EDF - Revision history

Hstraub at 15:08, 2 January 2026

2026-01-02T15:08:42Z

← Older revision		Revision as of 15:08, 2 January 2026
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	{{interwiki		{{interwiki
	\|en = ~~EDT~~		\|en = EDF
	\|de = ~~Edt~~		\|de = Edf
	}}		}}
	The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2\|perfect fifth (3/2)]] in a certain number of [[equal]] steps.		The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2\|perfect fifth (3/2)]] in a certain number of [[equal]] steps.

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

2025-08-01T19:39:44Z

Removing from Category:Edonoi using Cat-a-lot

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

Fredg999 category edits: Moving from Category:Edfs to Category:Edf using Cat-a-lot

2025-08-01T19:08:05Z

Moving from Category:Edfs to Category:Edf using Cat-a-lot

← Older revision		Revision as of 19:08, 1 August 2025
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	* [[Ed9/4]]		* [[Ed9/4]]

	[[Category:~~Edfs~~\| ]]		[[Category:Edf\| ]]
	<!-- main article -->		<!-- main article -->
	[[Category:Edonoi]]		[[Category:Edonoi]]

Fredg999 category edits: Moving from Category:Edf to Category:Edfs using Cat-a-lot

2025-08-01T19:00:27Z

Moving from Category:Edf to Category:Edfs using Cat-a-lot

← Older revision		Revision as of 19:00, 1 August 2025
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	* [[Ed9/4]]		* [[Ed9/4]]

	[[Category:~~Edf~~\| ]] <!-- main article -->		[[Category:Edfs\| ]]
			<!-- main article -->
	[[Category:Edonoi]]		[[Category:Edonoi]]
	[[Category:Lists of scales]]		[[Category:Lists of scales]]

BudjarnLambeth: Typo

2025-05-22T03:37:05Z

Typo

← Older revision		Revision as of 03:37, 22 May 2025
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	Division of 3/2 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, EDF scales have a perceptually important [[Pseudo-octave\|false octave]], with various degrees of accuracy.		Division of 3/2 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, EDF scales have a perceptually important [[Pseudo-octave\|false octave]], with various degrees of accuracy.

	Arguments for the utility of 3/2 as a base - whether an equivalence or just a [[period]] - are its being one of the strongest [[consonance]]s after the [[octave]], as well as its use to ~~fork~~ structurally important [[pentachord]]s in many musical traditions past and present.		Arguments for the utility of 3/2 as a base - whether an equivalence or just a [[period]] - are its being one of the strongest [[consonance]]s after the [[octave]], as well as its use to form structurally important [[pentachord]]s in many musical traditions past and present.

	Perhaps the first to divide the perfect fifth was [[Wendy Carlos]] ([http://www.wendycarlos.com/resources/pitch.html ''Three Asymmetric divisions of the octave'']). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.		Perhaps the first to divide the perfect fifth was [[Wendy Carlos]] ([http://www.wendycarlos.com/resources/pitch.html ''Three Asymmetric divisions of the octave'']). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.

BudjarnLambeth at 03:11, 22 May 2025

2025-05-22T03:11:19Z

← Older revision		Revision as of 03:11, 22 May 2025
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	== See also ==		== See also ==
	* [[Relative errors of small EDFs]]		* [[Relative errors of small EDFs]]
			* [[Ed9/4]]

	[[Category:Edf\| ]] <!-- main article -->		[[Category:Edf\| ]] <!-- main article -->
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	[[Category:Lists of scales]]		[[Category:Lists of scales]]
	[[Category:Acronyms]]		[[Category:Acronyms]]


			{{todo\|inline=1\|cleanup\|improve layout}}

BudjarnLambeth: Adapted wording from ed16/15

2025-04-25T01:25:11Z

Adapted wording from ed16/15

← Older revision		Revision as of 01:25, 25 April 2025
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	The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2\|perfect fifth (3/2)]] in a certain number of [[equal]] steps.		The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2\|perfect fifth (3/2)]] in a certain number of [[equal]] steps.

	Division of ~~the~~ 3/2 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. ~~The question~~ of ~~equivalence is still in its infancy~~. ~~The~~ utility of 3/2 as a base ~~though, is apparent by~~ being one of the strongest [[consonance]]s after the [[octave]]~~. Many~~, ~~though not all, of these scales have a perceptually~~ important ~~false octave, with various degrees of accuracy~~.		Division of 3/2 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, EDF scales have a perceptually important [[Pseudo-octave\|false octave]], with various degrees of accuracy.

			Arguments for the utility of 3/2 as a base - whether an equivalence or just a [[period]] - are its being one of the strongest [[consonance]]s after the [[octave]], as well as its use to fork structurally important [[pentachord]]s in many musical traditions past and present.

	Perhaps the first to divide the perfect fifth was [[Wendy Carlos]] ([http://www.wendycarlos.com/resources/pitch.html ''Three Asymmetric divisions of the octave'']). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.		Perhaps the first to divide the perfect fifth was [[Wendy Carlos]] ([http://www.wendycarlos.com/resources/pitch.html ''Three Asymmetric divisions of the octave'']). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.

	~~Incidentally, one~~ way to ~~treat 3/2 as an equivalence~~ is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.		One way to approach some EDF tunings is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.

	Alternatively, [[User:CompactStar\|CompactStar]] has also suggeted the usage of [[Half-prime subgroup\|half-prime]] (such as 3/2.5/2.7/2.11/2.…) subgroups for a JI/RTT-based interpretation of EDFs. But such a system, even for the simplest case of 3/2.5/2.7/2, would require very high odd-limit intervals if we want everything to fit within 3/2. The simplest chord in the 7/2-limit which fits inside 3/2 is already quite complex as 1-[[28/27]]-[[10/9]] (27:28:30) and that is a very dense tone cluster–to have a non-tone cluster it is required to go up to 1-[[10/9]]-[[7/5]] (45:50:63). However this approach has the advantage, or disadvantage depending on your compositional approach, of completely avoiding octaves similar to no-twos subgroups that are used for [[EDT]]s.		Alternatively, [[User:CompactStar\|CompactStar]] has also suggeted the usage of [[Half-prime subgroup\|half-prime]] (such as 3/2.5/2.7/2.11/2.…) subgroups for a JI/RTT-based interpretation of EDFs. But such a system, even for the simplest case of 3/2.5/2.7/2, would require very high odd-limit intervals if we want everything to fit within 3/2. The simplest chord in the 7/2-limit which fits inside 3/2 is already quite complex as 1-[[28/27]]-[[10/9]] (27:28:30) and that is a very dense tone cluster–to have a non-tone cluster it is required to go up to 1-[[10/9]]-[[7/5]] (45:50:63). However this approach has the advantage, or disadvantage depending on your compositional approach, of completely avoiding octaves similar to no-twos subgroups that are used for [[EDT]]s.

BudjarnLambeth: Add link

2024-12-26T01:15:20Z

Add link

← Older revision		Revision as of 01:15, 26 December 2024
Line 11:		Line 11:
	Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.		Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.

	Alternatively, [[User:CompactStar\|CompactStar]] has also suggeted the usage of half-prime (such as 3/2.5/2.7/2.11/2.…) subgroups for a JI/RTT-based interpretation of EDFs. But such a system, even for the simplest case of 3/2.5/2.7/2, would require very high odd-limit intervals if we want everything to fit within 3/2. The simplest chord in the 7/2-limit which fits inside 3/2 is already quite complex as 1-[[28/27]]-[[10/9]] (27:28:30) and that is a very dense tone cluster–to have a non-tone cluster it is required to go up to 1-[[10/9]]-[[7/5]] (45:50:63). However this approach has the advantage, or disadvantage depending on your compositional approach, of completely avoiding octaves similar to no-twos subgroups that are used for [[EDT]]s.		Alternatively, [[User:CompactStar\|CompactStar]] has also suggeted the usage of [[Half-prime subgroup\|half-prime]] (such as 3/2.5/2.7/2.11/2.…) subgroups for a JI/RTT-based interpretation of EDFs. But such a system, even for the simplest case of 3/2.5/2.7/2, would require very high odd-limit intervals if we want everything to fit within 3/2. The simplest chord in the 7/2-limit which fits inside 3/2 is already quite complex as 1-[[28/27]]-[[10/9]] (27:28:30) and that is a very dense tone cluster–to have a non-tone cluster it is required to go up to 1-[[10/9]]-[[7/5]] (45:50:63). However this approach has the advantage, or disadvantage depending on your compositional approach, of completely avoiding octaves similar to no-twos subgroups that are used for [[EDT]]s.

	== Individual pages for EDFs ==		== Individual pages for EDFs ==

FloraC: Recategorize

2024-05-19T08:59:27Z

Recategorize

← Older revision		Revision as of 08:59, 19 May 2024
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	* [[Relative errors of small EDFs]]		* [[Relative errors of small EDFs]]

	~~[[Category:Equal-step tuning]]~~
	[[Category:Edf\| ]] <!-- main article -->		[[Category:Edf\| ]] <!-- main article -->
			[[Category:Edonoi]]
			[[Category:Lists of scales]]
	[[Category:Acronyms]]		[[Category:Acronyms]]

FloraC: Rework the trivial statement into something more informative and improve linking

2024-05-18T15:18:33Z

Rework the trivial statement into something more informative and improve linking

← Older revision		Revision as of 15:18, 18 May 2024
Line 3:		Line 3:
	\|de = Edt		\|de = Edt
	}}		}}
	The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2\|perfect fifth]] in a certain number of [[equal]] steps.		The '''equal division of the fifth''' ('''EDF''' or '''ED3/2''') is a [[tuning]] obtained by dividing the [[3/2\|perfect fifth (3/2)]] in a certain number of [[equal]] steps.

	Division of the 3:2 into equal parts ~~can be conceived of as to~~ directly ~~use~~ this interval as an equivalence~~, or not~~. The question of [[equivalence]] is still in its infancy. The utility of 3:2 as a base though, is apparent by being one of the strongest ~~consonances~~ after the octave. Many, if not all, of these scales have a perceptually important ~~pseudo (~~false) octave, with various degrees of accuracy.		Division of the 3/2 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. The question of equivalence is still in its infancy. The utility of 3/2 as a base though, is apparent by being one of the strongest [[consonance]]s after the [[octave]]. Many, though not all, of these scales have a perceptually important false octave, with various degrees of accuracy.

	Perhaps the first to divide the perfect fifth was [[Wendy Carlos]] ([http://www.wendycarlos.com/resources/pitch.html ''Three Asymmetric divisions of the octave'']). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.		Perhaps the first to divide the perfect fifth was [[Wendy Carlos]] ([http://www.wendycarlos.com/resources/pitch.html ''Three Asymmetric divisions of the octave'']). [[Carlo Serafini]] has also made much use of the alpha, beta and gamma scales.
Line 11:		Line 11:
	Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.		Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone, and conversely one way to treat secundal chords (relative to scales where the large step is no larger than 253¢) as the one true type of triad is the use of 3/2 as the (formal) equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an [http://www.youtube.com/watch?v=x_HSMND6RnA example] of it.

	Alternatively, [[User:CompactStar\|CompactStar]] has also suggeted the usage of half-prime (such as 3/2.5/2.7/2.11/2...) subgroups for a JI/RTT-based interpretation of EDFs. But such a system, even for the simplest case of 3/2.5/2.7/2, would require very high odd-limit intervals if we want everything to fit within 3/2. The simplest chord in the 7/2-limit which fits inside 3/2 is already quite complex as 1-[[28/27]]-[[10/9]] (27:28:30) and that is a very dense tone cluster–to have a non-tone cluster it is required to go up to 1-[[10/9]]-[[7/5]] (45:50:63). However this approach has the advantage, or disadvantage depending on your compositional approach, of completely avoiding octaves similar to no-twos subgroups that are used for [[EDT]]s.		Alternatively, [[User:CompactStar\|CompactStar]] has also suggeted the usage of half-prime (such as 3/2.5/2.7/2.11/2.…) subgroups for a JI/RTT-based interpretation of EDFs. But such a system, even for the simplest case of 3/2.5/2.7/2, would require very high odd-limit intervals if we want everything to fit within 3/2. The simplest chord in the 7/2-limit which fits inside 3/2 is already quite complex as 1-[[28/27]]-[[10/9]] (27:28:30) and that is a very dense tone cluster–to have a non-tone cluster it is required to go up to 1-[[10/9]]-[[7/5]] (45:50:63). However this approach has the advantage, or disadvantage depending on your compositional approach, of completely avoiding octaves similar to no-twos subgroups that are used for [[EDT]]s.

	== Individual pages for EDFs ==		== Individual pages for EDFs ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"