Monday, April 25, 2011

The Equally Divided Tetrachord

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.

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!


  1. One can approximate this quite well in 11-limit
    JI by 11/10 x 11/10 x 400/363. Avicenna lists
    12/11 x 12/11 x 121/108,13/12 x 13/12...14/13 x 14/13, etc. and Al-Farabi gives 10/9 x 10/9 x 27/25, I've been unable to trace this virtually ET tetrachord further. It sounds quite nice, though not as good, IMO, as Ptolemy's 12/11 x 11/10 x 10/9.

  2. #477 in your catalog! ;-)

    I really wish "Divisions of the Tetrachord" wasn't out of print. I saw that Larry Polansky had scans of it online, and it was actually that and the Polychrome Triangular Graph ( ) which inspired me to work out what I posted above. Thank you for your work!

    I've been informed that this sort of thing is the basis of Porcupine temperament, so that's yet another tuning I'm adding to the list of what I need to explore further. ;-)