A recent post in my husband’s blog (Steven Cornelius) describes his disciplined approach to ear training. At one point he mentions the overtone series. I can relate. This is currently at the center of my musical existence as my quartet continues to record Composer Ben Johnston‘s string quartet cycle. It also happens to be at the root of our Western harmonic language.
What is an overtone? They are beautifully synchronized pitches that are neatly folded inside almost every note we hear.
The pitch of a string is determined by the rate at which it vibrates. On the violin, for example, the D string vibrates in a single arc from one fixed end to the other at a rate of 293 cycles a second. That perceived pitch is the first partial or harmonic, aka the fundamental. However, the string is simultaneously vibrating in smaller divisions (halves, thirds etc.). Each of these segments creates its own pitch as well.
Overtones vibrate in direct proportional relationship with the fundamental. For example, each half vibrates twice as fast as the whole (2:1), which produces the first overtone, the octave. Three equal divisions of the string creates a vibration ratio of 3:2. From this you get the second overtone, the fifth (one octave higher). Each subsequent division produces another overtone, always higher in pitch than the previous one. Because the vibrations of the overtones are perfectly synchronized with the fundamental, there are no interference patterns in the sound waves. The intervals are pure; their sound is smooth and harmonious.
Steven embedded a chart of the overtone series in his post. Certain overtones are marked as being perceived as “out of tune.” The plain fact is, if you are using equal temperament (ET) as your litmus test, every single overtone will be out of tune, except those that produce octaves of the fundamental (2nd, 4th, 8th 16h overtones and so on). The ones marked as out of tune in the chart are just the most extreme examples. The 7th partial is about 31 cents lower than ET (about 1/6 of a step), the 11th is 53 cents higher (approximately a quarter tone) and the 13th is 27 cents higher than ET.
Ben Johnston has chosen to move outside the limitations of ET and inhabit this expanded pitch world of pure relationships – the world of just intonation. Imagine a chord with the 13th partial sounding in one of the voices, and then imagine modulating to that partial – building chords using the overtones it generates. Welcome to Ben’s world.
Right now Kepler is tackling Johnston’s String Quartet No. 7 – a piece that has, to our knowledge, never been played. Small wonder, as it has a vocabulary of over 1,200 pitches to the octave. I have been compiling a chart of the accidentals, and so far (first movement only) have found 85 distinct combinations of Ben’s various modifiers.
During yesterday’s rehearsal, we managed to pick through 5 (five) measures of music – figuring out the actual pitches, their functions, and listening to them. A few more days and we will be ready to bring the first movement to Ben. He will answer our questions, and we will begin the long road to the recording studio.
Wish us Bon Voyage…