RE: [tuning] Hello and a Question

Hi Keenan Pepper,

The main reason your metric doesn't "work" for me is that it assigns the
same distance for 15/8 as for 5/3. 5/3 is much more consonant than 15/8 and
thus should receive a shorter distance. The triangular lattice you see a lot
of on this list is more appropriate than the rectangular lattice you suggest
for this reason.

Partch diamonds and Wilson CPS scales are the most "compact" structures on
this triangular lattice and its higher-dimensional analogues.

Alternatively, if you are willing to allow octave specificity, you could
stick to the rectangular lattice but be sure to retain all powers of two,
not excluding them the way you did. Then 15/8 would come out as longer than
5/3 since the former involves a move of 3 steps along the 2-axis while the
latter involves a move of no steps along the 2-axis.

I'd also ask you to consider carefully whether a city-block metric (one is
restriced to move along the lines of the lattice, no "shortcuts" allowed) is
not in fact a more appropriate measure than the Euclidean metric you're
using.

Finally, I'd suggest that the log of n, rather than n, should be the length
of each rung along the n-axis. As evidence, I believe that ratios of 9 are
more dissonant than ratios of 7, etc.

If you agree with my suggestions and go the rectangular, octave-specific
route, you'll be using Tenney's Harmonic Distance function. If you prefer
octave-specificity, you'll want to look in the archives at the discussions
between Paul Hahn, Carl Lumma, and myself on triangular lattice metrics.

-Paul