17-limit - Revision history

Haidilao: In color notation, when "so" appears by itself, we need to write "iso" to differenciate it from solfege "so/sol"

2026-04-18T16:31:27Z

In color notation, when "so" appears by itself, we need to write "iso" to differenciate it from solfege "so/sol"

← Older revision		Revision as of 16:31, 18 April 2026
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	\| 104.955		\| 104.955
	\| 17o2		\| 17o2
	\| so 2nd		\| iso 2nd
	\| large septendecimal semitone		\| large septendecimal semitone
	\|-		\|-
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	\| 603.000		\| 603.000
	\| 17o5		\| 17o5
	\| so 5th		\| iso 5th
	\| greater septendecimal tritone		\| greater septendecimal tritone
	\|-		\|-
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	\| 1101.045		\| 1101.045
	\| 17o8		\| 17o8
	\| so octave		\| iso octave
	\| large septendecimal major seventh		\| large septendecimal major seventh
	\|}		\|}

Overthink: /* Edo approximation */ caps

2026-01-12T07:36:27Z

Edo approximation: caps

← Older revision		Revision as of 07:36, 12 January 2026
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	Here is a list of edos which provides relatively good tunings for 17-limit intervals ([[TE relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.		Here is a list of edos which provides relatively good tunings for 17-limit intervals ([[TE relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.

	{{Note\| [[~~wart~~ notation]] is used to specify the [[val]] chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17. }}		{{Note\| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17. }}

	== Intervals ==		== Intervals ==

FloraC: /* Edo approximation */ adopt template

2025-12-27T17:44:12Z

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 17-limit intervals ([[TE relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.		Here is a list of edos which provides relatively good tunings for 17-limit intervals ([[TE relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.

	~~: '''~~Note~~''':~~ [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17.		{{Note\| [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17. }}

	== Intervals ==		== Intervals ==

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

2025-08-20T16:25:12Z

Edo approximation: calibrate the list according to my research results

← Older revision		Revision as of 16:25, 20 August 2025
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	== Edo approximation ==		== Edo approximation ==
	Here is a list of [[edo]]s with progressively better tunings for 17-limit intervals (decreasing [[TE error]]): {{EDOs\| 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954 }} and so on.		Here is a list of [[edo]]s with progressively better tunings for 17-limit intervals ([[monotonicity limit]] ≥ 17 and decreasing [[TE error]]): {{EDOs\| 31, 38df, 41, 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954 }} 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 17-limit intervals ([[TE relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.		Here is a list of edos which provides relatively good tunings for 17-limit intervals ([[TE relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.

FloraC: /* Terminology and notation */ rework to sort it topically

2025-07-13T13:44:16Z

Terminology and notation: rework to sort it topically

← Older revision		Revision as of 13:44, 13 July 2025
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	== Terminology and notation ==		== Terminology and notation ==
	Conceptualization systems disagree on whether 17/16 should be a [[diatonic semitone]] or a [[chromatic semitone]], and as a result the disagreement propagates to all intervals of [[harmonic class\|HC17]].		Conceptualization systems disagree on whether 17/16 should be a [[diatonic semitone]] or a [[chromatic semitone]], and as a result the disagreement propagates to all intervals of [[harmonic class\|HC17]].
			* In [[Functional Just System]], 17/16 is a diatonic semitone, separated by [[4131/4096]] from [[256/243]], the Pythagorean diatonic semitone.
			* In [[Helmholtz–Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone.

	~~In [[Functional Just System]], 17/16 is a diatonic semitone, separated by [[4131/4096]] from [[256/243]], the Pythagorean diatonic semitone.~~ The case for it ~~being a diatonic semitone includes~~:		The case for mapping it to either category may include:
	* The diatonic semitone is simpler than the chromatic semitone in the [[chain of fifths]], being -5 steps as opposed to +7 steps, and the ~~associated [[comma]]~~ 4131/4096 is small enough ~~to be considered a comma~~ which ~~does~~ not alter the interval category.		* Number of steps in the chain of fifths. The diatonic semitone is simpler than the chromatic semitone in the [[chain of fifths]], being -5 steps as opposed to +7 steps.
	* If [[7/4]] is known to be a seventh, assigning 17/16 to a second will make intervals [[17/14]] and [[21/17]] thirds. This is favorable because 17/14 and 21/17 are important building blocks of {{w\|tertian harmony}}.		* Size of the associated formal commas. The formal comma of the chromatic mapping, 2187/2176, is simpler and smaller than that of the diatonic mapping, 4131/4096, though both are generally considered small enough as commas which do not alter the interval category. The chromatic mapping has the advantage of keeping the Pythagorean order of diatonic semitone < chromatic semitone in the intervals of 17.
			* Interactions with other primes. On one hand, if [[7/4]] is known to be a seventh, assigning 17/16 to a second will make intervals [[17/14]] and [[21/17]] thirds. This is favorable because 17/14 and 21/17 are important building blocks of {{w\|tertian harmony}}. On the other hand, if [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for harmonics to be positive and subharmonics to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.
	~~In [[Helmholtz-Ellis notation]], 17/16 is a chromatic semitone~~, ~~separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone. The case for it being a chromatic semitone includes:~~
	* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for harmonics to be positive and subharmonics to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.

	In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.		In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.

FloraC at 15:23, 17 December 2024

2024-12-17T15:23:55Z

← Older revision		Revision as of 15:23, 17 December 2024
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	== See also ==		== See also ==
	* [[17-odd-limit]]
	* [[Seventeen limit tetrads]]		* [[Seventeen limit tetrads]]

	[[Category:17-limit\| ]] <!-- main article -->		[[Category:17-limit\| ]] <!-- main article -->
	[[Category:Listen]]		[[Category:Listen]]

FloraC: +relation to odd limits and harmonic/subharmonic modes

2024-12-17T15:22:03Z

+relation to odd limits and harmonic/subharmonic modes

← Older revision		Revision as of 15:22, 17 December 2024
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	{{Prime limit navigation\|17}}		{{Prime limit navigation\|17}}
	The '''17-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 17. It is the 7th [[prime limit]] and is ~~thus~~ a superset of the [[13-limit]] and a subset of the [[19-limit]]. It adds to the [[13-limit]] a semitone of about 105¢ – [[17/16]] – and several other intervals between the 17th [[harmonic]] and the lower ones.		The '''17-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 17. It is the 7th [[prime limit]] and is a superset of the [[13-limit]] and a subset of the [[19-limit]]. It adds to the [[13-limit]] a semitone of about 105¢ – [[17/16]] – and several other intervals between the 17th [[harmonic]] and the lower ones.

	The 17-limit is a [[Rank and codimension\|rank-7]] system, and can be modeled in a 6-dimensional [[lattice]], with the primes 3, 5, 7, 11, 13, and 17 represented by each dimension. The prime 2 does not appear in the typical 17-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is needed.		The 17-limit is a [[Rank and codimension\|rank-7]] system, and can be modeled in a 6-dimensional [[lattice]], with the primes 3, 5, 7, 11, 13, and 17 represented by each dimension. The prime 2 does not appear in the typical 17-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is needed.

			These things are contained by the 17-limit, but not the 13-limit:
			* The [[17-odd-limit]];
			* Mode 9 of the harmonic or subharmonic series.

	== Terminology and notation ==		== Terminology and notation ==

FloraC: /* Terminology and notation */ "otonal intervals" -> "harmonics" and "utonal intervals" -> "subharmonics"

2024-11-24T09:19:55Z

Terminology and notation: "otonal intervals" -> "harmonics" and "utonal intervals" -> "subharmonics"

← Older revision		Revision as of 09:19, 24 November 2024
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	In [[Helmholtz-Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone. The case for it being a chromatic semitone includes:		In [[Helmholtz-Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone. The case for it being a chromatic semitone includes:
	* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for ~~otonal intervals~~ to be positive and ~~utonal intervals~~ to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.		* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for harmonics to be positive and subharmonics to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.

	In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.		In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.

FloraC: /* Edo approximations */ wording

2024-09-02T10:42:00Z

Edo approximations: wording

← Older revision		Revision as of 10:42, 2 September 2024
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	The names tabulated in [[#Intervals]] are common names and do not follow this discussion yet.		The names tabulated in [[#Intervals]] are common names and do not follow this discussion yet.

	== Edo ~~approximations~~ ==		== Edo approximation ==
	A list of [[edo]]s with progressively better tunings for 17-limit intervals: {{EDOs\| 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954 }} and so on.		Here is a list of [[edo]]s with progressively better tunings for 17-limit intervals (decreasing [[TE error]]): {{EDOs\| 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954 }} and so on.

	~~Another~~ list of edos which provides relatively good tunings for 17-limit intervals ([[TE ~~relative error\|~~relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.		Here is a list of edos which provides relatively good tunings for 17-limit intervals ([[TE relative error]] < 5.4%): {{EDOs\| 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 }} and so on.

	: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17.		: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17.

	== Intervals ==		== Intervals ==

FloraC: /* Terminology and notation */ rework

2024-07-25T15:48:25Z

Terminology and notation: rework

← Older revision		Revision as of 15:48, 25 July 2024
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	In [[Functional Just System]], 17/16 is a diatonic semitone, separated by [[4131/4096]] from [[256/243]], the Pythagorean diatonic semitone. The case for it being a diatonic semitone includes:		In [[Functional Just System]], 17/16 is a diatonic semitone, separated by [[4131/4096]] from [[256/243]], the Pythagorean diatonic semitone. The case for it being a diatonic semitone includes:
	* The diatonic semitone is simpler than the chromatic semitone in the [[chain of fifths]], being -5 steps as opposed to +7 steps, and the associated [[comma]] 4131/4096 is small enough to be considered a comma which does not alter the interval category.		* The diatonic semitone is simpler than the chromatic semitone in the [[chain of fifths]], being -5 steps as opposed to +7 steps, and the associated [[comma]] 4131/4096 is small enough to be considered a comma which does not alter the interval category.
	* It is ~~preferable for the~~ intervals [[17/14]] and [[21/17]] ~~to be~~ thirds ~~since they work well in tertian harmony~~. ~~If [[7/4]]~~ is ~~known to be a seventh, then~~ 17/~~16 must be a second~~.		* If [[7/4]] is known to be a seventh, assigning 17/16 to a second will make intervals [[17/14]] and [[21/17]] thirds. This is favorable because 17/14 and 21/17 are important building blocks of {{w\|tertian harmony}}.

	In [[Helmholtz-Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone. The case for it being a chromatic semitone includes:		In [[Helmholtz-Ellis notation]], 17/16 is a chromatic semitone, separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone. The case for it being a chromatic semitone includes:
	* It is ~~preferable for otonal intervals~~ to be ~~positive and utonal intervals to be negative in the chain of fifths.~~		* If [[5/4]] is known to be a third, then 17/16 being a unison will make [[17/15]] a second and [[20/17]] a third. This is favorable because 17/15 is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. The HEJI authors find it generally favorable for otonal intervals to be positive and utonal intervals to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.
	* It is preferable for the interval [[17/15]] ~~to be~~ a ~~major~~ second ~~since it~~ is the [[mediant]] of major seconds of [[9/8]] and [[8/7]]. ~~If [[5/4]] is known~~ to be ~~a major third~~, ~~then 17/16 must be an augmented unison~~.

	In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.		In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded [[chain-of-fifths notation]] is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.