Guess What? Tuned Pianos Aren't Really in Tune (Music lecture #2)
Admitedly this topic has less practical value for you than the last post on music. But it's something that fascinated (and surprised) me when I first found out about it, so maybe you'll be interested as well.

***Note Names and Octaves***
We have to start with a couple of basic facts. A note is a frequency, distinct for its number of vibrations per second. 440 vibrations/sec, famously, is known as "A" in our letter-based naming system. What we consider "higher" and "lower" pitches are really faster and slower vibrations--the faster the vibration the "higher" the pitch, although there is nothing literally high or low about them. We have simply graphed sound onto a spatial metaphor of up-down.

So, how/why does our musical alphabet start over once pitches achieve a certain height? What makes all the "A"s sound so much the same, whether they're "high" A's pr "Low"???

Though it's a bit more involved than this, and there is some controversy over the psychoacoustics/neurobiology of how it all works in our brains, the basic reason is that the waves/frequencies of the 2 A's, or any two pitches with the same name, are related by the simplest of ratios, 2:1, known as the "octave." 440 vib/sec and 880 vib/sec (the next "A" up) are going to have much in common:

1) a sound wave "emits" (sort of) resonating frequencies above the fundamental pitch (called overtones) at 2, 3, 4.. times its frequency. So in a very real sense the 880 A is "contained" in and is a part of the 440 A.

2) Their similar waves at a 2:1 ratio will create a regular, periodic "beat" 440 times a second. No other frequency above 440 will create so many regular events as 880 will, just like no other frequency slower then 440 will create so many regular events per second as 220 (the next "A" down) will. The sound we hear in their combination is a fairly simple wave because of that.

Tuning, then, is a matter of achieving the desired ratio between frequencies, finding those that have enough in periodic common (or close enough that we can't tell the difference, which is really really close) to be experienced "in tune" with one another. But we can't just use 2:1 to tune an entire piano. Starting with A and using 2:1 ratios up and down will merely tune all the A's to each other. What will we do about the other pitches?

***The Perfect Fifth***
The next ratio within a 2:1 octave with the most consonance between waves/overtones is 3:2, experienced in our naming system as the distance of a "perfect fifth." It is the interval found between the first note of a major scale and its fifth note. The same distance separates the first and fifth notes of a minor scale. Because of its consonance, almost every mode we ever use (with very very rare exception) employs this interval as its 5th. Even other cultures that use completely different tuning systems, or scales with more pitches than we do, will still likely make prominent use of the 3:2 interval. Like the 2:1 octave, it is such a consonant "in tune" sounding interval (you can hear it between C and G), that some hearers will not recognize the presence of 2 distinct pitches when they are played together.

***Octaves and Fifths within our our note-naming/scale/key system***
Our naming system on modern keyboards works like this: there are 12 different pitch names (pitch classes): ABCDEFG, interspersed with 5 "black keys." The smallest distance between any 2 pitches by this system is called a "half step," that is the interval between any key on a piano and the very next adjacent key (or the very next fret on a guitar). Each half-step is theoretically the same "distance," which really means the same "ratio" between the 2 frequencies. Once you "ascend" 12 of those distances, through all of the pitch classes, you arrive at the octave (2:1), or the same pitch-name you started with, and the pattern continues--theoretically forever. So our tuning system (equal temperament, it's called) operates under the principle that every octave (2:1) is identically divided into 12 equal intervals/ratios (half-steps). It is the reality that allows us to play any tune in any key, and for it to sound exactly the same in any case.

The perfect fifth is found on the keyboard as a distance of 7 half-steps (7 frets). As you know, 12 and 7 do not have a common multiple, except for 12 x 7. Likewise, if you start with any pitch, say A, and ascend a perfect 5th to E, and then ascend a perfect 5th from E to B and then from B and so on, you will eventually arrive back at an A, but not until you have landed on each of the other pitches one time. It is called the "circle of fifths." It begins on any one pitch and ascends (or descends) the distance of a perfect fifth until it arrives back on the pitch it started with.

***The Plan***
So, theoretically, you should be able to tune a piano--after all the A's are in place--by tuning all the fifths up from A (E) at a 3:2 ratio, and then all the fifths up from E (B) and so on, until all the fifths are perfectly in tune and the circle has been completed back to A.

***Here's the problem.***
If you're quick with math, or just a good guesser, you may have anticipated the problem. By the time we get to the end of that process and are ready to make the circle complete with the A's we started with, a 3:2 interval would not put us back there. The truth is that tuning in natural 3:2 fifths will not land you on pitches/frequencies that allow the octave to be divided equally among 12 half-steps. And you would not arrive, after 12 3:2 intervals, at a pitch we would recognize as the same on which you started; not even really close enough you could fool people. But our theoretical naming system, the one that allows us to play the same tune in any of the 12 keys, requires that the fifths do ultimately arrive at the starting point, so that, all octaves (2:1) being considered equivalent pitches, there are only 12 different notes.

Imagine there is a perfect size for a slice of pie. You cut into the pie and carve out perfect slices until the last part left is too small or too large to allow even perfect slices at the end. That is the piano tuner's dilemma. Perfect fifths must create a complete circle to make our system of notation work on a single keyboard. But tune properly, they don't.

So what do we do? The same thing we would do if we were pie-cutters and 12 people expected correct-sized slices. We cheat and hope they don't notice.

Our perfect fifths are fudged slightly smaller than 3:2. They are not in tune in any natural sense. They are close enough that we can't really much tell the difference, especially now that we're all used to it. But multiply that difference by 12 and you've got something much much worse than the Mayberry band. By squeezing each fifth only slightly, we allow the smallest interval (the half-step) to divide the octave in 12 equal parts. That makes all the keys sound identical, even if none of them are perfectly in natural tune. So, the next time some hoity-toity musician tells you your piano isn't in tune, you tell them theirs isn't either.

***Addendum***
Keyboards have not always been the way they are now. In fact, today's equal temperament didn't come around until the 19th century. Prior to that, well-tempered tuning allowed music to be played in all the keys on a single instrument, but they sounded different from one another because some of the fifths were perfect, but not all. In Bach's younger days (1700), a keyboard was designed to be played in only a few closely related keys, so most keyboard instruments had more than one keyboard, to facilitate playing in non-standard keys. The compromise equal-tempered tuning system we use today allows total flexibility for the instrument but was met with more than a little controversy. Tuning by whole number natural ratios was considered to be the harmony of God and the universe. Like most technical innovations, this one was met with some resistance. A fascinating recent book entitled Temperament details the story of its slowly evolving, bumpy acceptance.

At any rate, tempering is one reason why Bach wrote The Well-Tempered Clavier, a keyboard work of preludes and fugues in both the major and minor modes and in all 12 keys: because with this new instrument, he could.

But the compromise is not all good news. We miss out on some beautiful (and challenging) sounds. Many composers, beginning in the last half of the 20th century, have experimented with tuning systems that depart from equal temperament and employ ratios based in "Pythagorean", whole number, "Just Intonation." Harry Partch was a renegade American composer who designed and built his own instruments to accomodate his compositions that used such tuning systems. (A fabulous site about him and his instruments is here--scroll down and press the "play instrument link underneath the diamond marimba, or the "boo.") If you want an introduction to music that is based on Just Intonation, try the "Harp of New Albion" by Terry Riley, a solo piano work from 1986, in which the piano is tuned at whole number intervals from the tonal center. You can hear a juicy excerpt here (2 MB file), but you must promise to turn it up and listen closely...and note all the high ringing pitches that emerge over time--those are overtones, excited by the tuning precision that allows the vibration of many strings up the piano, even those that are not struck with the keys. We can achieve something similar on our equal tuned pianos with the pedal down (allowing the sounds to ring and encouraging some basic overtones), but nothing with the clarity and brilliance of this.