31edo - Revision history

Overthink: /* Theory */ I don't think that part's needed, actually

2026-06-10T00:03:22Z

Theory: I don't think that part's needed, actually

← Older revision		Revision as of 00:03, 10 June 2026
Line 21:		Line 21:
	One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.		One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.

	In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)\|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1\|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds each, which is much more than the two varieties available ~~for each~~ in 12edo.		In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)\|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1\|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds each, which is much more than the two varieties available in 12edo.

	=== Prime harmonics ===		=== Prime harmonics ===

Overthink: /* Theory */ clarify a little

2026-06-10T00:02:11Z

Theory: clarify a little

← Older revision		Revision as of 00:02, 10 June 2026
Line 21:		Line 21:
	One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.		One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.

	In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)\|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1\|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds, which is much more than the two varieties in 12edo.		In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)\|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1\|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds each, which is much more than the two varieties available for each in 12edo.

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: 1. Ratios don't have otonality or utonality. 2. Ratios here should have some significance per se to start with. 3. Replace 175/128 and 256/175 with 48/35 and 35/24 for sanity

2026-05-10T18:58:33Z

1. Ratios don't have otonality or utonality. 2. Ratios here should have some significance per se to start with. 3. Replace 175/128 and 256/175 with 48/35 and 35/24 for sanity

← Older revision		Revision as of 18:58, 10 May 2026
Line 147:		Line 147:
	\| 541.9		\| 541.9
	\| Superfourth		\| Superfourth
	\| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[~~175~~/~~128~~]]		\| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[48/35]]
	\| {{UDnote\|step=14}}		\| {{UDnote\|step=14}}
	\|-		\|-
Line 165:		Line 165:
	\| 658.1		\| 658.1
	\| Subfifth		\| Subfifth
	\| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[~~256~~/~~175~~]]		\| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[35/24]]
	\| {{UDnote\|step=17}}		\| {{UDnote\|step=17}}
	\|-		\|-

Godtone: /* Intervals / some important otonal and utonal ratios of the extremely accurate 2.5.7 interpretation were removed at some point. specifically, 31edo is a circle of 35/32's, so as the less accurate 55=25 is included, so should 77=49 and 355=175. these are more important than dubious 13-limit interpretations imo. also, why is there no diesis?

2026-05-10T15:21:54Z

Intervals: some important otonal and utonal ratios of the extremely accurate 2.5.7 interpretation were removed at some point. specifically, 31edo is a circle of 35/32's, so as the less accurate 5*5=25 is included, so should 7*7=49 and 35*5=175. these are more important than dubious 13-limit interpretations imo. also, why is there no diesis?

← Older revision		Revision as of 15:21, 10 May 2026
Line 69:		Line 69:
	\| 38.7		\| 38.7
	\| Super-unison		\| Super-unison
	\| [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]]		\| [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]], [[128/125]]
	\| {{UDnote\|step=1}}		\| {{UDnote\|step=1}}
	\|-		\|-
Line 87:		Line 87:
	\| 154.8		\| 154.8
	\| Neutral second		\| Neutral second
	\| [[11/10]], [[12/11]], [[13/12]]		\| [[11/10]], [[12/11]], [[13/12]], [[35/32]]
	\| {{UDnote\|step=4}}		\| {{UDnote\|step=4}}
	\|-		\|-
Line 147:		Line 147:
	\| 541.9		\| 541.9
	\| Superfourth		\| Superfourth
	\| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]''		\| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[175/128]]
	\| {{UDnote\|step=14}}		\| {{UDnote\|step=14}}
	\|-		\|-
Line 165:		Line 165:
	\| 658.1		\| 658.1
	\| Subfifth		\| Subfifth
	\| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]''		\| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[256/175]]
	\| {{UDnote\|step=17}}		\| {{UDnote\|step=17}}
	\|-		\|-
Line 225:		Line 225:
	\| 1045.2		\| 1045.2
	\| Neutral seventh		\| Neutral seventh
	\| [[11/6]], [[20/11]], [[24/13]]		\| [[11/6]], [[20/11]], [[24/13]], [[64/35]]
	\| {{UDnote\|step=27}}		\| {{UDnote\|step=27}}
	\|-		\|-
Line 243:		Line 243:
	\| 1161.3		\| 1161.3
	\| Sub-octave		\| Sub-octave
	\| [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]]		\| [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]], [[125/64]]
	\| {{UDnote\|step=30}}		\| {{UDnote\|step=30}}
	\|-		\|-

Overthink: /* Other Instruments */ link

2026-05-04T01:54:17Z

Other Instruments: link

← Older revision		Revision as of 01:54, 4 May 2026
Line 1,518:		Line 1,518:

	=== Other Instruments ===		=== Other Instruments ===
	[[File:31edo array kalimba.jpg\|none\|thumb\|640x640px\|31edo array kalimba built by Tristan Bay; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning]]		[[File:31edo array kalimba.jpg\|none\|thumb\|640x640px\|31edo array kalimba built by [[Tristan Bay]]; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning]]

	=== Lumatone ===		=== Lumatone ===

Overthink: /* Skip fretting */ - extra newlines

2026-05-04T01:53:47Z

Skip fretting: - extra newlines

← Older revision		Revision as of 01:53, 4 May 2026
Line 1,525:		Line 1,525:
	=== Skip fretting ===		=== Skip fretting ===
	'''[[Skip fretting system 31 2 9]]''' is a [[skip fretting]] system for 31edo.		'''[[Skip fretting system 31 2 9]]''' is a [[skip fretting]] system for 31edo.


	'''[[Skip fretting system 31 3 7]]''' is another skip fretting system for 31edo.		'''[[Skip fretting system 31 3 7]]''' is another skip fretting system for 31edo.


	'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].		'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].

Overthink: /* As a tuning of other temperaments */ expand section

2026-04-24T22:57:40Z

As a tuning of other temperaments: expand section

← Older revision		Revision as of 22:57, 24 April 2026
Line 28:		Line 28:

	=== As a tuning of other temperaments ===		=== As a tuning of other temperaments ===
	Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]] or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts~~. Mohajira and its alternative, called [[migration]], merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system~~. In fact, 31edo can be defined as the unique temperament that [[tempering out\|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].		Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]] or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. In fact, 31edo can be defined as the unique temperament that [[tempering out\|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].

	31edo also [[support]]s [[orwell]], which splits the [[3/1\|perfect twelfth]] into seven equal parts of ~7/6. Another notable temperament it supports is [[myna]], which sets the ~~structure of~~ [[7/6]], [[6/5]], 11/9~[[16/13]], 5/4, and 9/7 being equidistant, ~~and thus 31edo's~~ five interval categories. ~~Finally, it~~ supports [[squares]], which splits the [[8/3\|perfect eleventh]] into four equal parts, each representing [[14/11]]~9/7, two of which make [[18/11]], and four of which make [[8/3]]. Then, prime [[5/1\|5]] is found by tempering out [[81/80]], completing the 11-limit.		If we split the meantone [[generator]] of ~3/2 into two neutral thirds, each representing [[11/9]]~[[27/22]], then we get the [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] temperament [[mohaha]], tempering out [[121/120]] and [[243/242]]. We can then map [[7/4]] to the semi-diminished seventh (-13 generators), tempering out [[385/384]], to get the full 11-limit mohajira temperament, which maps 7/6, 6/5, 11/9, 5/4, and 9/7 equidistant from each other. Alternatively, we can use the septimal meantone mapping of 7/4 (+20 generators) to get [[migration]]. Mohajira and [[migration]] merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system.

			The supermajor second [[8/7]] is mapped to a third of the perfect fifth in 31edo, thus tempering out [[1029/1024]], supporting [[slendric]] in the [[2.3.7 subgroup\|2.3.7-subgroup]]. Slendric is a [[cluster temperament]] with 5 clusters of notes in an octave, each with nearby intervals separated by the interval found at -5 generators, or 1 step of 31edo, representing [[49/48]]~[[64/63]]. For example, 9/8, 8/7, and 7/6 are one step apart from each other, as well as 9/7, 21/16, and 4/3. 31edo supports the full 7-limit extension mothra, which tempers out 81/80, thus equating the 49/48~64/63 spacer with [[36/35]], so that 9/8~10/9, 8/7, 7/6, and 6/5 are all mapped equidistantly, as well as 5/4, 9/7, 21/16, and 4/3. Mothra splits into two 11-limit extensions: [[Gamelismic clan#Undecimal mothra\|undecimal mothra]] (26 & 31) tempering out [[99/98]], and [[mosura]] (31 & 36) tempering out [[176/175]].

			[[Miracle]] temperament splits the slendric generator in two parts and the perfect fifth in six, each representing [[15/14]]~[[16/15]], thus tempering out [[225/224]], so that 5/4 is found at -7 generators. The 11-limit version of miracle sets 11/9 to the neutral third, with prime 11 mapped at +15 generators. While 31edo supports miracle, a more accurate tuning is [[72edo]]. [[Valentine]] temperament splits the slendric generator in three parts and the perfect fifth in nine, each representing [[21/20]], tempering out [[126/125]]. Valentine can also be seen as [[Carlos Alpha]] but with octaves added. The canonical 11-limit extension equates the step with [[22/21]], thus tempering out [[121/120]], [[176/175]], and [[441/440]].

			31edo also [[support]]s [[orwell]], which splits the [[3/1\|perfect twelfth]] into seven equal parts of ~7/6. Three of these reach [[8/5]], and two reach [[11/8]], with 1–7/6–11/8–8/5 being the [[orwell tetrad]]. Commas tempered out by orwell include [[99/98]], [[121/120]], [[176/175]], and [[385/384]], among others.

			Another notable temperament it supports is [[myna]], which is generated by the minor third, and sets the intervals [[7/6]], [[6/5]], 11/9~[[16/13]], 5/4, and 9/7 being equidistant. Like mohajira, it creates five interval categories, but with 126/125 tempered out instead of 81/80.

			31edo also supports [[squares]], which splits the [[8/3\|perfect eleventh]] into four equal parts, each representing [[14/11]]~9/7, two of which make [[18/11]], and four of which make [[8/3]]. The [[2.3.7.11-subgroup\|2.3.7.11 subgroup]] version of this temperament is sometimes known as ''skwares'', tempering out 99/98 and 243/242. Then, prime [[5/1\|5]] is found by tempering out [[81/80]], completing the 11-limit.

			Another temperament supported by 31edo is [[würschmidt]], which is generated by 5/4, such that 8 intervals of 5/4 reach [[6/1]]. Würschmidt extends to the 7- and 11-limit through the skwares mapping, also creating 5 interval categories, with the thirds being 7/6, 6/5, 11/9, 5/4, and 14/11~9/7, each equidistant from each other.

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

Overthink: /* Theory */ explain interval categories

2026-04-24T22:12:03Z

Theory: explain interval categories

← Older revision		Revision as of 22:12, 24 April 2026
Line 19:		Line 19:
	Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit\|7-]], [[9-odd-limit\|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.		Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit\|7-]], [[9-odd-limit\|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.

	One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.		One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.

			In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)\|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1\|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds, which is much more than the two varieties in 12edo.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 28:		Line 30:
	Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]] or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. Mohajira and its alternative, called [[migration]], merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system. In fact, 31edo can be defined as the unique temperament that [[tempering out\|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].		Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]] or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. Mohajira and its alternative, called [[migration]], merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system. In fact, 31edo can be defined as the unique temperament that [[tempering out\|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].

	31edo also [[support]]s [[orwell]], which splits the [[3/1\|perfect twelfth]] into seven equal parts of ~7/6. Another notable temperament it supports is [[myna]], which ~~divides~~ the ~~category~~ of ~~thirds into five different intervals: subminor, minor, neutral, major, and supermajor, representing~~ [[7/6]], [[6/5]], 11/9~[[16/13]], 5/4, and 9/7, ~~respectively~~. Finally, it supports [[squares]], which splits the [[8/3\|perfect eleventh]] into four equal parts, each representing [[14/11]]~9/7, two of which make [[18/11]], and four of which make [[8/3]]. Then, prime [[5/1\|5]] is found by tempering out [[81/80]], completing the 11-limit.		31edo also [[support]]s [[orwell]], which splits the [[3/1\|perfect twelfth]] into seven equal parts of ~7/6. Another notable temperament it supports is [[myna]], which sets the structure of [[7/6]], [[6/5]], 11/9~[[16/13]], 5/4, and 9/7 being equidistant, and thus 31edo's five interval categories. Finally, it supports [[squares]], which splits the [[8/3\|perfect eleventh]] into four equal parts, each representing [[14/11]]~9/7, two of which make [[18/11]], and four of which make [[8/3]]. Then, prime [[5/1\|5]] is found by tempering out [[81/80]], completing the 11-limit.

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

FloraC: Interwiki

2026-03-24T11:09:22Z

Interwiki

← Older revision		Revision as of 11:09, 24 March 2026
Line 1:		Line 1:
	{{~~interwiki~~		{{Interwiki
			\| en = 31edo
	\| de = 31-EDO		\| de = 31-EDO
	~~\| en = 31edo~~
	\| es = 31 EDO		\| es = 31 EDO
	\| ja = 31平均律		\| ja = 31平均律
			\| zh = 31平均律
	}}		}}
	{{Infobox ET}}		{{Infobox ET}}

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:17Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
Line 1,328:		Line 1,328:
	\| (P8, ccP4/5)		\| (P8, ccP4/5)
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Octave stretch or compression ==		== Octave stretch or compression ==