19-limit - Revision history

Overthink: /* Edo approximation */ caps

2026-01-11T20:42:15Z

Edo approximation: caps

← Older revision		Revision as of 20:42, 11 January 2026
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	Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.		Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.

	{{Note\| [[~~wart~~ notation]] is used to specify the [[val]] chosen for the edo. In the above list, "34dh" means taking the second closest approximations of harmonics 7 and 19. }}		{{Note\| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "34dh" means taking the second closest approximations of harmonics 7 and 19. }}

	== Intervals ==		== Intervals ==

FloraC: /* Edo approximation */ adopt template

2025-12-27T17:44:43Z

Edo approximation: adopt template

← Older revision		Revision as of 17:44, 27 December 2025
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	Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.		Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.

	~~: '''~~Note~~''':~~ [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "34dh" means taking the second closest approximations of harmonics 7 and 19.		{{Note\| [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "34dh" means taking the second closest approximations of harmonics 7 and 19. }}

	== Intervals ==		== Intervals ==

FloraC: /* Edo approximation */ calibrate the list according to my research results

2025-08-20T16:23:17Z

Edo approximation: calibrate the list according to my research results

← Older revision		Revision as of 16:23, 20 August 2025
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	== Edo approximation ==		== Edo approximation ==
	Here is a list of [[edo]]s with progressively better tunings for 19-limit intervals (decreasing [[TE error]]): {{EDOs\| 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on.		Here is a list of [[edo]]s with progressively better tunings for 19-limit intervals ([[monotonicity limit]] ≥ 19 and decreasing [[TE error]]): {{EDOs\| 34dh, 38df, 41, 50, 53, 58h, 68, 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

	Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.		Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.

	: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "~~152fg~~" means taking the second closest ~~approximation~~ of harmonics 13 and 17.		: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "34dh" means taking the second closest approximations of harmonics 7 and 19.

	== Intervals ==		== Intervals ==

Francium: /* Music */ +music

2024-12-27T18:56:35Z

Music: +music

← Older revision		Revision as of 18:56, 27 December 2024
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	; [[Domin]]		; [[Domin]]
	* [https://www.youtube.com/watch?v=WTo5YihoLqs ''Asuttan''] (2024)		* [https://www.youtube.com/watch?v=WTo5YihoLqs ''Asuttan''] (2024)
			* [https://www.youtube.com/watch?v=OPt3Y9VSliU ''Asuttan Bouta''] (2024)

	; [[Joseph Monzo]]		; [[Joseph Monzo]]

FloraC at 15:36, 17 December 2024

2024-12-17T15:36:08Z

← Older revision		Revision as of 15:36, 17 December 2024
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	{{Prime limit navigation\|19}}		{{Prime limit navigation\|19}}
	The '''19-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 19. It is the 8th [[prime limit]] and is ~~thus~~ a superset of the [[17-limit]] and a subset of the [[23-limit]].		The '''19-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 19. It is the 8th [[prime limit]] and is a superset of the [[17-limit]] and a subset of the [[23-limit]].

	The 19-limit is a [[~~Rank~~ and codimension\|rank-8]] system, and can be modeled in a 7-dimensional [[lattice]], with the primes 3, 5, 7, 11, 13, 17, and 19 represented by each dimension. The prime 2 does not appear in the typical 19-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, an eighth dimension is ~~need~~.		The 19-limit is a [[rank and codimension\|rank-8]] system, and can be modeled in a 7-dimensional [[lattice]], with the primes 3, 5, 7, 11, 13, 17, and 19 represented by each dimension. The prime 2 does not appear in the typical 19-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, an eighth dimension is needed.

	These things are contained by the 19-limit, but not the 17-limit:		These things are contained by the 19-limit, but not the 17-limit:

FloraC at 15:23, 17 December 2024

2024-12-17T15:23:35Z

← Older revision		Revision as of 15:23, 17 December 2024
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	The 19-limit is a [[Rank and codimension\|rank-8]] system, and can be modeled in a 7-dimensional [[lattice]], with the primes 3, 5, 7, 11, 13, 17, and 19 represented by each dimension. The prime 2 does not appear in the typical 19-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, an eighth dimension is need.		The 19-limit is a [[Rank and codimension\|rank-8]] system, and can be modeled in a 7-dimensional [[lattice]], with the primes 3, 5, 7, 11, 13, 17, and 19 represented by each dimension. The prime 2 does not appear in the typical 19-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, an eighth dimension is need.

			These things are contained by the 19-limit, but not the 17-limit:
			* The [[19-odd-limit\|19-]] and [[21-odd-limit]];
			* Mode 10 and 11 of the harmonic or subharmonic series.

	== Terminology and notation ==		== Terminology and notation ==
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	; [[Joseph Monzo]]		; [[Joseph Monzo]]
	* [https://www.youtube.com/watch?v=it5avwRE8PI ''Theme from Invisible Haircut''] (1990)		* [https://www.youtube.com/watch?v=it5avwRE8PI ''Theme from Invisible Haircut''] (1990)

	~~== See also ==~~
	* [[19-odd-limit]]

	[[Category:19-limit\| ]] <!-- main article -->		[[Category:19-limit\| ]] <!-- main article -->

Recentlymaterialized: /* Terminology and notation */

2024-11-24T16:25:25Z

Terminology and notation

← Older revision		Revision as of 16:25, 24 November 2024
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	== Terminology and notation ==		== Terminology and notation ==
	~~Intervals~~ categories of [[harmonic class\|HC19]] are relatively clear. [[19/16]] is most commonly considered a minor third, as 1–19/16–3/2 is an important {{w\|tertian}} chord~~. The~~ [[Functional Just System]] and [[Helmholtz–Ellis notation]] agree. However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.		[[Interval_region\|Interval categories]] of [[harmonic class\|HC19]] are relatively clear. [[19/16]] is most commonly considered a minor third, as 1–19/16–3/2 is an important {{w\|tertian}} chord (the [[Functional Just System]] and [[Helmholtz–Ellis notation]] agree). However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.

	== Edo approximation ==		== Edo approximation ==

FloraC: /* Terminology and notation */ wording and style

2024-11-24T11:16:43Z

Terminology and notation: wording and style

← Older revision		Revision as of 11:16, 24 November 2024
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	== Terminology and notation ==		== Terminology and notation ==
	Intervals of [[harmonic class\|HC19]] are relatively clear ~~in terms of their categories~~. [[19/16]] is most commonly considered a minor third, as ~~1-19~~/~~16-3~~/2 is an important {{w\|tertian}} chord. The [[Functional Just System]] and [[~~Helmholtz-Ellis~~ notation]] agree. However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.		Intervals categories of [[harmonic class\|HC19]] are relatively clear. [[19/16]] is most commonly considered a minor third, as 1–19/16–3/2 is an important {{w\|tertian}} chord. The [[Functional Just System]] and [[Helmholtz–Ellis notation]] agree. However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.

	== Edo approximation ==		== Edo approximation ==

FloraC: /* Edo approximations */ wording

2024-09-02T10:43:18Z

Edo approximations: wording

← Older revision		Revision as of 10:43, 2 September 2024
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	Intervals of [[harmonic class\|HC19]] are relatively clear in terms of their categories. [[19/16]] is most commonly considered a minor third, as 1-19/16-3/2 is an important {{w\|tertian}} chord. The [[Functional Just System]] and [[Helmholtz-Ellis notation]] agree. However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.		Intervals of [[harmonic class\|HC19]] are relatively clear in terms of their categories. [[19/16]] is most commonly considered a minor third, as 1-19/16-3/2 is an important {{w\|tertian}} chord. The [[Functional Just System]] and [[Helmholtz-Ellis notation]] agree. However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.

	== Edo ~~approximations~~ ==		== Edo approximation ==
	A list of [[edo]]s with progressively better tunings for 19-limit intervals: {{EDOs\| 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on.		Here is a list of [[edo]]s with progressively better tunings for 19-limit intervals (decreasing [[TE error]]): {{EDOs\| 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on.

	~~Another~~ list of edos which provides relatively good tunings for 19-limit intervals ([[TE ~~relative error\|~~relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.		Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs\| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.

	: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "152fg" means taking the second closest approximation of harmonics 13 and 17.		: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "152fg" means taking the second closest approximation of harmonics 13 and 17.

	== Intervals ==		== Intervals ==

Francium: /* Music */ +music

2024-08-16T13:52:42Z

Music: +music

← Older revision		Revision as of 13:52, 16 August 2024
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	== Music ==		== Music ==
			; [[Domin]]
			* [https://www.youtube.com/watch?v=WTo5YihoLqs ''Asuttan''] (2024)

	; [[Joseph Monzo]]		; [[Joseph Monzo]]
	* [https://www.youtube.com/watch?v=it5avwRE8PI ''Theme from Invisible Haircut''] (1990)		* [https://www.youtube.com/watch?v=it5avwRE8PI ''Theme from Invisible Haircut''] (1990)