It’s risky for science to analyze the Arts because they are so — well, Artistic. But here’s a piece of the gigantic field of music where a bit of science brings us some understanding.
We can only eat an elephant one bite at a time, so let me lead you to the bite I have in mind, limiting ourselves to what’s called Western music — notes and chords like you might play on a piano, or a flute, or a guitar.
Frequencies, Overtones and the G String
If we pluck the G string on a guitar we can hear the musical tone it produces, which is technically called G3, three white keys below the middle C on a piano. We can also measure the frequency of vibration, which turns out to be about 196 vibrations per second, called Hertz and abbreviated Hz.
If you pull the guitar string gently with your finger, halfway between its ends, and let go, the entire string moves together in a smooth motion and you will hear a pretty pure G3 sound. However, if you use a pick, and especially if you pluck the string near one end, you will hear a crisp, more metallic sound. The tone is still G3, but it sounds “edgy” to us because the string is vibrating at several frequencies at once. The string is no longer moving smoothly back and forth, but rather it is rippling in some complicated shape. The actual sound of a plucked string is very complicated, but here’s an approximate description: in addition to 196 Hz, the string is making other sounds that are exact multiples of this frequency: 392 Hz (2 times), 588 Hz (3 times), and so forth. Our ear hears both the fundamental tone, the G3, but also the other notes, which are called overtones.
Now let’s move from the guitar, which needs to be tuned, to the piano, which we will hope is already tuned. If you play a white key and the adjacent black key at the same time, most people in our musical tradition think those sound dissonant or harsh together. But if you play what’s called a “fifth”, say middle C and the G four white keys to the right of it, those notes sound good together — we say they are consonant. So what’s going on, anyway?
The secret is revealed when we look at the frequencies: middle C (let’s give it its proper name, C4) has a frequency of about 262 Hz. The next G up, G4, is at 392 Hz. When you hit a piano key, a felt-covered wooden “hammer” hits the corresponding string. Because it strikes the string rather than stroking it, it produces not only the fundamental tone but also lots of overtones, as high as our ears can hear. And look at what happens: 3 times 262 is 786, which is almost equal to 2 times 392 is 784. So the second overtone of the C4 is approximately the same note as the first overtone of the G4. The actual note is G5, one octave above G4, but our ears don’t hear it as a separate note. Instead, what we hear is that when C4 and G4 are played together, the combined sound sounds natural, or good, or restful to us.
Equal Temperament and the Wolf
What musicians know, but most non-musicians don’t, is that the notes of the piano represent a hard-fought compromise in how music should sound. We saw above that the overtones of C4 and G4 almost match, but not quite. Wouldn’t they sound better if we tuned the G slightly up to get a perfect match?
The answer is, yes, the two notes sound even better together when you do that. So now you could fill in the other notes by moving a fifth at a time. You start with C, go up a fifth, and tune G to match it. Then up another fifth to D and make that match. This way, you can fill in all twelve notes of the scale, each a fifth higher than the one before: C, G, D, A, E, B, F#, C#, G#, D#, A#, F and back to C. Unfortunately, when you do this, music with nice-sounding chords sounds very good in some keys but out-of-tune in other keys. You are limited to playing musical compositions only in certain keys, because if you transpose them to a different key you can wind up with what are called “wolf intervals”. These are pairs of notes that should go well together but in fact sound awful — they have a nasty wavering pitch, the “beat note” between their overtones, that some people compare to the howling of a wolf.
Unless we want to re-tune all our instruments when we pick up a piece of music written in a different key, we need a compromise. Therefore, Western musical instruments are most often tuned to what is called “equal temperament”: every two adjacent notes on the piano have exactly the same frequency ratio. The ratio is chosen to be the twelfth root of two, so that after you step up from C to C# to D and so on, by the time you get to C one octave up, it is exactly twice the frequency of the C you started with. There are instruments that can be tuned as you play them — you can bend the pitch when you play a guitar, or “lip” the pitch on a saxophone — however, most of the time an instrumental group sticks with equal temperament when they tune up their instruments and when they play them.
Just Tuning and Dolly Parton
With the human voice, however, we don’t need to compromise. We can, if we choose, adjust the pitch of our singing to exactly harmonize two or more voices. And we don’t need a frequency meter to do it, we can do it simply by listening to each other when we sing together. This is called just tuning or just intonation, the practice of adjusting the pitch until two notes sound perfect together or, what is the same thing, the overtones exactly match each other.
To achieve just tuning, the singers need to be able to hear each other without distraction. When we hear singers whose voices seem to blend perfectly — for example, listen to the album Trio by Dolly Parton, Linda Ronstadt and Emmylou Harris — the instrumental accompaniments are typically in a lower register (lower pitched notes), so as not confuse the singers.
The most striking examples of just tuning occur when singers are harmonizing without any instruments at all. Consider the practice of Barbershop Singing. The trademark sound of Barbershop is a “harmonic seventh” chord or “ringing chord”. This is a “seventh” chord — such as C, E, G and B flat — in which the notes are tuned so that as many overtones as possible exactly match. In particular, the B flat is slightly flatter than it would be, for example, on a piano.
When the four voices are well balanced and accurately tuned, the listener has the illusion that there are actually five people singing. This occurs because there are approximately twice as many perfectly matched overtones in a Barbershop Seventh chord as you would expect from two pairs of harmonizing singers. Our ears interpret the extra overtones as an extra singer on the stage.
Back when I was a choral singer I had the pleasure of singing some compositions by the Estonian composer Arvo Pärt, and I still greatly enjoy listening to them. His music is partly inspired by Gregorian Chant and Russian Orthodox Church music, but the chant tradition is unison singing — all singers singing the same note. Pärt takes similar musical lines but writes parts for several singers at once. And here’s an amazing trick he pulls: he writes notes to be sung together that would normally be considered dissonant, but in performance (such as Paul Hillier’s CD of Arvo Pärt: De Profundis) the singers so accurately tune their voices that dissonant pairs of tones suddenly sound “right” because the high harmonics perfectly match one another. I don’t know whether to credit Pärt or Hillier or his singers, but the effect is amazing. I must caution you however that you will like Pärt’s music better if you have had the experience of singing in harmony and trying to achieve that perfect synchrony of blended voices such as Dolly Parton’s Trio achieves.
Science Speculation: “Just Tuning” is a pretty esoteric topic for this blog. If you’re a musician, you may be annoyed by my attempt to “scientize” an art form. If you’re not a musician, you may wonder whether this matters at all!
But I say, there is great joy to be gained from music, and if science can lead us from Dolly Parton to Barbershop to Gregorian Chant to an Estonian composer and then back again, that’s a journey that I’m happy to take!
Just tuning, where voices are perfectly blended, is a prominent factor in some vocal music. Do you have favorite performances where the blend of the voices seems out of this world?
Drawing Credit: gustavorezende, on openclipart.org