User:Zhenlige/Logical tempering: Difference between revisions

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'''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.

'''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.

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 ==

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

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

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 ~~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~~.

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'''{{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.
+'''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.
+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 ==
 [[Octave equivalence]] can be thought as logically tempering out [[2/1]].
-The [[5L 2s|diatonic]] degrees of [[3-limit]] intervals (seconds, thirds etc.) are the result of logical tempering into [[7edo|7et]]. Similarly, all [[MOS]]-based degree systems can be thought as a logical temperament based on the equalized scale.
+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 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.
+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]].