Merciful intonation: Difference between revisions

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Specifically, the noble (phi-weighted) [[mediant]] between two nearby<ref>For the phi-weighted mediant to give a noble number, it is necessary that the two ratios <math>\frac{n_1}{d_1}</math> and <math>\frac{n_2}{d_2}</math> satisfy the equation <math>|n_{1}d_{2}-d_{1}n_{2}| = 1</math>.

</ref> JI intervals is taken. For example, the noble mediant between <math>\frac54</math> ≈ 386.3¢ and <math>\frac97</math> ≈ 435.1¢ is equal to <math>\frac{5 + 9\phi}{4 + 7\phi}</math> ≈ 422.5¢.

</ref> JI intervals is taken. Phi as a factor for weighting is applied to the more complex of the two intervals, which Keenan suggests may be the point of maximum complexity between the two. For example, the noble mediant between <math>\frac54</math> ≈ 386.3¢ and <math>\frac97</math> ≈ 435.1¢ is equal to <math>\frac{5 + 9\phi}{4 + 7\phi}</math> ≈ 422.5¢.

An MI interval may also be referred to as a "metastable interval", by analogy with [https://en.wikipedia.org/wiki/Metastability the physics concept] whereby opposing gravitational pulls on an object are balanced, insofar as the noble mediant is the interval which is placed at the precise point between the two JI intervals such that it sensed as equally distant from both of them. Such intervals are often near [[harmonic entropy]] maxima.

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 Specifically, the noble (phi-weighted) [[mediant]] between two nearby<ref>For the phi-weighted mediant to give a noble number, it is necessary that the two ratios <span><math>\frac{n_1}{d_1}</math></span> and <span><math>\frac{n_2}{d_2}</math></span> satisfy the equation <span><math>|n_{1}d_{2}-d_{1}n_{2}| = 1</math></span>.
-</ref> JI intervals is taken. For example, the noble mediant between <span><math>\frac54</math></span> ≈ 386.3¢ and <span><math>\frac97</math></span> ≈ 435.1¢ is equal to <span><math>\frac{5 + 9\phi}{4 + 7\phi}</math></span> ≈ 422.5¢.
+</ref> JI intervals is taken. Phi as a factor for weighting is applied to the more complex of the two intervals, which Keenan suggests may be the point of maximum complexity between the two. For example, the noble mediant between <span><math>\frac54</math></span> ≈ 386.3¢ and <span><math>\frac97</math></span> ≈ 435.1¢ is equal to <span><math>\frac{5 + 9\phi}{4 + 7\phi}</math></span> ≈ 422.5¢.
 An MI interval may also be referred to as a "metastable interval", by analogy with [https://en.wikipedia.org/wiki/Metastability the physics concept] whereby opposing gravitational pulls on an object are balanced, insofar as the noble mediant is the interval which is placed at the precise point between the two JI intervals such that it sensed as equally distant from both of them. Such intervals are often near [[harmonic entropy]] maxima.