DKW theory: Difference between revisions

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== Ratios of signature intervals and DKW coordinates ==

Given a signature for a particular tonality diamond ([[tonality diamond#Relation to subgroups|not precisely]] the same as a particular subgroup), there are three logarithmic ratios of its intervals. The closer the tuning system represents these ratios, the closer it represents the signature, the partition of the octave, and therefore the consonances of the diamond. We call the ratios '''C:B''', '''C:A''', and '''B:A''' ''diaschismian'', ''kleismian'', and ''interdiesian'' ratios. Setting any of these ratios to a particular value implies that a [[comma]] is tempered out - e.g. '''C:A''' = 3:1 means the comma '''C'''/'''A'''<sup>3</sup> is tempered - and in fact, the names of the ratios derive from 5-limit commas of this type, being the [[diaschisma]] (C:B = 2:1), [[15625/15552|kleisma]] (C:A = 3:1), and [[393216/~~390265~~|wurschmidt comma]] (B:A = 3:2) respectively. In this way, [[projective tuning space]] in any three-prime subgroup is given a grid consisting of these three families of ''fundamental commas'', with all other commas being expressible in terms of these families, and points corresponding to edos lying at intersections of particular commas of these families.

Given a signature for a particular tonality diamond ([[tonality diamond#Relation to subgroups|not precisely]] the same as a particular subgroup), there are three logarithmic ratios of its intervals. The closer the tuning system represents these ratios, the closer it represents the signature, the partition of the octave, and therefore the consonances of the diamond. We call the ratios '''C:B''', '''C:A''', and '''B:A''' ''diaschismian'', ''kleismian'', and ''interdiesian'' ratios. Setting any of these ratios to a particular value implies that a [[comma]] is tempered out - e.g. '''C:A''' = 3:1 means the comma '''C'''/'''A'''<sup>3</sup> is tempered - and in fact, the names of the ratios derive from 5-limit commas of this type, being the [[diaschisma]] (C:B = 2:1), [[15625/15552|kleisma]] (C:A = 3:1), and [[393216/390625|wurschmidt comma]] (B:A = 3:2) respectively. In this way, [[projective tuning space]] in any three-prime subgroup is given a grid consisting of these three families of ''fundamental commas'', with all other commas being expressible in terms of these families, and points corresponding to edos lying at intersections of particular commas of these families.

If '''A''', '''B''', and '''C''' are expressed in [[cents]] or logarithmic units, we can define the ''DKW coordinates'' to be '''D''' = ('''C'''-'''B''')/('''C'''+'''B'''), '''K''' = ('''C'''-'''A''')/('''C'''+'''A'''), and '''W''' = ('''B'''-'''A''')/('''B'''+'''A'''). The use of these fractions is so that a ratio of ''x:y'' will be as negative as ''y:x'' is positive. Note that only any two of these ratios are linearly independent from one another, meaning that in theory the third coordinate is redundant; it is included here for the sake of symmetry.

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.

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 == Ratios of signature intervals and DKW coordinates ==
-Given a signature for a particular tonality diamond ([[tonality diamond#Relation to subgroups|not precisely]] the same as a particular subgroup), there are three logarithmic ratios of its intervals. The closer the tuning system represents these ratios, the closer it represents the signature, the partition of the octave, and therefore the consonances of the diamond. We call the ratios '''C:B''', '''C:A''', and '''B:A''' ''diaschismian'', ''kleismian'', and ''interdiesian'' ratios. Setting any of these ratios to a particular value implies that a [[comma]] is tempered out - e.g. '''C:A''' = 3:1 means the comma '''C'''/'''A'''<sup>3</sup> is tempered - and in fact, the names of the ratios derive from 5-limit commas of this type, being the [[diaschisma]] (C:B = 2:1), [[15625/15552|kleisma]] (C:A = 3:1), and [[393216/390265|wurschmidt comma]] (B:A = 3:2) respectively. In this way, [[projective tuning space]] in any three-prime subgroup is given a grid consisting of these three families of ''fundamental commas'', with all other commas being expressible in terms of these families, and points corresponding to edos lying at intersections of particular commas of these families.
+Given a signature for a particular tonality diamond ([[tonality diamond#Relation to subgroups|not precisely]] the same as a particular subgroup), there are three logarithmic ratios of its intervals. The closer the tuning system represents these ratios, the closer it represents the signature, the partition of the octave, and therefore the consonances of the diamond. We call the ratios '''C:B''', '''C:A''', and '''B:A''' ''diaschismian'', ''kleismian'', and ''interdiesian'' ratios. Setting any of these ratios to a particular value implies that a [[comma]] is tempered out - e.g. '''C:A''' = 3:1 means the comma '''C'''/'''A'''<sup>3</sup> is tempered - and in fact, the names of the ratios derive from 5-limit commas of this type, being the [[diaschisma]] (C:B = 2:1), [[15625/15552|kleisma]] (C:A = 3:1), and [[393216/390625|wurschmidt comma]] (B:A = 3:2) respectively. In this way, [[projective tuning space]] in any three-prime subgroup is given a grid consisting of these three families of ''fundamental commas'', with all other commas being expressible in terms of these families, and points corresponding to edos lying at intersections of particular commas of these families.
 If '''A''', '''B''', and '''C''' are expressed in [[cents]] or logarithmic units, we can define the ''DKW coordinates'' to be '''D''' = ('''C'''-'''B''')/('''C'''+'''B'''), '''K''' = ('''C'''-'''A''')/('''C'''+'''A'''), and '''W''' = ('''B'''-'''A''')/('''B'''+'''A'''). The use of these fractions is so that a ratio of ''x:y'' will be as negative as ''y:x'' is positive. Note that only any two of these ratios are linearly independent from one another, meaning that in theory the third coordinate is redundant; it is included here for the sake of symmetry.
 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.