13edt - Revision history

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

ArrowHead294 at 12:57, 10 March 2025

2025-03-10T12:57:39Z

← Older revision		Revision as of 12:57, 10 March 2025
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	[[Category:Macrotonal]]		[[Category:Macrotonal]]
	[[Category:Nonoctave]]		[[Category:Nonoctave]]
	[[Category:~~Bohlen-Pierce~~]]		[[Category:Bohlen–Pierce]]

ArrowHead294 at 19:10, 4 February 2025

2025-02-04T19:10:33Z

← Older revision		Revision as of 19:10, 4 February 2025
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	[[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.

	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.		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 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]].
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| class="wikitable center-all right-3 left-5"		{\| class="wikitable center-all right-3 left-5"
	\|+Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	! Periods<br>per tritave		\|-
	! Generator<br>(reduced)		! Periods<br />per tritave
	! Cents<br>(reduced)		! Generator<br />(reduced)
	! Associated<br>ratio		! Cents<br />(reduced)
			! Associated<br />ratio
	! Temperament		! Temperament
	\|-		\|-
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	\| [[Canopus]]		\| [[Canopus]]
	\|-		\|-
	\|1		\| 1
	\|5\13		\| 5\13
	\|731.63		\| 731.63
	\|75/49		\| 75/49
	\|		\|
	\|-		\|-

ArrowHead294: /* Intervals */

2025-01-29T19:04:10Z

Intervals

← Older revision		Revision as of 19:04, 29 January 2025
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	! [[Cent]]s		! [[Cent]]s
	! [[Hekt]]s		! [[Hekt]]s
	! [[4L 5s (3/1-equivalent)\|Enneatonic]] degree		! [[4L 5s (3/1-equivalent)\|Enneatonic]]<br />degree
	! Corresponding		! Corresponding<br />3.5.7 subgroup<br />intervals
	3.5.7 subgroup <br>		! [[Lambda ups and downs notation\|Lambda]] (sLsLsLsLs,<br />E = 1/1)
	intervals
	! [[Lambda ups and downs notation\|Lambda]]
	(sLsLsLsLs, <br>
	E = 1/1)
	\|-		\|-
	\| 0		\| 0
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	\| 100		\| 100
	\| A1/m2		\| A1/m2
	\| [[49/45]] (-1~~.1c~~); [[27/25]] (+13.1c)		\| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
	\| F		\| F
	\|-		\|-
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	\| 200		\| 200
	\| M2/d3		\| M2/d3
	\| [[25/21]] (-9.2c)		\| [[25/21]] (−9.2{{c}})
	\| F#, Gb		\| F#, Gb
	\|-		\|-
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	\| 300		\| 300
	\| A2/P3/d4		\| A2/P3/d4
	\| [[9/7]] (+3.8c)		\| [[9/7]] (+3.8{{c}})
	\| G		\| G
	\|-		\|-
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	\| 400		\| 400
	\| A3/m4/d5		\| A3/m4/d5
	\| [[7/5]] (+2.7c)		\| [[7/5]] (+2.7{{c}})
	\| H		\| H
	\|-		\|-
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	\| 500		\| 500
	\| M4/m5		\| M4/m5
	\| [[75/49]] (-5.4c)		\| [[75/49]] (−5.4{{c}})
	\| H#, Jb		\| H#, Jb
	\|-		\|-
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	\| 600		\| 600
	\| A4/M5		\| A4/M5
	\| [[5/3]] (-6.5c)		\| [[5/3]] (−6.5{{c}})
	\| J		\| J
	\|-		\|-
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	\| 700		\| 700
	\| A5/m6/d7		\| A5/m6/d7
	\| [[9/5]] (+6.5c)		\| [[9/5]] (+6.5{{c}})
	\| A		\| A
	\|-		\|-
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	\| 800		\| 800
	\| M6/m7		\| M6/m7
	\| [[49/25]] (+5.4c)		\| [[49/25]] (+5.4{{c}})
	\| A#, Bb		\| A#, Bb
	\|-		\|-
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	\| 900		\| 900
	\| A6/M7/d8		\| A6/M7/d8
	\| [[15/7]] (-2.7c)		\| [[15/7]] (−2.7{{c}})
	\| B		\| B
	\|-		\|-
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	\| 1000		\| 1000
	\| P8/d9		\| P8/d9
	\| [[7/3]] (-3.8c)		\| [[7/3]] (−3.8{{c}})
	\| C		\| C
	\|-		\|-
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	\| 1100		\| 1100
	\| A8/m9		\| A8/m9
	\| [[63/25]] (+9.2c)		\| [[63/25]] (+9.2{{c}})
	\| C#, Db		\| C#, Db
	\|-		\|-
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	\| 1200		\| 1200
	\| M9/d10		\| M9/d10
	\| [[135/49]] (+1.1c); [[25/9]] (~~-13~~.1c)		\| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
	\| D		\| D
	\|-		\|-

BudjarnLambeth: /* Theory */ Extend odd harmonics table, remove integer harmonics table

2024-10-06T06:30:45Z

Theory: Extend odd harmonics table, remove integer harmonics table

← Older revision		Revision as of 06:30, 6 October 2024
Line 11:		Line 11:

	== Theory ==		== Theory ==
	{{Harmonics in equal\|13\|3\|1\|prec=2}}		{{Harmonics in equal\|13\|3\|1\|prec=2\|intervals=odd}}
	{{Harmonics in equal\|13\|3\|1\|prec=2\|intervals=odd\|~~columns~~=16}}		{{Harmonics in equal\|13\|3\|1\|prec=2\|intervals=odd\|start=12}}

	* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]		* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]

Lériendil at 22:05, 5 October 2024

2024-10-05T22:05:36Z

← Older revision		Revision as of 22:05, 5 October 2024
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	{{Main\|Intervals of BP}}		{{Main\|Intervals of BP}}

	{\| class="wikitable center-1 right-2 right-3"		{\| class="wikitable center-all right-2 right-3"
	\|-		\|-
	! Steps		! Steps
	! [[Cent]]s		! [[Cent]]s
	! [[Hekt]]s		! [[Hekt]]s
	! ~~BP enneatonic~~ degree		! [[4L 5s (3/1-equivalent)\|Enneatonic]] degree
	! Corresponding 3.5.7 subgroup intervals		! Corresponding
	~~! Comments~~		3.5.7 subgroup <br>
	~~! Generator for...~~		intervals
	! [[~~4L 5s (3/1-equivalent)#Notation~~\|~~BPS enneatonic notation~~]] (J = 1/1)		! [[Lambda ups and downs notation\|Lambda]]
			(sLsLsLsLs, <br>
			E = 1/1)
	\|-		\|-
	\| 0		\| 0
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	\| P1		\| P1
	\| 1/1		\| 1/1
	\|		\| E
	\|
	~~\| J~~
	\|-		\|-
	\| 1		\| 1
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	\| 100		\| 100
	\| A1/m2		\| A1/m2
	\| ~~[[27/25]],~~ [[49/45]]		\| [[49/45]] (-1.1c); [[27/25]] (+13.1c)
	~~\| BP "semitone"~~		\| F
	\| [[~~Procyon~~]]
	\| K
	\|-		\|-
	\| 2		\| 2
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	\| 200		\| 200
	\| M2/d3		\| M2/d3
	\| [[25/21]]		\| [[25/21]] (-9.2c)
	\| ~~BP "wholetone"~~		\| F#, Gb
	~~\| [[Sirius]]~~
	~~\| K~~#, Lb
	\|-		\|-
	\| 3		\| 3
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	\| 300		\| 300
	\| A2/P3/d4		\| A2/P3/d4
	\| [[9/7]]		\| [[9/7]] (+3.8c)
	\|		\| G
	~~\| [[Bohlen-Pierce-Stearns\|Linear BP]]~~
	~~\| L~~
	\|-		\|-
	\| 4		\| 4
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	\| 400		\| 400
	\| A3/m4/d5		\| A3/m4/d5
	\| [[7/5]]		\| [[7/5]] (+2.7c)
	\|		\| H
	~~\| [[Canopus]]~~
	~~\| M~~
	\|-		\|-
	\| 5		\| 5
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	\| 500		\| 500
	\| M4/m5		\| M4/m5
	\| [[75/49]]		\| [[75/49]] (-5.4c)
	\| ~~false 3/2~~		\| H#, Jb
	~~\| false Father~~
	~~\| M~~#, Nb
	\|-		\|-
	\| 6		\| 6
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	\| 600		\| 600
	\| A4/M5		\| A4/M5
	\| [[5/3]]		\| [[5/3]] (-6.5c)
	\|		\| J
	~~\| [[Arcturus]]~~
	~~\| N~~
	\|-		\|-
	\| 7		\| 7
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	\| 700		\| 700
	\| A5/m6/d7		\| A5/m6/d7
	\| [[9/5]]		\| [[9/5]] (+6.5c)
	\|		\| A
	~~\| Arcturus~~
	~~\| O~~
	\|-		\|-
	\| 8		\| 8
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	\| 800		\| 800
	\| M6/m7		\| M6/m7
	\| [[49/25]]		\| [[49/25]] (+5.4c)
	\| ~~false 2/1~~		\| A#, Bb
	~~\| false Father~~
	~~\| O~~#, Pb
	\|-		\|-
	\| 9		\| 9
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	\| 900		\| 900
	\| A6/M7/d8		\| A6/M7/d8
	\| [[15/7]]		\| [[15/7]] (-2.7c)
	\|		\| B
	~~\| Canopus~~
	~~\| P~~
	\|-		\|-
	\| 10		\| 10
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	\| 1000		\| 1000
	\| P8/d9		\| P8/d9
	\| [[7/3]]		\| [[7/3]] (-3.8c)
	\|		\| C
	~~\| Linear BP~~
	~~\| Q~~
	\|-		\|-
	\| 11		\| 11
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	\| 1100		\| 1100
	\| A8/m9		\| A8/m9
	\| [[63/25]]		\| [[63/25]] (+9.2c)
	\|		\| C#, Db
	~~\| Sirius~~
	~~\| Q~~#, Rb
	\|-		\|-
	\| 12		\| 12
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	\| 1200		\| 1200
	\| M9/d10		\| M9/d10
	\| [[25/9]], [[~~135~~/49]]		\| [[135/49]] (+1.1c); [[25/9]] (-13.1c)
	\|		\| D
	~~\| Procyon~~
	~~\| R~~
	\|-		\|-
	\| 13		\| 13
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	\| A9/P10		\| A9/P10
	\| [[3/1]]		\| [[3/1]]
	\| ~~Tritave~~		\| E
	\|
	~~\| J~~
	\|}		\|}