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 ratio of frequency deltas between the major third and minor third. The minor third can be written as <math>g-g^4/4</math>, and the major third can be written as <math>g^4/4-1</math>. Therefore we must find the roots 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.~~49453~~, 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 = 1.496 = 697.3c.

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 ratio of frequency deltas between the major third and minor third. The minor third can be written as <math>g-g^4/4</math>, and the major third can be written as <math>g^4/4-1</math>. Therefore we must find the roots 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 1.495, 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 = 1.496 = 697.3c.

Below is a list of temperaments and their various optimizations for proportionally beating chords. They are ordered by highest power in the polynomial, with ties broken by leading coefficients, then 2nd term coefficients, 3rd term coefficients, 4th term coefficients..., etc. In the case of negative coefficients, only the absolute value is considered.

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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 ratio of frequency deltas between the major third and minor third. The minor third can be written as <math>g-g^4/4</math>, and the major third can be written as <math>g^4/4-1</math>. Therefore we must find the roots 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 1.49453, 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 = 1.496 = 697.3c. <!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->
+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 ratio of frequency deltas between the major third and minor third. The minor third can be written as <math>g-g^4/4</math>, and the major third can be written as <math>g^4/4-1</math>. Therefore we must find the roots 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 1.495, 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 = 1.496 = 697.3c. <!--Essentially tempered [[Dyadic chord|dyadic]] triads are also more difficult to tune with simple delta-signatures, since they lack simple JI preimages.-->
 Below is a list of temperaments and their various optimizations for proportionally beating chords. They are ordered by highest power in the polynomial, with ties broken by leading coefficients, then 2nd term coefficients, 3rd term coefficients, 4th term coefficients..., etc. In the case of negative coefficients, only the absolute value is considered.