Ed6 - Revision history

FloraC: /* As generator chains for temperaments */ - fake entries

2026-04-20T13:12:36Z

As generator chains for temperaments: - fake entries

← Older revision		Revision as of 13:12, 20 April 2026
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	* [[16ed6]] – [[hemiwürschmidt]] generator (with octaves)		* [[16ed6]] – [[hemiwürschmidt]] generator (with octaves)
	* [[17ed6]] – [[minortone]] generator (with octaves)		* [[17ed6]] – [[minortone]] generator (with octaves)
	* [[19ed6]] – [[porcupine]] generator (with octaves)
	* [[21ed6]] – [[progression]] generator (with octaves)
	* [[24ed6]] – [[twothirdtonic]] generator (with octaves)
	* [[26ed6]] – [[septidiasemi]] generator (with octaves)		* [[26ed6]] – [[septidiasemi]] generator (with octaves)
	* [[35ed6]] – [[octacot]] generator (with octaves)
	* [[40ed6]] – [[valentine]] generator (with octaves)

	== Individual pages for ed6's ==		== Individual pages for ed6's ==

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

2025-08-01T19:39:41Z

Removing from Category:Edonoi using Cat-a-lot

← Older revision		Revision as of 19:39, 1 August 2025
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	[[Category:Ed6's\| ]]		[[Category:Ed6's\| ]]
	<!-- main article -->		<!-- main article -->
	~~[[Category:Edonoi]]~~
	[[Category:Lists of scales]]		[[Category:Lists of scales]]

Fredg999 category edits: Moving from Category:Ed6 to Category:Ed6's using Cat-a-lot

2025-08-01T18:42:57Z

Moving from Category:Ed6 to Category:Ed6's using Cat-a-lot

← Older revision		Revision as of 18:42, 1 August 2025
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	* [[Ed12]]		* [[Ed12]]

	[[Category:Ed6\| ]] <!-- main article -->		[[Category:Ed6's\| ]]
			<!-- main article -->
	[[Category:Edonoi]]		[[Category:Edonoi]]
	[[Category:Lists of scales]]		[[Category:Lists of scales]]

FloraC: /* Ed6-edo correspondence */ cleanup

2025-05-28T13:48:52Z

Ed6-edo correspondence: cleanup

← Older revision		Revision as of 13:48, 28 May 2025
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	* [[256ed6\|256]], [[269ed6\|269]], [[287ed6\|287]], [[323ed6\|323]], [[512ed6\|512]]		* [[256ed6\|256]], [[269ed6\|269]], [[287ed6\|287]], [[323ed6\|323]], [[512ed6\|512]]

	== ~~Ed6-edo~~ correspondence ==		== Ed6–edo correspondence ==
	Following ed6's (up to 200) contain good correspondences to edo tunings<ref>Edo with relative error of 6th harmonic below 1/3</ref>.		Following ed6's (up to 200) contain good correspondences to edo tunings<ref>Edo with relative error of 6th harmonic below 1/3</ref>.

	{\| class="wikitable"		{\| class="wikitable center-1 center-2"
	\|-		\|-
	! Ed6		! Ed6
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	\| [[13ed6]]		\| [[13ed6]]
	\| [[5edo]]		\| [[5edo]]
	\| 13ed6 is 5edo with ~6.9 cent compressed octaves.~~<br />~~Equivalently, 5edo is 13ed6 with hexataves stretched by ~18 cents.~~<br />~~Patent vals match through the 13-limit.		\| 13ed6 is 5edo with ~6.9 cent compressed octaves. Equivalently, 5edo is 13ed6 with hexataves stretched by ~18 cents. Patent vals match through the 13-limit.
	\|-		\|-
	\| [[18ed6]]		\| [[18ed6]]

FloraC: Address how it optimizes for a 3-limit chord. Put the generator chain approach to its own section

2025-05-22T11:28:03Z

Address how it optimizes for a 3-limit chord. Put the generator chain approach to its own section

← Older revision		Revision as of 11:28, 22 May 2025
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	The 6th harmonic, sextuple, or hexatave, is particularly wide as far as [[interval of equivalence\|equivalence]]s go, and there are only about 3.9 instances of the 6th harmonic in the [[human hearing range]]. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the [[10/1\|10th]], and to a lesser extent, the [[12/1\|12th]] share this property).		The 6th harmonic, sextuple, or hexatave, is particularly wide as far as [[interval of equivalence\|equivalence]]s go, and there are only about 3.9 instances of the 6th harmonic in the [[human hearing range]]. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the [[10/1\|10th]], and to a lesser extent, the [[12/1\|12th]] share this property).

	However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1\|octave]] and [[3/1\|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. ~~This is not to say~~ ed6's ~~not supporting this should be dismissed out of hand as entirely worthless,~~ for ~~to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking~~ the ~~''n''-th root of~~ 6 ~~is itself an approach to finding temperaments like squares, tritonic~~, and ~~sensi. This approach can~~ of ~~course be used indiscriminately~~.		However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1\|octave]] and [[3/1\|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. In fact, ed6's optimize for the 1:2:3:6 chord, with equal magnitudes and opposite signs of [[error]] on 2 and 3.

	~~Some equal divisions~~ of ~~the 6th harmonic~~ serve as ~~generators~~ for ~~octave~~ temperaments:		== As generator chains for temperaments ==
			Taking the ''n''-th root of 6 is itself an approach to finding [[regular temperament\|temperaments]] like [[squares]], [[tritonic]], and [[sensi]]. This approach can of course be used indiscriminately. The ed6's serve as generator chains for the temperaments:
	* [[4ed6]] – [[squares]] generator (with octaves)		* [[4ed6]] – [[squares]] generator (with octaves)
	* [[5ed6]] – [[tritonic]] generator (with octaves)		* [[5ed6]] – [[tritonic]] generator (with octaves)

FloraC: - typo

2025-05-22T10:58:13Z

- typo

← Older revision		Revision as of 10:58, 22 May 2025
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	The 6th harmonic, sextuple, or hexatave, is particularly wide as far as [[interval of equivalence\|equivalence]]s go, and there are only about 3.9 instances of the 6th harmonic in the [[human hearing range]]. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the [[10/1\|10th]], and to a lesser extent, the [[12/1\|12th]] share this property).		The 6th harmonic, sextuple, or hexatave, is particularly wide as far as [[interval of equivalence\|equivalence]]s go, and there are only about 3.9 instances of the 6th harmonic in the [[human hearing range]]. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the [[10/1\|10th]], and to a lesser extent, the [[12/1\|12th]] share this property).

	However, ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1\|octave]] and [[3/1\|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. This is not to say ed6's not supporting this should be dismissed out of hand as entirely worthless, for to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking the ''n''-th root of 6 is itself an approach to finding temperaments like squares, tritonic, and sensi. This approach can of course be used indiscriminately.		However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1\|octave]] and [[3/1\|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. This is not to say ed6's not supporting this should be dismissed out of hand as entirely worthless, for to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking the ''n''-th root of 6 is itself an approach to finding temperaments like squares, tritonic, and sensi. This approach can of course be used indiscriminately.

	Some equal divisions of the 6th harmonic serve as generators for octave temperaments:		Some equal divisions of the 6th harmonic serve as generators for octave temperaments:

FloraC: Rework

2025-05-21T11:11:21Z

Rework

← Older revision		Revision as of 11:11, 21 May 2025
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	'''Equal divisions of the 6th harmonic''' ('''~~ED6~~''') are [[tuning]]s obtained by dividing the [[6/1\|6th harmonic]] in a certain number of [[equal]] steps.		'''Equal divisions of the 6th harmonic''' ('''ed6''') are [[tuning]]s obtained by dividing the [[6/1\|6th harmonic]] in a certain number of [[equal]] steps.

	~~== Introduction ==~~		The 6th harmonic, sextuple, or hexatave, is particularly wide as far as [[interval of equivalence\|equivalence]]s go, and there are only about 3.9 instances of the 6th harmonic in the [[human hearing range]]. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the [[10/1\|10th]], and to a lesser extent, the [[12/1\|12th]] share this property).
	The 6th harmonic, or ''hexatave'', is particularly wide as far as ~~equivalences~~ go, and there are only about 3.9 ~~hexataves~~ in the [[human hearing range]]. If one does indeed deal with ~~hexatave~~ equivalence, this will alter one's musical approach dramatically. Even so, the ~~hexatave~~ is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the 10th, and to a lesser extent, the 12th share this property). Following this, the quintessential reason for using a hexatave based tuning is that it will split the difference between octave and tritave based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world (see [[44ed6]] and [[49ed6]]). However, this is not to say of ed6's not supporting this important 13 & 18 temperament that they can be dismissed out of hand as entirely worthless, for to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking the ''n''-th root of 6 is itself an approach to finding temperaments like squares, tritonic, and sensi. This approach can of course be used indiscriminately.

	~~Some equal divisions~~ of the ~~hexatave serve~~ as ~~generators~~ for ~~octave~~ temperaments:		However, ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1\|octave]] and [[3/1\|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. This is not to say ed6's not supporting this should be dismissed out of hand as entirely worthless, for to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking the ''n''-th root of 6 is itself an approach to finding temperaments like squares, tritonic, and sensi. This approach can of course be used indiscriminately.

	* [[4ed6]] [[squares]] generator (with octaves)		Some equal divisions of the 6th harmonic serve as generators for octave temperaments:
	* [[5ed6]] [[tritonic]] generator (with octaves)		* [[4ed6]] – [[squares]] generator (with octaves)
	* [[6ed6]] [[harry]] generator (with octaves)		* [[5ed6]] – [[tritonic]] generator (with octaves)
	* [[7ed6]] [[sensi]] generator (with octaves)		* [[6ed6]] – [[harry]] generator (with octaves)
	* [[8ed6]] [[würschmidt]] generator (with octaves)		* [[7ed6]] – [[sensi]] generator (with octaves)
	* [[10ed6]] [[myna]] generator (with octaves)		* [[8ed6]] – [[würschmidt]] generator (with octaves)
	* [[14ed6]] [[~~Sensipent family\|~~hemisensi]] generator (with octaves)		* [[10ed6]] – [[myna]] generator (with octaves)
	* [[16ed6]] [[~~Würschmidt family\|hemiwürschimdt~~]] generator (with octaves)		* [[14ed6]] – [[hemisensi]] generator (with octaves)
	* [[17ed6]] [[~~Minortonic family\|Minortonic~~]] generator (with octaves)		* [[16ed6]] – [[hemiwürschmidt]] generator (with octaves)
	* [[19ed6]] [[~~Porcupine~~]] generator (with octaves)		* [[17ed6]] – [[minortone]] generator (with octaves)
	* [[21ed6]] [[progression]] generator (with octaves)		* [[19ed6]] – [[porcupine]] generator (with octaves)
	* [[24ed6]] [[twothirdtonic]] generator (with octaves)		* [[21ed6]] – [[progression]] generator (with octaves)
	* [[26ed6]] [[septidiasemi]] generator (with octaves)		* [[24ed6]] – [[twothirdtonic]] generator (with octaves)
	* [[35ed6]] [[octacot]] generator (with octaves)		* [[26ed6]] – [[septidiasemi]] generator (with octaves)
	* [[40ed6]] [[valentine]] generator (with octaves)		* [[35ed6]] – [[octacot]] generator (with octaves)
			* [[40ed6]] – [[valentine]] generator (with octaves)

	== Individual pages for ed6's ==		== Individual pages for ed6's ==

FloraC: /* Individual pages for ed6's */ +512

2025-04-19T12:26:35Z

Individual pages for ed6's: +512

← Older revision		Revision as of 12:26, 19 April 2025
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	; 200 and beyond		; 200 and beyond
	* [[256ed6\|256]], [[269ed6\|269]], [[287ed6\|287]], [[323ed6\|323]]		* [[256ed6\|256]], [[269ed6\|269]], [[287ed6\|287]], [[323ed6\|323]], [[512ed6\|512]]

	== Ed6-edo correspondence ==		== Ed6-edo correspondence ==

FloraC: /* Individual pages for ed6's */ +256

2025-04-18T13:20:54Z

Individual pages for ed6's: +256

← Older revision		Revision as of 13:20, 18 April 2025
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	; 200 and beyond		; 200 and beyond
	* [[269ed6\|269]], [[287ed6\|287]], [[323ed6\|323]]		* [[256ed6\|256]], [[269ed6\|269]], [[287ed6\|287]], [[323ed6\|323]]

	== Ed6-edo correspondence ==		== Ed6-edo correspondence ==

FloraC: /* Individual pages for ed6's */ +323

2025-04-15T12:31:00Z

Individual pages for ed6's: +323

← Older revision		Revision as of 12:31, 15 April 2025
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	; 200 and beyond		; 200 and beyond
	* [[269ed6]], [[287ed6]]		* [[269ed6\|269]], [[287ed6\|287]], [[323ed6\|323]]

	== Ed6-edo correspondence ==		== Ed6-edo correspondence ==