13edt - Revision history

FloraC: Fix weird placement of the theory section title. Various other cleanup

2026-10-12T06:07:09Z

Fix weird placement of the theory section title. Various other cleanup

← Older revision		Revision as of 06:07, 12 October 2026
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	{{Infobox ET}}		{{Infobox ET}}
			{{ED intro}} It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.
	[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[Riemann zeta function#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]		[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[Riemann zeta function#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
	[[File:pts-3-5-7-e3-twtop.jpg\|thumb\|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]		[[File:pts-3-5-7-e3-twtop.jpg\|thumb\|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]

	'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-~~tempered version of the [[Bohlen–Pierce]] scale,~~ and ~~therefore has received by far the most attention among equal divisions of the tritave.~~		== Theory ==
			{{See also\| Relationship between Bohlen-Pierce and octave-ful temperaments }}

	It provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as [[~~Sensamagic_clan#Bohpier\|~~bohpier ~~temperament~~]]. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.		13edt provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as the rank-2 [[bohpier]] temperament. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.

	13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].		13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].

	In the [[3.5.7 subgroup]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].		In the 3.5.7 subgroup, 13edt maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

	~~== Theory ==~~
	~~{{Harmonics in equal\|13\|3\|1\|prec=2\|intervals=odd}}~~
	~~{{Harmonics in equal\|13\|3\|1\|prec=2\|intervals=odd\|start=12}}~~

	* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]		=== Odd harmonics ===
			{{Harmonics in equal\|13\|3\|1\|prec=2\|intervals=odd\|columns=11}}
			{{Harmonics in equal\|13\|3\|1\|prec=2\|intervals=odd\|columns=12\|start=12\|collapsed=true\|title=Approximation of odd harmonics in 13edt (continued)}}

	== Intervals ==		== Intervals ==

CompactStar: /* Music */

2026-10-11T22:19:26Z

Music

← Older revision		Revision as of 22:19, 11 October 2026
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	== Music ==		== Music ==
			See also: [[Bohlen–Pierce scale/Music]]

	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/shorts/dKFwJsxE3VU ''13ed3''] (2026)		* [https://www.youtube.com/shorts/dKFwJsxE3VU ''13ed3''] (2026)

CompactStar at 23:26, 20 September 2026

2026-09-20T23:26:05Z

← Older revision		Revision as of 23:26, 20 September 2026
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	13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].		13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].

	In the ~~[[no-2]]~~ [[3~~/1]]-[[equave]]-[[~~7~~-limit~~]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].		In the [[3.5.7 subgroup]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

	== Theory ==		== Theory ==

Lucius Chiaraviglio: /* See also */ Insert Music section before this, starting with Bryan Deiter's ''13ed3'' (2026)

2026-08-22T15:43:57Z

See also: Insert Music section before this, starting with Bryan Deiter's ''13ed3'' (2026)

← Older revision		Revision as of 15:43, 22 August 2026
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	\| [[Arcturus]]		\| [[Arcturus]]
	\|}		\|}

			== Music ==
			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/dKFwJsxE3VU ''13ed3''] (2026)

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

Fredg999: Text replacement - "[[the Riemann zeta function and tuning" to "[[Riemann zeta function"

2026-07-16T19:21:40Z

Text replacement - "[[the Riemann zeta function and tuning" to "[[Riemann zeta function"

← Older revision		Revision as of 19:21, 16 July 2026
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	{{Infobox ET}}		{{Infobox ET}}
	[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[~~the~~ Riemann zeta function ~~and tuning~~#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]		[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[Riemann zeta function#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
	[[File:pts-3-5-7-e3-twtop.jpg\|thumb\|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]		[[File:pts-3-5-7-e3-twtop.jpg\|thumb\|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]

Squib: add 3.5.7 projective tuning space image

2026-06-24T16:16:13Z

add 3.5.7 projective tuning space image

← Older revision		Revision as of 16:16, 24 June 2026
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	{{Infobox ET}}		{{Infobox ET}}
	[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[the Riemann zeta function and tuning#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]		[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[the Riemann zeta function and tuning#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
			[[File:pts-3-5-7-e3-twtop.jpg\|thumb\|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]

	'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.		'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.

Zhenlige: “3/1-equave-7-limit” doesn't worth a page

2026-05-22T16:47:26Z

“3/1-equave-7-limit” doesn't worth a page

← Older revision		Revision as of 16:47, 22 May 2026
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	13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].		13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].

	In the [[no-2]] [[3/1-equave-7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].		In the [[no-2]] [[3/1]]-[[equave]]-[[7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

	== Theory ==		== Theory ==

Fredg999 category edits: Removing from Category:Edt using Cat-a-lot

2025-07-31T15:31:26Z

Removing from Category:Edt using Cat-a-lot

← Older revision		Revision as of 15:31, 31 July 2025
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	* [[23ed7\|23ED7]]: relative ED7		* [[23ed7\|23ED7]]: relative ED7

	~~[[Category:Edt]]~~
	[[Category:Tritave]]		[[Category:Tritave]]
	[[Category:Macrotonal]]		[[Category:Macrotonal]]
	[[Category:Nonoctave]]		[[Category:Nonoctave]]
	[[Category:Bohlen–Pierce]]		[[Category:Bohlen–Pierce]]

ArrowHead294 at 13:01, 10 April 2025

2025-04-10T13:01:50Z

← Older revision		Revision as of 13:01, 10 April 2025
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	{{Infobox ET}}		{{Infobox ET}}
	[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[~~The_Riemann_Zeta_Function_and_Tuning~~#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]		[[File:13edt.png\|thumb\|alt=13edt.png\|A plot of the [[the Riemann zeta function and tuning#Removing primes\|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]

	'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.		'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.

ArrowHead294: /* Intervals */

2025-03-14T13:50:42Z

Intervals

← Older revision		Revision as of 13:50, 14 March 2025
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	! [[4L 5s (3/1-equivalent)\|Enneatonic]]<br />degree		! [[4L 5s (3/1-equivalent)\|Enneatonic]]<br />degree
	! Corresponding<br />3.5.7 subgroup<br />intervals		! Corresponding<br />3.5.7 subgroup<br />intervals
	! [[Lambda ups and downs notation\|Lambda]] ~~(sLsLsLsLs,~~<br />E = 1/1)		! [[Lambda ups and downs notation\|Lambda]]<br />(sLsLsLsLs, {{nowrap\|E {{=}} 1/1}})
	\|-		\|-
	\| 0		\| 0