RE: [tuning] re: Trines and triads (Paul Erlich and Carl Lumma)

>Are we sure that 4:5:6 will not have higher tonalness than 3:4:5, when
>triadic harmonic entropy appears?

Yeah, you can sort of see that from the voronoi diagrams (of triads in the
first N harmonics).

>Further, since tonalness is defined as a measure of the "clarity" of the
>fundamental, should rootedness be a new concept, or should our tonalness
>metric include a fix for this phenomenon (the fundamental is more clearly
>resolved when the lowest note sounding is octave-equivalent to it)?

I'd keep it separate, as it may turn out to be easier mathematically to
operate with tonalnesses ("adding" or "multiplying" them in various ways)
before rootedness is taken into account. Rootedness, as Sethares suggests,
may be a result of the masking phenomenon (low frequencies obscure nearby
high ones, but not vice versa).

>Lastly, would removing all factors of 2 from a chord, except those
>which preserve the ordering of its identities (leaving the factors
>whose removal would invert the chord) before performing tonalness
>measurements be a workable fix?

Ugly.