Delta-rational chord: Difference between revisions

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== DR and RTT ==

One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the root and letting <math>g</math> be the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of <math>g-g^4/4</math>, and the major third in the same triad has a delta of <math>g^4/4-1</math>. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial <math>g^4-2g-2</math> (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.

One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the triad's root and letting <math>g</math> be the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of <math>g-g^4/4</math>, and the major third in the same triad has a delta of <math>g^4/4-1</math>. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial <math>g^4-2g-2</math> (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.

However, the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: <math>g^4+2g-8=0.</math> The latter equation has solution g = 1.4960 = 697.3c.

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 == DR and RTT ==
-One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the root and letting <math>g</math> be the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of <math>g-g^4/4</math>, and the major third in the same triad has a delta of <math>g^4/4-1</math>. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial <math>g^4-2g-2</math> (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.
+One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the triad's root and letting <math>g</math> be the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of <math>g-g^4/4</math>, and the major third in the same triad has a delta of <math>g^4/4-1</math>. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial <math>g^4-2g-2</math> (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.
 However, the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: <math>g^4+2g-8=0.</math> The latter equation has solution g = 1.4960 = 697.3c. <!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->