7-odd-limit - Revision history

Overthink: remove WIP banner

2026-04-11T04:25:17Z

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	~~{{WIP}}~~
	{{Odd-limit navigation}}		{{Odd-limit navigation}}
	The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. Reduced to an octave, these are:		The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. Reduced to an octave, these are:

Overthink: /* Intervals of the 7-odd-limit */ fill in later sections

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Intervals of the 7-odd-limit: fill in later sections

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	=== 10/7 ===		=== 10/7 ===
			The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how 7/5 is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

	=== 12/7 ===		=== 12/7 ===
			The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of 7/6, and the twelfth complement of 7/4. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

			It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

	=== 7/4 ===		=== 7/4 ===
			The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

			Though often considered a minor seventh, it also has some qualities of a sixth. For example, the 7/5 interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being 12/7.

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	=== 12/7 ===		=== 12/7 ===
	=== 7/4 ===		=== 7/4 ===

	~~[[Category:Odd limits]]~~

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			{{Odd-limit navigation}}
	The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. Reduced to an octave, these are:		The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. Reduced to an octave, these are:

Overthink: /* 7/5 */ add content

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7/5: add content

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	=== 7/5 ===		=== 7/5 ===
			The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is 10/7, its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

			In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between a major third 5/4 and a minor seventh 7/4. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

			It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

	=== 10/7 ===		=== 10/7 ===
	=== 12/7 ===		=== 12/7 ===

Overthink: /* Approximation by edos */ add diagram

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Approximation by edos: add diagram

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	== Approximation by edos ==		== Approximation by edos ==
			[[File:7-odd-limit in edos.png\|thumb\|right\|A diagram showing the approximation of the 7-odd-limit by various edos.]]
	The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)\|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.		The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)\|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

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	== ~~Consistent EDOs~~ ==		== Approximation by edos ==
			The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)\|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.
	The first ~~EDO to be~~ consistent to the 7-odd-limit is 4edo, and the first ~~EDO~~ to ~~be distinctly consistent~~ to the 7-odd-limit is [[27edo]].

	== Intervals of the 7-odd-limit ==		== Intervals of the 7-odd-limit ==

Overthink: /* Triads dividing the perfect fourth */ move sentence; include complements

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Triads dividing the perfect fourth: move sentence; include complements

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	Harmonically, 8/7 can be seen as contrasting with 7/6, differing from it by 49/48 (35.7 cents). As such, we can build triads by stacking 7/6 and 8/7, such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but the septimal triads split the [[perfect fourth]], rather than splitting the [[perfect fifth]] like pental triads do. As such, it can be considered a form of [[chthonic harmony\|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.		Harmonically, 8/7 can be seen as contrasting with 7/6, differing from it by 49/48 (35.7 cents). As such, we can build triads by stacking 7/6 and 8/7, such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but the septimal triads split the [[perfect fourth]], rather than splitting the [[perfect fifth]] like pental triads do. As such, it can be considered a form of [[chthonic harmony\|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

	Here, 8/7 is a type of minor interval, and 7/6 is a type of major interval. This contrasts with diatonic, where 8/7 is a supermajor second, ~~and~~ 7/6 is a subminor third~~. Their octave complements can be classified accordingly~~, ~~with~~ 12/7 ~~being~~ a ~~minor interval~~, and 7/4 ~~being~~ a ~~major interval~~.		Here, 8/7 is a type of minor interval, and 7/6 is a type of major interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This contrasts with diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

	=== 7/6 ===		=== 7/6 ===

Overthink: /* 7/6 */ note 9/7 as supermajor third

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7/6: note 9/7 as supermajor third

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	=== 7/6 ===		=== 7/6 ===
	The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the perfect fifth, that being 1–7/6–3/2. ~~However, note that the~~ interval between 7/6 and 3/2 is [[9/7]], which ~~is not in~~ the 7~~-odd-limit~~, ~~but rather the~~ [[9-odd-limit]]. We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.		The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the perfect fifth, that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

	As described above in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with 8/7 in triads such as 1–7/6–4/3, with 7/6 major counterpart of 8/7.		As described above in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with 8/7 in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

	=== 7/5 ===		=== 7/5 ===

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-==Consistent EDOs==
+== Consistent EDOs ==
 The first EDO to be consistent to the 7-odd-limit is 4edo, and the first EDO to be distinctly consistent to the 7-odd-limit is [[27edo]].