DKW theory: Difference between revisions

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'''DKW theory''' is a theory developed by Lériendil of the efficacy of the representation of three-prime ~~subgroups~~ of [[JI]] by [[edo]]s and [[rank-2 temperament]]s, though generalizations should be possible for multi-prime subgroups. The fundamental element of DKW theory is the ''diharmonic tonality diamond'' (DTD), the simplest case of generalized [[tonality diamond]] with two prime harmonics.

'''DKW theory''' is a theory developed by [[User:Lériendil|Lériendil]] of the efficacy of the representation of three-prime [[subgroup]]s of [[JI]] by [[edo]]s and [[rank-2 temperament]]s, though generalizations should be possible for multi-prime subgroups. The fundamental element of DKW theory is the ''diharmonic tonality diamond'' (DTD), the simplest case of generalized [[tonality diamond]] with two prime harmonics.

== The structure of DTDs ==

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With that in mind, we can define the ''DKW error'' of a given tuning with particular valuations for the primes within the subgroup (e.g. a [[val]] for an [[equal temperament]], or the [[Majestazic system|tempered-primes definition]] of a tuning of a [[regular temperament]]) as the squared distance between the DKW coordinates of the representation of the signature in this valuation, and the just DKW coordinates (though it is just as well possible to target a non-just valuation for the primes such as might be obtained from a [[rank-3 temperament]] that one is working within) - and in the case of rank-2 temperaments or [[stretched octave|equave stretches]] of an EDO, an optimal value for the one free parameter as the one that minimizes DKW error.

[[Category:Regular temperament tuning]]

[[Category:Diamond]]

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-'''DKW theory''' is a theory developed by Lériendil of the efficacy of the representation of three-prime subgroups of [[JI]] by [[edo]]s and [[rank-2 temperament]]s, though generalizations should be possible for multi-prime subgroups. The fundamental element of DKW theory is the ''diharmonic tonality diamond'' (DTD), the simplest case of generalized [[tonality diamond]] with two prime harmonics.
+'''DKW theory''' is a theory developed by [[User:Lériendil|Lériendil]] of the efficacy of the representation of three-prime [[subgroup]]s of [[JI]] by [[edo]]s and [[rank-2 temperament]]s, though generalizations should be possible for multi-prime subgroups. The fundamental element of DKW theory is the ''diharmonic tonality diamond'' (DTD), the simplest case of generalized [[tonality diamond]] with two prime harmonics.
 == The structure of DTDs ==
@@ Line 29: / Line 29: @@
 With that in mind, we can define the ''DKW error'' of a given tuning with particular valuations for the primes within the subgroup (e.g. a [[val]] for an [[equal temperament]], or the [[Majestazic system|tempered-primes definition]] of a tuning of a [[regular temperament]]) as the squared distance between the DKW coordinates of the representation of the signature in this valuation, and the just DKW coordinates (though it is just as well possible to target a non-just valuation for the primes such as might be obtained from a [[rank-3 temperament]] that one is working within) - and in the case of rank-2 temperaments or [[stretched octave|equave stretches]] of an EDO, an optimal value for the one free parameter as the one that minimizes DKW error.
+[[Category:Regular temperament tuning]]
+[[Category:Diamond]]