Talk:385/384

"deep holes"

In addition to equating 77/64 and 6/5, tempering out the keenanisma equates 48/35 with 11/8, 35/24 with 16/11, and 12/11 with 35/32, which are 7-limit intervals of low complexity, lying across from 1/1 in the hexanies 8/7–6/5–48/35–8/5–12/7–2 and 7/6–5/4–35/24–5/3–7/4–2. Hence keenanismic tempering allows the hexany to be viewed as containing some 11-limit harmony. The hexany is a fundamental construct in the 3D lattice of 7-limit pitch classes, the "deep holes" of the lattice as opposed to the "holes" represented by major and minor tetrads, and in terms of the cubic lattice of 7-limit tetrads, the otonal tetrad with root 11 (or 11/8) is represented by [-2 0 0]: 1–6/5–48/35–12/7–2. In terms of 7-limit chord relationships, this complexity is as low as possible for an 11-limit projection comma, equaling the [0 1 -1] of 56/55 and less than the other alternatives. Since keenanismic temperament is also quite accurate, this singles it out as being of special interest.

Can anyone explain what this means, practically? – Sintel🎏 (talk) 23:19, 27 April 2025 (UTC)

Granted, it looks like some of the statements in the opening part of the paragraph could use some clarification, especially to make sure that 11-limit intervals are not called 7-limit intervals or vice versa, but the gist of it- at least to my understanding- is that when the keenanisma is tempered out, 11-limit intervals arise in what would otherwise be fairly simple 7-limit harmonies, and on top of that, keenanismic temperament is actually quite accurate- though I would add the caveat that the accuracy bit is only really true for tuning systems in which the keenanisma is less than half the size of an EDO-step. Hope that helps. --Aura (talk) 00:20, 28 April 2025 (UTC)

I got roughly as far. I was hoping for some intuition on the whole lattice thing. Also, what are "tetrads" in this context. It can't be just any random collection of four 7-limit intervals. – Sintel🎏 (talk) 00:48, 28 April 2025 (UTC)

Ah. There's much about lattices that I don't know or understand- at least in this context. That said, from the looks of things a tetrad is a type of four-note chord. --Aura (talk) 00:59, 28 April 2025 (UTC)