I've recently been studying tetrachords as well as equally divided octaves (EDOs). This morning, I was inspired to equally divide the just perfect fourth (4:3) into various sizes, starting with the tetrachord.
The basic step size in the equally divided tetrachord (EDT) works out to 166 cents. This yields a second that is -34 cents from that of the standard 12-EDO major, and -37.9 cents from the just major tone (9:8), but only -1.5 cents from the 29-EDO neutral second. Likewise, the EDT also yields a third that is +32 cents from the 12-EDO minor, and +16.4 from the 5-limit minor (6:5) or +37.9 from the Pythagorean minor (32:27), but only +1 cent from the 29-EDO neutral third. As the perfect fourth of 29-EDO is only -1.4 cents from just (4:3), clearly a subset of 29-EDO makes an excellent approximation of the EDT. 36-EDO offers an even better approximation, and if its 500-cent perfect fourth is used as the basis instead, the EDT becomes an exact subset of 36-EDO.
Regardless of whether the EDT is played straight or tempered to an EDO, it features what can best be described as a neutral second that leans toward major and a neutral third that leans toward minor. Adding a disjunct EDT from the just perfect fifth (3:2) to round out an octave (additional pitches at 702, 868, 1034, and 1200 cents), the resulting pattern of the interval "leanings" (M2, m3, M6, m7) suggests the Dorian mode. Alternatively, adding a conjunct EDT from the fourth (additional pitches at 664, 830, 996, and 1200 cents), the fifth degree turns out very flat, -36 cents from 12-EDO and -37.9 from just (3:2), though still only +1.9 from the nearest degree of 29-EDO; the sixth is neutral leaning toward minor, +16.4 cents from just (8:5) and +2.4 cents from 29-EDO, while the seventh is the Pythagorean minor (16:9), +3 cents from 29-EDO.
This study has heightened my interest in 29-EDO, and I plan to explore it in greater detail. Also, of course, the perfect fourth can be divided equally into other step sizes as well; I've done the basic calculations for several of these, and a few look very interesting!
| CENTS | ROLE |
| 0 | Hypate (Tonic) |
| 166 | Parhypate (Second) |
| 332 | Lichanos (Third) |
| 498 | Mese (Fourth) |
The basic step size in the equally divided tetrachord (EDT) works out to 166 cents. This yields a second that is -34 cents from that of the standard 12-EDO major, and -37.9 cents from the just major tone (9:8), but only -1.5 cents from the 29-EDO neutral second. Likewise, the EDT also yields a third that is +32 cents from the 12-EDO minor, and +16.4 from the 5-limit minor (6:5) or +37.9 from the Pythagorean minor (32:27), but only +1 cent from the 29-EDO neutral third. As the perfect fourth of 29-EDO is only -1.4 cents from just (4:3), clearly a subset of 29-EDO makes an excellent approximation of the EDT. 36-EDO offers an even better approximation, and if its 500-cent perfect fourth is used as the basis instead, the EDT becomes an exact subset of 36-EDO.
Regardless of whether the EDT is played straight or tempered to an EDO, it features what can best be described as a neutral second that leans toward major and a neutral third that leans toward minor. Adding a disjunct EDT from the just perfect fifth (3:2) to round out an octave (additional pitches at 702, 868, 1034, and 1200 cents), the resulting pattern of the interval "leanings" (M2, m3, M6, m7) suggests the Dorian mode. Alternatively, adding a conjunct EDT from the fourth (additional pitches at 664, 830, 996, and 1200 cents), the fifth degree turns out very flat, -36 cents from 12-EDO and -37.9 from just (3:2), though still only +1.9 from the nearest degree of 29-EDO; the sixth is neutral leaning toward minor, +16.4 cents from just (8:5) and +2.4 cents from 29-EDO, while the seventh is the Pythagorean minor (16:9), +3 cents from 29-EDO.
This study has heightened my interest in 29-EDO, and I plan to explore it in greater detail. Also, of course, the perfect fourth can be divided equally into other step sizes as well; I've done the basic calculations for several of these, and a few look very interesting!
