idio}}) to get “abstract” notes and intervals, but not directly the actual played notes. Different from “real” tempering, a logical temperament with [[torsion]] is allowed. The commas tempered out in the logical temperament are sometimes called “chromas”, which is a generalization of the concept of MOS [[chroma]]s.

Multiple logical temperaments can be used at the same time. For example, one may first categorize intervals as alterations of [[3-limit]] intervals, and then logically temper out [[2187/2048]] to get [[5L 2s|diatonic]] degrees. Alternatively, one may use the 5-, 7- and 12-form degree systems in the same piece.

When the actual notes are already tempered, to get unambiguous results, the logical temperament should be based on the tempered notes i.e. it should [[support]] the actual temperament.

== Examples ==

The conventional categorization of [[7-limit]] intervals treats [[5/4]] as a major third and [[7/4]] as a minor seventh, which implies [[dominant (temperament)|dominant]] as the logical temperament. A piece using [[adaptive JI]] with that framework can be thought as a dynamic [[detempering]] of a piece in dominant.

[[Octave equivalence]] can be thought as logically tempering out [[2/1]].

The [[5L 2s|diatonic]] degrees (seconds, thirds etc.) of [[3-limit]] intervals are the result of logical tempering into [[7edo|7et]], tempering out the diatonic [[chroma]] [[2187/2048]]. Similarly, every [[MOS]]-based degree system can be thought as a logical temperament based on the equalized scale.

The conventional categorization of [[7-limit]] intervals treats [[5/4]] as a major third and [[7/4]] as a minor seventh, which implies [[dominant (temperament)|dominant]] as the logical temperament. A piece using [[adaptive JI]] with that framework can be thought as a dynamic [[detempering]] of a piece in dominant. Alternatively, 7/4 can be considered as an augmented sixth, giving [[septimal meantone]] logic.

The classical triads are made by stacking [[5/4]] and [[6/5]] intervals, meaning their difference [[25/24]] is a chroma, giving a [[dicot]] framework. Merging it with an actual [[5-limit]] [[rank]]-2 temperament gives a unique MOS structure. For example, [[meantone]] gives the famous [[5L 2s]] diatonic scale, and [[diaschismic]] gives [[2L 8s]].

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-'''Logical tempering'''{{idio}} means using a [[regular temperament]] (called the '''logical temperament''') as a category system. For example, diatonic interval degrees are the result of logical tempering [[3-limit]] into [[7edo|7et]], and [[User:Zhenlige/Chromatic notation|chromatic degrees]] are the result of [[12edo|12et]].
+{{../en2zh}}
+'''Logical tempering'''{{idio}} means using a [[regular temperament]] (called the '''logical temperament'''{{idio}}) to get “abstract” notes and intervals, but not directly the actual played notes. Different from “real” tempering, a logical temperament with [[torsion]] is allowed. The commas tempered out in the logical temperament are sometimes called “chromas”, which is a generalization of the concept of MOS [[chroma]]s.
+Multiple logical temperaments can be used at the same time. For example, one may first categorize intervals as alterations of [[3-limit]] intervals, and then logically temper out [[2187/2048]] to get [[5L 2s|diatonic]] degrees. Alternatively, one may use the 5-, 7- and 12-form degree systems in the same piece.
+When the actual notes are already tempered, to get unambiguous results, the logical temperament should be based on the tempered notes i.e. it should [[support]] the actual temperament.
 == Examples ==
-The conventional categorization of [[7-limit]] intervals treats [[5/4]] as a major third and [[7/4]] as a minor seventh, which implies [[dominant (temperament)|dominant]] as the logical temperament. A piece using [[adaptive JI]] with that framework can be thought as a dynamic [[detempering]] of a piece in dominant.
+[[Octave equivalence]] can be thought as logically tempering out [[2/1]].
+The [[5L 2s|diatonic]] degrees (seconds, thirds etc.) of [[3-limit]] intervals are the result of logical tempering into [[7edo|7et]], tempering out the diatonic [[chroma]] [[2187/2048]]. Similarly, every [[MOS]]-based degree system can be thought as a logical temperament based on the equalized scale.
+The conventional categorization of [[7-limit]] intervals treats [[5/4]] as a major third and [[7/4]] as a minor seventh, which implies [[dominant (temperament)|dominant]] as the logical temperament. A piece using [[adaptive JI]] with that framework can be thought as a dynamic [[detempering]] of a piece in dominant. Alternatively, 7/4 can be considered as an augmented sixth, giving [[septimal meantone]] logic.
+The classical triads are made by stacking [[5/4]] and [[6/5]] intervals, meaning their difference [[25/24]] is a chroma, giving a [[dicot]] framework. Merging it with an actual [[5-limit]] [[rank]]-2 temperament gives a unique MOS structure. For example, [[meantone]] gives the famous [[5L 2s]] diatonic scale, and [[diaschismic]] gives [[2L 8s]].