7-odd-limit

 Odd-limits:

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• 5-odd-limit
• 7-odd-limit
• 9-odd-limit
• 11-odd-limit
• 13-odd-limit
• 15-odd-limit
• 17-odd-limit
• 19-odd-limit
• 21-odd-limit
• 23-odd-limit
• 25-odd-limit
• 27-odd-limit

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:

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

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 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 3/2, and it is mapped as such in the Slendric temperament.

Triads dividing the perfect fourth

Main article: Chthonic 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 "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. 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

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 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 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

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.