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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, | The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. It is the smallest odd-limit containing intervals of the [[7-limit|7-prime-limit]], thus creating xenharmonic categories not found in traditional music theory. In a 7-prime-limit system, all the ratios of the 7- or [[9-odd-limit]] can be treated as consonances. | ||
== Table of 7-odd-limit intervals == | |||
Reduced to an octave, the intervals of the 7-odd-limit are: | |||
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== Intervals of the 7-odd-limit == | == Intervals of the 7-odd-limit == | ||
=== 8/7 === | === 8/7 === | ||
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the perfect fifth | The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the [[perfect fifth]], and it is mapped as such in the [[Slendric]] temperament. | ||
Here, | It can also be considered in an ambiguous category of "semifourths" between seconds and thirds. Here, it is a minor interval, with 7/6, its fourth complement, being the corresponding major interval. The 7/6 and 8/7 intervals can be stacked to make fourth-bound triads; see [[#Triads dividing the perfect fourth]]. | ||
=== 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. 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. | 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 | As described in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with [[#8/7|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 === | ||
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. | 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|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 | In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between [[5/4]], which is a major third, and [[#7/4|7/4]], which is a minor seventh. 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. | 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 === | ||
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. | The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how [[#7/5|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. | 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|7/6]], and the twelfth complement of [[#7/4|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. | 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. | ||
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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. | 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. | Though often considered a minor seventh, it also has some qualities of a sixth. For example, the [[#7/5|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|12/7]]. | ||
== Harmony in the 7-odd-limit == | |||
=== Tetradic harmony === | |||
The 7-odd-limit is where tetrads start to get more prevalent. For example, we can build the 1–5/4–3/2–7/4 "harmonic seventh chord" by adding 7/4 on top of a 1–5/4–3/2 major triad. The harmonic seventh chord sounds somewhat similar to the dominant seventh chord, except it is more consonant and resolved. It can also be called the "major tetrad", similarly to how 1–5/4–3/2 is called the major triad. | |||
The 5-limit minor triad 1–6/5–3/2 can be derived by reflecting every note about the midpoint of the root and the fifth. If we do the same for the harmonic seventh chord and reduce the steps to an octave, then we get 1–6/5–3/2–12/7 "subharmonic seventh chord" or "minor tetrad". | |||
=== Triads dividing the perfect fourth === | |||
{{Main|Chthonic harmony}} | |||
The 7/6 and 8/7 intervals can be seen as contrasting with each other, differing from each other by 49/48 (35.7 cents). We can build triads by stacking 7/6 and 8/7 that span the [[perfect fourth]], 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 these septimal triads split the perfect fourth, rather than splitting the [[perfect fifth]] like 5-limit 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, 7/6 is a type of major interval, and 8/7 is a type of minor interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This is different from diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh. | |||
Latest revision as of 04:59, 12 April 2026
The 7-odd-limit consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. It is the smallest odd-limit containing intervals of the 7-prime-limit, thus creating xenharmonic categories not found in traditional music theory. In a 7-prime-limit system, all the ratios of the 7- or 9-odd-limit can be treated as consonances.
Table of 7-odd-limit intervals
Reduced to an octave, the intervals of the 7-odd-limit are:
| Interval | Cents | Name |
|---|---|---|
| 1/1 | 0.0 | Unison |
| 8/7 | 231.2 | Septimal major 2nd |
| 7/6 | 266.9 | Septimal minor 3rd |
| 6/5 | 315.6 | Classical minor 3rd |
| 5/4 | 386.4 | Classical major 3rd |
| 4/3 | 498.0 | Perfect 4th |
| 7/5 | 582.5 | Lesser septimal tritone |
| 10/7 | 617.5 | Greater septimal tritone |
| 3/2 | 702.0 | Perfect 5th |
| 8/5 | 813.6 | Classical minor 6th |
| 5/3 | 884.4 | Classical major 6th |
| 12/7 | 933.1 | Septimal major 6th |
| 7/4 | 968.8 | Septimal minor 7th |
| 2/1 | 1200.0 | Octave |
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 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.
Intervals of the 7-odd-limit
8/7
The 8/7 interval can be considered the septimal major second, or supermajor second, by diatonic interval classification, in the sense that it is slightly wider than the 9/8 major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the perfect fifth, and it is mapped as such in the Slendric temperament.
It can also be considered in an ambiguous category of "semifourths" between seconds and thirds. Here, it is a minor interval, with 7/6, its fourth complement, being the corresponding major interval. The 7/6 and 8/7 intervals can be stacked to make fourth-bound triads; see #Triads dividing the perfect fourth.
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. 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 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
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 5/4, which is a major third, and 7/4, which is a minor seventh. 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
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
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
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.
Harmony in the 7-odd-limit
Tetradic harmony
The 7-odd-limit is where tetrads start to get more prevalent. For example, we can build the 1–5/4–3/2–7/4 "harmonic seventh chord" by adding 7/4 on top of a 1–5/4–3/2 major triad. The harmonic seventh chord sounds somewhat similar to the dominant seventh chord, except it is more consonant and resolved. It can also be called the "major tetrad", similarly to how 1–5/4–3/2 is called the major triad.
The 5-limit minor triad 1–6/5–3/2 can be derived by reflecting every note about the midpoint of the root and the fifth. If we do the same for the harmonic seventh chord and reduce the steps to an octave, then we get 1–6/5–3/2–12/7 "subharmonic seventh chord" or "minor tetrad".
Triads dividing the perfect fourth
The 7/6 and 8/7 intervals can be seen as contrasting with each other, differing from each other by 49/48 (35.7 cents). We can build triads by stacking 7/6 and 8/7 that span the perfect fourth, 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 these septimal triads split the perfect fourth, rather than splitting the perfect fifth like 5-limit triads do. As such, it can be considered a form of "semiquartal" or "chthonic" harmony, which is one approach to septimal harmony.
Here, 7/6 is a type of major interval, and 8/7 is a type of minor interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This is different from diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.
