In Wednesdayâ€™s class we began by pairing up and experimenting with long ropes in order to visualize the vibration of a string. One person stood holding their end in front of them while the opposite person swung the rope at different speeds. We attempted to create, at first, one broad swing of the rope (like you might see in a game of jump-rope). Then we doubled the speed so that the rope was divided into two equal parts, each rotating conversely (while one side swung upwards, the other rotated downwards). This increase in speed was continued until it wasn’t possible to divide the rope into any smaller sections (usually occuring around five divisions of the rope). Each dividing point between rotating sections is considered a ‘node’, or a place where the vibration is zero.
We then gathered into a circle in the classroom and used a monochord (an instrument consisting of a single string) to discover the specific ratios that create each interval above the tonic pitch. We began by splitting the chord in half (done by lightly touching in the center of the vibrating string) so that each section of the string was vibrating at twice its original speed. This is the same as what we had just experienced with the rope when we doubled our initial speed in order to create two vibrating sections. This time with the string of the monochord, an octave occured above the original pitch (shown by the ratio 2:1, where the higher pitch is vibrating two times for each one vibration in the lower note). We continued to use this same method to achieve the 5th (ratio of 3:2), the 4th (4:3), and so on through each of the twelve intervals. We discussed that frequency ratios always come in pairs that add up to an octave. For instance, the ratio 3:2 will be paired with the ratio 4:3 (a 5th plus a 4th equaling an octave).
The class reminded me of Stuart Isacoffâ€™s book â€œTemperamentâ€ which addresses the history, problems, and evolution of tempering the Western scale. After the class, I went back and read the section concerning Pythagoras and his original discovery of the geometry of music. Pythagoras, who invented the monochord, stated that â€œmusicâ€™s rules are simply the geometry governing things in motion: not only vibrating strings but also celestial bodies and the human soul.â€ Pythagoras believed that the most pleasing of harmonies arose from the simplest of proportions and that complexity would insight chaos. What is fascinating about this is that behind his discoveries of pure musical geometry there lies a forbidden and volatile darkness. He found that pure octaves and fifths, according to his ratios, are incommensurate (also referred to in Greek as â€˜alogonâ€™ meaning â€˜the unutterableâ€™). Fifths will never complete a perfect circle (as suggested by the widely accepted circle-of-fifths), but will reach toward infinity in an unending spiral. This essentially boils down to the fact that octaves are based upon multiples of 2 (2:1) while fifths are based upon multiples of 3 (3:2). In this case, no multiple of 2 will ever meet a multiple of 3. If one were to compare the pitch achieved by an octave and that achieved from the completion of a circle of fifths, they would be very similar yet â€œout of tuneâ€. This spiraling phenomenon hints at a more complex mathematic sequence, that of the golden ratio. Even so, these simple ratios were believed to be an expression of the divine. It is easy to find similar ratios present within nature. Saint Augustine, in fact, believed that churches and cathedrals were to be more than just shrines, and instructed that proper proportions were to be used in their construction. Thus the heights, lengths, and depths of the structures formed the proportions of Pythagorasâ€™s â€œcelestial harmoniesâ€ (1:1, 2:1, 2:3, and 3:4).
So what difference does this make to us, as musicians and as people? What effect does this really have on our performance? I think it is crucial to understand the fundamentals of the creation of sound, of pitch, especially when such things are taken for granted everyday. I remember the feeling I had when I first discovered the ratios involved in music. Once I got past the initial migraine acquired from my first lecture on equal temperament, I began to look a bit into proportions. It made perfect sense (and also supplied an interesting and practical perspective to my high school math classes). This is the real foundation of what I do every day, of each note I play. It is a fundamental that comes before technique, before fingerings and musicality. In a sense it is the DNA of music (more specifically of pitch). Yet as crucial as these fundamentals are, an understanding of them is not essential for the enjoyment of music. Recently, Warren mentioned a workshop that he was conducting years ago. During the course of the class, he plucked two notes on a string, the second a fifth higher than the first. Soon after, a young boy came running into the room exclaiming “What was that beautiful music?!”. Like the young boy, a single, simple fifth can produce a level of joy bordering on ecstasy. Warren also noted that infants are particularly drawn to simple intervals. This has been quite a meal for my thoughts (even just thinking back to our class sends my head spinning!). Every time I try to find a solution to these musical systems I find that I develop more and more questions. It is truely amazing how much chaos lies within order!