Many years ago in college, I engaged in a long trail of mathematics courses. Gradually I accepted the notion — mathematics can be used to express ideas in the Arts. This notion included music. You may recall I have suggested that music can be a nonverbal form of communication reaching past the human intellect directly into the soul. Nevertheless, I have not commented on the connection between this primal level of communication with the soul and math.
Consider this: the physical air surrounding us has a gentle ambient force or influence upon us at approximately 15 pounds per square inch. Our ear can measure minor disturbances in the ambient pressure of air. Our ears translate the disturbances as sound if they occur within the audible frequency range. Much of my previous blogs have focused on the philosophical and theistic nature of Art. This little patch of prose will focus on the physical aspects of Music. Let us begin…
Pipe organs in churches are a wonderful mover of air at very low frequencies. Watch a woofer (speaker) re-create the sound of a kick drum, notice how it vibrates, place your hand close to the speaker, notice the movement of air. You can feel the sound. You can feel the air moving. Place you hand over the air vent (if the cabinet has one), notice the air movement. The movement is sound, we feel it, and we talk about ‘hearing’ it. At the end of the day, all hearing is ‘feeling’ — the movement of the little bones and hairs in our inner ear. This movement of air is the beginning of communicating our emotions in word and feeling.
Let us traverse boundaries between physics and music. Getting the most out of your instrument depends upon your understanding of the basics of physics and sound — how do we control the movement of air. Let us explore the universals: pitch, octaves, volume, overtones, and timbre. Let us go where few musicians tread…
Pitch. If we measure vibrations in the air, we can calculate the speed (frequency) of movement and assign a name (the note, a verbal handle describing the event) to the vibration when it falls within the range of sensitivity our ears possess, and we can hear it. Our ears perceive sound from about 16Hz (low pitch) to about 16KHrs (high pitch). We feel higher frequency sound, but do not hear the sound
Octaves. The lowest (thickest) string of a guitar vibrates at about 82 KHz (when the string is set to scale). The music community has a generally accepted name for this frequency — ‘E’. The bass E string vibrates at about 41 KHz (notice the difference in frequency). If we look at the pitch of the bass E, one octave lower, we note it vibrates at ½ the frequency. If we increase the pitch of the guitar E-string to the next octave, then the vibration is approximately 164 KHz (twice the frequency). This suggests if we increase pitch by one octave, we double the frequency (increase in pitch). If we decrease the pitch by one octave, we cut the frequency in half. This suggests there are boundaries to low pitch sound (zero vibration). We are not far from this lower limit when we are pushing out a driving rhythm on the ‘open’ bass E string.
Volume. Pluck a guitar string. The vibration of the string occupies a defined space. Watch it closely with your eye, the string seems to get thicker (during a state of vibration). Pluck the string harder and the defined space of the vibration increases. This consumption of this space is very similar to a sine wave; we represent amplitude of a sine wave as the physical distance from the center (zero) of the sine wave to the outer extremity of the sine wave. In fact, if you watch very close, at the exact middle of the string, the vibration is very slight. The max amount of string variance occurs approximately at the 1/4 point of the total string length and the 3/4 point of the total string length. As the vibration (amplitude) increases, the volume increases. The frequency remains stable, but amplitude (energy level) is magnified (loudness).
Overtones. The vibration of the string produces overtones (harmonics) that exist above and below the pitch of the string. Overtones are the sounds we often describe as ‘warmth’ when verbal defining a sound. These harmonics continue into virtual infinity above or below the fundamental frequency, but become less and less loud as they increase in frequency. In a good acoustic guitar, we hear the harmonics present as a subtle reverb.
Timbre. By manipulating the production of overtones and harmonics, we can change how the instrument sounds (deep crisp lower tone; high clear tone; nasal voice; clear voice; etc). This changing of the voice for the instrument is changing the timbre. The selection of construction materials, changes the timbre of the instrument.
Oh my — too much technical stuff! Enough of my pattering on pitch, octaves, volume, overtones, timbre and other such gibberish. I once read in a 16th century book that the presence of God is much like air, God is all around us. We are never far from the influence of God. My point? Mathematics runs through the middle of music, they need each other. Using mathematics, we are able to express musical concepts in terms that are palatable to artists, we can define the nuances that separate songs one from another.