11edo - Revision history

Vector at 20:45, 16 May 2026

2026-05-16T20:45:27Z

@@ Line 154: / Line 154: @@
 === Scales ===
 As a subset of 22edo, 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset. Firstly, you may note that in 11edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo's patent val.)
 Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 1-2-1-2-1-2-2.{{Navbox EDO}}
 {{Cat|Edos}}

Vector at 20:36, 16 May 2026

2026-05-16T20:36:05Z

← Older revision		Revision as of 20:36, 16 May 2026
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	=== Scales ===		=== Scales ===
	~~Arguably~~ the ~~main MOS~~ scale in 11edo is ~~the smitonic Scale~~ 2-1-2-2-1-2-1. The ~~smitonic scale is melodically rather similar in some regards to the 12edo diatonic scale. However, it is harmonically totally different due~~ to the ~~absence~~ of ~~harmonics~~ 3 ~~and~~ 5 in 11edo.		As a subset of 22edo, 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset. Firstly, you may note that in 11edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo's patent val.)

	~~There are others including~~ the ~~checkertonic~~ scale 2-1-1-2-1-1-2~~-1, which can be viewed as a stretched augmented/tcherepnin scale~~.		Another temperament that resides in 11edo is called ''orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 1-2-1-2-1-2-2.{{Navbox EDO}}

	{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Vector at 19:54, 6 May 2026

2026-05-06T19:54:15Z

← Older revision		Revision as of 19:54, 6 May 2026
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	This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is [[smitonic]] (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.		This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is [[smitonic]] (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.

	For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.		For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system. This is in accordance with KISS notation.

	"Interval categories" are based on 22edo usage of the [[ADIN]] system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and [[7-limit]] qualities.		"Interval categories" are based on 22edo usage of the [[ADIN]] system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and [[7-limit]] qualities.

Hotcrystal0: cent values to one decimal place look better

2026-02-26T12:28:50Z

cent values to one decimal place look better

← Older revision		Revision as of 12:28, 26 February 2026
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	\|-		\|-
	\|1		\|1
	\|109		\|109.1
	\|[16/15], 15/14, 18/17, ['''17/16''']		\|[16/15], 15/14, 18/17, ['''17/16''']
	\|vC#, ^Db		\|vC#, ^Db
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	\|-		\|-
	\|2		\|2
	\|218		\|218.2
	\|8/7, '''9/8''', [17/15], [25/22]		\|8/7, '''9/8''', [17/15], [25/22]
	\|D		\|D
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	\|-		\|-
	\|3		\|3
	\|327		\|327.3
	\|6/5, 11/9, 17/14		\|6/5, 11/9, 17/14
	\|vD#, ^Eb		\|vD#, ^Eb
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	\|-		\|-
	\|4		\|4
	\|436		\|436.4
	\|[9/7], 14/11, 32/25		\|[9/7], 14/11, 32/25
	\|E		\|E
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	\|-		\|-
	\|5		\|5
	\|545		\|545.5
	\|'''11/8''', 15/11		\|'''11/8''', 15/11
	\|^F, Gb		\|^F, Gb
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	\|-		\|-
	\|6		\|6
	\|~~655~~		\|654.5
	\|16/11, 22/15		\|16/11, 22/15
	\|F#, vG		\|F#, vG
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	\|-		\|-
	\|7		\|7
	\|~~764~~		\|763.6
	\|[14/9], 11/7, '''25/16'''		\|[14/9], 11/7, '''25/16'''
	\|^G, Ab		\|^G, Ab
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	\|-		\|-
	\|8		\|8
	\|~~873~~		\|872.7
	\|5/3, 18/11, 28/17		\|5/3, 18/11, 28/17
	\|G#, vA		\|G#, vA
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	\|-		\|-
	\|9		\|9
	\|~~982~~		\|981.8
	\|'''7/4''', 16/9, [30/17], [44/25]		\|'''7/4''', 16/9, [30/17], [44/25]
	\|^A, Bb		\|^A, Bb
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	\|-		\|-
	\|10		\|10
	\|~~1091~~		\|1090.9
	\|['''15/8'''], 28/15, 17/9, [32/17]		\|['''15/8'''], 28/15, 17/9, [32/17]
	\|A#, vB		\|A#, vB

Inthar: /* Scales */

2026-02-23T04:56:03Z

Scales

← Older revision		Revision as of 04:56, 23 February 2026
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	Arguably the main MOS scale in 11edo is the smitonic Scale 2-1-2-2-1-2-1. The smitonic scale is melodically rather similar in some regards to the 12edo diatonic scale. However, it is harmonically totally different due to the absence of harmonics 3 and 5 in 11edo.		Arguably the main MOS scale in 11edo is the smitonic Scale 2-1-2-2-1-2-1. The smitonic scale is melodically rather similar in some regards to the 12edo diatonic scale. However, it is harmonically totally different due to the absence of harmonics 3 and 5 in 11edo.

			There are others including the checkertonic scale 2-1-1-2-1-1-2-1, which can be viewed as a stretched augmented/tcherepnin scale.

	~~There are others including the Checkertonic scale 2-1-1-2-1-1-2-1.~~		{{Navbox EDO}}

	{{Cat\|Edos}}		{{Cat\|Edos}}

Lériendil: /* Compositional theory */

2026-02-19T00:48:40Z

Compositional theory

← Older revision		Revision as of 00:48, 19 February 2026
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	One approach to harmony in 11edo is to take advantage of the dual fifths. 11edo has two fifth-like intervals roughly equidistant from 3/2, allowing for triadic-like harmony to function by changing the quality of the fifth while the third remains constant, the exact opposite of diatonic systems where the third changes quality and the fifth remains constant. If 3\11 (the nearminor third) is chosen as the third in question, these chords are present in the smitonic scale, for which 3\11 serves as the generator, and the chords that this begets are [0 3 6] and [0 3 7]\11. Notably, these can be inverted to [0 3 8], [0 5 8], and [0 4 8]\11, all of which are triads bounded by 8\11, which represents the consonance 5/3. The last of these also constitutes a stack of the very well-tuned interval 9/7.		One approach to harmony in 11edo is to take advantage of the dual fifths. 11edo has two fifth-like intervals roughly equidistant from 3/2, allowing for triadic-like harmony to function by changing the quality of the fifth while the third remains constant, the exact opposite of diatonic systems where the third changes quality and the fifth remains constant. If 3\11 (the nearminor third) is chosen as the third in question, these chords are present in the smitonic scale, for which 3\11 serves as the generator, and the chords that this begets are [0 3 6] and [0 3 7]\11. Notably, these can be inverted to [0 3 8], [0 5 8], and [0 4 8]\11, all of which are triads bounded by 8\11, which represents the consonance 5/3. The last of these also constitutes a stack of the very well-tuned interval 9/7.

			Another important bounding interval in 11edo is 7/4, represented by 9\11, and important chords bounded by it include [0 5 9] and [0 4 9]\11, which represent 8:11:14 and its complement.

	=== Scales ===		=== Scales ===

Lériendil: /* Chords */

2026-02-19T00:39:36Z

Chords

← Older revision		Revision as of 00:39, 19 February 2026
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	== Compositional theory ==		== Compositional theory ==
	=== Chords ===		=== Chords ===
	11edo has numerous xenharmonic sounding chords. An approach to ~~chords~~ in 11edo ~~is stacking the interval 9/7 for very accurate 14:18:23 chords.~~		11edo has numerous xenharmonic sounding chords.

	~~Another~~ is to take advantage of the dual fifths. 11edo has two fifth-like intervals roughly equidistant from 3/2, allowing for triadic-like harmony to function by changing the quality of the fifth while the third remains constant, the exact opposite of diatonic systems where the third changes quality and the fifth remains constant. ~~The~~ chords that this begets are [0 3 11] and [0 7 11].		One approach to harmony in 11edo is to take advantage of the dual fifths. 11edo has two fifth-like intervals roughly equidistant from 3/2, allowing for triadic-like harmony to function by changing the quality of the fifth while the third remains constant, the exact opposite of diatonic systems where the third changes quality and the fifth remains constant. If 3\11 (the nearminor third) is chosen as the third in question, these chords are present in the smitonic scale, for which 3\11 serves as the generator, and the chords that this begets are [0 3 6] and [0 3 7]\11. Notably, these can be inverted to [0 3 8], [0 5 8], and [0 4 8]\11, all of which are triads bounded by 8\11, which represents the consonance 5/3. The last of these also constitutes a stack of the very well-tuned interval 9/7.

	=== Scales ===		=== Scales ===

Lériendil: /* Intervals and notation */

2026-02-17T16:02:00Z

Intervals and notation

← Older revision		Revision as of 16:02, 17 February 2026
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	For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.		For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.

	Interval ~~qualities~~ are based on 22edo usage of the [[ADIN]] system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and [[7-limit]] qualities.		"Interval categories" are based on 22edo usage of the [[ADIN]] system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and [[7-limit]] qualities.

	JI approximations (within the 25-odd-limit) of steps in 11edo, as well as the aforementioned ways of notating 11edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.		JI approximations (within the 25-odd-limit) of steps in 11edo, as well as the aforementioned ways of notating 11edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.

Lériendil at 16:00, 17 February 2026

2026-02-17T16:00:34Z

← Older revision		Revision as of 16:00, 17 February 2026
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	'''11edo''', or 11 equal divisions of the octave (sometimes called '''11-TET''' or '''11-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/11) ~= 109.09 cents, 11 of which stack to the perfect octave 2/1.		'''11edo''', or 11 equal divisions of the octave (sometimes called '''11-TET''' or '''11-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/11) ~= 109.09 cents, 11 of which stack to the perfect octave 2/1.

	11edo lacks a diatonic ([[5L 2s]]) [[perfect fifth\|fifth]], or even an [[~~7L 2s\|~~armotonic]] or [[~~5L 3s\|~~oneirotonic]] fifth, with the 3rd harmonic nearly halfway between its steps. The [[5/4\|5th harmonic]], as well, lies right in between its steps. However, the errors cancel to allow 11edo representations of intervals such as [[9/8]], [[5/3]], and [[15/8]].		11edo lacks a diatonic ([[5L 2s]]) [[perfect fifth\|fifth]], or even an [[armotonic]] or [[oneirotonic]] fifth, with the 3rd harmonic nearly halfway between its steps. The [[5/4\|5th harmonic]], as well, lies right in between its steps. However, the errors cancel to allow 11edo representations of intervals such as [[9/8]], [[5/3]], and [[15/8]].

	11edo can also be considered to approximate the 7th, 11th, and 17th harmonics, with 9\11 approximating [[7/4]], 5\11 approximating [[11/8]], and 1\11 approximating [[17/16]]. In fact, [[22edo]]'s approximations to intervals of 7, 9, 11, 15, and 17 all come from 11edo. Given the logic of 22edo as an [[archy]] system, the same interval represents both 9/8 and [[8/7]].		11edo can also be considered to approximate the 7th, 11th, and 17th harmonics, with 9\11 approximating [[7/4]], 5\11 approximating [[11/8]], and 1\11 approximating [[17/16]]. In fact, [[22edo]]'s approximations to intervals of 7, 9, 11, 15, and 17 all come from 11edo. Given the logic of 22edo as an [[archy]] system, the same interval represents both 9/8 and [[8/7]].

	The result of all this is that while 11edo forces harmony to arrange in ways extremely alien to a perspective based on the [[3-limit\|3-]] or [[5-limit]], it still retains a breadth of approximation that allows for a complete tonal system to be built around 11edo's properties. Smitonic ([[4L 3s]]), generated by 3\11 (representing [[6/5]] and [[11/9]]), can serve as a useful basis scale for navigating 11edo. The simple [[JI]] interval approximated best by 11edo is [[9/7]], at 1.3{{c}} sharp, while 5/3 and 7/4 can be used as bounding consonances for chords.		The result of all this is that while 11edo forces harmony to arrange in ways extremely alien to a perspective based on the [[3-limit\|3-]] or [[5-limit]], it still retains a breadth of approximation that allows for a complete tonal system to be built around 11edo's properties. Smitonic (4L 3s), generated by 3\11 (representing [[6/5]] and [[11/9]]), can serve as a useful basis scale for navigating 11edo. The simple [[JI]] interval approximated best by 11edo is [[9/7]], at 1.3{{c}} sharp, while 5/3 and 7/4 can be used as bounding consonances for chords.

	== General theory ==		== General theory ==
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	In 22edo, as the diatonic MOS has [[hardness]] 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C.		In 22edo, as the diatonic MOS has [[hardness]] 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C.

	This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is smitonic (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.		This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is [[smitonic]] (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.

	For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.		For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.

Lériendil: /* Edostep interpretations */

2026-02-17T15:55:19Z

Edostep interpretations

← Older revision		Revision as of 15:55, 17 February 2026
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	=== Edostep interpretations ===		=== Edostep interpretations ===
	One step of 11edo can be interpreted in the 2.9.15.7.17 subgroup as ~~[[18/17]], 17/16, [[16/15]], and [[15/14]].~~		One step of 11edo can be interpreted in the 2.9.15.7.17 subgroup as:
	* [[18/17]] (the interval between 17/16 and 9/8, or [[17/14]] and 9/7)		* [[18/17]] (the interval between 17/16 and 9/8, or [[17/14]] and 9/7)
	* 17/16 (the octave-reduced 17th harmonic)		* 17/16 (the octave-reduced 17th harmonic)