Are the "Errors" of Equal Temperament Trivial?

It is sometimes thought of as a dirty little secret that twelve tone equal temperament (the tuning that is used for the vast majority of modern music) contains no purely tuned intervals, except for the octave.

It is widely believed that the perfect fifth, for example, should express the frequency ratio of 3/2. In twelve tone equal temperament, it is about two cents flat. It is often believed that the major third should express the frequency ratio of 5/4. In equal temperament, this interval is about fourteen cents sharp. Similarly, all the intervals in equal temperament (except the octave) are detuned, or tempered, away from "purely" tuned intervals.

Some people are a little upset about this. I have discussed this controversy in several posts in this blog. Today, I would like to discuss just one small portion of this controversy.

Some people claim that these "errors" of equal temperament just aren't big enough to matter. I like equal temperament, but I can't agree with this.

The difference between a tempered fifth and a purely tuned fifth is pretty small, only about two cents. (A cent is 1/100 of a semitone or 1/1200 of an octave) Is this big enough to be detectable? It depends. Many instruments have tuning errors that are much bigger than two cents, but sound fine. However, there are cases where a difference of two cents can be pretty noticeable, such as when you have two tones sounding at the same time.

I would like to point out that something might not be noticeable on a conscious level, but could still have an effect. Our brains soak in millions of tiny details that we are not consciously aware of, but do affect our overall impressions. I'm not too concerned about a difference of two cents, but I'm not prepared to claim that it doesn't have an effect.

A fifth can be thought of as a building block within a scale. The other intervals in equal temperament can be formed by producing a circle of fifths. So a fifth above C is G. A fifth above G is D, and so on until you get back to G. This makes a small two cent discrepancy much more important because this discrepancy increases. If the G is two cents flat, then the D is four cents flat (from 9/8).

The tempered major third is either about eight cents flat (from the Pythagorean 81/64 major third) or about 14 cents sharp (from the 5/4 just intonation major third), depending on your perspective. Either way, it's enough to be noticeable in many situations.

Equal temperament was adopted for largely practical reasons. Some believe that you can just get used to its tempered intervals. It is true that just intonation sometimes sounds out of tune to someone used to tempered tuning. Of course, some people ask why you would ever want to allow your ears to get used to these detuned intervals.

Another factor is that the brain tries to find patterns and structure in music. The acoustic differences between equal temperament and just intonation hint at important structural differences. These structural differences may have a profound impact on how the music is interpreted. This may also partially explain why just intonation sounds out of tune to some people. It's not out of tune, but it may present an unfamiliar structural organization to the brain of the listener.

It seems that it does both just intonation and equal temperament a disservice to suggest that the differences between them are not important. Purely tuned intervals are precisely defined and even small deviations from them can have a profound effect. There are good reasons for exploring the fascinating world of just intonation.

There are also tempered scales with more than twelve notes that have closer approximations to purely tuned intervals. These may be worth exploring, but I would once again suggest that these intervals are not the same as purely tuned intervals.

Tempered intervals have their own charm. They're not proper substitutes for purely tuned intervals, but they have enjoyed an enormous amount of success in certain contexts. There are times when I prefer them. (Again, it's largely a matter of context.) In my mind, one of the key questions of microtonal music is why they can be used so successfully. I discussed this somewhat in The Measurement of Pitch, but I plan on discussing it in greater detail at a later time.

See also Is Dissonance as Bad as people Think?


Anonymous said...

I just don't buy that anything in which the fundamental pitches are not related in just intonation is an 'error'. It's kind of an absurd premise, in my opinion. The ear does not have integer detectors. With the exception of those with damaged hearing, people can not even recognize when two sine waves are related in a 2:1 ratio at all, much less tune them by ear.

sting ray said...

Interesting. Although I have perfect pitch, on many occasions after tuning my guitar by comparing adjacent strings, not all chords that I play would sound in-tune.

(BTW, I got here via a link from bestestblogofalltime.blogspot.com)

I've just created my own music weblog too. It's still new, but eventually there will be a lot of music-related content. Tke a look when you're free :)

The Astral Technician said...

Haven't had a chance to read your site yet, but I'm interested in the use of microtonal music to encourage altered states of awareness, and its potential use in biofeedback.

I've bookmarked your blog and I'll be reading more in depth shortly.


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