4edt - Revision history

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

2025-08-01T19:23:08Z

Removing from Category:Edonoi using Cat-a-lot

← Older revision		Revision as of 19:23, 1 August 2025
Line 53:		Line 53:
	4EDT is a generator of the [[Vulture family\|vulture temperament]], which tempers out 10485760000/10460353203 in the 5-limit.		4EDT is a generator of the [[Vulture family\|vulture temperament]], which tempers out 10485760000/10460353203 in the 5-limit.

	~~[[Category:Edonoi]]~~
	[[Category:Macrotonal]]		[[Category:Macrotonal]]

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

2025-07-31T15:31:34Z

Removing from Category:Edt using Cat-a-lot

← Older revision		Revision as of 15:31, 31 July 2025
Line 53:		Line 53:
	4EDT is a generator of the [[Vulture family\|vulture temperament]], which tempers out 10485760000/10460353203 in the 5-limit.		4EDT is a generator of the [[Vulture family\|vulture temperament]], which tempers out 10485760000/10460353203 in the 5-limit.

	~~[[Category:Edt]]~~
	[[Category:Edonoi]]		[[Category:Edonoi]]
	[[Category:Macrotonal]]		[[Category:Macrotonal]]

ArrowHead294 at 18:25, 13 March 2025

2025-03-13T18:25:54Z

← Older revision		Revision as of 18:25, 13 March 2025
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	4edt fails to approximate a lot of low prime harmonics well—the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) that these intervals see practical use.		4edt fails to approximate a lot of low prime harmonics well—the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) that these intervals see practical use.

	4edt can be viewed as a "collapsed" version of the [[~~Bohlen-Pierce~~]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].		4edt can be viewed as a "collapsed" version of the [[Bohlen–Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].

	=== Harmonics ===		=== Harmonics ===

ArrowHead294 at 15:15, 20 February 2025

2025-02-20T15:15:47Z

← Older revision		Revision as of 15:15, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==
			4edt fails to approximate a lot of low prime harmonics well—the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) that these intervals see practical use.
	4edt fails to approximate a lot of low prime harmonics ~~well -- the~~ first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) that these intervals see practical use.

	4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].		4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].
Line 12:		Line 11:

	=== Approximation of intervals ===		=== Approximation of intervals ===

	{\| class="wikitable right-all left-4 left-5"		{\| class="wikitable right-all left-4 left-5"
	\|-		\|-
Line 18:		Line 16:
	! Cents		! Cents
	! [[Hekt]]s		! [[Hekt]]s
	! Corresponding <br>JI intervals		! Corresponding<br>JI intervals
	! Comments		! Comments
	\|-		\|-

Lériendil: /* Harmonics */

2024-09-25T02:00:51Z

Harmonics

← Older revision		Revision as of 02:00, 25 September 2024
Line 9:		Line 9:

	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|4\|3\|1}}		{{Harmonics in equal\|4\|3\|1\|columns=15}}


	=== Approximation of intervals ===		=== Approximation of intervals ===

Akselai at 13:07, 28 April 2024

2024-04-28T13:07:34Z

← Older revision		Revision as of 13:07, 28 April 2024
Line 4:		Line 4:
	== Theory ==		== Theory ==

	4edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) ~~until~~ these intervals see practical use.		4edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) that these intervals see practical use.

	4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].		4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].

Akselai: /* Theory */

2024-04-28T13:07:08Z

Theory

← Older revision		Revision as of 13:07, 28 April 2024
Line 4:		Line 4:
	== Theory ==		== Theory ==

	4edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until further multiples of ~~4edt~~ until these intervals see practical use.		4edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until edts with further multiples of 4 (i.e. [[8edt]], [[12edt]], etc.) until these intervals see practical use.

	4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].		4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].

Akselai at 13:03, 28 April 2024

2024-04-28T13:03:13Z

← Older revision		Revision as of 13:03, 28 April 2024
Line 8:		Line 8:
	4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].		4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].

	=== ~~Odd harmonics~~ ===		=== Harmonics ===
	{{Harmonics in equal\|4\|3\|1~~\|intervals=odd~~}}		{{Harmonics in equal\|4\|3\|1}}

Akselai: clean up

2024-04-28T12:57:07Z

clean up

← Older revision		Revision as of 12:57, 28 April 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	~~'''4EDT''' is the [[Edt\|equal division of the third harmonic]] into four parts of 475.4888 [[cent]]s each, corresponding to 2.5237 [[edo]].~~		{{EDO intro}}

	~~The 4th root~~ of 3, ~~might~~ be viewed ~~alternately~~ as a ~~degenerate form or a fundamental building block~~ of Bohlen-Pierce ~~harmony~~, analogous to how 5edo ~~relates to~~ diatonic ~~music. The situation is different however, as in this case both 5 and 7 are relatively well represented (opposed to just 3 in 5edo)~~. While the ~~approximations~~ may seem excessively vague, ~~and~~ some might say ~~impossible~~, they are nevertheless categorically important to the perception of the scale~~, and, may even be heard as those harmonies given~~ the width of the "scale". It is doubtful ~~however,~~ that this scale could ~~recieve~~ much melodic treatment, and is more useful as a harmonic entity, either to ~~demonstrate~~ BP harmony, or as a ~~component~~ of scales like [[8edt]].		== Theory ==

			4edt fails to approximate a lot of low prime harmonics well -- the first prime harmonic that is approximated by 4edt within 10 cents is 47. (Compare this to [[3edt]], which has the 13th harmonic, and [[5edt]], which has the 5th harmonic.) Nevertheless, in terms of [[convergent]]s, 4edt manages to accurately approximate [[25/19]] with one step, and a less accurate [[16/7]] with three steps (hence [[21/16]] with one step). However, it is not until further multiples of 4edt until these intervals see practical use.

			4edt can be viewed as a "collapsed" version of the [[Bohlen-Pierce]] [[4L 5s (3/1-equivalent)\|lambda scale]], analogous to how 5edo is a [[collapsed]] version of the [[diatonic]] scale. While the approximation for the 5th and 7th harmonics by 4edt may seem excessively vague (or even impossibly vague, as some might say), they are nevertheless categorically important to the perception of the scale{{clarify}}. Given the width of the "scale", 4edt can even be perceived as within the modal logic of Bohlen-Pierce harmony. However, it is doubtful that this scale could receive much melodic treatment, and is more useful as an abstract harmonic entity, either to skeletonize BP harmony, or serving as a subset of scales like [[8edt]].

			=== Odd harmonics ===
			{{Harmonics in equal\|4\|3\|1\|intervals=odd}}


			=== Approximation of intervals ===

	{\| class="wikitable right-all left-4 left-5"		{\| class="wikitable right-all left-4 left-5"
Line 8:		Line 18:
	! Steps		! Steps
	! Cents		! Cents
	! ~~Hekts~~		! [[Hekt]]s
	! Corresponding <br>JI intervals		! Corresponding <br>JI intervals
	! Comments		! Comments
Line 16:		Line 26:
	\| 0		\| 0
	\| [[1/1]]		\| [[1/1]]
	\|		\| perfect unison
	\|-		\|-
	\| 1		\| 1
	\| 475.4888		\| 475.4888
	\| 325		\| 325
	\| [[17/13]], [[21/16]], 25/19, 33/25		\| [[17/13]], [[21/16]], [[25/19]], 33/25
	\|		\|
	\|-		\|-
	\| 2		\| 2
Line 28:		Line 38:
	\| 650		\| 650
	\| [[19/11]], 45/26, [[26/15]], (85/49), 33/19		\| [[19/11]], 45/26, [[26/15]], (85/49), 33/19
	\|		\| one step of [[2edt]]
	\|-		\|-
	\| 3		\| 3

Plumtree: Infobox ET added

2022-10-05T19:42:00Z

Infobox ET added

← Older revision		Revision as of 19:42, 5 October 2022
Line 1:		Line 1:
			{{Infobox ET}}
	'''4EDT''' is the [[Edt\|equal division of the third harmonic]] into four parts of 475.4888 [[cent]]s each, corresponding to 2.5237 [[edo]].		'''4EDT''' is the [[Edt\|equal division of the third harmonic]] into four parts of 475.4888 [[cent]]s each, corresponding to 2.5237 [[edo]].