Just Intonation versus Equal Temperament

OK, I'm a geek. I'll admit it. I like science and math. I like to write programs. I even learn new programming languages just for the fun of it. Maybe that's why I'm so fascinated by the math behind the fabulous sound of a properly-tuned chord.

In this post, I'll give a very basic introduction to just intonation, the reason that a cappella music (especially barbershop) sounds so good.

The ear perceives harmony as a function of the relative frequencies of the pitches involved. If the ratio is 2 to 1, we hear an octave; if 3 to 2, a perfect fifth; if 4 to 3, a perfect fourth; if 5 to 4, a major third, and so on. The reason these simple ratios sound so pleasing has to do with the minimization of "beats" which occur when frequencies are close together, but not identical. As an example, sounding a tone of 100 Hz along with one of 105 Hz produces a noticeable periodic variation in volume at 5 cycles per second. These beats occur not only between the fundamental frequencies, but also between the overtones of the two pitches.

What's an "overtone"? you may ask. Nearly every musical sound contains multiple frequencies, at various amplitudes (I'll explain why in another post). The lowest frequency (called the fundamental) is what we perceive as the pitch of the sound. The higher frequencies are normally integer multiples of the fundamental, so a note that we would call an "A" above middle C, if played on a violin, will contain the fundamental (440 Hz), plus 880 Hz, 1320 Hz, 1760 Hz, 2200 Hz, etc., at diminishing amplitudes. These higher frequencies are called overtones or harmonics. Confusingly, harmonics are numbered differently from overtones. The fundamental corresponds to the first harmonic, while the first overtone is the same as the second harmonic.

Back to beats. When we sing a perfectly tuned fifth (say A-220 and E-330), the second harmonic of the E (2 x 330 = 660 Hz) lines up with the third harmonic of the A (3 x 220 = 660 Hz) and reinforces it. Other, higher harmonics also reinforce each other and provide a pleasing sound. When the two notes are not perfectly tuned, the harmonics do not line up, and we perceive a noisy sound because of the beats formed by their close-but-not-quite-perfect alignment.

So, what is "just intonation" and how does it differ from "equal temperament"? Just intonation (JI) simply means that the intervals between the notes in a chord are tuned in small-whole-number ratios. A cappella singers naturally sing in JI -- it's intuitive. But the problem with JI is that you need several different frequencies of a given note, based on the function of that note in the chord. For example, the E above middle C is 330 Hz when it's the fifth (3/2) above an A-220, but only 327 Hz if it's the major third (5/4) above middle C (261.6 Hz). Ear singers naturally adjust their pitch to maximize the consonance of each chord, but a keyboard instrument can't do that. You only get one frequency for each note.

That's why equal temperament was invented. That system tunes each note such that all half steps are equal (i.e., the 12th root of 2). It yields a set of pitches that cannot make up any JI interval except octaves, but that allow you to play equally well (or badly) in every key. If you were to tune a piano to the key of C major in JI, the C major triad would sound great, as would F major, G major, A minor, and E minor, but the D-A "perfect" fifth would not be in tune. In ET, no chord is perfectly in tune, but the difference is not too bad. In fact, we have gotten so used to hearing "almost-in-tune" chords, that we hardly notice the imperfections unless we think about it.

I've got a lot more to say about JI, but it'll have to wait for another day.

Singing -- what's it all about?

Singing is fun. Singing in a quartet, especially so. Singing in a large group can be enjoyable, but nothing beats the thrill of four guys finding that perfect blend, balance, and tuning that makes your hair stand on end. One guy put it like this:

That last chord, I just didn't want to let go! It was so sweet!

What makes a cappella singing such a kick? There's something visceral about music. It's more than an intellectual appreciation for the beauty of the melody and harmony, more than the emotional tug of a well-crafted lyric, more than the artistic expression of thoughts in rhyme and rhythm. A good song, well sung, will touch you deep in your innermost being. It will lift you to heights of joy, or bring a lump to your throat and a tear to your eye. And sharing that feeling with three other like-minded guys just amplifies the experience. Being a part of producing that reaction in others brings a fulfillment found in few other pastimes.

That's why I sing. How about you?