## The Well-Tempered Scale

© Copyright 1998 & 1999, Jim Loy

Pythagoras & music:

Pythagoras discovered the ratios of frequencies that make up the musical scale. Play a note on a stretched string (as on a guitar). The string is vibrating, and it is this vibration that we hear. The string vibrates, causing the air to vibrate, the vibration in the air moves outward (in waves) from the source of the sound, and makes your ear drum vibrate, and you have a way of detecting this vibration. Now, press down on the string at its center, holding it against the neck of the guitar (or whatever), making it a string half as long. This shorter string plays a note an octave higher than the full string did, and vibrates twice as fast. If the original note was a 60 cycle (Hz) tone, then this shorter string is producing a 120 cycle tone. This second note sounds very much the same as the first note. In fact some people cannot tell the difference. Even musicians have confused notes an octave apart, when played on different instruments.

Instead of holding the string in the center, let's hold it so that it is 2/3 its original length. This string now plays a note a "fifth" higher than the original note. The string is vibrating 3/2 as fast as it was originally. If the original tone was 60 cycle, then this is 90 cycles. A note and its fifth sound very good when played simultaneously. Here are other ratios of vibrations:

 octave 2 fifth 3/2 fourth 4/3 Major 3rd 5/4 Minor 3rd 6/5 Major 6th 5/3 Minor 6th 8/5

You may know that when you go up a fifth, then up a fourth from there, you have gone up an octave, all together. Let's see if that works: Up a fifth=3/2, up a fourth from there=(3/2)(4/3)=2, which is an octave. It works great.

But, there's a flaw in this musical scale. Let's start at C, and keep going up a fifth, until we find another C. That is C-G-D-A-E-B-F#-C#-G#-D#-A#-F-C. This C vibrates (3/2) to the twelfth power faster than the original C. Those of you who are familiar with numbers may have guessed what has just gone wrong. This C is not an even number of octaves above our original C. It's close. Instead of vibrating 128 times as fast (seven octaves higher), it vibrates 129.7463379 times as fast. It is obvious to the ear that these notes are not both C. For the same reason, F# is not the same as G-flat (going down from C), even though they are the same on a piano.

Let's try something different. Besides octaves and fifths, major thirds are very important. So, let's start with C and go up three major thirds, to the next C (C-E-G#-C on a piano). We get (5/4)(5/4)(5/4)=125/64. That is close to an octave, but not exact. It is a little short (flat) of the octave. So, we have shown that pure major thirds, and pure fifths, and pure octaves are not compatible. They don't come out even. So how can we ever tune a piano?

Pythagorean tempering:

We can tune our piano, using both pure fifths and pure octaves. We start at C, and tune the fifths like we did above (C-G-D-A-E-B-F#-C#-G#-D#-A#-F). And we stop at F (E#), since the next fifth ruins an octave. We can tune the whole piano, with those 12 notes. This is called Pythagorean tuning. We have eleven pure fifths, and one really terrible fifth (F-C). Such a terrible sound is called a "wolf," by the way. The major thirds are not pure. In fact they do not sound very good.

Another Pythagorean tuning is to start at a key other than C, so that the really bad sounds are all distant from C. In the above tuning scheme, F (our bad fifth) is closely related to C, and causes problems. So, we could start at F# (or B or some key like that) and tune pure fifths from there. And we would get nice chords in C and related keys. Again, we would get one really bad fifth.

We do not use the above scales. A C scale based on this sounds fairly good. But, a D scale sounds slightly out of tune. And various notes on every scale sound slightly out of tune. In fact, every scale sounds different. The situation is that we are all used to a different scale, instead of this one.

Meantone tempering:

Later, an attempt was made to produce pure octaves, and as many almost pure major triads (chords like C-E-G) as possible. This kind of tuning is called "meantone." There were several different meantone tuning schemes. Each involved slight adjustments to the sizes of the major thirds and fifths. As we saw above, pure major thirds, pure fifths, and pure octaves are not really compatible. So, with meantone, we end up with relatively few usable keys. About eight of the 12 major keys are usable. And there are several "wolves," the terrible sounds.

Tempering is involved here. Tempering involves adjusting some of the intervals. In this case, we adjust them, so they fit into an octave. That is what we did with major thirds, in Pythagorean tuning. We made them bigger, so that our fifths could be pure.

Well tempering:

Near the beginning of the 18th Century, "well tempering" became popular. This was a little more complicated, in a way. But, every key became usable. And there were no wolves. There were several "well tempering" tuning schemes. Essentially, all octaves were pure. Keys related to C had nearly pure major thirds and fifths. Keys distant from C had much less pure sounds, but were not too bad. And the sequence, from a pure C triad to the impure distant triads, was gradual.

None of the scales or chords sounded bad. In fact every major and minor key sounded different. C sounded placid and fairly uninteresting. The more distant keys sounded more interesting. You might call some keys harsh, or agitated, or tense. And so, music could be written to suit the mood (or color) of each key.

Bach wrote the Well-Tempered Klavier, 48 prelude and fugues, two in each of the 12 major and 12 minor keys.

Equal tempering:

Since the middle of the 19th Century, the most popular form of tuning has been Equal Temperament (there is no word spelled "temperment," by the way). This is a scale with pure octaves, but with the other notes spaced evenly between the octaves. So, the interval between adjacent notes is an identical minor 2nd. And, every scale sounds the same, just higher or lower. None of the scales sounds out of tune.

But, you can throw away Pythagoras' ratios. All of the thirds and fifths are tempered the same amount. They are close to the Pythagorean sounds. But, only the octaves are pure. All of the other intervals are impure. No key sounds bad. No key sounds pure. All keys are somewhat interesting. But, they are all the same. Most music, from the middle of the 19th Century until now, has been written for equal tempering. England stayed with well tempering, long after the rest of Europe had switched to equal tempering.

The expression "well tempered" can apply to equal tempering, as it essentially means that all keys are available. But traditionally, "well tempering" applies to the various methods of unequal tempering which made all of the keys available.

So, we now live in a world of equal tempered scales? Not exactly. We now live in a world where Bach will probably be played on a well tempered instrument. Medieval music may be performed using a Pythagorean scale. And, Chopin will probably be played on an equal tempered piano. We now lean toward authenticity.

Note: When I began writing this article, I thought that "well temperament" was the same as "equal temperament." Apparently, lots of people have made the same mistake. And I always misspelled "temperament."

Also see Harmony and Sine Waves.