Merciful intonation: Difference between revisions
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'''Merciful Intonation (MI)''' is intonation | '''Merciful Intonation (MI)''' is a kind of '''anti-just intonation (anti-JI)'''. Merciful intonation stands in contrast to [[Just Intonation|Just Intonation (JI)]] by using [https://mathworld.wolfram.com/NobleNumber.html 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<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>. | Specifically, the noble (phi-weighted) [[mediant]] between two nearby<ref group="note">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 + | </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. | 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. | ||
The name was inspired by [[Erv Wilson]]'s earlier description of the [[Acoustic phi|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. | The name was inspired by [[Erv Wilson]]'s earlier description of the [[Acoustic phi|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. | ||
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== Listening == | == Listening == | ||
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* https://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt | * https://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt | ||
* [http://anaphoria.com/temes.PDF A relevant letter from Lorne Temes to Erv Wilson] | * [http://anaphoria.com/temes.PDF A relevant letter from Lorne Temes to Erv Wilson] | ||
* [https://forum.sagittal.org/viewtopic.php?p=4604#p4604 The Feudal Manifesto] on the Sagittal forum | |||
== See also == | |||
* [[Nobletone]] | |||
== Notes == | |||
<references group="note"/> | |||
[[Category:Method]] | |||
[[Category:Terms]] | |||
[[Category:Golden ratio]] | |||