5-odd-limit

The 5-odd-limit is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered consonances in standard Western music theory, and include as a subset the intervals of the 3-limit, which are the perfect consonances. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino.

Table of 5-odd-limit intervals

Reduced to an octave, the intervals of the 5-odd-limit are:

Interval	Cents	Name	Type
1/1	0.0	Unison	Equivalence
6/5	315.6	Classical minor 3rd	Imperfect consonance
5/4	386.4	Classical major 3rd	Imperfect consonance
4/3	498.0	Perfect 4th	Perfect consonance
3/2	702.0	Perfect 5th	Perfect consonance
8/5	813.6	Classical minor 6th	Imperfect consonance
5/3	884.4	Classical major 6th	Imperfect consonance
2/1	1200.0	Octave	Equivalence

Approximation by edos

The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, Dicot, does not lead to categorical conflicts the same way 3edo's Father does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.

The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. The second edo to do so, 19edo, is a Meantone tuning like 12edo though more accurate; the arithmetic of 5-limit intervals may lead to non-12edo results, such as five major thirds stacking to a fifth. 22edo, a non-Meantone tuning, has the opposite tuning tendencies to 12edo.

Intervals of the 5-odd-limit

Perfect consonances

Perfect fourth (4/3)

Main article: Perfect fourth

The perfect fourth is a perfect consonance and the bounding interval for chthonic harmony. It also exists between the fifth over the root and the root an octave up. It is the dark generator of the diatonic scale; stacking it produces the Locrian mode.

In certain triadic musical traditions that use 4:5:6 as a consonant chord, the perfect fourth over the root can be considered dissonant, as it resolves downwards to the major third.

Perfect fifth (3/2)

Main article: Perfect fifth

The perfect fifth is an unambiguous perfect consonance. It appears in musical systems worldwide, and can be easily tuned by ear. This is the reason behind the prevalence of the Pythagorean system of tuning.

In the context of 5-limit consonances, the perfect fifth serves as the bounding interval of the triads 10:12:15 (minor) and 4:5:6 (major), which utilize the other 5-odd-limit consonances 5/4 and 6/5.

Imperfect consonances

Classical major third (5/4)

The major third 5/4 serves primarily as a component in tertian chords like 4:5:6. It is the most consonant "third" interval. Building a scale by stacking 4:5:6 triads produces the Zarlino diatonic major scale. The 4:5:6 triad is well-represented in 15edo (which has a stretched triad), 19edo, 22edo, 31edo, 34edo, 41edo, 46edo, and 53edo.

5/4 is also the octave-reduced generator of the prime 5 axis in lattice-based just intonation.

Classical minor third (6/5)

The classical minor third 6/5 is the fifth complement of 5/4. The distinction between the two leads to the paradigm of major vs. minor in interval classification and in triadic harmony; it is why the "third" category of intervals exists at all, and additionally why thirds are often considered the "default" example of interval qualities.

One quality of 6/5 worth noting is that chords with 6/5 as a lower interval are, as a rule, not "rooted" (in that their root note is not a power of 2 in the harmonic series). The significance of this is debated by xenharmonic theorists; Lamplight uses it as a model for the different "feels" of the chords 4:5:6 and 10:12:15.

Classical major sixth (5/3)

The classical major sixth is the octave complement of 6/5. It is according to some the next most consonant interval within the octave after 4/3; Leriendil sees it as an important target interval on the level of 4/3 and it is also the bounding interval of the chord 3:4:5, which may be seen as an inversion of 4:5:6 or as the primary focus of 5-limit "/3" harmony (such as in Kleismic).

Classical minor sixth (8/5)

The classical minor sixth is, while consonant on its own, unusually dissonant for a 5-odd-limit consonance in certain contexts; the result is likely a combination of factors. First is its complex ratio - it is the only 5-odd-limit interval in the octave that uses 8 in the numerator. Second is its proximity to the golden ratio, which serves as a distinctly dissonant target (similar to the semioctave's influence on 7/5). Third is its proximity to 3/2, which produces a 'zone' of dissonance around it. Also of relevance to the discussion is the 12edo augmented triad, which contains a note tuned the same way as the classical minor sixth (and which may be voiced as it in JI depending on interpretation) yet is considered a dissonant chord.

This and the major sixth mainly show up in chords in Western harmony as the bounding intervals of triads in certain inversions.