24edo
24edo, or 24 equal divisions of the octave, is the equal tuning featuring steps of (1200/24) = 50 cents, 24 of which stack to the perfect octave 2/1. It is arguably one of the most common entry points into microtonality due to containing the familiar pitches of 12edo.
Theory
24edo is rather underappreciated due to its history of being used in atonal music.
JI approximation
Although 24edo inherits its approximations of the 5-limit from 12edo, it doesn't allow one to stack more than one instance of 5/4 without excessive error accumulation. Furthermore, despite having a more accurate 7/4 than 12edo in terms of absolute error, the 7th harmonic suffers from the same problem, as well as worse problems.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -2.0 | +13.7 | -18.8 | -1.3 | +9.5 | -5.0 | +2.5 | +21.7 | +20.4 | +5.0 |
| Relative (%) | 0.0 | -3.9 | +27.4 | -37.7 | -2.6 | +18.9 | -9.9 | +5.0 | +43.5 | +40.8 | +9.9 | |
| Steps
(reduced) |
24
(0) |
38
(14) |
56
(8) |
67
(19) |
83
(11) |
89
(17) |
98
(2) |
102
(6) |
109
(13) |
117
(21) |
119
(23) | |
| Quality | Inframinor | Minor | Neutral | Major | Ultramajor |
|---|---|---|---|---|---|
| Cents | 250 | 300 | 350 | 400 | 450 |
| Just interpretation | 15/13 | 19/16, 6/5 | 11/9 | 24/19, 5/4 | 13/10 |
Diatonic thirds are bolded.
Chords
Because it contains 12edo and 8edo as subsets, 24edo has the capacity for all the same types of chords as those edos. As if that weren't enough, careful use of a set of rules known as the dinner party rules helps to add more viable chords to the list- every chord must be comprised of a chain of friends in which each note is a "friend" to every other note, no note can have an "enemy", and, there must not be any crowding except in tension chords.
Examples of friends in this system are a major third, a minor third, a neutral third, an inframinor third/ultramajor second, a paramajor fourth (~11/8), a paraminor fifth (~16/11) and, of course, the perfect fourth and perfect fifth. Examples of enemies are an ultraprime/inframinor second, and an infraoctave/ultramajor seventh. The most notable "frenemies"- that is, intervals that act as both "friends" and "enemies" at the same time- are a tritone, as well as a minor second, a major seventh, an ultramajor third/paraminor fourth (~13/10) and an inframinor sixth/paramajor fifth (~20/13). Crowding in this system is caused by intervals smaller than or equal to a major second relative to the unison or octave.
