Saddle chord

A saddle chord is a chord that represents a saddle point in the harmonic entropy surface, rather than a local minimum or maximum. Because saddle points only occur in two-dimensional or higher surfaces, a saddle chord cannot be a dyad (since the harmonic entropy graph for dyads is a one-dimensional curve). It must be a triad, tetrad or higher.

Chords at or near local minima sound "clean" and have a single primary approximation just intonation approximation. For example, the justly intoned major chord 4:5:6 is a local minimum, and its approximation in 12edo is close by.

In contrast, a chord at or near a local maximum sounds especially "dirty" and discordant. "Dirty" dyads include many quarter tone-based intervals. There are also dirty triads, for example, the quarter tone triad {0,1,2} in 24edo with each note a quarter tone apart.

For triads and higher, though, there are other possibilities, corresponding to monkey saddle and horse saddle-type points. These chords do not have a single just approximation but rather are a compromise between multiple ones, and as such their harmonic entropy is not a local minimum (at least not in all directions). They are intermediate between consonant and "dirty" in terms of sounds.

Todo: add examples