29-limit - Revision history

Francium: /* Music */ +music

2026-03-07T12:18:15Z

Music: +music

← Older revision		Revision as of 12:18, 7 March 2026
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	== Music ==		== Music ==
			; [[Francium]]
			* [https://www.youtube.com/watch?v=PwvKS0RhTgs ''Spring Your Miracle''] (2026)

	; [[Randy Wells]]		; [[Randy Wells]]
	* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)		* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)

	[[Category:29-limit\| ]] <!-- main article -->		[[Category:29-limit\| ]] <!-- main article -->

Overthink: /* Edo approximations */ proper english

2026-02-04T05:04:00Z

Edo approximations: proper english

← Older revision		Revision as of 05:04, 4 February 2026
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	== Edo approximations ==		== Edo approximations ==
	[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. ~~Intervals~~ [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).		[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. The intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).

	Edos with increasingly better approximations of the 29-limit ([[monotonicity limit]] ≥ 29 and decreasing [[TE error]]): {{EDOs\| 72, 77, 99ef, 118, 121i, 130, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 1395, 1578 }}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].		Edos with increasingly better approximations of the 29-limit ([[monotonicity limit]] ≥ 29 and decreasing [[TE error]]): {{EDOs\| 72, 77, 99ef, 118, 121i, 130, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 1395, 1578 }}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

Overthink: note octave complements, a couple of fixes

2026-02-04T05:02:13Z

note octave complements, a couple of fixes

← Older revision		Revision as of 05:02, 4 February 2026
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	* Mode 15 of the harmonic or subharmonic series.		* Mode 15 of the harmonic or subharmonic series.

	The 29-limit intervals of the 2.3.29 subgroup are [[submajor and supraminor]], with [[29/27]] being a supraminor second, [[32/29]] a submajor second, [[29/24]] a supraminor third, and [[36/29]] a submajor third. While ~~lower-limit~~ supraminor and submajor intervals ~~exist~~, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1\|29th harmonic]] is thus quite simple to classify by [[5L 2s\|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1\|5]], [[7/1]], etc. Primes [[17/1\|17]] and [[23/1\|23]] are not so friendly in terms of interval categorization, and may be considered discordant to the fundamental, being a semitone and a tritone when [[octave reduced]] respectively. Thus many people wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.		The 29-limit intervals of the 2.3.29 subgroup are [[submajor and supraminor]], with [[29/27]] being a supraminor second, [[32/29]] a submajor second, [[29/24]] a supraminor third, and [[36/29]] a submajor third, with their [[octave complement]]s classified accordingly. While supraminor and submajor intervals occur in lower limits, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1\|29th harmonic]] is thus quite simple to classify by [[5L 2s\|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1\|5]], [[7/1\|7]], etc. Primes [[17/1\|17]] and [[23/1\|23]] are not so friendly in terms of interval categorization, and may be considered discordant to the fundamental, being a semitone and a tritone when [[octave reduced]] respectively. Thus many people wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.

	However, the 29-limit approaches the point where [[consonance]] stops being registered, and intervals become very close to each other, such as [[29/28]] only being wider than [[30/29]] by [[841/840]], a comma of 2.06{{c}}. This difference is [[JND\|unnoticeable]] melodically, and very difficult to hear harmonically.		However, the 29-limit approaches the point where [[consonance]] stops being registered, and intervals become very close to each other, such as [[29/28]] only being wider than [[30/29]] by [[841/840]], a comma of 2.06{{c}}. This difference is [[JND\|unnoticeable]] melodically, and very difficult to hear harmonically.

FloraC: /* Edo approximations */ - bold since similar pages don't have them. + note

2026-01-17T12:17:47Z

Edo approximations: - bold since similar pages don't have them. + note

← Older revision		Revision as of 12:17, 17 January 2026
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	[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. Intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).		[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. Intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).

	~~EDOs~~ with increasingly better approximations of the 29-limit ([[monotonicity limit]] ≥ 29 and decreasing [[TE error]]): {{EDOs\| 72, 77, 99ef, 118, 121i, 130, ~~'''~~140~~'''~~, 152fgj, ~~'''~~159~~'''~~, ~~'''~~183~~'''~~, 217, ~~'''~~243e~~'''~~, 270, 282, ~~'''~~311~~'''~~, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, ~~'''~~954hj~~'''~~, 1106, 1282, 1308, 1323, 1395, ~~'''~~1578~~'''~~}}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].		Edos with increasingly better approximations of the 29-limit ([[monotonicity limit]] ≥ 29 and decreasing [[TE error]]): {{EDOs\| 72, 77, 99ef, 118, 121i, 130, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 1395, 1578 }}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

			{{Note\| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "99ef" means taking the second closest approximations of harmonics 11 and 13. }}

	== Music ==		== Music ==

Overthink: /* Edo approximations */ add lower error ET sequence

2026-01-16T23:54:20Z

Edo approximations: add lower error ET sequence

← Older revision		Revision as of 23:54, 16 January 2026
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	== Edo approximations ==		== Edo approximations ==
	[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. Intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).		[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. Intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).

			EDOs with increasingly better approximations of the 29-limit ([[monotonicity limit]] ≥ 29 and decreasing [[TE error]]): {{EDOs\| 72, 77, 99ef, 118, 121i, 130, '''140''', 152fgj, '''159''', '''183''', 217, '''243e''', 270, 282, '''311''', 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, '''954hj''', 1106, 1282, 1308, 1323, 1395, '''1578'''}}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

	== Music ==		== Music ==

Overthink: note potential discordance of 17 and 23

2026-01-16T23:08:14Z

note potential discordance of 17 and 23

← Older revision		Revision as of 23:08, 16 January 2026
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	* Mode 15 of the harmonic or subharmonic series.		* Mode 15 of the harmonic or subharmonic series.

	The 29-limit intervals of the 2.3.29 subgroup are [[submajor and supraminor]], with [[29/27]] being a supraminor second, [[32/29]] a submajor second, [[29/24]] a supraminor third, and [[36/29]] a submajor third. While lower-limit supraminor and submajor intervals exist, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1\|29th harmonic]] is thus quite simple to classify by [[5L 2s\|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1\|5]], [[7/1]], etc. ~~Note that primes~~ [[17/1\|17]] and [[23/1\|23]] are not so friendly in terms of interval categorization, and many wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.		The 29-limit intervals of the 2.3.29 subgroup are [[submajor and supraminor]], with [[29/27]] being a supraminor second, [[32/29]] a submajor second, [[29/24]] a supraminor third, and [[36/29]] a submajor third. While lower-limit supraminor and submajor intervals exist, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1\|29th harmonic]] is thus quite simple to classify by [[5L 2s\|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1\|5]], [[7/1]], etc. Primes [[17/1\|17]] and [[23/1\|23]] are not so friendly in terms of interval categorization, and may be considered discordant to the fundamental, being a semitone and a tritone when [[octave reduced]] respectively. Thus many people wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.

	However, the 29-limit approaches the point where [[consonance]] stops being registered, and intervals become very close to each other, such as [[29/28]] only being wider than [[30/29]] by [[841/840]], a comma of 2.06{{c}}. This difference is [[JND\|unnoticeable]] melodically, and very difficult to hear harmonically.		However, the 29-limit approaches the point where [[consonance]] stops being registered, and intervals become very close to each other, such as [[29/28]] only being wider than [[30/29]] by [[841/840]], a comma of 2.06{{c}}. This difference is [[JND\|unnoticeable]] melodically, and very difficult to hear harmonically.

Overthink: Add a proper description

2026-01-16T23:05:25Z

Add a proper description

← Older revision		Revision as of 23:05, 16 January 2026
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	These things are contained by the 29-limit, but not the 23-limit:		These things are contained by the 29-limit, but not the 23-limit:
	* The [[29-odd-limit]];		* The [[29-odd-limit]];
	* Mode 15 of the harmonic or subharmonic series.		* Mode 15 of the harmonic or subharmonic series.

			The 29-limit intervals of the 2.3.29 subgroup are [[submajor and supraminor]], with [[29/27]] being a supraminor second, [[32/29]] a submajor second, [[29/24]] a supraminor third, and [[36/29]] a submajor third. While lower-limit supraminor and submajor intervals exist, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1\|29th harmonic]] is thus quite simple to classify by [[5L 2s\|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1\|5]], [[7/1]], etc. Note that primes [[17/1\|17]] and [[23/1\|23]] are not so friendly in terms of interval categorization, and many wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.

			However, the 29-limit approaches the point where [[consonance]] stops being registered, and intervals become very close to each other, such as [[29/28]] only being wider than [[30/29]] by [[841/840]], a comma of 2.06{{c}}. This difference is [[JND\|unnoticeable]] melodically, and very difficult to hear harmonically.

	== Edo approximations ==		== Edo approximations ==
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	; [[Randy Wells]]		; [[Randy Wells]]
	* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)		* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)

	~~{{stub}}~~

	[[Category:29-limit\| ]] <!-- main article -->		[[Category:29-limit\| ]] <!-- main article -->

Lériendil at 15:45, 13 March 2025

2025-03-13T15:45:36Z

← Older revision		Revision as of 15:45, 13 March 2025
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	; [[Randy Wells]]		; [[Randy Wells]]
	* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)		* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)

			{{stub}}

	[[Category:29-limit\| ]] <!-- main article -->		[[Category:29-limit\| ]] <!-- main article -->

FloraC: +rank, lattice representation, relation to odd limits and harmonic/subharmonic modes

2024-12-17T15:39:26Z

+rank, lattice representation, relation to odd limits and harmonic/subharmonic modes

← Older revision		Revision as of 15:39, 17 December 2024
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	{{Prime limit navigation\|29}}		{{Prime limit navigation\|29}}
	The '''29-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 29. It is the 10th [[prime limit]] and is ~~thus~~ a superset of the [[23-limit]] and a subset of the [[31-limit]]. The prime 29 is notable as being the prime that ends a record prime gap starting at 23. Thus, the 29-limit is in some sense analogous to the [[11-limit]] as both include the prime ending a record prime gap.		The '''29-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 29. It is the 10th [[prime limit]] and is a superset of the [[23-limit]] and a subset of the [[31-limit]]. The prime 29 is notable as being the prime that ends a record prime gap starting at 23. Thus, the 29-limit is in some sense analogous to the [[11-limit]] as both include the prime ending a record prime gap.

			The 29-limit is a rank-10 system, and can be modeled in a 9-dimensional lattice, with the primes 3 to 29 represented by each dimension. The prime 2 does not appear in the typical 29-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a tenth dimension is needed.

			These things are contained by the 29-limit, but not the 23-limit:
			* The [[29-odd-limit]];
			* Mode 15 of the harmonic or subharmonic series.

	== Edo approximations ==		== Edo approximations ==
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	; [[Randy Wells]]		; [[Randy Wells]]
	* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)		* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)

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

	[[Category:29-limit\| ]] <!-- main article -->		[[Category:29-limit\| ]] <!-- main article -->

FloraC: Move from "Music in just intonation" page

2024-01-07T10:35:21Z

Move from "Music in just intonation" page

← Older revision		Revision as of 10:35, 7 January 2024
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	== Edo approximations ==		== Edo approximations ==
	[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. Intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).		[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. Intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).

			== Music ==
			; [[Randy Wells]]
			* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)

	== See also ==		== See also ==