The Space Between

As many may already be aware, Pythagoras spent much of his time exploring the roots of harmonic theory in hopes of creating what he would call a Music of the Spheres, or a system of musical ratios based on rational numbers in order to express the perfection and plan of the cosmos (a term he coined which gave rise to "Universe").  As Nichomachus relays the story, the key was revealed to Pythagoras as he was deep in thought and strolled past a blacksmith shop where he heard these perfect ratios being rung out with hammers upon their anvils.  Pythagoras apparently barged into the shop, seized the hammers and proceeded to suss out the meaning behind what he heard.  This led to his discovery of string length and tension being of equal proportion to tone and ratio of tone.  

Pythagoras proceeded to develop a tuning system based upon these ratios consisting of two tetrachords adding together to make the octave.  E F G A  & B C D E', each consisting of a perfect fourth and separated by a sesquioctave (semitone).  Nichomachus cleverly sidesteps the issue in his accounts of these tunings and for good reason:  Turns out, Plato had (centuries before Nichomachus)  developed a 4 octave tuning system based solely on mathematics and the ratios therein.  Again this system created the diatonic scale (dia - two, tonic - tone) separated by a semitone.  Problem was, this semitone turns out to be an irrational number!  The square root of 2 when simplified. Pythagoras and his followers did not like irrational numbers; they found them troublesome and poor examples of the perfection of the cosmos.  So the issue was skirted.

Pythagoras had encountered this number before, however; in his theorem of right triangles where the sum of the squares of the two short sides of a right triangle equal the square of the hypotenuse.  The theorem works beautifully when the sides are 3, 4  for instance whereupon the hypotenuse will come to the square root of 25 or 5.  What if, though, both sides are 1?  We now end up with the square root of 2.  

Let us jump ahead a few centuries to the discovery of the golden mean.  The golden mean is found in all things that we as human beings find proportionately perfect: architecture, paintings, the shell of a nautilus and the proportions of the body.  The golden mean is based on a very mysterious number: the square root of 2.  

My query then; I am forced to wonder, did Pythagoras and his followers accidently stumble upon the very number around which all perfect proportion rests?  How odd that this mysterious "space between" the tetrachords turned out to be the very number that allowed these notes to swirl infinitely through space and create perfect beauty.  

Welcome.

Welcome, friends, to my blog.  Though at the moment there is nothing here (besides this disclaimer), it will be a spot for ruminations on music, cosmology, art and ecology and other beauty that crosses my path.