Temperament

(entry for 11/22/24)

Having described modes and talked about major scales and the three kinds of minor, it’s time now to turn our attention to something that happened about 500 years ago that has had a profound effect on all music ever since. Namely, the invention of temperament.

No, we’re not talking about temper tantrums (though some musicians do seem to have those when discussing this subject!).

We’re talking about the difference between ‘real’ notes (pure notes) and the ones on a piano or fretted instruments or anything with sound holes or keys. Which, if you didn’t already know, are NOT pure! (Except for the notes named A.)

Let’s go back for a moment to the notion of sharp notes and flat notes. We talked in an earlier post about the ‘invention’ of ‘black notes.’ We mentioned that these pitches were invented in order to provide intervals of a ‘perfect fifth’ for pitches that didn’t naturally have them.

In order to explain all this we need to do a bit of math for a moment. (Yes, we’ve been saying that math is not music and that music is not math, but in this one case music is math!)

We’ve defined the ‘perfect’ fifth in terms of how many half-steps it ‘contains.’ Now we need to find a better definition— one that is based in science.

To go up a perfect fifth from any given note, you need to multiply the speed of vibration of that note by 1.5.  It’s just that simple.

A bit of an explanation is in order. What’s an octave? We’ve said that it’s when two notes have the same exact note-name. The scientific answer is: you double the vibration speed of a note to go up an octave, or you cut it in half to go down an octave.

Let’s take the famous A440 for example. This is the note that for the last couple hundred years or so has defined all other notes. What it means is that the A above middle C vibrates at exactly 440 vibrations per second. If you play that note on a violin, the string (the ‘open’ A string, open meaning not touched or shortened) is going back and forth exactly 440 times every second, as the bow drags across it, or as you pluck it. If you blow that note on a flute, it means the air column inside the flute is vibrating 440 times every second. If you hit that note on a marimba, the wooden ‘plank’ for that note vibrates exactly 440 times every second. Etc.

Going back to the violin for a second, if you keep exactly the same amount of tension on the string, but touch it at exactly half way between the two ends, you double the speed. The sound you now hear is 880 vibrations per second. This is the A exactly an octave higher than the first A! If you keep the tension exactly the same, but double the length of the string, you get A an octave lower than A440. Which is A220. (The open A-string on a ‘cello, for example.)

There's an illustration of these three pitches in notation form at the top of this post.

But let's say you want to play the E between those first two As-- the A440 and the A880. That equals a fifth above the lower A. (Why? A B C D E, equals five letter names, therefore a fifth.) To get that note, instead of dividing the A string in half, you touch it a third of the way from one end, and play the longer piece (the piece that is two-thirds the length of the open string). This gives a vibration speed that is exactly 1.5 times the second speed. 440 times 1.5 equals 660. Therefore the E a fifth above A440 is E660.

If life were simple, that would be true. But life isn’t simple. If you play that E on a fretless instrument such as a violin, yes, that E is 660. But if you play it on a guitar or a piano or a flute or a trumpet, it isn’t. Now it’s 659.255 vibrations per second, if the instrument is properly tuned. NOT 660!

How can that be, you ask?

Here’s how: When we were inventing those black notes, we discovered the hard way that A-sharp and B-flat were not actually the same note. That F-sharp and G-flat are not the same note. They’re close, but they’re not the same. Just as the E on the piano is very close to the E on the violin, but not exactly the same.

It turns out that, when you use ‘pure’ tuning, as in a fretless instrument, A-sharp is a slightly higher pitch than B-flat. That D-sharp is a slightly higher pitch than E-flat. Etc.

Important note:  Open string pitches on a fretless instrument are pure only if they are tuned by ear.  If they're tuned by using an electronic tuning device, they're tempered!

If you were making a piano-type instrument in which all the pitches in an octave were exactly correct, you’d have to have over 30 different keys in the octave rather than the 12 different keys we have now. That would make playing the piano absolutely impossible rather than merely difficult! In addition to making keyboard instruments unplayable, there’s another effect. When you get into keys that use lots of sharps or flats, the scales sound weirder and weirder. By the time you get to six flats or six sharps, the discrepancies are quite striking, and even people with un-educated ears can hear the difference.

The actual math involved is extremely complicated (not to mention boring), and is beyond the scope of these posts. At the moment you’re just going to have to take my word for it, but if you want all the painful nitty-gritty of the whole subject, you can Google three different phrases: Pure scale, Just scale, and Tempered scale.

I’ll save you the trouble. My definitions are way more simple than those you’ll get when you Google it, but they’re easier to understand, too. The pure scale and the just scale are the ones you get when you play the violin. (Or slide trombone. Or any other fretless or non-keyed instrument. Or when singers sing unaccompanied.) The tempered scale is the one you get when you play the piano, or guitar, or clarinet, or any other keyed or fretted instrument.

The tempered scale was invented so that G-sharp and A-flat could be the same note. Etc. So we only need twelve pitches per octave. The compromises are tiny, and most of us have ears that are incapable of hearing the difference, but the difference is there. If you measure the notes with an oscilloscope, a machine that makes sound waves visible, you can see the difference and measure it.

The tempered scale was invented by two different people, a year apart from each other. Neither one knew about the other. The time was just right for the invention to happen, so it did. A Chinese musician and member of that country's royal family named Zhu Zaiyu invented it in 1584. A year later, an English engineer named Simon Stevin invented it again, thinking he was being original. It was many years before Europe and Asia realized that members of their continents had both invented the same thing! Without knowing each other!

A lot of people believe that J.S. Bach invented the tempered scale. Not true. It was invented 101 years before he was born. He did help to popularize it. His set of compositions entitled The Well-Tempered Clavier showed the world that pieces composed in all of the available keys were of equal value to each other. From then on, there was little controversy about it. There are still people today who insist on composing in the ‘pure’ or ‘just’ tonalities, but they are in the vast minority.

Before we leave the subject, however, there is an interesting fact that most people are not aware of. In fact, most musicians are not aware of it. Namely, people who play un-fretted instruments, or who sing, automatically adjust their pitches to match the piano or whatever else is playing at the same time, to make the sounds blend better. They don’t think about it, they usually aren’t even aware they’re doing it. They just do it. When a singer sings the note E660, if there’s no accompaniment they sing E660. If they are being accompanied by a piano or other tempered-scale instrument, whether or not they realize they are doing it, they sing E659.255, to match it! That’s why many professional choirs insist on singing acapella. They want their pitches to be pure!

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