The good, the bad, and the ugly all exist in music. One way they reveal themselves to us is how combined musical pitches seem to either get along with each other or want to fight. Today we take a quick look at how our ears make these judgments based on what at first glance look like cold, uncaring numbers.
When we hear music, what’s happening in the physical world? Well, sound waves are happening. A sound wave is a variation in air pressure. If you could see them, they might look a little like these:
See how more short wavelength waves (high frequency) can fit into the same space that fewer long wavelength waves (low frequency) can fit into? You hear the top figure as a higher note and the bottom figure as a lower note. So frequency determines pitch.
We describe a musical note’s frequency by how many of those waves occur per second (Hertz). Humans can hear between 20 and 20,000 Hertz (Hz). A really low tuba note is near the very bottom of this range; a really high violin note is around 2600 Hz (or even 5200 Hz using harmonics). To get near 20,000, the sound is an unmusical hearing test type whine.
When we listen to music, though, we aren’t just listening to a single note. We listen to different notes played simultaneously. Some combinations of notes sound “good” or consonant, and some combinations sound “bad” or dissonant. Why? How is sweet different than sour? When is some “sour” a good thing? We can wade in to the answer from three directions: “just intonation,” “good” versus “bad” intervals, and “out-of-tuneness.”
Trombone Envy: Just Intonation and the Hopeless Dreams of Keyboard Instruments
Just why is it that a trombone section, by listening to each other and “playing to the pocket,” can create perfect chords pianos and organs can only dream of? Nature, and what our ears perceive, reveals itself through numbers. Let’s take an example of three notes that sound good together: B flat, D natural, and F natural — the three members of a B flat major chord. If we make the B flat the one below middle C on the piano, and play the chord, let’s say, in a trombone section, the Hertz measurements of a perfect major chord (“just intonation”) will be: 233.08, 291.35, and 349.62. A full, resonant, precisely in tune major chord, like this one, has a numerical (Hertz number) ratio of 5:4 between the D and the B flat (the major third) , and of 3:2 between the F and the B flat (the perfect fifth). WOW! Exact ratios of 5:4 and 3:2 (small integers) are pleasing to the ear! And it’s no coincidence that these ratios correspond exactly to a note’s position in its harmonic series (mentioned in my last article).
If we play the EXACT same chord on the piano, B flat below middle C and the D natural and F natural above that (assuming the piano has been tuned properly), the Hertz numbers come out like this: 233.08, 293.66, and 349.23. WHOA! That’s not a perfect major chord! Sounds okay, but that’s not 5:4 and 3:2! The fifth is pretty close, but the major third is way high! Poor keyboard. For practical reasons, a piano has to be tuned so that every octave of every note is perfectly in tune (equal temperament). In that arrangement, you can’t have every chord perfectly tuned.
When a trombone (or any non-fixed pitch instrument’s) section “plays to the pocket,” the players listen to each other, adjust pitch, and recognize when that special round full sound characteristic of a perfectly tuned major chord happens. In particular, the player with the third plays it a little lower, for the reasons described above.
Bad Interval! Bad!
Some intervals, the musical distance from one note to another (like the sweet major third and perfect fifth we discussed), sound sour, not sweet, like they just don’t want to be there and need to move on. Our ears hear that restlessness, and composers use that feeling to lead one chord to the next or spice up an otherwise bland chord. Let’s pick two and figure out why they sound the way they do.
If we pair our middle B flat with the E natural above it, we get a tritone, just a half-step below the perfect fifth. Stings the ear a little. If we look back at those easy-on-the-ear small integer ratios of the major third (5:4) and perfect fifth (3:2), how does the tritone compare? It’s 64:45! Those are big integers, and we hear that as dissonance. Another interval our ears consider dissonant is the minor second or half-step, which is as close as two notes can be in western music, like the A natural below our middle B flat. That ratio is 16:15, also pretty big numbers. We want these intervals to resolve; composers know that and use the tension to their advantage.
SKREEEEEEE!!! Tune That Up!
Why do instruments played “out of tune” with each other sound like fingernails on a chalkboard? When two notes that are supposed to be the same, but aren’t quite there, are played simultaneously, you can actually hear a pulse trying to drive you mad. The further out of tune the notes are, the faster the pulse (wawawawawa). As you bring the notes into tune, the pulse slows (waaaaaaawaaaaawaaaaaa), and, hopefully, disappears. What’s going on? I fear we may have waded too far into the weeds here, and there is actually a mathematical formula in analytic geometry that describes this phenomenon. In short, the turbulence caused when the slightly out of tune sound waves are completely out of phase creates its own wave, and that new wave is the pulse you hear. There is an easy formula, however, that tells us how often this happens per second and piano tuners use it every day. If you can count the number of beats per second, that’s how many Hertz the two instruments are out of tune by! For instance, if trombone player 1 is playing B flat at 233 Hz and player 2 is playing at 236 Hz, you will hear three wa’s (wawawa) every second. As player 2 pulls out the tuning slide, the wa’s go away.
The good, the bad, the sweet, the sour: all necessary to music to give us the rich palette of sound we love. But let’s leave those ugly wa’s out of it!
By Tim Walker (Trombone)
Walnut Creek Concert Band, Lafayette Studio Big Band, Pacific Brass Quintet, Diablo Symphony Orchestra