2.3.7 subgroup - Revision history

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\|#]]

FloraC: Recategorize

2026-02-06T08:21:54Z

Recategorize

← Older revision		Revision as of 08:21, 6 February 2026
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	[[Category:Subgroup]]		[[Category:Subgroup]]
			[[Category:Rank-3 temperaments]]
	[[Category:7-limit]]		[[Category:7-limit]]
	~~[[Category:Rank 3]]~~
	[[Category:Lists of scales]]		[[Category:Lists of scales]]

Squib: /* Dieses and rank-2 exotemperaments */

2025-10-24T04:07:45Z

Dieses and rank-2 exotemperaments

← Older revision		Revision as of 04:07, 24 October 2025
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	{\| class="wikitable center-1 center-2 center-3 center-4"		{\| class="wikitable center-1 center-2 center-3 center-4"
	\|+ style="font-size: 105%;" \| Trienstonian (~~1029~~/~~1024~~)		\|+ style="font-size: 105%;" \| Trienstonian (28/27)
	\|-		\|-
	! rowspan="2" \| Interval !! rowspan="2" \| Just tuning !! colspan="2" \| Tunings*		! rowspan="2" \| Interval !! rowspan="2" \| Just tuning !! colspan="2" \| Tunings*

Overthink: Completed trienstonian section

2025-10-15T04:42:16Z

Completed trienstonian section

@@ Line 162: / Line 162: @@
 ==== Trienstonian ====
-{{Todo|expand|inline=1}}
+'''Trienstonian''' temperament tempers out 28/27 in the 2.3.7 subgroup, mapping 7/4 to +3 fifths, and equating 9/8 with 7/6 and 9/7 with 4/3. This temperament equates the outer intervals in each cluster of three intervals without setting the middle to them. Due to its inaccuracy, it can reasonably be considered an exotemperament, which in a pentatonic system can be considered an analog to [[mavila]].
 == Music ==

Overthink: added exotemperament section like with 2.5.7

2025-09-21T19:30:51Z

added exotemperament section like with 2.5.7

← Older revision		Revision as of 19:30, 21 September 2025
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	\| 16/9 \|\| 996.090 \|\| 1001.747 \|\| +5.657		\| 16/9 \|\| 996.090 \|\| 1001.747 \|\| +5.657
	\|}		\|}
	<nowiki />* In 2.3.7-targeted DKW tuning		<nowiki>*</nowiki> In 2.3.7-targeted DKW tuning

			=== Dieses and rank-2 [[Exotemperament\|exotemperaments]] ===

			==== Trienstonian ====
			{{Todo\|expand\|inline=1}}

	== Music ==		== Music ==

← 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.