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metrics: tenney, barlow, etc...

🔗Paul Fly <pfly@neuron.net>

6/2/2000 7:52:45 AM

hello, i've recently discovered and enjoyed carter scholz's piece
"lattice (3327)" -- off the compilation CD "hallways". since i
am interested in algorithmic music and tuning, i wanted to know
more about the math and algorithms behind this piece.
http://tesla.csuhayward.edu/history/Computer/composers/Scholz/Lattice_notes.htmlhas in-depth notes on the piece, but there are a few things i
don't understand -- partially because i don't know HMSL code,
partially because my math is weak, and partially because i
keep suspecting there are typos on this page.

specifically, i'm wondering if anyone can explain or give pointers
to explainations of the "barlow metric" and "indigestibility".
the scholz URL says that a prime has "indigestibility" =
ind(p) - (2 . (p _ 1)^2)/p
i'm unsure what the '.' and the '_' are in that formula though.
is the '_' is a typo for '-'?

the barlow metric (distance between two ratios x and y) is said to be
D(x,y) = 2 . ISUFC(i=1,n, |xi _ yi| . F(((pi _ 1)^2,pi))
this formula uses terms defined elsewhere in the URL, but has
HMSL code i don't understand (what is 'ISUFC'?), and also seems
to have potential typos ('.'? '_'?). there is also a code example
of the barlow metric used on a group of ratios (instead of just
two), but it's just as confusing to me. so i'm quite lost on
understanding the barlow metric...

another metric described that i have difficulty grasping is the
'harmonic metric'. this is "an original function that attempts to
find closeness of one pitch in the lattice to a presumed harmonic
series. the fundamental of the series is presumed to be the ratio
expressed by the minima of all dimensions". this is shown in HMSL
code as D(x) = ISUFC(i=0,n, wi . |xi _ min(xi)|)
and D(L) = ISUFC(j=0,m, D(xj))
but i am at a loss to understand; what is the 'minima of all
dimensions' in a lattice?

finally, the notes in this URL end with some unrealized ideas.
one is the idea that a lattice space could be realized as a
continuous rather than discrete space (allowing, i suppose,
irrational numbers in addition to rational ones). in a
continuous space you could move around harmonically by
'gliding' rather than using discrete steps in pitch.
i'm curious if anyone has tried this or has any thoughts on it.

thanks! pfly