Posted by on Nov 1, 2013 in Off Topic | 0 comments

A Theory of Everything

Hey, shouldn’t everyone have one?

I’ve suggested many times on this blog that number is at the heart of all things. Here’s one article: Pythagoras’ Epiphany. I think the beauty of music, especially harmony, is similar to the beauty of math, but happening in real time. It slightly parts the veil, deepening the view of what is most basic and true about our universe. I believe that this connection with the deeper reality gives us a sensation of beauty. Here are a few more articles that go into detail, with examples and illustrations.

Saturn’s Rings

Beauty Is Truth

Prime Numbers and the Big Bang

If the Big Bang actually happened, then our universe blossomed outward from a point of infinite density, a singularity. What existed before that?

I assert that it was number. The Big Bang was a mathematical event.

What space did it occur in?

One of the gnarliest problems in physics is the issue of reconciling gravity with the rest of the basic forces. There are four: gravity, electromagnetism, and the strong and weak nuclear forces. Gravity is the odd one. It is far weaker than the others. It appears to operate in a continuum, a smooth field, while the others seem to be quantized — that is, they come in little discrete packets. Even space and time themselves may be quantized.

Math is like gravity, in that it works in a continuum. In the world of mathematics, there is no smallest anything. You can zoom in forever.

Here’s an example. If I measure the circumference and diameter of a circle very accurately, and divide one by the other, I always get the same number, pi, about 3.14. If I measure more and more accurately, I can get the value of pi to quite a few decimal places, but even with the best equipment imaginable, I will eventually run into the sizes of atoms themselves, and the calculation can’t get any finer.

But in the ideal world of math itself, pi goes out to as many digits as you please. There is no point where it runs into the grainy nature of reality.

It’s as though in the real world, there are pixels, and when you blow things up enough they start to show, but math itself has no such trouble. Things are infinitely divisible.

———————–

Einstein’s relativity and quantum mechanics are famously incompatible. Each describes the universe beautifully within our ability to measure so far. But it’s really hard to come up with a theory of reality that allows both to be true. Part of the problem is that relativity assumes a smooth continuum, and quantum mechanics assumes that on a tiny, tiny scale, everything happens in jumps, rather than smoothly. A theory of everything would have to explain how both relativity and quantum mechanics can be so true.

The deepest reality, the substrate, upon which our universe is based, is simply the world of number.

The universe we live in, with its stars and planets and galaxies and people, is a mathematical object, like the Mandelbrot Set or a cellular automaton, that grew from this substrate.

Here’s a tiny, tiny piece of the Mandelbrot Set. Click to enlarge for full glory! This is a mathematical object, at least as big as our universe, and if our universe is one too, then maybe the beauty of this object is related once again to the beauty of music, or of Keats’ Grecian Urn — it shows us, a little bit, the nature of Creation.

If the deep reality is a continuum, and the immediate reality of stars and planets sprang from this continuum, then maybe gravity is different from the other three forces because it is a feature of the deep reality. It is a manifestation of the shape of space-time. It is working in the substrate.

Relativity and gravity are happening in the basic reality, the continuum, the world of number.

Quantum mechanics, electromagnetism, light, nuclear forces, matter and energy are all happening in the particular reality that came about when the singularity happened.

Relativity and quantum mechanics can’t be reconciled because they actually operate in different realities — gravity in the basic reality, and the other forces in the immediate universe.

If this is true, it may offer insight into the nature of dark matter.

Dark matter has not been observed directly within our quantized universe. Its existence has been deduced, or conjectured, because galaxies move and rotate as though there is a lot of mass there that we cannot see. The idea is that dark matter interacts with “our” universe only through gravity, and not through the other forces. That is why we can’t see it, because seeing requires light.

What if dark matter is something that exists in the basic reality, rather than in our particular Big-Bang-generated one? The only link between the realities would be gravity.

There is no reason why our particular singularity should be the only one.

Perhaps what we call dark matter is just the gravitational shadow of other universes.

Here’s a scenario:

1. Ours is one of many universes, each one starting with a different set of “seed” values.
2. Ours is of course perfectly designed for us to exist, and the other universes are also “coincidentally” perfect for whatever exists in them.
3. The universes all exist in a space-time continuum, in which gravity is the only “force,” being actually a distortion of space-time as Einstein described.
4. The universes can attract each other through gravity, and so they tend to clump in the same places.

The ratio of dark matter to ordinary matter (about 5:1) may turn out to be an important number. Maybe “nearby” universes (those with similar seed values) attract each other more strongly than more “distant” ones (those with more different seed values), and the 5:1 ratio is the result of an infinite sum — the total pull of all those other universes, fading off into the “distance.”

Here’s a terrific article on dark matter from April 2013 if you’d like to explore further.

Posted by on Dec 11, 2012 in Just Intonation, Septimal Harmony | 0 comments

Prime Numbers and the Big Bang

Every prime number generates a new musical universe.

Prime numbers are numbers, greater than one, that can only be evenly divided by themselves and 1. All other numbers are composite — that is, they can be made by multiplying two or more primes.

1) Multiplying by 1 does nothing. 1 is a singularity, the universe before the big bang, the anvil upon which the music is forged.

2) Two starts off the explosion. Multiplying by two creates a universe of octaves, an endless, sterile line of equally spaced mile markers on the road to harmony.

Reminds me of the first chapter of Genesis, where everything is formless until the Creator starts differentiating stuff, day from night, water from sky, land from water, animals from plants, and people from animals. Start multiplying by two and before you know it you have a universe!

I’m also reminded of the current theory as to how our own universe came to be. Here’s a nice summary I found on a physics message board. It’s by Joel Novicio, an undergraduate physics student at the time.

The Big Bang singularity is a point of zero volume, but very high mass, which makes the density infinite. This singularity contained all of the matter and energy in the Universe. The initial moment of the cyclopean explosion very well remains a mystery — however, astronomers and physicists believe that after the tiniest fraction of a second, the strong nuclear force and the electromagnetic force separated, which probably caused the Universe to begin inflating. The Big Bang itself created space, time, and all of the matter and energy we know today.

OK, maybe I’m getting a little bit woo-woo here, but really I don’t think this is a trivial or accidental connection. The musical universe arises from the numbers. So does the physical one, at its deepest levels. I think that’s why we perceive music as beautiful.

I am stretching it now, but guess what is thought to have happened next after the splitting of the forces? Quarks! Quarks are the building blocks of protons and neutrons, almost all the matter we’re familiar with. And they come in threes.

3) Three makes it interesting. Keep multiplying and dividing by 3 and you can get an equivalent for every key on the keyboard, and many more. The notes never repeat, as you multiply and divide, so this universe is infinite as well.

This is the central spine of the lattice. The crucial notes 4 (perfect fourth, 4/3) and 5 (perfect fifth, 3/2) are multiples of 3. They are the backbone of music, and in my opinion, the fact that these are almost exactly in tune in equal temperament is a big reason why ET has been able to be so successful. If the 4 and the 5 were as far out of tune as the major third is, I don’t think ET could ever have been adopted.

Pythagoras based his musical scale entirely on 3 and 2. His followers expanded this, compounding it many times into what is now called Pythagorean tuning.

The first few notes generated by this tuning are beautiful. The 5 (x3) and 4 (÷3) are perfect consonances. The 2 (3×3) is really sweet. I personally like the Pythagorean sixth (3x3x3, 6+ on my map). But apparently the ear can’t follow compounds of 3 forever. By the time you get to the Pythagorean major third (3x3x3x3) you have a dissonant note. It’s on the central spine of the lattice, just off the border of my map, to the east of the 6+.

Here’s a 5/4 major third, with the tonic, and then in the context of a major chord.

just 3

Now here it is in Pythagorean tuning. It’s even sharper than the equal tempered version. Ouch!

pythagorean 3

The universe of threes is infinite, but still somewhat limited musically.

4) Four doesn’t add anything new, it’s just two octaves, every second mile marker.

5) Five, on the other hand, combines with three to create a vast and wondrous universe, the world of the lattice, and adds many more flavors of consonance, dissonance and beauty. The twelve tones I’ve just described, and virtually all of European classical music, can be found in this universe.

6) Six, like four, adds nothing fundamental. It’s 3×2, and generates only Pythagorean intervals.

7) Aha.

America doesn’t export much any more. Except culture. American music, and the movies, have spread worldwide.

Strange turn of events considering that 100 years ago, America was pretty raw. It imported much of its culture from Europe. But when it imported the music of Africa, and combined it with the music of Europe, blues and jazz and rock and roll were born, and the world’s music is still ringing like a bell. Go Johnny go!

In my opinion, the great advance in this music (harmonically, at least), is the incorporation of the prime number seven.

Next: Seven

Posted by on Nov 20, 2012 in The Notes | 2 comments

A Reciprocal Note: The Fourth

All the notes I’ve discussed so far are found above and to the right of the tonic, in the northeast quadrant of the map. These notes are generated by multiplication alone.

What about the notes that are generated by division? These are found to the left and down on the lattice.

The closer a note is to the tonic, the smaller the numbers are, and the easier it is for the ear to tell where it is. I’ll cover this much more in later posts, but I think the character of an individual note, its unique harmonic color, is largely determined by two signals it sends to the ear and mind:

1) How far away is home (the tonic)?

2) What direction is it?

And the closer the note is to home, the clearer the signal is.

The perfect fourth is the same distance from the center as the 5 is, but in the exact opposite direction: divide-by-3 instead of multiply-by-3. So it sends a signal of equal strength and opposite direction. How does this mirror-fifth sound?

I hear beauty, and tremendous tension. Something has to happen here, and soon — it feels like a pencil balancing on its point, unstable equilibrium.

Here are two of the most powerful phenomena in music: tension and resolution.

One resolution is right next door: the major third. It’s only a half step lower, and it is a point of stable equilibrium.

Aaaaahhhh.

The ratio of the perfect fourth is 1/3. This can be octave-reduced (octave-expanded) by moving it up two octaves, to 4/3.

The energy of the fourth, the division energy, has had a number of names. Harry Partch, a major composer and explorer of music in just intonation, called the quality of the right-and-up harmonies (the ones you get to by multiplication) otonality, from overtone. He called the energy of division-based harmony utonality, for undertone.

Once again I’m going with Mathieu on this one. In Harmonic Experience, he gives an excellent rationale for calling this energy reciprocal. I think he’s right. Each overtonal note has its mirror twin, and the twins are identical, just upside down from each other — reciprocals. The fourth is the reciprocal of the fifth.

Next: Mixed Messages

Posted by on Nov 17, 2012 in The Lattice | 0 comments

The Tonic Major Chord

The tonic is the center of the lattice. A drone note on the tonic establishes the center of that particular musical universe.

Adding a major third and a perfect fifth (5/4 and 3/2) further reinforces the center and starts to carve out some territory on the map.

This is the tonic major chord:

In my view, the tonic major helps the ear grab onto the center, by adding two notes that point directly at it. The ear has more information to work with.

The mind has amazing real time mathematical ability. Maybe a more accurate way to say this is that the mind has an amazing ability to quickly analyze and predict physical phenomena. The physical phenomena can be described by math. I don’t think the mind is working with arithmetic calculations at blinding speed, like a computer. It’s more of a massively parallel, holistic analog processor, that achieves a similar result.

Willie Mays used to catch fly balls with his back to the plate. Here’s a famous one:

Mays watches the ball start its flight, calculates the parabola it will follow (fine tuned by the conditions that day), and sets out at top speed for the spot, 400+ feet deep in center field, where he knows it’s going to land. He doesn’t (can’t!) look at the ball until it’s almost upon him. Marvelous.

So the ear hears a note, another one at 3x the frequency (remember octaves don’t count, 3/2 works like 3/1 in this regard), and another one at 5x. All three notes are direct signposts, pointing exactly at the tonic. Here we are, says the mind.

This may be why the equal-tempered major third gives me that slight queasy feeling. The tonic is the tonic, all right, but that equal-tempered third doesn’t point right at it! It’s close enough that the ear correctly identifies it, but it’s actually pointing at a note about 1% sharp of the tonic, and something sounds subtly off, like day-old sushi.

Here it is again: pure third, ET third, pure third. The middle note, the ET third, has a ratio of about 5.04/4.

JI3 vs ET3

Is it slight tonal vertigo? Where is home?

Next: Compound Notes

Posted by on Nov 5, 2012 in Background, Just Intonation | 0 comments

Beauty is Truth

It’s probably Keats’ most famous pair of lines:

I believe he’s right on the money. I think that when we experience beauty, it’s because we have seen a little deeper into the nature of things.

This seems especially true of mathematical beauty. I had a college friend who found math exquisitely beautiful. He bought a blackboard for his room, and stayed up until all hours, glorying in the work. Elegance, simplicity (but not too much!),  and beauty are important guidelines to the rightness of a solution or direction of research. A sense of beauty guides the scientist as well as the artist. I’m really familiar with this from my engineering career.

So there is the nugget of my own epiphany:

The beauty of music is the beauty of mathematics, perceived in real time.

We see this in its visual manifestations all the time. The curve of the cables of the Golden Gate Bridge, the pattern of seeds in the sunflower, the rings of Saturn — all clear manifestations of the way the universe works, that can be described by math, and that we find beautiful.

Music presents a pure, distilled form of this: beauty created by small, whole numbers and their relationships to each other.

Next: Notes and Intervals