The physical world of sound was first written about in ancient Greece, most extensively by Pythagoras about 2,500 years ago. Pythagoras was obsessed with numbers. He was a member of a secret society that dealt with numbers and saw numbers everywhere. Among other things, he observed that when you hit two resonant objects, if one is exactly two times the length of the other, the relationship sound is extremely clean. The relationship, in fact, is what we today call an octave. This phenomenon continues geometrically. Pythagoras didn't quantify most of the geometry you were tortured with in high school after all. In other words, objects whose lengths relates to each other with power of two whole number ratios; twice, four times, eight times as long as the others and so on, they all sound like clear higher octaves when struck. The reason this is important is objects with much less geometrically clean ratios sound much less clean, more dissonant. Let me play you an example. Here is an octave. Now, listen to this non-octave. That second combination of two pitches sounds less clean and more physically dissonant. The sound waves don't fit into one another and what results is an interference pattern. By the way, we call that an octave just because when we count up the most common western scale, when we get to the eighth node, it sounds the same, so it's an eighth and that's an octave, and then the scale seems to start again. Anyway, this whole phenomenon is also related to what is called the harmonic series. The harmonic series describes the added vibrations that co-exist above the fundamental, which is the loudest pitch we hear. Think about it this way. With a jump rope, if you add some energy to its swinging, you can maybe get to waves in the same length, and there maybe three in that same distance of rope. Those are the higher harmonics. With a little more energy, you can get that rope to have multiple wavelengths. Harmonic overtones are just like that except, they actually happen simultaneously. In fact, there's almost no pitch that we hear that does not contain those other higher pitches in the harmonic series. They're almost always there. Guess what? The overtone that is usually the loudest is the first one on the harmonic series, the one that is the exactly half the wavelength, twice the frequencies. So the other node is already there, which means when you combine two pitches where one is twice the frequency as the other, it sounds very consonant. It's sort of a reaffirmation of something that's already there. So boy, is that physically consonant? Let me try to show you how the upper octave is already there and it just requires you to listen. Listen to this pitch carefully. See if you can hear the upper octave. It's actually there in this pitch. Now, I'm going to add that other note, that upper octave. You almost can't hear play it first, it fits in so very cleanly. Now, asking you to hear this as a little like asking you to notice that you're breathing air. It's something in the physical environment that is always there but easily ignored. It's something that once you notice it, it can drive you crazy, thinking about how you never notice it. But, oh my goodness, it's there all the time. Pythagoras noted that the sounds of other clean ratios, which can be expressed in low whole numbers. Remember, frequency in the length of the things being used to produce sound usually have a direct relationship given the same material. The longer the thing, the lower the frequency. So he noticed that in these things the 3:2 ratio sounds like what we today call a perfect fifth. That's a geometrically clean ratio, and ratios of 4:3 produce perfect fourths. We call those intervals perfect because they are simple ratios and less physically dissonant. It's exactly related to the fact that if you build with two blocks that are each half the size of one larger one, the two small ones will fit perfectly on the larger one. Whereas, if the ratios are slightly off the fit and the building will be a little clunky. It's, of course, much more complicated in that, mostly because math is not linear but exponential. The deeper you get into the physics of sound, the more interesting it becomes, and composers can spend entire careers thinking about ratios. In fact, if you want to enter into the truly gorgeous Alice in Wonderland landscapes of harmonics and pure ratio music, there's some great links in the resources. But beware, you can get lost in there. Here's just one example. A piece by Wolfgang Von Schwenitzs that uses complicated combinations of whole number intervals. Some maybe more consonant than others but all built from a simple geometrical way of hearing the world that blossoms into a much more complicated geometrical flower. At first, this might sound foreign. But after awhile, you can become drawn into this music of the spheres. Guess who coined the term Music of the spheres to draw a direct relationship between the celestial, orbits and music? That's right, Pythagoras. For him, music wasn't just part of nature, it was nature. You can sense echoes of that sentiment in many modern composers like Von Schwenitzs who use clean Pythagorean ratios in their music. Anyway, for our purposes right now, just remember; two sounds whose frequencies have low number ratios sound cleaner, and thus more easily consonant according to the physical world.
