4edo - Revision history

Overthink: Undo revision 230820 by MisterShafXen (talk) Only 19, 27, and perhaps 45 seem accurate enough to be usable. I don't know of anyone who uses 3edo for high-limit stuff.

2026-05-24T17:46:11Z

Undo revision 230820 by MisterShafXen (talk) Only 19, 27, and perhaps 45 seem accurate enough to be usable. I don't know of anyone who uses 3edo for high-limit stuff.

← Older revision		Revision as of 17:46, 24 May 2026
Line 20:		Line 20:

	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|4}}~~Odd harmonics 3-23 sorted by error (greatest to least): 21, 15, 17, 3, 9, 5, 7, 13, 11, 23, 19~~		{{Harmonics in equal\|4}}
	~~{{Harmonics in equal\|steps=4\|start=12\|columns=9\|collapsed=yes}}~~
	~~Odd harmonics 25-41 sorted by error: 35, 29, 41, 25, 31, 33, 37, 39, 27~~
	~~{{Harmonics in equal\|steps=4\|start=21\|columns=9\|collapsed=yes}}~~
	~~Odd harmonics 43-59 sorted by error: 59, 49, 57, 51, 43, 47, 55, 53, 45~~

	=== Subsets and supersets ===		=== Subsets and supersets ===

MisterShafXen: added more harmonics

2026-05-24T11:45:07Z

added more harmonics

← Older revision		Revision as of 11:45, 24 May 2026
Line 20:		Line 20:

	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|4}}		{{Harmonics in equal\|4}}Odd harmonics 3-23 sorted by error (greatest to least): 21, 15, 17, 3, 9, 5, 7, 13, 11, 23, 19
			{{Harmonics in equal\|steps=4\|start=12\|columns=9\|collapsed=yes}}
			Odd harmonics 25-41 sorted by error: 35, 29, 41, 25, 31, 33, 37, 39, 27
			{{Harmonics in equal\|steps=4\|start=21\|columns=9\|collapsed=yes}}
			Odd harmonics 43-59 sorted by error: 59, 49, 57, 51, 43, 47, 55, 53, 45

	=== Subsets and supersets ===		=== Subsets and supersets ===

Overthink: /* Theory */ write some things better

2026-02-12T04:28:24Z

Theory: write some things better

← Older revision		Revision as of 04:28, 12 February 2026
Line 9:		Line 9:

	== Theory ==		== Theory ==
	Like [[3edo]], 4edo is already familiar as a chord of [[12edo]]. Not only that, but 4edo establishes tonality in much the same ways that 3edo does—with only two notes at a time as opposed to three aside from octave reduplications of the tonic, though the Tonic-Antitonic contrast from [[2edo]] also works. Also like with 3edo, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit [[mapping]], or [[val]], for 4edo goes {{val\| 4 6 9 11 }}, all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3edo goes with triadic harmony, mapping the [[7-limit]] [[consistent]]ly, and sending 15/14, 21/20, 25/24, and 36/35 to the unison. Somewhat confusingly, the patent mapping of 4edo sees 9/8 mapped to the unison also, leading to [[Very low accuracy temperaments #Antitonic\|antitonic]], though this can be traced to both 3/2 and 4/3 being mapped to 2\4.		Like [[3edo]], 4edo is already familiar as a chord of [[12edo]]. Not only that, but 4edo establishes tonality in much the same ways that 3edo does—with only two notes at a time as opposed to three aside from octave reduplications of the tonic, though the Tonic-Antitonic contrast from [[2edo]] also works. Also like with 3edo, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit [[mapping]], or [[val]], for 4edo goes {{val\| 4 6 9 11 }}, all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3edo goes with triadic harmony, mapping the [[7-limit]] [[consistent]]ly, and sending [[15/14]], [[21/20]], [[25/24]], and [[36/35]] to the unison. Somewhat confusingly, the patent mapping of 4edo sees [[9/8]] mapped to the unison also, leading to [[Very low accuracy temperaments #Antitonic\|antitonic]], though this can be traced to both [[3/2]] and [[4/3]] being mapped to 2\4.

	By putting together the triples of integers which uniquely represent 7-limit tetrads in the [[The Seven Limit Symmetrical Lattices\|7-limit cubic lattice of tetrads]] with the number of 4edo steps returned by the {{val\| 4 6 9 11 }} we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4edo skeletons of another.		By putting together the triples of integers which uniquely represent 7-limit tetrads in the [[The Seven Limit Symmetrical Lattices\|7-limit cubic lattice of tetrads]] with the number of 4edo steps returned by the {{val\| 4 6 9 11 }} we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4edo skeletons of another.
Line 15:		Line 15:
	We can also add more kinds of chords, for instance the subminor ([[6:7:9:10\|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126\|1–9/7–3/2–9/5]]) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.		We can also add more kinds of chords, for instance the subminor ([[6:7:9:10\|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126\|1–9/7–3/2–9/5]]) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.

	4edo can be viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone ~~is the~~ the [[patent val]] while the major sixth is in 4b~~-edo~~ (using [[wart notation]]). 4b~~-edo~~ has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].		4edo can be viewed as a [[dual-fifth]] system (the smallest in fact, besides the trivial [[1edo]]), with the tritone and major sixth as the flat and sharp "fifths". The tritone represents 3/2 in the [[patent val]], while the major sixth represents 3/2 in the 4b val (using [[wart notation]]). The 4b val has one of the sharpest mappings of 3/2 of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].

	4edo can be seen as a trivial tuning of the [[diminished (temperament)\|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first edo that is [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet\|gariberttet]] tuning.		4edo can be seen as a trivial tuning of the [[diminished (temperament)\|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves; 4edo is the first edo that is [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet\|gariberttet]] tuning.

	=== Odd harmonics ===		=== Odd harmonics ===

Overthink: /* Theory */ fix typo

2025-12-19T03:19:33Z

Theory: fix typo

← Older revision		Revision as of 03:19, 19 December 2025
Line 15:		Line 15:
	We can also add more kinds of chords, for instance the subminor ([[6:7:9:10\|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126\|1–9/7–3/2–9/5]]) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.		We can also add more kinds of chords, for instance the subminor ([[6:7:9:10\|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126\|1–9/7–3/2–9/5]]) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.

	4edo can ne viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].		4edo can be viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].

	4edo can be seen as a trivial tuning of the [[diminished (temperament)\|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first edo that is [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet\|gariberttet]] tuning.		4edo can be seen as a trivial tuning of the [[diminished (temperament)\|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first edo that is [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet\|gariberttet]] tuning.

FloraC: /* Theory */ +subsets and supersets

2025-05-23T12:57:22Z

Theory: +subsets and supersets

← Older revision		Revision as of 12:57, 23 May 2025
Line 21:		Line 21:
	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|4}}		{{Harmonics in equal\|4}}

			=== Subsets and supersets ===
			4edo is the first composite edo, containing [[2edo]] as the only nontrivial subset edo.

	== Intervals ==		== Intervals ==

FloraC: -redundant category

2025-02-23T13:25:50Z

-redundant category

← Older revision		Revision as of 13:25, 23 February 2025
Line 183:		Line 183:

	[[Category:7-limit]]		[[Category:7-limit]]
	~~[[Category:Macrotonal]]~~
	[[Category:Listen]]		[[Category:Listen]]

FloraC: /* Theory */ cleanup

2025-02-20T15:54:12Z

Theory: cleanup

← Older revision		Revision as of 15:54, 20 February 2025
Line 9:		Line 9:

	== Theory ==		== Theory ==
	Like [[3edo]], 4edo is already familiar as a chord of [[12edo]]. Not only that, but 4edo establishes tonality in much the same ways that 3edo does—with only two notes at a time as opposed to three aside from octave reduplications of the tonic, though the Tonic-Antitonic contrast from [[2edo]] also works. Also like with 3edo, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit [[mapping]], or [[~~Vals and_Tuning Space\|~~val]], for 4edo goes {{val\|4 6 9 11}}, all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3edo goes with triadic harmony, mapping the [[7-limit]] [[consistent]]ly, and sending 15/14, 21/20, 25/24, and 36/35 to the unison. Somewhat confusingly, the patent mapping of 4edo sees 9/8 mapped to the unison also, leading to [[Very low accuracy temperaments #Antitonic\|antitonic]], though this can be traced to both 3/2 and 4/3 being mapped to 2\4.		Like [[3edo]], 4edo is already familiar as a chord of [[12edo]]. Not only that, but 4edo establishes tonality in much the same ways that 3edo does—with only two notes at a time as opposed to three aside from octave reduplications of the tonic, though the Tonic-Antitonic contrast from [[2edo]] also works. Also like with 3edo, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit [[mapping]], or [[val]], for 4edo goes {{val\| 4 6 9 11 }}, all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3edo goes with triadic harmony, mapping the [[7-limit]] [[consistent]]ly, and sending 15/14, 21/20, 25/24, and 36/35 to the unison. Somewhat confusingly, the patent mapping of 4edo sees 9/8 mapped to the unison also, leading to [[Very low accuracy temperaments #Antitonic\|antitonic]], though this can be traced to both 3/2 and 4/3 being mapped to 2\4.

	By putting together the triples of integers which uniquely represent 7-limit tetrads in the [[~~The_Seven_Limit_Symmetrical_Lattices~~\|7-limit cubic lattice of tetrads]] with the number of 4edo steps returned by the {{val\|4 6 9 11}} we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4edo skeletons of another.		By putting together the triples of integers which uniquely represent 7-limit tetrads in the [[The Seven Limit Symmetrical Lattices\|7-limit cubic lattice of tetrads]] with the number of 4edo steps returned by the {{val\| 4 6 9 11 }} we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4edo skeletons of another.

	We can also add more kinds of chords, for instance the subminor (1-7/~~6-3~~/~~2-5~~/3) and supermajor (~~1-9~~/~~7-3~~/~~2-9~~/5) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.		We can also add more kinds of chords, for instance the subminor ([[6:7:9:10\|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126\|1–9/7–3/2–9/5]]) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.

	4edo can ne viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating ET, only outmatched by that of [[1edo]], and even falling outside of the 600-800 cent range of [[2L 1s]].		4edo can ne viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].

	~~When viewed from a [[regular temperament]] perspective,~~ 4edo can be seen as a tuning of the [[~~Dimipent family #Dimipent~~\|~~dimipent temperament~~]], since it tempers [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] ~~scale~~, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first ~~EDO~~ that is [[~~The_Riemann_zeta_function_and_tuning~~#~~Zeta_EDO_lists~~\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two ~~EDOs~~. If you compress the octaves instead, it can be interpreted as critically sharp [[~~Subgroup~~ temperaments#Gariberttet\|~~Gariberttet~~]].		4edo can be seen as a trivial tuning of the [[diminished (temperament)\|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first edo that is [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet\|gariberttet]] tuning.

	=== ~~Harmonics~~ ===		=== Odd harmonics ===
	{{Harmonics in equal\|4}}		{{Harmonics in equal\|4}}

ArrowHead294 at 15:24, 20 February 2025

2025-02-20T15:24:51Z

← Older revision		Revision as of 15:24, 20 February 2025
Line 6:		Line 6:
	}}		}}
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro\|4}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 15:14, 20 February 2025

2025-02-20T15:14:58Z

← Older revision		Revision as of 15:14, 20 February 2025
Line 9:		Line 9:

	== Theory ==		== Theory ==
	Like [[3edo]], 4edo is already familiar as a chord of [[12edo]]. Not only that, but 4edo establishes tonality in much the same ways that 3edo ~~does — with~~ only two notes at a time as opposed to three aside from octave reduplications of the tonic, though the Tonic-Antitonic contrast from [[2edo]] also works. Also like with 3edo, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit [[mapping]], or [[Vals and_Tuning Space\|val]], for 4edo goes {{val\|4 6 9 11}}, all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3edo goes with triadic harmony, mapping the [[7-limit]] [[consistent]]ly, and sending 15/14, 21/20, 25/24, and 36/35 to the unison. Somewhat confusingly, the patent mapping of 4edo sees 9/8 mapped to the unison also, leading to [[Very low accuracy temperaments #Antitonic\|antitonic]], though this can be traced to both 3/2 and 4/3 being mapped to 2\4.		Like [[3edo]], 4edo is already familiar as a chord of [[12edo]]. Not only that, but 4edo establishes tonality in much the same ways that 3edo does—with only two notes at a time as opposed to three aside from octave reduplications of the tonic, though the Tonic-Antitonic contrast from [[2edo]] also works. Also like with 3edo, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit [[mapping]], or [[Vals and_Tuning Space\|val]], for 4edo goes {{val\|4 6 9 11}}, all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3edo goes with triadic harmony, mapping the [[7-limit]] [[consistent]]ly, and sending 15/14, 21/20, 25/24, and 36/35 to the unison. Somewhat confusingly, the patent mapping of 4edo sees 9/8 mapped to the unison also, leading to [[Very low accuracy temperaments #Antitonic\|antitonic]], though this can be traced to both 3/2 and 4/3 being mapped to 2\4.

	By putting together the triples of integers which uniquely represent 7-limit tetrads in the [[The_Seven_Limit_Symmetrical_Lattices\|7-limit cubic lattice of tetrads]] with the number of 4edo steps returned by the {{val\|4 6 9 11}} we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4edo skeletons of another.		By putting together the triples of integers which uniquely represent 7-limit tetrads in the [[The_Seven_Limit_Symmetrical_Lattices\|7-limit cubic lattice of tetrads]] with the number of 4edo steps returned by the {{val\|4 6 9 11}} we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4edo skeletons of another.
Line 24:		Line 24:
	== Intervals ==		== Intervals ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Intervals of 4edo		\|+ style="font-size: 105%;" \| Intervals of 4edo
			\|-
	! rowspan="2" \| [[Degree]]		! rowspan="2" \| [[Degree]]
	! rowspan="2" \| [[Cent]]s		! rowspan="2" \| [[Cent]]s
Line 81:		Line 82:
	\| [[File:piano_1_1edo.mp3]]		\| [[File:piano_1_1edo.mp3]]
	\|}		\|}
			<nowiki />* Based on treating 4edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
	<nowiki>*~~</nowiki> based~~ on treating 4edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

	== Notation ==		== Notation ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Notation of 4edo		\|+ style="font-size: 105%;" \| Notation of 4edo
			\|-
	! rowspan="2" \| [[Degree]]		! rowspan="2" \| [[Degree]]
	! rowspan="2" \| [[Cent]]s		! rowspan="2" \| [[Cent]]s
Line 122:		Line 123:
	In 4edo:		In 4edo:
	* [[ups and downs notation]] is identical to standard notation;		* [[ups and downs notation]] is identical to standard notation;
	* ~~mixed~~ [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.		* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

	== Solfege ==		== Solfege ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Solfege of 4edo		\|+ style="font-size: 105%;" \| Solfege of 4edo
			\|-
	! [[Degree]]		! [[Degree]]
	! [[Cents]]		! [[Cents]]

Lériendil: chromatic pairs link

2024-12-20T04:55:55Z

chromatic pairs link

← Older revision		Revision as of 04:55, 20 December 2024
Line 17:		Line 17:
	4edo can ne viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating ET, only outmatched by that of [[1edo]], and even falling outside of the 600-800 cent range of [[2L 1s]].		4edo can ne viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating ET, only outmatched by that of [[1edo]], and even falling outside of the 600-800 cent range of [[2L 1s]].

	When viewed from a [[regular temperament]] perspective, 4edo can be seen as a tuning of the [[Dimipent family #Dimipent\|dimipent temperament]], since it tempers [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] scale, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first EDO that is [[The_Riemann_zeta_function_and_tuning#Zeta_EDO_lists\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two EDOs. If you compress the octaves instead, it can be interpreted as critically sharp [[~~Chromatic_pairs~~#Gariberttet\|Gariberttet]].		When viewed from a [[regular temperament]] perspective, 4edo can be seen as a tuning of the [[Dimipent family #Dimipent\|dimipent temperament]], since it tempers [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] scale, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first EDO that is [[The_Riemann_zeta_function_and_tuning#Zeta_EDO_lists\|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two EDOs. If you compress the octaves instead, it can be interpreted as critically sharp [[Subgroup temperaments#Gariberttet\|Gariberttet]].

	=== Harmonics ===		=== Harmonics ===