7-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
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
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.
