2.5.7 subgroup - Revision history

Overthink: /* Scales */ proper todo

2026-03-17T22:12:33Z

Scales: proper todo

← Older revision		Revision as of 22:12, 17 March 2026
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	=== Scales ===		=== Scales ===
	{{Todo\|~~expand~~\|inline=1}}		{{Todo\|complete section\|inline=1}}

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

FloraC: + chords and harmony

2026-03-17T09:18:44Z

+ chords and harmony

← Older revision		Revision as of 09:18, 17 March 2026
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	In [[color notation]], this subgroup may be called '''yaza nowa''', which means that it is the intersection of 2.3.5 and 2.3.7 ("yaza"), but without 3 ("nowa").		In [[color notation]], this subgroup may be called '''yaza nowa''', which means that it is the intersection of 2.3.5 and 2.3.7 ("yaza"), but without 3 ("nowa").

			== Chords and harmony ==
			The fundamental consonances of this subgroup may be taken to be [[4:7:10]] and [[14:20:35]]. The otonal chord is [[DR]] and the utonal chord is the inverse of the otonal chord, similar to [[4:5:6]] and [[10:12:15]] in the [[5-limit]]. They are bounded by [[5/2]] and the middle voices contrast by [[49/40]].

			4:7:10 may be extended to [[4:7:10:13]], preserving the DR property and implying adding prime [[13/1\|13]] to this subgroup.

	== Properties ==		== Properties ==

FloraC: /* Regular temperaments */ use TE error like other pages. Remove "birds" as a section title (questionable relevance)

2026-03-17T09:11:13Z

Regular temperaments: use TE error like other pages. Remove "birds" as a section title (questionable relevance)

← Older revision		Revision as of 09:11, 17 March 2026
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	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	A list of edos with progressively better~~<sup>*</sup>~~ tunings for the 2.5.7 subgroup: {{EDOs\| 6, 15, 16, 21, 25, ~~27,~~ 31, ~~68, 103~~, ~~134~~, 140, 171, 239, 379, 410, ~~550, 618,~~ 789~~, 5902~~, 6691, 7480, 8269, 9058, 9847 }}~~, and so on~~.		A list of edos with progressively better tunings for the 2.5.7 subgroup (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''6''', 15, 16, 21, 25, '''31''', 99, 109, 140, 171, 208, 239, 348, '''379''', 410, '''789''', 6691, 7480, 8269, 9058, 9847, … }}.

	~~Another list of edos is those with progressively smaller relative error for the 2.5.7 subgroup: {{EDOs\| 1, 2, 4, 6, 31, 379, 789, 103169 }}, and so on.~~		As [[31edo]] is very strong in the 2.5.7 subgroup so that it is a [[weakly consistent circle]] of [[5/4]]'s and [[7/4]]'s (and thus [[8/5]]'s and [[8/7]]'s) and a [[strongly consistent circle]] of [[35/32]]'s (and thus [[64/35]]'s), it makes sense for those interested in high-complexity [[fractional-octave temperaments]] to consider [[31st-octave temperaments]] (temperaments with a 1\31 [[period]]) that preserve this representation of 2.5.7, which can be seen as combining the simplificatory logics of [[didacus]], [[rainy]] and [[mercy]], which is the 2.5.7-subgroup restriction of [[miracle]]. See [[31st-octave temperaments #Birds]] for details on the canonical extension of it to the full [[19-limit]] that utilises 31edo's good approximation of the interval [[11/9]] and of the 13:17:19 chord.

	~~<sup>*</sup> in absolute DKW distance~~

	~~=== Birds ===~~
	As [[31edo]] is very strong in the 2.5.7 subgroup so that it is a [[weakly consistent circle]] of [[5/4]]'s and [[7/4]]'s (and thus [[8/5]]'s and [[8/7]]'s) and a [[strongly consistent circle]] of [[35/32]]'s (and thus [[64/35]]'s), it makes sense for those interested in high-complexity [[fractional-octave temperaments]] to consider [[31st-octave temperaments]] (temperaments with a 1\31 [[period]]) that preserve this representation of 2.5.7, which can be seen as combining the simplificatory logics of [[didacus]], [[rainy]] and ~~[[Quince clan\|quince]]/~~[[mercy]], which is the 2.5.7-subgroup restriction of [[miracle]]. See [[31st-~~octave_temperaments~~#Birds]] for details on the canonical extension of it to the full [[19-limit]] that utilises 31edo's good approximation of the interval [[11/9]] and of the 13:17:19 chord.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

FloraC: Re-organize (there's only one exotemp mentioned)

2026-03-17T09:01:45Z

Re-organize (there's only one exotemp mentioned)

← Older revision		Revision as of 09:01, 17 March 2026
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{Main\|Tour of regular temperaments#Clans defined by a 2.5.7 (yaza nowa) comma}}		{{Main\|Tour of regular temperaments#Clans defined by a 2.5.7 (yaza nowa) comma}}

			Some important [[diesis\|dieses]] in the 2.5.7 subgroup are:
			* [[128/125\|2/(5/4)<sup>3</sup>]], the [[enharmonic diesis]] (or just "diesis"), leading to [[augmented]] if tempered out.
			* [[16807/16384\|2/(8/7)<sup>5</sup>]], the [[cloudy comma]], leading to [[cloudy]] if tempered, which can be likened in structural role to the enharmonic diesis but for prime 7 instead of prime 5. Equating it with the enharmonic diesis to form a general-purpose diesis results in [[rainy]] temperament, as their difference is the [[rainy comma]].
			* [[50/49\|(10/7)/(7/5)]], the [[septimal tritonic diesis]], leading to [[jubilic]] if tempered out.
			* [[256/245\|(8/7)<sup>2</sup>/(5/4)]], which can be likened in structural role to the septimal tritonic diesis, leading to [[bapbo clan\|bapbo]] if tempered out, considered below:

	==== Didacus (3136/3125) ====		==== Didacus (3136/3125) ====
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	<nowiki />* In 2.5.7-targeted DKW tuning		<nowiki />* In 2.5.7-targeted DKW tuning

	==== ~~Dieses & rank-2 exotemperaments ====~~		==== Exotemperaments ====
	~~Some important [[diesis\|dieses]] in the 2.5.7 subgroup are:~~
	* [[128/125\|2/(5/4)<sup>3</sup>]], the [[enharmonic diesis]] (or just "diesis"), leading to [[augmented]] if tempered out.
	* [[16807/16384\|2/(8/7)<sup>5</sup>]], the [[cloudy comma]], leading to [[cloudy]] if tempered, which can be likened in structural role to the enharmonic diesis but for prime 7 instead of prime 5. Equating it with the enharmonic diesis to form a general-purpose diesis results in [[rainy]] temperament, as their difference is the [[rainy comma]].
	* [[50/49\|(10/7)/(7/5)]], the [[septimal tritonic diesis]], leading to [[jubilic]] if tempered out.
	* [[256/245\|(8/7)<sup>2</sup>/(5/4)]], which can be likened in structural role to the septimal tritonic diesis, leading to [[bapbo clan\|bapbo]] if tempered out, considered below:

	~~==== Bapbo (256/245)~~ ====
	The [[bapbo clan\|bapbo]] [[exotemperament]] tempers out [[256/245]] = S6⋅S8 = S4/S7, equating [[35/32]] with [[8/7]], and making [[5/4]] the square of 8/7, which hence serves as the generator. This comma is over 70 cents, and pathologically equates the outer edges of the clusters around 1\6 and 5\6 to each other without also equating the middle interval with them (if this is done, the tuning reduces to 6edo). Therefore, bapbo can safely be considered an exotemperament – perhaps an analogue to [[mavila]].		The [[bapbo clan\|bapbo]] [[exotemperament]] tempers out [[256/245]] = S6⋅S8 = S4/S7, equating [[35/32]] with [[8/7]], and making [[5/4]] the square of 8/7, which hence serves as the generator. This comma is over 70 cents, and pathologically equates the outer edges of the clusters around 1\6 and 5\6 to each other without also equating the middle interval with them (if this is done, the tuning reduces to 6edo). Therefore, bapbo can safely be considered an exotemperament – perhaps an analogue to [[mavila]].

FloraC: Generic cleanup

2026-03-17T08:50:17Z

Generic cleanup

Show changes

FloraC at 10:23, 11 February 2026

2026-02-11T10:23:54Z

← Older revision		Revision as of 10:23, 11 February 2026
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	{{todo\|expand\|inline=1}}		{{todo\|expand\|inline=1}}

	[[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

2026-02-06T08:22:24Z

Recategorize

← Older revision		Revision as of 08:22, 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]]

VectorGraphics at 04:31, 6 June 2025

2025-06-06T04:31:55Z

← Older revision		Revision as of 04:31, 6 June 2025
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	The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' ~~('''yaza nowa''' in [[color notation]])~~ is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on.		The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on.

	The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11.		The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11.
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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 5 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 5 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]].

			In [[color notation]], this subgroup may be called '''yaza nowa''', which means that it is the intersection of 2.3.5 and 2.3.7 ("yaza"), but without 3 ("nowa").

	== Properties ==		== Properties ==

ArrowHead294 at 03:34, 20 October 2024

2024-10-20T03:34:39Z

← Older revision		Revision as of 03:34, 20 October 2024
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	\|-		\|-
	\| 64/35 \|\| 1044.860 \|\| 1041.573 \|\| -3.288		\| 64/35 \|\| 1044.860 \|\| 1041.573 \|\| -3.288
	~~{{table notes\|cols=4~~
	~~\| In 2.5.7-targeted DKW tuning~~
	}}
	\|}		\|}
			<nowiki />* In 2.5.7-targeted DKW tuning

	=== [[Rainy]] ([[2100875/2097152]]) ===		=== [[Rainy]] ([[2100875/2097152]]) ===
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	\|-		\|-
	\| 64/35 \|\| 1044.860 \|\| 1018.297 \|\| -26.563		\| 64/35 \|\| 1044.860 \|\| 1018.297 \|\| -26.563
	~~{{table notes\|cols=4~~
	~~\| In 2.5.7-targeted DKW tuning~~
	}}
	\|}		\|}
			<nowiki />* In 2.5.7-targeted DKW tuning

	=== [[Jubilic]] ([[50/49]]) ===		=== [[Jubilic]] ([[50/49]]) ===
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	\|-		\|-
	\| 64/35 \|\| 1044.860 \|\| 1029.995 \|\| -14.865		\| 64/35 \|\| 1044.860 \|\| 1029.995 \|\| -14.865
	~~{{table notes\|cols=4~~
	~~\| In 2.5.7-targeted DKW tuning~~
	}}
	\|}		\|}
			<nowiki />* In 2.5.7-targeted DKW tuning

	== Dieses & rank-2 [[exotemperament]]s ==		== Dieses & rank-2 [[exotemperament]]s ==
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	\|-		\|-
	\| 64/35 \|\| 1044.860 \|\| 994.614 \|\| -50.247		\| 64/35 \|\| 1044.860 \|\| 994.614 \|\| -50.247
	~~{{table notes\|cols=4~~
	~~\| In 2.5.7-targeted DKW tuning~~
	}}
	\|}		\|}
			<nowiki />* In 2.5.7-targeted DKW tuning

	== Music ==		== Music ==

Godtone: note simple relation to 7-odd-limit

2024-09-07T16:50:08Z

note simple relation to 7-odd-limit

← Older revision		Revision as of 16:50, 7 September 2024
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	The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11.		The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11.

	A notable subset of the 2.5.7 subgroup is the 1.5.7 [[tonality diamond]], comprised of all intervals in which 1, 5 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.5.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[5/4]], [[7/5]], [[10/7]], [[8/5]], [[7/4]], and [[2/1]].		A notable subset of the 2.5.7 subgroup is the 1.5.7 [[tonality diamond]], comprised of all intervals in which 1, 5 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.5.7 tonality diamond (which is the 7-odd-limit (1.3.5.7) with intervals of 3 removed) within the octave is [[1/1]], [[8/7]], [[5/4]], [[7/5]], [[10/7]], [[8/5]], [[7/4]], and [[2/1]].

	Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: [[25/16]], [[25/14]], [[35/32]], [[64/35]], [[28/25]], and [[32/25]].		Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: [[25/16]], [[25/14]], [[35/32]], [[64/35]], [[28/25]], and [[32/25]].