Merciful intonation

Merciful Intonation (MI) is intonation which stands in contrast to Just Intonation (JI), by using noble number frequency ratios as a way of maximally avoiding coinciding harmonics. It was first systematically described by Margo Schulter and Dave Keenan.

Specifically, the noble (phi-weighted) mediant between two nearby[1] JI intervals is taken. For example, the noble mediant between $\frac54$ ≈ 386.3¢ and $\frac97$ ≈ 435.1¢ is equal to $\frac{5 + 9φ}{4 + 7φ}$ ≈ 422.5¢.

An MI interval may also be referred to as a "metastable interval", by analogy with 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.

The name was inspired by Erv Wilson's earlier description of the phi neutral sixth (833¢) as, "the worstest of the worst — and yet somehow with divinity imbued, Lord have mercy!" There is also an analogy to be found in rational : just :: noble : merciful, where the left halves are favorable human qualities, while the right halves are related ethical virtues; simultaneously, the left halves are mathematical properties while the right halves are musical qualities possessed by the simpler examples of intervals with those mathematical properties.

1. For the phi-weighted mediant to give a noble number, it is necessary that the two ratios $\frac{n_1}{d_1}$ and $\frac{n_2}{d_2}$ satisfy the equation $|n_{1}d_{2}-d_{1}n_{2}| = 1$.