Both pieces have 256 measures. The earlier piece had the pieces arranged in a 16x16 square, which provides plenty of room for wandering around the tuning space. The 4x4x4x4 arrangement of the new piece constrains things to be more orderly, to provide more structure for the ear to recognize. This piece is also a snapshot of the thermodynamic evolution at a somewhat cooler point relative to the phase transition, which should also provide more structure.
My fascination with phase transitions goes back to my sophomore year in college. Professor Stephen Schnatterly gave a wonderful demo of an inverted pendulum showing a transition and spontaneous symmetry breaking, as an analogy for a phase transition. That got me to look into Stanley's book Introduction to Phase Transitions and Critical Phenomena. In that book there are some images from a computer simulation of the Ising model. Nowadays, of course, computer simulations are not so exotic as they were in the early 1970s! Professor Dan Schroeder has a nice one on the web: Ising Model Simulation.
My interests in software and phase transitions led me to work under Professor Elliott Lieb for my first semester junior independent research. Lieb's idea was to look at the partition functions for lattice gasses. Lattice gasses are similar to the Ising model. Their partition functions are polynomials. If the zeroes of these polynomials approach the positive real axis as the size of the system increases, that would be a sign of a phase transition. Lieb pointed to a theorem in Marden's book Geometry of Polynomials: one can construct a set of matrices from the coefficients of a polynomial; if the determinants of these matrices all have, hmmm, some particular sign, then the zeroes of the polynomial will be in the negative half plane, i.e. will not be anywhere near the positive real line. If I could compute these determinants from the partition functions of larger and larger lattice gasses, and they all showed zeroes in the negative half plane, well, that'd be good evidence that lattice gasses don't have phase transitions! Ha, I do wonder how much of this am I remembering correctly!
So my research project was to compute determinants for a bunch of largish matrices. I had told Lieb that I was a computer programmer. Well, one with very limited skills, it turned out! The textbook formula for determinants has a daunting computational complexity, growing as the factorial of the size of the matrix. That's the formula I ended up using, which limited me to very small matrices. That was pretty much the end of my physics career!
Here's a curious later development, if anyone wants to pick up the ball. Some 25 years after that disastrous semester, I found myself once again in the land of large matrices. I don't remember the exact details, but we were computing reached states in finite state machines, using binary decision diagrams. Professor Edmund Clarke noticed a similarity to Gaussian elimination in sparse matrices. His observation led me to the theory of Tree Decomposition, a part of graph theory.
Gaussian elimination is how one should properly compute determinants. Gaussian elimination can transform a matrix to half-diagonal form, and then one simply multiplies the matrix elements along the diagonal. Gaussian elimination can still be a bit costly for large matrices. If the matrix can be kept sparse the cost can stay low. Tree decomposition is a way to see how the sparsity of a matrix can be preserved during Gaussian elimination. Tree width is the measure of this preservable sparsity.
So here is a grand research proposal: those matrices of Marden, whose determinants I was to compute - how does their tree width grow as the size of the lattice gas system grow? Ha, I am still trying to salvage my physics career, fifty years later!
Here is someone poking around in this general territory, as a starting point: Phase Transition of Tractability in Constraint Satisfaction and Bayesian Network Inference
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