Odd-limit

An odd-limit is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-odd-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

It may also be useful to restrict an odd-limit further to within a specific prime limit or subgroup, and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the 7-prime-limited 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the 2.3.5.11 subgroup 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a scale, is also known as a tonality diamond, and tonality diamonds generalize the idea of odd-limits.

Proper odd-limit is a stricter classification. The proper n-odd-limit consists of all intervals in the n-odd-limit that are in no lower odd-limits.

Distinction and monotonicity

Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the n-odd-limit is most commonly the nth square superparticular, S(n) = n²/(n² - 1) (though certain odd-limits can have a gap of size (n² + 1)/n² instead: notably 7-odd, with 50/49 between 7/5 and 10/7; or 21-odd, with 442/441 between 21/17 and 26/21). Therefore, for a tuning system such as an EDO to distinguish the n-odd-limit, it must be able to emancipate commas the size of S(n): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

Another important property having to do with representations of odd-limits is monotonicity: i.e. a tuning is monotone in the n-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as dyads, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like 10/9 to exhibit its "structural" role as (4/3)/(6/5) even if it is tuned sharp of 9/8 or flat of 11/10, for example). Odd-limit monotonicity, in the context of equal-step tunings, is also a necessary (but not sufficient) requirement for consistency in that odd-limit. If a tuning is monotonic in the n-odd-limit and also distinguishes all intervals within that set, we can say it is strictly monotone in the n-odd-limit.

Individual odd-limits

• 3-odd-limit

The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third.)

• 5-odd-limit

The 5-odd-limit expands the range to include imperfect consonances, which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the 5-limit major and minor thirds and sixths, approximated in 12edo. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

• 7-odd-limit
• 9-odd-limit

The 7- and 9-odd limit contain more "exotic" septimal consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

• 11-odd-limit

More info is on each individual odd-limit page.