A Nice Piece in Seventeen Tone Equal Temperament

I know from personal experience that seventeen tone equal temperament (seventeen equally spaced tones per octave) can be a challenging tuning. It sometimes comes across as harsh and dissonant. However, it also has a softer side. I think the producers of this piece did an excellent job of showing how pleasant music can emerge from what might seem like an unpromising tuning system. I think their choice of instrument and style is a good match for the unique possibilities of this particular tuning.

Seventeen Dragon Dreams (Music and Video by Sethares and Crowly)

Here is one of my compositions in seventeen tone equal temperament.

Slow Dance in Thirteen Tone Equal Temperament

What happens if you add an extra note to the common twelve tone scale? Well, if you space these thirteen notes equally within the octave, you get a tuning that is very dissonant and very unlike anything you may be familiar with.

Some have even concluded that this infamous thirteen tone equal temperament is the worst possible tuning! This kind of assertion just makes me want to explore this tuning even more. I really don't believe it's the worst. I actually consider it to be one of my favorites, although I'm willing to admit it can be difficult to work with at times.

Slow Dance in thirteen tone equal temperament is one of the first microtonal pieces I have worked on. I have tried to make a soft and gentle piece that contrasts with the ideas that many people have about this tuning.

I played this once for a friend. It was actually a version played with a different sound, so it sounded more dissonant than this one. He claimed he could hear it with his deaf ear. We were both pretty shocked by this. We tried several other microtonal pieces, but none of them had this same effect.

I have spent a ridiculous amount of time working on this piece. It's still not how I want it. I find the strange dissonances to be compelling, but they make for difficulties with production. I would like to add extra parts, but they always seem to clash with what is already there. Maybe, after I've gained more experience, I'll be able to return to it and expand upon it.

In the meantime, I hope you will enjoy this little glimpse into a very exotic tuning. Click here to listen.

X.J. Scott has written some excellent pieces in thirteen tone equal temperament. You can find them near the bottom of this page.

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?

Is Dissonance as Bad as People Think?

I was playing a recording of some of my microtonal music to a friend. Suddenly a very peculiar expression came upon his face. He then placed a finger into one of his ears. I didn't know how to interpret this. If he had placed both fingers into both ears I would have understood and turned down the music or turned it off completely. Well, it turns out he was deaf in the other ear, but to his surprise, he could actually hear this song with his bad ear. This was the first time he could hear anything with that ear for many years. He plugged his good ear to analyze better the hearing in his bad ear. He reported that he could hear portions of the song pretty clearly.

Why is this? Of course I don't know for sure. I'm not a doctor. But I have some ideas based on my study of music. The piece being played was written in thirteen note equal temperament. The vast majority of music we hear today is written in twelve tone tone equal temperament, a tuning that is considered to be fairly harmonious or consonant. Thirteen note tuning is, on the other hand, extremely dissonant. Some people consider it to be one of the worst possible tunings and, based on conventional harmonic theory, it is simply awful. I disagree with this assessment, but I would rather discuss, for now, possible reasons for this specific tuning's peculiar effect on this individual.

Most tones from an instrument are actually made up multiple overtones all playing at the same time. This is one reason why different instruments often sound so different even when they are playing the same note. The fundamental notes may be the same but the overtones are different or have different intensities.

Things get interesting when tones are combined. They interact in complicated ways. New combination tones are created and the new waveforms that result can be very complex.

The amount of complexity that results from tones interacting with each other can be reduced if the frequencies of the tones are simple ratios of each other. We consider these tones to be harmonious, whereas tones that interact in a more complicated way are considered dissonant.

Now I should point out something obvious. Since dissonant interactions are more complicated, they are harder to understand and, therefore, less well understood. Dissonance has also been a neglected subject. Harmony has been the subject and goal of most music theory. Harmony is a far easier subject to study and is often considered to be a lofty goal in itself. Harmony is tied up up with all sorts of romantic notions of what is pure and beautiful about the universe.

Dissonance, however, is vilified, often with no real understanding of what dissonance really is.

What's remarkable is that our modern tuning system of twelve equally spaced semitones in an octave ever caught on. Early on it faced passionate opposition because it deviated from so called pure and natural intervals. But it does have technical advantages and was eventually adopted, largely as a matter of convenience.

This is a strange irony. Our romantic notions of harmony really haven't changed much since ancient times, but most people view our modern tuning system with its introduced dissonances as a manifestion of perfect harmony, a realization of their romantic ideals. Of course this irony goes largely unrecognised because most people don't realise the true nature of our modern tuning system.

Of course some people are fully aware of this contradiction. The elimination or reduction of the dissonances in modern twelve tone tuning is a major force behind the modern microtonal movement. But it turns out that dissonance is harder to eliminate than it might seem. Different scales often just introduce different dissonances. (I'll save that story for another day) Others consider our modern tuning to be the best possible compromise. They may lament the impure intervals but consider them to be necessary imperfections.

Do we have to be content with this state of affairs? Is dissonance just a necessary imperfection, an unavoidable wart of nature? This is too big of a question to answer today, but if we return to the beginning we can at least give ourselves some food for thought. I believe one of the reasons my friend could hear that song in a strange tuning is because of the peculiar dissonances of that tuning. Somehow, it appears that the complex vibrations of that tuning interacted with his damaged ear in a complex way and caused it to send messages to his brain.

I have said almost nothing about the possible beauty or musical uses of dissonance, but I hope that I have at least demonstrated some of the power and mystery of it.

I know I have raised more questions than answers, but don't despair, this blog is new and there's a lot more coming. Stay tuned!