I'm now realizing this and I totally agree that hiding the octave transpositions might make things confusing. Maybe this could be clarified by having a first l

I'm now realizing this and I totally agree that hiding the octave transpositions might make things confusing. Maybe this could be clarified by having a first lattice where this adjustment is not made, and everyone can observe how frequencies rapidly become extremely low or high, maybe to the point of being inaudible. Then we can ask "how can this problem be solved?" and introduce the concept of pitch class lattices where everything is transposed into one octave. Regarding intervals, I've always seen the irony and should probably mention it. Yes, the classical concept of fifth is related to the 3-ratio, while thirds are related to the 5-ratio. I'd argue this falls under the "empty your cup" category, but yes, leaving it untold probably doesn't help with understanding.