2.3.7 subgroup - Revision history

Domin: /* Music */

2026-08-14T09:43:27Z

Music

← Older revision		Revision as of 09:43, 14 August 2026
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	== Music ==		== Music ==
	; [[Domin]]		; [[Domin]]
	* [https://www.youtube.com/watch?v=~~5bzoSu0gVNE~~ ''Pnong''] (2024)		* [https://www.youtube.com/watch?v=0CoFmqptcGU ''Pnong (2.3.7 JI)''] (2024)
			* [https://www.youtube.com/watch?v=JU_e2AK3cQ0 ''Ngaat (2.3.7 JI non octave)''] (2024)
			* [https://www.youtube.com/watch?v=_UFNfu7angg ''Xiticix (2.3.7 JI)''] (2024)
			* [https://www.youtube.com/watch?v=uFwJzgj49bU ''Traudn (2.3.7 JI)''] (2024)

	; [[Michael Harrison]]		; [[Michael Harrison]]

Domin: /* Music */

2026-08-14T09:24:28Z

Music

← Older revision		Revision as of 09:24, 14 August 2026
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	== Music ==		== Music ==
	; [[Domin]]		; [[Domin]]
	* [https://www.youtube.com/watch?v=5bzoSu0gVNE ''~~Ctun~~''] (2024)		* [https://www.youtube.com/watch?v=5bzoSu0gVNE ''Pnong''] (2024)

	; [[Michael Harrison]]		; [[Michael Harrison]]

Domin: /* Music */

2026-08-14T09:24:16Z

Music

← Older revision		Revision as of 09:24, 14 August 2026
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	== Music ==		== Music ==
			; [[Domin]]
			* [https://www.youtube.com/watch?v=5bzoSu0gVNE ''Ctun''] (2024)

	; [[Michael Harrison]]		; [[Michael Harrison]]
	* From Ancient Worlds (for harmonic piano), 1992		* From Ancient Worlds (for harmonic piano), 1992

FloraC: /* Scales */ move some scales here

2026-08-09T07:50:11Z

Scales: move some scales here

← Older revision		Revision as of 07:50, 9 August 2026
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	==== Misc ====		==== Misc ====
	* [[Diasem]]/Tas[9] ([[chirality\|left-handed]]): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1		* [[Diasem]]/Tas[9] ([[chirality\|left-handed]]): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1
			* Archytas Diatonic: 8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1
			* Safi al-Din Septimal: 8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1

	== Regular temperaments ==		== Regular temperaments ==

FloraC: /* Chords and harmony */ examples of using 21/16

2026-03-17T12:02:20Z

Chords and harmony: examples of using 21/16

← Older revision		Revision as of 12:02, 17 March 2026
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	== Chords and harmony ==		== Chords and harmony ==
	There are a number of ways to approach harmony in this subgroup, most of which are discussed in [[Superpyth #Chords and harmony]]. The basic forms of chords include the triad [[6:7:9]], the tetrad [[6:7:8:9]], and their utonal inverses. ~~Similarly, the~~ fourth-spanning triads [[6:7:8]] and [[21:24:28]] can be used, as well as their wide voicing 4:7:12 and 7:12:21. ~~Examples of extensions~~ of these chords are [[12:14:18:21]] and [[14:18:21:24]]. Like in [[5-limit]] [[JI]], one quickly runs into [[wolf interval]]s without care, but the 2.3.7 wolf being [[21/16]] or [[32/21]] may be considered less discordant, and useful in its own ways.		There are a number of ways to approach harmony in this subgroup, most of which are discussed in [[Superpyth #Chords and harmony]]. The basic forms of chords include the {{w\|tertian harmony\|tertian}} triad [[6:7:9]], the tetrad [[6:7:8:9]], and their utonal inverses. The fourth-spanning triads [[6:7:8]] and [[21:24:28]] can also be used, as well as their wide voicing 4:7:12 and 7:12:21. Extensions of these chords include but are not limited to the [[9-odd-limit]] [[anomalous saturated suspension\|saturated suspensions]], [[12:14:18:21]] and [[14:18:21:24]].

			Like in [[5-limit]] [[JI]], one quickly runs into [[wolf interval]]s without care, but the 2.3.7 wolf being [[21/16]] or [[32/21]] may be considered less discordant, and useful in its own ways. For example, 1–9/8–21/16–3/2 and 1–8/7–4/3–3/2 may be considered [[21-odd-limit]] concords. They are conflated in superpyth as the sus2-4 chord, but subtlely contrast each other in JI.

	== Properties ==		== Properties ==

FloraC: /* Chords and harmony */

2026-03-17T09:54:46Z

Chords and harmony

← Older revision		Revision as of 09:54, 17 March 2026
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	== Chords and harmony ==		== Chords and harmony ==
	There are a number of ways to approach harmony in this subgroup, most of which are discussed in [[Superpyth #Chords and harmony]]. The basic forms of chords include the triad [[6:7:9]], the tetrad [[6:7:8:9]], and their utonal inverses. Similarly, the fourth-spanning ~~triad~~ [[6:7:8]] and [[21:24:28]] can be used, as well as their wide voicing 4:7:12 and 7:12:21. Examples of extensions of these chords are [[12:14:18:21]] and [[14:18:21:24]]. Like in [[5-limit]] [[JI]], one quickly runs into [[wolf interval]]s without care, but the 2.3.7 wolf being [[21/16]] or [[32/21]] may be considered less discordant, and useful in its own ways.		There are a number of ways to approach harmony in this subgroup, most of which are discussed in [[Superpyth #Chords and harmony]]. The basic forms of chords include the triad [[6:7:9]], the tetrad [[6:7:8:9]], and their utonal inverses. Similarly, the fourth-spanning triads [[6:7:8]] and [[21:24:28]] can be used, as well as their wide voicing 4:7:12 and 7:12:21. Examples of extensions of these chords are [[12:14:18:21]] and [[14:18:21:24]]. Like in [[5-limit]] [[JI]], one quickly runs into [[wolf interval]]s without care, but the 2.3.7 wolf being [[21/16]] or [[32/21]] may be considered less discordant, and useful in its own ways.

	== Properties ==		== Properties ==

FloraC: + chords and harmony

2026-03-17T09:53:53Z

+ chords and harmony

← Older revision		Revision as of 09:53, 17 March 2026
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	When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].		When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].

			== Chords and harmony ==
			There are a number of ways to approach harmony in this subgroup, most of which are discussed in [[Superpyth #Chords and harmony]]. The basic forms of chords include the triad [[6:7:9]], the tetrad [[6:7:8:9]], and their utonal inverses. Similarly, the fourth-spanning triad [[6:7:8]] and [[21:24:28]] can be used, as well as their wide voicing 4:7:12 and 7:12:21. Examples of extensions of these chords are [[12:14:18:21]] and [[14:18:21:24]]. Like in [[5-limit]] [[JI]], one quickly runs into [[wolf interval]]s without care, but the 2.3.7 wolf being [[21/16]] or [[32/21]] may be considered less discordant, and useful in its own ways.

	== Properties ==		== Properties ==

FloraC: /* Regular temperaments */ bold for relative error. Fix formatting

2026-03-17T09:05:43Z

Regular temperaments: bold for relative error. Fix formatting

← Older revision		Revision as of 09:05, 17 March 2026
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	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	A list of edos with progressively better tunings for the 2.3.7 subgroup: {{EDOs\| 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 }} ~~and so on.~~		A list of edos with progressively better tunings for the 2.3.7 subgroup (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''5''', 12, 14, 17, 22, 31, '''36''', 77, 94, 130, '''135''', 171, 265, 306, 400, '''571''', 706, '''1277''', … }}

	Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs\| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.		Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs\| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.
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	{{Main\| Semaphore and godzilla }}		{{Main\| Semaphore and godzilla }}

	Semaphore tempers out the comma [[49/48]] {{S\|7}} in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].		Semaphore tempers out the comma [[49/48]] ({{S\|7}}) in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].

	The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230{{c}}; a chart of mistunings of simple intervals is below.		The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230{{c}}; a chart of mistunings of simple intervals is below.
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	{{Main\| Superpyth }}		{{Main\| Superpyth }}

	Archy tempers out the comma [[64/63]] {{S\|8}} in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.		Archy tempers out the comma [[64/63]] ({{S\|8}}) in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.

	The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585{{c}} though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.		The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585{{c}} though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.

FloraC: Generic cleanup

2026-03-17T08:58:35Z

Generic cleanup

Show changes

FloraC at 10:23, 11 February 2026

2026-02-11T10:23:25Z

← Older revision		Revision as of 10:23, 11 February 2026
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	<references group="note"/>		<references group="note"/>

	[[Category:~~Subgroup~~]]		[[Category:Just intonation subgroups\|#]]
	[[Category:Rank-3 temperaments]]		[[Category:Rank-3 temperaments\|#]]
	[[Category:7-limit]]		[[Category:7-limit\|#]]
	[[Category:Lists of scales]]		[[Category:Lists of scales\|#]]

← Older revision		Revision as of 09:05, 17 March 2026
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	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	A list of edos with progressively better tunings for the 2.3.7 subgroup: {{EDOs\| 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 }} ~~and so on.~~		A list of edos with progressively better tunings for the 2.3.7 subgroup (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''5''', 12, 14, 17, 22, 31, '''36''', 77, 94, 130, '''135''', 171, 265, 306, 400, '''571''', 706, '''1277''', … }}

	Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs\| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.		Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs\| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.
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	{{Main\| Semaphore and godzilla }}		{{Main\| Semaphore and godzilla }}

	Semaphore tempers out the comma [[49/48]] {{S\|7}} in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].		Semaphore tempers out the comma [[49/48]] ({{S\|7}}) in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].

	The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230{{c}}; a chart of mistunings of simple intervals is below.		The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230{{c}}; a chart of mistunings of simple intervals is below.
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	{{Main\| Superpyth }}		{{Main\| Superpyth }}

	Archy tempers out the comma [[64/63]] {{S\|8}} in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.		Archy tempers out the comma [[64/63]] ({{S\|8}}) in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.

	The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585{{c}} though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.		The [[DKW theory\|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585{{c}} though most other optimizations tune this a few cents flatter; a chart of mistunings of simple intervals is below.