Ed8 - Revision history

FloraC: Tabulate individual ed8's

2026-02-18T09:19:13Z

Tabulate individual ed8's

← Older revision		Revision as of 09:19, 18 February 2026
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	== Individual pages for ed8's ==		== Individual pages for ed8's ==
	* [[1ed8]]		{\| class="wikitable center-all"
	* [[2ed8]]		\|+ style=white-space:nowrap \| 1…59
	* [[4ed8]]		\|
	* [[5ed8]]		\| [[1ed8\|1]]
	* [[7ed8]]		\| [[2ed8\|2]]
	* [[8ed8]]		\|
	* [[10ed8]]		\| [[4ed8\|4]]
	* [[11ed8]]		\| [[5ed8\|5]]
	* [[13ed8]]		\|
	* [[14ed8]]		\| [[7ed8\|7]]
	* [[16ed8]]		\| [[8ed8\|8]]
	* [[17ed8]]		\|
	* [[19ed8]]		\|-
	* [[20ed8]]		\| [[10ed8\|10]]
	* [[22ed8]]		\| [[11ed8\|11]]
	* [[23ed8]]		\|
	* [[25ed8]]		\| [[13ed8\|13]]
	* [[26ed8]]		\| [[14ed8\|14]]
	* [[28ed8]]		\|
	* [[29ed8]]		\| [[16ed8\|16]]
	* [[31ed8]]		\| [[17ed8\|17]]
	* [[32ed8]]		\|
	* [[34ed8]]		\| [[19ed8\|19]]
	* [[35ed8]]		\|-
	* [[37ed8]]		\| [[20ed8\|20]]
	* [[38ed8]]		\|
	* [[40ed8]]		\| [[22ed8\|22]]
	* [[41ed8]]		\| [[23ed8\|23]]
	* [[43ed8]]		\|
	* [[44ed8]]		\| [[25ed8\|25]]
	* [[46ed8]]		\| [[26ed8\|26]]
	* [[47ed8]]		\|
	* [[49ed8]]		\| [[28ed8\|28]]
	* [[50ed8]]		\| [[29ed8\|29]]
	* [[52ed8]]		\|-
	* [[53ed8]]		\|
	* [[55ed8]]		\| [[31ed8\|31]]
	* [[56ed8]]		\| [[32ed8\|32]]
	* [[58ed8]]		\|
	* [[59ed8]]		\| [[34ed8\|34]]
			\| [[35ed8\|35]]
			\|
			\| [[37ed8\|37]]
			\| [[38ed8\|38]]
			\|
			\|-
			\| [[40ed8\|40]]
			\| [[41ed8\|41]]
			\|
			\| [[43ed8\|43]]
			\| [[44ed8\|44]]
			\|
			\| [[46ed8\|46]]
			\| [[47ed8\|47]]
			\|
			\| [[49ed8\|49]]
			\|-
			\| [[50ed8\|50]]
			\|
			\| [[52ed8\|52]]
			\| [[53ed8\|53]]
			\|
			\| [[55ed8\|55]]
			\| [[56ed8\|56]]
			\|
			\| [[58ed8\|58]]
			\| [[59ed8\|59]]
			\|}

			; 60 and beyond
	* [[124ed8]]		* [[124ed8]]

	[[Category:Ed8's\| ]] <!-- main article -->		[[Category:Ed8's\| ]] <!-- main article -->

FloraC: General cleanup

2026-02-18T09:11:49Z

General cleanup

@@ Line 1: / Line 1: @@
-'''Equal divisions of the triple octave''' (frequency ratio [[8/1]], aka "octuple") are closely related to equal divisions of the octave (frequency ratio 2/1, aka "duple"), in other words, ED2 or [[EDO]] scales. Given any number which is coprime to 3 for ED2, an ED8 can be generated by taking every third tone of the ED2. For example, given 5ED2 (aka [[5edo]]), three octaves of which, in cents are:
+An '''equal division of the eighth harmonic''' or '''the triple octave''' is a [[tuning]] obtained by dividing the triple octave (frequency ratio [[8/1]], a.k.a. "octuple") in a certain number of [[equal]] steps. They are closely related to [[EDO|equal divisions of the octave]] (frequency ratio [[2/1]], a.k.a. "duple"). Given any number which is coprime to 3 for edo, an ed8 can be generated by taking every third tone of the edo. For example, given [[5edo]] , three octaves of which, in cents are:
-* 0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600...
+* 0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600 …
-...taking every third tone yields:
+Taking every third tone yields:
-* '''0''' 240 480 '''720''' 960 1200 '''1440''' 1680 1920 '''2160''' 2400 2640 '''2880''' 3120 3360 '''3600'''...
+* '''0''' 240 480 '''720''' 960 1200 '''1440''' 1680 1920 '''2160''' 2400 2640 '''2880''' 3120 3360 '''3600''' …
-* '''0 720 1440 2160 2880 3600...'''
+The resultant tuning is [[5ed8]].
-The resultant scale we can call 5ED8.
+This approach yields more useful scales starting with edo systems which are larger, where a composer might decide a single degree is too small to be useful. As one example, consider [[31edo]], which is well known to be a consistent temperament in the [[11-odd-limit]], but whose single degree, approximately 38.7{{c}}, might be too small in some contexts (e.g. guitar frets). Taking every third step of 31edo produces [[31ed8]], an equal-stepped scale which repeats at 8/1, the triple octave, and has a single step of 116.1{{c}}.
-This approach yields more useful scales starting with ED2 systems which are larger, where a composer might decide a single degree is too small to be useful. As one example, consider 31ED2 (aka [[31edo]]), which is well known to be a consistent temperament in the 11-limit, but whose single degree, approximately 38.7¢, might be "too small" in some contexts (e.g. guitar frets). Taking every third step of 31ED2 produces [[31ed8|31ED8]], an equal-stepped scale which repeats at 8/1, the triple octave, and has a single step of 116.1¢.
+Ed8 scales also have the feature that they ascend the pitch continuum three times as fast as edo systems. 31 steps of 31edo is one octave, while 31 steps of 31ed8 is three octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.
-ED8 scales also have the feature that they ascend the pitch continuum three times as fast as ED2 systems. 31 tones of 31ED2 is one octave, while 31 tones of 31ED8 is three octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.
+== Individual pages for ed8's ==
-Ed8s are a type of [[equal-step tuning]].
+[[Category:Ed8's| ]] <!-- main article -->

Overthink: /* Individual pages for ED8s */ no descriptions here

2026-01-27T04:06:30Z

Individual pages for ED8s: no descriptions here

← Older revision		Revision as of 04:06, 27 January 2026
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	*[[58ed8]]		*[[58ed8]]
	*[[59ed8]]		*[[59ed8]]
	*[[124ed8]]~~, a strong 5.7.8.9.11.13.17.23 tuning. Optimum is [[143ed11]].~~		*[[124ed8]]

	[[Category:Ed8's\| ]]		[[Category:Ed8's\| ]]

Contribution: /* Individual pages for ED8s */

2025-10-27T17:40:29Z

Individual pages for ED8s

← Older revision		Revision as of 17:40, 27 October 2025
Line 56:		Line 56:
	*[[58ed8]]		*[[58ed8]]
	*[[59ed8]]		*[[59ed8]]
	*[[124ed8]]		*[[124ed8]], a strong 5.7.8.9.11.13.17.23 tuning. Optimum is [[143ed11]].

	[[Category:Ed8's\| ]]		[[Category:Ed8's\| ]]

Contribution at 17:37, 27 October 2025

2025-10-27T17:37:34Z

← Older revision		Revision as of 17:37, 27 October 2025
Line 56:		Line 56:
	*[[58ed8]]		*[[58ed8]]
	*[[59ed8]]		*[[59ed8]]
			*[[124ed8]]

	[[Category:Ed8's\| ]]		[[Category:Ed8's\| ]]

Fredg999 category edits: Moving from Category:Ed8 to Category:Ed8's using Cat-a-lot

2025-08-01T18:44:52Z

Moving from Category:Ed8 to Category:Ed8's using Cat-a-lot

← Older revision		Revision as of 18:44, 1 August 2025
Line 57:		Line 57:
	*[[59ed8]]		*[[59ed8]]

	[[Category:Ed8\| ]]		[[Category:Ed8's\| ]]

Fredg999 category edits: Removing from Category:Equal-step tuning using Cat-a-lot

2025-08-01T15:32:04Z

Removing from Category:Equal-step tuning using Cat-a-lot

← Older revision		Revision as of 15:32, 1 August 2025
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	[[Category:Ed8\| ]]		[[Category:Ed8\| ]]
	~~[[Category:Equal-step tuning]]~~

BudjarnLambeth at 03:19, 22 May 2025

2025-05-22T03:19:40Z

← Older revision		Revision as of 03:19, 22 May 2025
Line 15:		Line 15:
	ED8 scales also have the feature that they ascend the pitch continuum three times as fast as ED2 systems. 31 tones of 31ED2 is one octave, while 31 tones of 31ED8 is three octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.		ED8 scales also have the feature that they ascend the pitch continuum three times as fast as ED2 systems. 31 tones of 31ED2 is one octave, while 31 tones of 31ED8 is three octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.

	~~See:~~ [[~~Equal~~-step tuning~~\|Equal Temperaments~~]]		Ed8s are a type of [[equal-step tuning]].

	=Individual pages for ED8s=		=Individual pages for ED8s=

BudjarnLambeth: Formatting

2024-03-02T08:26:20Z

Formatting

← Older revision		Revision as of 08:26, 2 March 2024
Line 1:		Line 1:
	Equal ~~Divisions~~ of the ~~Triple Octave --~~ frequency ratio [[8/1]], aka "~~Octuple~~" -- are closely related to ~~Equal Divisions~~ of the ~~Octave --~~ frequency ratio 2/1, aka "~~Duple~~" -- in other words, ED2 or [[EDO]] scales. Given any number which is coprime to 3 for ED2, an ED8 can be generated by taking every third tone of the ED2. For example, given 5ED2 (aka [[5edo]]), three octaves of which, in cents are:		'''Equal divisions of the triple octave''' (frequency ratio [[8/1]], aka "octuple") are closely related to equal divisions of the octave (frequency ratio 2/1, aka "duple"), in other words, ED2 or [[EDO]] scales. Given any number which is coprime to 3 for ED2, an ED8 can be generated by taking every third tone of the ED2. For example, given 5ED2 (aka [[5edo]]), three octaves of which, in cents are:

	0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600...		* 0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600...

	...taking every third tone yields:		...taking every third tone yields:

	'''0''' 240 480 '''720''' 960 1200 '''1440''' 1680 1920 '''2160''' 2400 2640 '''2880''' 3120 3360 '''3600'''...		* '''0''' 240 480 '''720''' 960 1200 '''1440''' 1680 1920 '''2160''' 2400 2640 '''2880''' 3120 3360 '''3600'''...

	'''0 720 1440 2160 2880 3600...'''		* '''0 720 1440 2160 2880 3600...'''

	The resultant scale we can call 5ED8.		The resultant scale we can call 5ED8.

	This approach yields more useful scales starting with ED2 systems which are larger, where a composer might decide a single degree is too small to be useful. As one example, consider 31ED2 (aka [[31edo]]), which is well known to be a consistent temperament in the 11-limit, but whose single degree, approximately 38.7¢, might be "too small" in some ~~context~~ (e.g. guitar frets). Taking every third step of 31ED2 produces [[31ed8\|31ED8]], an equal-stepped scale which repeats at 8/1, the triple octave, and has a single step of 116.1¢.		This approach yields more useful scales starting with ED2 systems which are larger, where a composer might decide a single degree is too small to be useful. As one example, consider 31ED2 (aka [[31edo]]), which is well known to be a consistent temperament in the 11-limit, but whose single degree, approximately 38.7¢, might be "too small" in some contexts (e.g. guitar frets). Taking every third step of 31ED2 produces [[31ed8\|31ED8]], an equal-stepped scale which repeats at 8/1, the triple octave, and has a single step of 116.1¢.

	ED8 scales also have the feature that they ascend the pitch continuum three times as fast as ED2 systems. 31 tones of 31ED2 is one octave, while 31 tones of 31ED8 is three octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.		ED8 scales also have the feature that they ascend the pitch continuum three times as fast as ED2 systems. 31 tones of 31ED2 is one octave, while 31 tones of 31ED8 is three octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.

BudjarnLambeth: Changed 8/1 to 8/1

2023-07-06T02:21:23Z

Changed 8/1 to 8/1

← Older revision		Revision as of 02:21, 6 July 2023
Line 1:		Line 1:
	Equal Divisions of the Triple Octave -- frequency ratio 8/1, aka "Octuple" -- are closely related to Equal Divisions of the Octave -- frequency ratio 2/1, aka "Duple" -- in other words, ED2 or [[EDO]] scales. Given any number which is coprime to 3 for ED2, an ED8 can be generated by taking every third tone of the ED2. For example, given 5ED2 (aka [[5edo]]), three octaves of which, in cents are:		Equal Divisions of the Triple Octave -- frequency ratio [[8/1]], aka "Octuple" -- are closely related to Equal Divisions of the Octave -- frequency ratio 2/1, aka "Duple" -- in other words, ED2 or [[EDO]] scales. Given any number which is coprime to 3 for ED2, an ED8 can be generated by taking every third tone of the ED2. For example, given 5ED2 (aka [[5edo]]), three octaves of which, in cents are:

	0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600...		0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600...