Merciful intonation - Revision history

Overthink: Section off note

2026-02-08T01:30:43Z

Section off note

← Older revision		Revision as of 01:30, 8 February 2026
Line 1:		Line 1:
	'''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]].		'''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. 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¢.		</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¢.

Line 7:		Line 7:

	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.
			__NOTOC__
	~~<references/>~~

	== Listening ==		== Listening ==

Line 24:		Line 22:
	== See also ==		== See also ==
	* [[Nobletone]]		* [[Nobletone]]

			== Notes ==
			<references group="note"/>

	[[Category:Method]]		[[Category:Method]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Golden ratio]]		[[Category:Golden ratio]]

Overthink: link to scale page

2026-02-08T01:27:16Z

link to scale page

← Older revision		Revision as of 01:27, 8 February 2026
Line 21:		Line 21:
	* [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		* [https://forum.sagittal.org/viewtopic.php?p=4604#p4604 The Feudal Manifesto] on the Sagittal forum

			== See also ==
			* [[Nobletone]]

	[[Category:Method]]		[[Category:Method]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Golden ratio]]		[[Category:Golden ratio]]

Pailiaq: added more context as to how the weighting formula works

2025-11-26T05:41:35Z

added more context as to how the weighting formula works

← Older revision		Revision as of 05:41, 26 November 2025
Line 2:		Line 2:

	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>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.		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.

Cmloegcmluin: include in pages relevant to golden ratio

2024-04-06T18:48:29Z

include in pages relevant to golden ratio

← Older revision		Revision as of 18:48, 6 April 2024
Line 24:		Line 24:
	[[Category:Method]]		[[Category:Method]]
	[[Category:Terms]]		[[Category:Terms]]
			[[Category:Golden ratio]]

BudjarnLambeth: Categorised this uncategorised page

2023-04-27T00:31:50Z

Categorised this uncategorised page

← Older revision		Revision as of 00:31, 27 April 2023
Line 21:		Line 21:
	* [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		* [https://forum.sagittal.org/viewtopic.php?p=4604#p4604 The Feudal Manifesto] on the Sagittal forum

			[[Category:Method]]
			[[Category:Terms]]

Fredg999: Added 1 external link

2023-03-24T21:04:11Z

Added 1 external link

← Older revision		Revision as of 21:04, 24 March 2023
Line 20:		Line 20:
	* 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

Dave Keenan: Changed from the Greek letter phi to the standard math symbol phi (straight, not curly).

2022-10-10T22:13:28Z

Changed from the Greek letter phi to the standard math symbol phi (straight, not curly).

← Older revision		Revision as of 22:13, 10 October 2022
Line 2:		Line 2:

	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>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 + 9φ}{4 + 7φ}</math></span> ≈ 422.5¢.		</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¢.

	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.

Dave Keenan: Introduced the idea that MI is a kind of anti-JI, thanks to Mike Battaglia

2022-10-10T21:59:57Z

Introduced the idea that MI is a kind of anti-JI, thanks to Mike Battaglia

← Older revision		Revision as of 21:59, 10 October 2022
Line 1:		Line 1:
	'''Merciful Intonation (MI)''' is intonation ~~which~~ 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]].		'''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>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>.

Cmloegcmluin: add footnote per Dave's request and spruce up the math expressions

2021-04-07T22:21:34Z

add footnote per Dave's request and spruce up the math expressions

← Older revision		Revision as of 22:21, 7 April 2021
Line 1:		Line 1:
	'''Merciful Intonation (MI)''' is intonation which 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]].		'''Merciful Intonation (MI)''' is intonation which 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 JI intervals is taken. For example, the noble mediant between 5/~~4 ≈386~~.3¢ and 9/~~7 ≈435~~.1¢ is equal to (5 + ~~9ɸ):(~~4 + ~~7ɸ) ≈422~~.5¢.		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 + 9φ}{4 + 7φ}</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.

			<references/>

	== Listening ==		== Listening ==

Dave Keenan: between two JI intervals -> between two nearby JI intervals

2021-04-07T21:27:28Z

between two JI intervals -> between two nearby JI intervals

← Older revision		Revision as of 21:27, 7 April 2021
Line 1:		Line 1:
	'''Merciful Intonation (MI)''' is intonation which 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]].		'''Merciful Intonation (MI)''' is intonation which 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 JI intervals is taken. For example, the noble mediant between 5/4 ≈386.3¢ and 9/7 ≈435.1¢ is equal to (5 + 9ɸ):(4 + 7ɸ) ≈422.5¢.		Specifically, the noble (phi-weighted) [[mediant]] between two nearby JI intervals is taken. For example, the noble mediant between 5/4 ≈386.3¢ and 9/7 ≈435.1¢ is equal to (5 + 9ɸ):(4 + 7ɸ) ≈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.