Music and Hyperspace
I recently started reading the book Hyperspace (1994) by Michio Kaku (whom I later realized was the host of Visions of the Future on the Science Channel and who has appeared on various other such programs), in which he describes more clearly than I have ever seen the background and evolution of a unified theory of physics, from Newton to Riemann to Einstein and on. He begins by explaining in depth how one can imagine the effects of higher dimensions on those below them and describes how the observed laws of the universe fit into the hyperspace theory.
In painting a picture of higher spacial dimensionality, Kaku describes that one way we might perceive fifth-dimensional objects (that is, the fourth spacial dimension; not to be confused with ‘the fourth dimension’ which refers to time) is by observing the ’shadows’ they cast into our third dimension. For example, the shadow cast by a three-dimensional wire frame cube onto a 2-D plane creates the figure of two squares connected at the corners by straight lines. However, when the cube is rotated in three dimensions, its shadow performs transformations that a two-dimensional being (’flatlander‘) would deem impossible, but which would not confuse a three-dimensional being in the slightest!
Later, in chapter 5, “Quantum Heresy”, Kaku discusses the advent and impact of Quantum Theory on the established physical models of the universe. He shows how the rise of the Standard Model, while unequivocally empirically accurate, was in stark contrast to the elegant and beautiful theories of its forebears. For example,
Einstein’s equations, written out in their entirety, are only about an inch long and wouldn’t even fill up one line of this book. From this one line of equations, we can go beyond Newton’s laws and derive the warping of space, the Big Bang, and other astronomically important phenomena. However, just to write down the Standard Model in its entirety would require two-thirds of this page and would look like a blizzard of complex symbols.
After showing the theoretical ‘ugliness’ of the Standard Model, Kaku provides the following anecdote, asking the question, “Is Beauty Necessary?”:
I once attended a concert in Boston, where people were visibly moved by the power and intensity of Beethoven’s Ninth Symphony. After the concert, … I noticed some people staring in wonder at the sheet music left by the musicians.
To the untrained eye, I thought, the musical score of even the most moving musical piece must appear to be a raw mass of unintelligible squiggles, bearing more resemblance to a chaotic jumble of scratches than a beautiful work of art. However, to the ear of a trained musician, this mass of bars, clefs, keys, sharps, flats, and notes comes alive and resonates in the mind. … A sheet of music, therefore, is more than just the sum of its lines.
This juxtaposition of ideas reminded me at once of an idea I had a few years ago on the topic of music when trying to articulate the relationship between a composer and a performer. I imagined a great composer, focusing his creativity and technical skill on creating a work of music, notating it onto paper. I knew, however, that the work was only half complete. The work was now in the hands of the performer who then focuses his own creativity and skill taking the notes on the sheet and bringing them once again into an existence akin to the composer’s first conception. I then realized that this process could be described in the same terms as mathematical projection.
When the composer imagines melodies or plays them on an instrument, then notates onto paper their musical representation, he is in a sense projecting them onto the ‘plane‘ that contains all possible musical notations. When the performer receives the piece of music and begins to play, he is projecting the musical ’shadow’ back into the higher dimension; into an existence parallel to that in which the composer first drew his composition.
This interpretation appears to satisfy the relationships between the composer and his imagination, the composer and the sheet of music, the performer and the sheet of music, and the performer and his performance. When the composer creates music, he is pulling ‘functions‘ from a universe of musical laws – that he himself has chosen – and projecting them into the universe of all possible musical notations. Except for some potential additional context or direction, this simple sheet is all that a performer starts with. Then he applies his skill, experience, and creativity toward the task of hurling the simple marks on the page back into a universe parallel – but certainly not identical to – the universe from which the composer first drew the building blocks of his composition.
Some may consider the act of describing art with mathematics as diminishing its humanity, but I believe that music can combine the beauty of human creativity with the beauty of mathematics which is a wondrous thing indeed.