13edo - Revision history

Overthink: /* Theory */ note error, restate proximity of 135/128 to 1\13 in last paragraph

2026-01-26T23:24:50Z

Theory: note error, restate proximity of 135/128 to 1\13 in last paragraph

← Older revision		Revision as of 23:24, 26 January 2026
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	The simplest JI interpretation of 13edo is in the 2.5.11 [[subgroup]], in which it approximates intervals such as [[11/10]], [[121/80]], and [[64/55]]. However, it notably has very good approximations to 13, 17, and 19 as well.		The simplest JI interpretation of 13edo is in the 2.5.11 [[subgroup]], in which it approximates intervals such as [[11/10]], [[121/80]], and [[64/55]]. However, it notably has very good approximations to 13, 17, and 19 as well.

	Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19~~.21~~ subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.		Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.9.5.21.11.13.17.19 subgroup being a particularly good example. In this subgroup, all 21-odd-limit intervals have less than 25% relative error (23.1{{c}}), except for 22/19 and its [[octave complement]], which barely miss with 25.045% relative error. It has a substantial repertoire of complex consonances for its small size.

	One step of 13edo is very close to [[135/128]] by direct approximation (135/128 is a [[Wikipedia:Continued_fraction\|semiconvergent]] to 2<sup>1/13</sup>).		One step of 13edo is very close to [[135/128]] by direct approximation (135/128 is a [[Wikipedia:Continued_fraction\|semiconvergent]] to 2<sup>1/13</sup>).
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	13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].		13edo is the sixth [[prime edo]], following [[11edo]] and coming before [[17edo]].

	The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes ~~this~~ proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.) A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.		The 5-limit [[aluminium]] temperament (supported by [[65edo]], [[494edo]] and [[1547edo]]) realizes the proximity of 135/128 to 1\13 through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI. (See [[regular temperament]] for more about what this means and how to use it.) A notable smaller superset is [[26edo]], which is a good [[flattone]] tuning, although its step has very high harmonic entropy. An intermediate-sized superset is [[39edo]], which corrects the 3rd and (marginally) 5th harmonics better (but is worse for the 7th harmonic) and has a step size with less harmonic entropy than [[26edo]], while having a more manageable number of notes than 65edo.

	== Intervals ==		== Intervals ==

Overthink: /* Intervals */ fix note

2026-01-22T04:17:46Z

Intervals: fix note

← Older revision		Revision as of 04:17, 22 January 2026
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	! #		! #
	! Cents		! Cents
	! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.</ref>		! Approximated 21-odd-limit Ratios<ref>Ratios are based on treating 13edo as a 2.9.5.21.11.13.17.19 subgroup temperament; other approaches are possible.</ref>
	![[Erv Wilson's Linear Notations\|Erv Wilson]]		![[Erv Wilson's Linear Notations\|Erv Wilson]]
	! Archaeotonic		! Archaeotonic

Overthink: /* Theory */ clarify

2026-01-15T04:28:43Z

Theory: clarify

← Older revision		Revision as of 04:28, 15 January 2026
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	Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.		Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.

	One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction\|semiconvergent]]).		One step of 13edo is very close to [[135/128]] by direct approximation (135/128 is a [[Wikipedia:Continued_fraction\|semiconvergent]] to 2<sup>1/13</sup>).

	In 13edo, the steps less than 600{{c}} are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.		In 13edo, the steps less than 600{{c}} are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.

Overthink: /* Harmony in 13edo */ fix typo

2026-01-10T22:15:17Z

Harmony in 13edo: fix typo

← Older revision		Revision as of 22:15, 10 January 2026
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	By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.		By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.

	The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u>[[13edo#top\|close]]</u> to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 ~~tetrad~~, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.		The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite <u>[[13edo#top\|close]]</u> to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 tetrads, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.

	Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.		Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.

Lucius Chiaraviglio: /* Theory */ Add links to scales and intervals and a couple of other things

2026-01-10T12:00:54Z

Theory: Add links to scales and intervals and a couple of other things

← Older revision		Revision as of 12:00, 10 January 2026
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	== Theory ==		== Theory ==
	13edo has a sharp fifth of 738c, which serves sort of an opposite role to 9edo's flat fifth of 667 cents (in fact, they are both separated from 3/2 by approximately the same amount in opposite directions). Notably for scale theory, this sharp fifth is extremely close to the [[Logarithmic phi\|golden generator]] of 741 cents, and so 13edo has the MOS scales 2L 1s, 3L 2s, and 5L 3s and functions as an equalized 8L 5s.		13edo has a sharp fifth of 738c, which serves sort of an opposite role to 9edo's flat fifth of 667 [[cents]] (in fact, they are both separated from [[3/2]] by approximately the same amount in opposite directions). Notably for scale theory, this sharp fifth is extremely close to the [[Logarithmic phi\|golden generator]] of 741 cents, and so 13edo has the MOS scales [[2L 1s]], [[3L 2s]], and [[5L 3s]] and functions as an equalized [[8L 5s]].

	The simplest JI interpretation of 13edo is in the 2.5.11 subgroup, in which it approximates intervals such as 11/10, 121/80, and 64/55. However, it notably has very good approximations to 13, 17, and 19 as well.		The simplest JI interpretation of 13edo is in the 2.5.11 [[subgroup]], in which it approximates intervals such as [[11/10]], [[121/80]], and [[64/55]]. However, it notably has very good approximations to 13, 17, and 19 as well.

	Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.		Additionally, 13edo has an excellent approximation to the 21st [[harmonic]], and a reasonable approximation to the 9th harmonic. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.
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	One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction\|semiconvergent]]).		One step of 13edo is very close to [[135/128]] by direct approximation (it is a [[Wikipedia:Continued_fraction\|semiconvergent]]).

	In 13edo, the steps less than ~~600¢~~ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.		In 13edo, the steps less than 600{{c}} are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.

	=== Odd harmonics ===		=== Odd harmonics ===

BudjarnLambeth: /* Other scales */ add 2

2025-12-04T03:28:30Z

Other scales: add 2

← Older revision		Revision as of 03:28, 4 December 2025
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	=== Other scales ===		=== Other scales ===
	* Ibex scale{{idio}}: 2 2 4 1 2 2 (''6-tone subset of [[archaeotonic]][7]'')		* Ibex scale{{idio}}: 2 2 4 1 2 2 (''6-tone subset of [[archaeotonic]][7]'')
			* 461.5cET: 5 5 5... (''nonoctave'')
			* 738.5cET: 8 8 8... (''nonoctave'')

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Other scales */

2025-12-04T02:46:30Z

Other scales

← Older revision		Revision as of 02:46, 4 December 2025
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	=== Other scales ===		=== Other scales ===
	* Ibex scale{{idio}}: 2 2 4 1 2 2 (''6-tone subset of [[archaeotonic]][7]]'')		* Ibex scale{{idio}}: 2 2 4 1 2 2 (''6-tone subset of [[archaeotonic]][7]'')

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Other scales */

2025-12-04T02:46:10Z

Other scales

← Older revision		Revision as of 02:46, 4 December 2025
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	=== Other scales ===		=== Other scales ===
	* Ibex scale{{idio}}: 2 2 4 1 2 2		* Ibex scale{{idio}}: 2 2 4 1 2 2 (''6-tone subset of [[archaeotonic]][7]]'')

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Moment of symmetry scales */

2025-12-04T02:44:18Z

Moment of symmetry scales

← Older revision		Revision as of 02:44, 4 December 2025
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	[[:File:13edo_horograms.pdf\|13edo horograms.pdf]]		[[:File:13edo_horograms.pdf\|13edo horograms.pdf]]

	~diagram by [[Andrew Heathwaite]], based on [[~~horagram~~]]s pioneered by [[Erv Wilson]]		~diagram by [[Andrew Heathwaite]], based on [[horogram]]s pioneered by [[Erv Wilson]]

	=== Near-12edo scales ===		=== Near-12edo scales ===

BudjarnLambeth: /* Moment of symmetry scales */ add links

2025-12-04T02:42:53Z

Moment of symmetry scales: add links

← Older revision		Revision as of 02:42, 4 December 2025
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	[[:File:13edo_horograms.pdf\|13edo horograms.pdf]]		[[:File:13edo_horograms.pdf\|13edo horograms.pdf]]

	~diagram by Andrew Heathwaite, based on ~~horagrams~~ pioneered by Erv Wilson		~diagram by [[Andrew Heathwaite]], based on [[horagram]]s pioneered by [[Erv Wilson]]

	=== Near-12edo scales ===		=== Near-12edo scales ===