DKW theory: Difference between revisions

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For a diharmonic diamond, the signature consists of three intervals, which we can call '''A''', '''B''', and '''C''' in this order. It turns out that either '''X1''' = '''B''', or '''X1''' = '''BC'''; and that the former is the case for all F-type diamonds, while the latter is the case for all N-type diamonds. Therefore, the intervals of the signature partition the equave into 7 parts (B:A:B:C:B:A:B) for an F-type diamond, or into 9 parts (C:B:A:B:C:B:A:B:C) for a N-type diamond. We can therefore equate how well a given three-prime subgroup of JI is represented in a given tuning system to how well the signature is represented in that tuning system.

=== The signature basis for commas ===

Any interval in a three-prime subgroup can be written as a vector with basis elements being the elements of the group's signature. These often serve as a much more useful basis for commas than using the primes directly as the basis, as many useful commas are formed by ratios of powers of two signature intervals (as discussed below), and measuring complexity this way prioritizes a set of intervals more usable as commas.

== 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'''3 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.

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.

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.

Setting any of the ratios '''C:B''', '''C:A''', and '''B:A''' to a particular value implies that a [[comma]] is tempered out - e.g. '''C:A''' = 3:1 means the comma '''C'''/'''A'''3 is tempered. We call the ratios '''C:B''', '''C:A''', and '''B:A''' ''diaschismian'', ''kleismian'', and ''interdiesian'' ratios, or simply ''D-ratios'', ''K-ratios'', and ''W-ratios'', with corresponding D-commas, K-commas, and W-commas (these names derive from the specific triad of commas of this type tempered out in [[34edo]]'s 5-limit approximation, the [[2048/2025|diaschisma]], the [[15625/15552|kleisma]], and the [[393216/390625|wurschmidt comma]]). 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 other logarithmic units, we can define the ''DKW coordinates'' of the subgroup's representation 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.

== DKW theory in the 5-limit ==

By far the most important three-prime subgroup is the [[5-limit]], i.e. the 2.3.5 subgroup. It turns out that the 2.3.5 tonality diamond's ordering is type ''N-VI'', and that its signature is '''C''' = [[9/8]] : '''B''' = [[16/15]] : '''A''' = [[25/24]].

Below are commas with simple expressions in terms of these intervals, with "complexity" measured as the sum of the absolute values of powers of '''A''', '''B''', and '''C'''.

Note all commas considered here (aside from complexity-1) are those that contain at least one minus sign in their signature expression.

{| class="wikitable center-all"

|-

! Comma

! Defined temperament

! Complexity

! Subgroup monzo (9/8.16/15.25/24)

|-

| [[9/8]]

| [[Very low accuracy temperaments #Antitonic|Antitonic]]

| 1

| {{Monzo| 1 0 0 }}

|-

| [[16/15]]

| [[Father]]

| 1

| {{Monzo| 0 1 0 }}

|-

| [[25/24]]

| [[Dicot]]

| 1

| {{Monzo| 0 0 1 }}

|-

| [[27/25]]

| [[Bug]]

| 2

| {{Monzo| 1 0 -1 }}

|-

| [[135/128]]

| [[Mavila]]

| 2

| {{Monzo| 1 -1 0 }}

|-

| [[128/125]]

| [[Augmented (temperament)|Augmented]]

| 2

| {{Monzo| 0 1 -1 }}

|-

| [[648/625]]

| [[Diminished (temperament)|Diminished]]

| 3

| {{Monzo| 1 0 -2 }}

|-

| [[2048/2025]]

| [[Diaschismic]]

| 3

| {{Monzo| -1 2 0 }}

|-

| [[3125/3072]]

| [[Magic]]

| 3

| {{Monzo| 0 -1 2 }}

|-

| [[81/80]]

| [[Meantone]]

| 3

| {{Monzo| 1 -1 -1 }}

|}

[[Category:Regular temperament tuning]]

[[Category:Diamond]]

@@ Line 22: / Line 22: @@
 For a diharmonic diamond, the signature consists of three intervals, which we can call '''A''', '''B''', and '''C''' in this order. It turns out that either '''X<sub>1</sub>''' = '''B''', or '''X<sub>1</sub>''' = '''BC'''; and that the former is the case for all F-type diamonds, while the latter is the case for all N-type diamonds. Therefore, the intervals of the signature partition the equave into 7 parts (B:A:B:C:B:A:B) for an F-type diamond, or into 9 parts (C:B:A:B:C:B:A:B:C) for a N-type diamond. We can therefore equate how well a given three-prime subgroup of JI is represented in a given tuning system to how well the signature is represented in that tuning system.
+=== The signature basis for commas ===
+Any interval in a three-prime subgroup can be written as a vector with basis elements being the elements of the group's signature. These often serve as a much more useful basis for commas than using the primes directly as the basis, as many useful commas are formed by ratios of powers of two signature intervals (as discussed below), and measuring complexity this way prioritizes a set of intervals more usable as commas.
 == 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 [[393216/390625|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.
+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.
-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.
+Setting any of the ratios '''C:B''', '''C:A''', and '''B:A''' 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. We call the ratios '''C:B''', '''C:A''', and '''B:A''' ''diaschismian'', ''kleismian'', and ''interdiesian'' ratios, or simply ''D-ratios'', ''K-ratios'', and ''W-ratios'', with corresponding D-commas, K-commas, and W-commas (these names derive from the specific triad of commas of this type tempered out in [[34edo]]'s 5-limit approximation, the [[2048/2025|diaschisma]], the [[15625/15552|kleisma]], and the [[393216/390625|wurschmidt comma]]). 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 other logarithmic units, we can define the ''DKW coordinates'' of the subgroup's representation 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.
+== DKW theory in the 5-limit ==
+By far the most important three-prime subgroup is the [[5-limit]], i.e. the 2.3.5 subgroup. It turns out that the 2.3.5 tonality diamond's ordering is type ''N-VI'', and that its signature is '''C''' = [[9/8]] : '''B''' = [[16/15]] : '''A''' = [[25/24]].
+Below are commas with simple expressions in terms of these intervals, with "complexity" measured as the sum of the absolute values of powers of '''A''', '''B''', and '''C'''.
+Note all commas considered here (aside from complexity-1) are those that contain at least one minus sign in their signature expression.
+{| class="wikitable center-all"
+|-
+! Comma
+! Defined<br>temperament
+! Complexity
+! Subgroup monzo<br>(9/8.16/15.25/24)
+|-
+| [[9/8]]
+| [[Very low accuracy temperaments #Antitonic|Antitonic]]
+| 1
+| {{Monzo| 1 0 0 }}
+|-
+| [[16/15]]
+| [[Father]]
+| 1
+| {{Monzo| 0 1 0 }}
+|-
+| [[25/24]]
+| [[Dicot]]
+| 1
+| {{Monzo| 0 0 1 }}
+|-
+| [[27/25]]
+| [[Bug]]
+| 2
+| {{Monzo| 1 0 -1 }}
+|-
+| [[135/128]]
+| [[Mavila]]
+| 2
+| {{Monzo| 1 -1 0 }}
+|-
+| [[128/125]]
+| [[Augmented (temperament)|Augmented]]
+| 2
+| {{Monzo| 0 1 -1 }}
+|-
+| [[648/625]]
+| [[Diminished (temperament)|Diminished]]
+| 3
+| {{Monzo| 1 0 -2 }}
+|-
+| [[2048/2025]]
+| [[Diaschismic]]
+| 3
+| {{Monzo| -1 2 0 }}
+|-
+| [[3125/3072]]
+| [[Magic]]
+| 3
+| {{Monzo| 0 -1 2 }}
+|-
+| [[81/80]]
+| [[Meantone]]
+| 3
+| {{Monzo| 1 -1 -1 }}
+|}
 [[Category:Regular temperament tuning]]
 [[Category:Diamond]]
 {{Todo| add examples }}