9edt - Revision history

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

2025-07-31T15:31:42Z

Removing from Category:Edt using Cat-a-lot

← Older revision		Revision as of 15:31, 31 July 2025
Line 78:		Line 78:

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

ArrowHead294 at 13:06, 10 April 2025

2025-04-10T13:06:49Z

← Older revision		Revision as of 13:06, 10 April 2025
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	Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)\|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap\|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.		Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)\|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap\|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

	9edt is the third [[~~The_Riemann_Zeta_Function_and_Tuning~~#Removing primes\|no-twos zeta peak edt]].		9edt is the third [[the Riemann zeta function and tuning#Removing primes\|no-twos zeta peak edt]].

	=== Relation to edos ===		=== Relation to edos ===

ArrowHead294 at 16:35, 20 February 2025

2025-02-20T16:35:49Z

← Older revision		Revision as of 16:35, 20 February 2025
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	! rowspan="2" \| Steps		! rowspan="2" \| Steps
	! colspan="2" \| Size		! colspan="2" \| Size
	! rowspan="2" \| Comparable intervals (~~in [[cent\|¢]]~~)		! rowspan="2" \| Comparable intervals (¢)
	\|-		\|-
	! ~~(in [[cent\|¢]])~~		! Cents
	! in [[~~hekt~~]]s		! [[Hekt]]s
	\|-		\|-
	! colspan="3" \| 0		! colspan="3" \| 0

ArrowHead294 at 16:35, 20 February 2025

2025-02-20T16:35:02Z

← Older revision		Revision as of 16:35, 20 February 2025
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	== Theory ==		== Theory ==
	It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6¢ flat. However, the [[5/1]] is ~~39¢~~ flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of ~~845¢~~, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].		It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6{{c}} flat. However, the [[5/1]] is 39{{c}} flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845{{c}}, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

	Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)\|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap\|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.		Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)\|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap\|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

ArrowHead294 at 16:34, 20 February 2025

2025-02-20T16:34:42Z

← Older revision		Revision as of 16:34, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 04:02, 21 December 2024

2024-12-21T04:02:21Z

← Older revision		Revision as of 04:02, 21 December 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
			It has a decent seventh harmonic ([[7/1]]) which is 12.4¢ sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6¢ flat. However, the [[5/1]] is 39¢ flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth—or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845¢, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]]—making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

	It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].		Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L 5s (3/1-equivalent)\|3/1-equivalent "lambda" scale]], of which 9edt offers the "equalized" interpretation of {{nowrap\|L {{=}} s}}, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

	Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[~~4L_5s_~~(3/1-equivalent)\|3/1-equivalent ~~4L 5s]], aka~~ lambda~~, [[moment of symmetry~~]] ~~scale~~, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

	9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].		9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].

	=== Relation to edos ===		=== Relation to edos ===

	9edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].		9edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].

Line 19:		Line 17:

	{\| class="wikitable"		{\| class="wikitable"
			\|-
	! rowspan="2" \| Steps		! rowspan="2" \| Steps
	! colspan="2" \| Size		! colspan="2" \| Size
	! rowspan="2" \| Comparable intervals (in [[cent\|¢]])		! rowspan="2" \| Comparable intervals (in [[cent\|¢]])
	\|-		\|-
	!(in [[cent\|¢]])		! (in [[cent\|¢]])
	!in [[hekt]]s		! in [[hekt]]s
	\|-		\|-
	! colspan="3" \| 0		! colspan="3" \| 0
Line 31:		Line 30:
	\| 1		\| 1
	\| 211.328		\| 211.328
	\|144.444		\| 144.444
	\| [[9/8]] (204)		\| [[9/8]] (204)
	\|-		\|-
	\| 2		\| 2
	\| 422.657		\| 422.657
	\|288.889		\| 288.889
	\| [[9/7]] (435)		\| [[9/7]] (435)
	\|-		\|-
	\| 3		\| 3
	\| 633.985		\| 633.985
	\|433.333		\| 433.333
	\| [[13/9]] (637)		\| [[13/9]] (637)
	\|-		\|-
	\| 4		\| 4
	\| 845.313		\| 845.313
	\|577.778		\| 577.778
	\| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)		\| [[13/8]] (841), [[5/3]] (884), [[8/5]] (814)
	\|-		\|-
	\| 5		\| 5
	\| 1056.642		\| 1056.642
	\|722.222		\| 722.222
	\| [[9/5]] (1018), [[11/6]] (1049)		\| [[9/5]] (1018), [[11/6]] (1049)
	\|-		\|-
	\| 6		\| 6
	\| 1267.970		\| 1267.970
	\|866.667		\| 866.667
	\| [[27/13]] (1265)		\| [[27/13]] (1265)
	\|-		\|-
	\| 7		\| 7
	\| 1479.298		\| 1479.298
	\|1011.111		\| 1011.111
	\| [[7/3]] (1467)		\| [[7/3]] (1467)
	\|-		\|-
	\| 8		\| 8
	\| 1690.627		\| 1690.627
	\|1155.556		\| 1155.556
	\| [[8/3]] (1698)		\| [[8/3]] (1698)
	\|-		\|-
	\| 9		\| 9
	\| 1901.955		\| 1901.955
	\|1300		\| 1300
	\| [[3/1]]		\| [[3/1]]
	\|}		\|}
Line 77:		Line 76:
	== Music ==		== Music ==
	* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]		* [https://www.youtube.com/watch?v=sEQP1AtjPrA Far Away From Them / Spazzystackers] by [[Mandrake]]

	[[Category:Macrotonal]]		[[Category:Macrotonal]]
	[[Category:Edt]]		[[Category:Edt]]

Lériendil at 00:07, 19 September 2024

2024-09-19T00:07:36Z

← Older revision		Revision as of 00:07, 19 September 2024
Line 6:		Line 6:
	It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].		It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

	Following [[4edt]], this is the next edt that supports [[~~4L_5s_(3/1-equivalent)\|lambda~~]] temperament. For small edts, this property is virtually the same as supporting a 3/1-equivalent 4L 5s [[moment of symmetry]] scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.		Following [[4edt]], this is the next edt that supports [[BPS]] temperament. For small edts, this property is virtually the same as supporting a [[4L_5s_(3/1-equivalent)\|3/1-equivalent 4L 5s]], aka lambda, [[moment of symmetry]] scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

	9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].		9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].

Akselai: clean up more

2024-05-20T09:23:47Z

clean up more

← Older revision		Revision as of 09:23, 20 May 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro}}		{{EDO intro}}

			== Theory ==

	It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].		It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

			Following [[4edt]], this is the next edt that supports [[4L_5s_(3/1-equivalent)\|lambda]] temperament. For small edts, this property is virtually the same as supporting a 3/1-equivalent 4L 5s [[moment of symmetry]] scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

	9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].		9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].

	~~Following~~ [[~~4edt~~]], ~~this is~~ the ~~next edt that supports [[4L_5s_(3/1~~-~~equivalent)\|lambda]] temperament. This property is virtually the same as supporting a 3/1~~-~~equivalent 4L 5s [[moment of symmetry]]~~ scale, ~~of which 9edt offers~~ the ~~"equalized" interpretation of L = s, analogous~~ to [[~~7edo~~]] ~~in diatonic ([[5L 2s]]) music~~.		=== Relation to edos ===

			9edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].

	~~9edt is also related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[~~3/1~~]].~~		=== Harmonics ===
			{{Harmonics in equal\|9\|3\|1\|}}
			{{Harmonics in equal\|9\|3\|1\|intervals=prime}}

	{\| class="wikitable"		{\| class="wikitable"
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	\| [[3/1]]		\| [[3/1]]
	\|}		\|}

	~~== Harmonics ==~~
	~~{{Harmonics in equal\|9\|3\|1\|}}~~
	~~{{Harmonics in equal\|9\|3\|1\|intervals=prime}}~~

	== Music ==		== Music ==

Akselai: clean up

2024-05-20T01:04:50Z

clean up

← Older revision		Revision as of 01:04, 20 May 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	The '''9 equal division of 3''', the [[tritave]], divides it into 9 equal steps of size 211.328 [[cent]]s each. It has a decent 7 and an excellent 13, but a 5 which is 39 cents flat; if octaves were added and it was a sixth, it would count as a [[neutral sixth]]. The corresponding interval for [[5/3]] is 845 cents, which is a neutral sixth between [[8/5]] and [[5/3]], which is really more of a [[13/8]], though this is allegedly a no-twos tuning. On the 3.7.13 [[subgroup]] it tempers out [[351/343]] and [[2197/2187]]. 9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].		{{EDO intro}}

	~~Following~~ [[~~4edt~~]], ~~this~~ is the ~~next "Lambda"~~ (~~BP related~~) ~~equal division~~ of ~~the tritave~~; ~~in a certain sense analogous~~ to [[~~7edo~~]] in ~~diatonic music~~.		It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]].

	This scale is also related to [[17edo]] by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].		9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes\|no-twos zeta peak edt]].

			Following [[4edt]], this is the next edt that supports [[4L_5s_(3/1-equivalent)\|lambda]] temperament. This property is virtually the same as supporting a 3/1-equivalent 4L 5s [[moment of symmetry]] scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music.

			9edt is also related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]].

	{\| class="wikitable"		{\| class="wikitable"
	! rowspan="2" \| Steps		! rowspan="2" \| Steps
	! colspan="2" \| Size		! colspan="2" \| Size
	! rowspan="2" \| Comparable intervals		! rowspan="2" \| Comparable intervals (in [[cent\|¢]])
	\|-		\|-
	!(in [[cent\|¢]])		!(in [[cent\|¢]])
	!in ~~hekts~~		!in [[hekt]]s
	\|-		\|-
	! colspan="3" \| 0		! colspan="3" \| 0
Line 63:		Line 67:
	\|}		\|}

	== ~~Prime harmonics~~ ==		== Harmonics ==
			{{Harmonics in equal\|9\|3\|1\|}}
	{{Harmonics in equal\|9\|3\|1\|intervals=prime}}		{{Harmonics in equal\|9\|3\|1\|intervals=prime}}

Akselai: /* Prime harmonics */

2024-01-17T09:40:07Z

Prime harmonics

← Older revision		Revision as of 09:40, 17 January 2024
Line 64:		Line 64:

	== Prime harmonics ==		== Prime harmonics ==
	{{Harmonics in equal\|9\|3\|1}}		{{Harmonics in equal\|9\|3\|1\|intervals=prime}}

	== Music ==		== Music ==