66edo - Revision history

BudjarnLambeth: /* Octave stretch or compression */ fix links

2025-09-19T08:44:48Z

Octave stretch or compression: fix links

← Older revision		Revision as of 08:44, 19 September 2025
Line 53:		Line 53:
	66edo tunes multiple-of-3 [[harmonic]]s quite sharp, but most other simple harmonics quite flat.		66edo tunes multiple-of-3 [[harmonic]]s quite sharp, but most other simple harmonics quite flat.

	Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi\|340zpi]], [[~~ed10~~\|219ed10]] & [[~~ed9~~\|209ed9]].		Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi\|340zpi]], [[equal tuning\|219ed10]] & [[equal tuning\|209ed9]].

	Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi\|338zpi]] and [[104edt]].		Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi\|338zpi]] and [[104edt]].

BudjarnLambeth: /* Octave stretch or compression */

2025-09-19T05:52:39Z

Octave stretch or compression

← Older revision		Revision as of 05:52, 19 September 2025
Line 57:		Line 57:
	Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi\|338zpi]] and [[104edt]].		Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi\|338zpi]] and [[104edt]].

	Moderate ''[[octave shrinking]]'' will instead improve upon 66edo's best mapping of prime 3 dramatically. It reduce the error on most other simple harmonics. It flips the mapping of many of them, but introduces less mapping inconsistencies than stretching. Moderately octave-compressed tunings for 66edo include [[105edt]] and [[zpi\|342zpi]].		Moderate ''[[octave shrinking]]'' will instead improve upon 66edo's best mapping of prime 3 dramatically. It reduce the error on most other simple harmonics. It flips the mapping of many of them, but introduces less internal mapping inconsistencies than stretching by the same amount. Moderately octave-compressed tunings for 66edo include [[105edt]] and [[zpi\|342zpi]].

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Octave stretch or compression */

2025-09-19T05:51:54Z

Octave stretch or compression

← Older revision		Revision as of 05:51, 19 September 2025
Line 51:		Line 51:

	== Octave stretch or compression ==		== Octave stretch or compression ==
	66edo tunes multiple-of-3 [[~~harmonics~~]] quite sharp, but most other simple harmonics quite flat.		66edo tunes multiple-of-3 [[harmonic]]s quite sharp, but most other simple harmonics quite flat.

	Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi\|340zpi]], [[ed10\|219ed10]] & [[ed9\|209ed9]].		Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi\|340zpi]], [[ed10\|219ed10]] & [[ed9\|209ed9]].

BudjarnLambeth: Stretch or compression (temporary until full version can be done)

2025-09-19T05:51:45Z

Stretch or compression (temporary until full version can be done)

← Older revision		Revision as of 05:51, 19 September 2025
Line 49:		Line 49:

	In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.		In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

			== Octave stretch or compression ==
			66edo tunes multiple-of-3 [[harmonics]] quite sharp, but most other simple harmonics quite flat.

			Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi\|340zpi]], [[ed10\|219ed10]] & [[ed9\|209ed9]].

			Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi\|338zpi]] and [[104edt]].

			Moderate ''[[octave shrinking]]'' will instead improve upon 66edo's best mapping of prime 3 dramatically. It reduce the error on most other simple harmonics. It flips the mapping of many of them, but introduces less mapping inconsistencies than stretching. Moderately octave-compressed tunings for 66edo include [[105edt]] and [[zpi\|342zpi]].

	== Instruments ==		== Instruments ==

Lucius Chiaraviglio: /* Instruments */ Insert music section after this, starting with 2 Bryan Deister short videos

2025-07-26T05:57:47Z

Instruments: Insert music section after this, starting with 2 Bryan Deister short videos

← Older revision		Revision as of 05:57, 26 July 2025
Line 80:		Line 80:

	31/16: string 2 fret 9		31/16: string 2 fret 9

			== Music ==
			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/jHwtQg0aR14 ''Spring Yard Zone (microtonal version in 66edo) - Sonic The Hedgehog''] (2022)
			* [https://www.youtube.com/shorts/Cn28RJSSgQ0 ''microtonal improvisation in 66edo''] (2025)

Yourmusic Productions: /* Instruments */ Add lumatone mapping link

2025-07-25T20:56:16Z

Instruments: Add lumatone mapping link

← Older revision		Revision as of 20:56, 25 July 2025
Line 51:		Line 51:

	== Instruments ==		== Instruments ==
			=== Lumatone ===
			[[Lumatone mapping for 66edo]]
	=== Skip fretting ===		=== Skip fretting ===
	'''Skip fretting system 66 7 11''' is a [[skip fretting]] system for [[66edo]]. All examples on this page are for 7-string [[guitar]].		'''Skip fretting system 66 7 11''' is a [[skip fretting]] system for [[66edo]]. All examples on this page are for 7-string [[guitar]].
Line 78:		Line 80:

	31/16: string 2 fret 9		31/16: string 2 fret 9

	~~[[Category:Ammonite]]~~

	~~{{Todo\|add lumatone mapping}}~~

Lucius Chiaraviglio: /* Instruments */ Todo|add lumatone mapping (this is now the only gap in the range 9edo through 97edo)

2025-07-21T10:03:27Z

Instruments: Todo|add lumatone mapping (this is now the only gap in the range 9edo through 97edo)

← Older revision		Revision as of 10:03, 21 July 2025
Line 80:		Line 80:

	[[Category:Ammonite]]		[[Category:Ammonite]]

			{{Todo\|add lumatone mapping}}

BudjarnLambeth: /* Skip fretting */ merged

2025-04-27T03:40:17Z

Skip fretting: merged

@@ Line 50: / Line 50: @@
 In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
-== Skip fretting ==
+== Instruments ==
-[[Skip fretting system 66 7 11]]
+=== Skip fretting ===
 [[Category:Ammonite]]

FloraC: No need to remind readers of what a regular temperament is everywhere

2025-04-11T11:06:23Z

No need to remind readers of what a regular temperament is everywhere

← Older revision		Revision as of 11:06, 11 April 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~		The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various [[primodal]] [[over-5]] scales.

	Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various [[over-5]] scales. ~~(See [[primodality]].)~~

	The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit.		The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit.

BudjarnLambeth: /* Theory */ (''See regular temperament for more about what all this means and how to use it.'')

2025-04-11T07:32:48Z

Theory: (''See regular temperament for more about what all this means and how to use it.'')

← Older revision		Revision as of 07:32, 11 April 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various [[over-5]] scales.		The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. (''See [[regular temperament]] for more about what all this means and how to use it.'')

			Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various [[over-5]] scales. (See [[primodality]].)

	The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit.		The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit.