The Harmonic Series
The reason Just intonation sounds so beautiful and pure, is that there is a phenomenon of nature that occurs in all vibrating bodies. When a string vibrates, it sounds not just it's fundamental sound, but also the notes that are at precise integer divisions of the string. This can easily be heard by pounding a low C on a grand piano and listening into the sound. What you are hearing are the overtones, the notes that sound along with the fundamental in any vibrating system. The same is true of wind instruments. A brass instrument, for example, can sound the tone that is the full length of its tubing, but it also sounds the notes of the overtone series. Brass players take advante of this fact and can sound the different notes of the series precisely through adjusting their embouchure and breath. This is analogous to the way a guitar player can lightly touch a string to produce harmonics, but in the case of brass instruments, the notes are actually quite strong.
Here is the harmonic series, based on the low tone of C. Stare at this for awhile—I'll be expecting you to know this intimately and to be able to write the series above any note.
What you will instantly recognize here is that this series contains exactly the intervals and ratios that we've been working with all along:
- The second degree of the series is an octave above the first (2:1)
- The third degree of the series is an octave and a fifth above the fundamental (3:1) and also forms the interal of the perfect fifth with the higher octave (3:2).
- Degrees 4:3 form the perfect fourth.
- Degrees 5:4 form a major third, tuned to the same principles as Just intonation.
- Degrees 6:5 form a minor third, again tuned to the same Just proportions.
Thus far, we have stopped at the ratio of 6, because we've been working within a 5-limit system (the senarius). The series above shows a partial reason why.
When we reach the seventh harmonic, it is noticeably "out of tune" with its neighboring tones when it comes to creating pure sounding triads. I will provide a sound file soon that shows off the sound of the series better than my words can describe it. Similarly, the F# of the series is lower than the F# we would want to use in forming a pure major triad.
<<The harmonic series soundfile goes here>>
Nevertheless, it has been a foundational concept since the experimental, scientific discovery of this series in mid-18th Century Europe that our major triad, CEG, appears to be embedded into the very fabric of vibrating bodies themselves.
Much, much more will be said about this factlet in the weeks to come. For moment, know the series, understand it to be of the nature of vibrating objects, and connect this series to the mathematical ratios that were known to musicians for a great many centuries before the scientific corroboration of the series a few short centuries ago.
Before we leave musical ratios, we must conclude with a discussion of the notion of the comma. For that, let's do one more page.