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.

A Rondo in Minor Modes for Two Pianos in Nineteen Tone Equal Temperament

Nineteen tone equal temperament (nineteen equally spaced notes per octave instead of the usual twelve) is a popular choice for microtonal composers. It has major and minor triads that are closer to just intonation than in twelve tone equal temperament. It is generally regarded as a more consonant tuning than standard twelve tone equal temperament.

Personally, I find Nineteen tone equal temperament to be somewhat unsettling to work with. Its fifth and major third are both flatter than purely tuned intervals and I usually prefer major thirds that are sharper than purely tuned (5/4) and closer to the Pythagorean major third (81/64).

In any case, I wanted to experiment with this temperament. I came up with a piece for two pianos that could actually be played in real life if the two pianos were tuned to share the 19 tones per octave. I know this is an awkward way to perform, but this is actually done successfully on occasion during concerts. I decided to explore this temperament's darker side. Here is my Rondo in minor modes for two pianos in nineteen tone equal temperament.

See also my Nocturne for two pianos in seventeen tone equal temperament.

Microtonal Ukulele

Here's a great video of a young musician playing a ukulele tuned to thirteen tone equal temperament. This is often thought of as a dissonant and unpleasant tuning, but I think he makes it sound wonderful.

Thirteen Tone Uke

Microtonal Speedlinking

Xenharmonic Wikispaces has a new equal temperaments section devoted to organising compositions and theory by individual temperaments. This is the first online attempt that I'm aware of to organise compositions in this manner. I hope this idea takes off. I think this could be a great resource for studying the repertoire of specific temperaments.

Neil Haverstick, the prolific microtonal guitarist, author and composer, was recently interviewed at Tokafi. The interview is well worth reading. It's nice to see one of the "Forgotten Greats and Unsung Heroes" (explained in the interview) get some well deserved recognition. (see also this article.)

The hang drum is a recently invented pitched percussion instrument that has been receiving a lot of attention lately. Some of them feature microtonal tunings. You can learn more about them at this page at Odd Music

How about a xenharmonic ukulele tuned to thirteen tone equal temperament and played by a thirteen year old? You can listen to it here. (I wish I had one of those.)

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?

The Measurement of Pitch

Microtonal music involves experimenting with collections of musical pitches that are different from what we are normally used to.

If we are to understand what microtonal music involves, it is helpful to be able to measure and describe what pitch is. This is not just a dry technical topic to get out of the way as soon as possible. The principles involved in the way we measure and describe pitch affects, not just the way we perceive the structure of music, but also the way we perceive and understand the universe itself.

We know that sound is caused by vibration and that this vibration is transmitted by air or water or some other object. This vibration can be measured easily by describing how many vibrations occur within a second. This unit of measurement is called hertz. Every pitch can be described in terms of hertz. The A above middle C, for example, is usually considered to have a measurement of 440 hertz (although this can vary depending on which standard or tuning is used.)

Hertz are useful for describing something called the harmonic series.

Consider the notes that are made by a horn without valves, like a hunting horn. Suppose that the lowest note that can be produced by the horn is 100 hertz. The horn can also produce notes at 200 hertz, 300 hertz, etc. This is an easy sequence to understand. You just keep adding 100 hertz to the proceeding number. This series of tones produces a harmonic series. A vibrating string also produces a harmonic series. The fundamental frequency is being produced at the same time as overtones that correspond to the progression of a harmonic series.

It has long been recognised that you can use notes in a harmonic series to produce music. It is interesting to consider how the brain tries to categorise these notes. If one note has twice the frequency of another note, they are heard as being the same note, even though they differ in pitch. The distance between these notes is called an octave. So how many hertz are there in an octave? It depends on the octave. If one octave spans the distance of 100 hertz, then the next octave above it will have 200 Hertz, then 400 hertz and so on. An octave represents a different way of measuring differences in pitch.

A series of octaves forms a logarithmic progression. Each new term is achieved by multiplying the preceding term by a specific amount (in this case by two). This is very different from the harmonic series. The harmonic series is a linear progression because each new term results from adding a certain amount to the preceding term.

The concept of the octave is nearly universal. You can make music that isn't based on octaves, but the human tendency to categorise pitches into octaves is extremely strong. The vast majority of music throughout the world and throughout history is based on octaves (The octaves may not always be exact because of something called inharmonicity).

A series of octaves has a sort of pure crystalline beauty. Every octave is exactly the same size (in a logarithmic sense). Octaves are a way to measure and map "musical space" that fits in very nicely with how we perceive pitch.

It is convenient to have a unit of measurement that is logarithmic, like the octave, but smaller. The equal tempered semitone is one such measurement. It is exactly one twelve of an octave. The semitone can also be broken down into one hundred cents.

Octaves are powerful because they represent a simple 2/1 ratio of frequencies. It has been the traditional approach to use other simple ratios to fill in the space within an octave. This causes conflict. Simple frequency ratios provide a high degree of consonance (they sound harmonious), but for mathematical reasons, ratios of whole numbers can't duplicate the crystalline regularity of spacing that already exists on a higher level among the octaves. To accomplish this regular spacing, you have to use irrational numbers (numbers that can't be expressed as the ratio of two whole numbers, you can't completely write them down because they go on forever.)

It is not my intention to say whether it is better to divide an octave into rational or irrational intervals. I do intend to point out that either option produces a kind of tension. I also believe that both options have a basis in how we perceive pitch. There is a natural response to the consonance of rationally tuned intervals. There also seems to be a natural response to regularly spaced intervals.

A simple demonstration of these competing forces can be shown by comparing a keyboard that is tuned to twelve note equal temperament with one that is tuned to small number, rational intervals (just intonation). If you play the same chord on both instruments, it will generally be agreed that the keyboard tuned to just intonation sounds more consonant. If you play an ascending chromatic scale, many people will prefer the even progression of the equal tempered keyboard.

Consonance and dissonance are sometimes thought of as fundamental competing forces in music. I would like to suggest the possibility that there are two competing forces that are even more fundamental. I would consider these to be the conflicting tendencies to measure nature in either linear or logarithmic terms. I would also suggest that this fundamental conflict plays a crucial role in the human perception of what constitutes harmony and dissonance.

It would appear this conflict is at least partially responsible for the wide variety of scales and tunings that have arisen through history. There doesn't seem to be any one optimum way to balance the forces that are at play.

With great difficulty, I can dimly perceive what a universe might be like if these opposing forces were not in conflict. In such a universe you could construct a tuning that features perfectly tuned intervals and equal spacings within an octave. This would require a rewriting of the very laws of mathematics. This is not only impossible within our own reality, I question whether it is possible in any conceivable reality.

If such a universe is possible, I can't say what it would be like. There is only one thing that I am fairly certain of. Its music would be incredibly dull.

See also Logarithmic Interval Measures

Is Equal Temperament a Danger to World Culture?

I have previously discussed some of the reasons why twelve tone equal temperament has become so common. See Equal Temperament.

Equal temperament has become so popular that, in many parts of the world, it is displacing other musical traditions and tunings. Some of these traditions are very ancient and offer an important link to the past.

I think equal temperament is a very useful tuning with obvious advantages in instrument building and as a standard for multiple instrumentalists to play together. I am glad that so many people have the option of using equal temperament. On the other hand, it's sad to see the decline of so many other rich musical traditions.

There's much more involved than the potential loss of beautiful and exotic music. Even if a musical tradition was recorded and documented for posterity, it would still be a great loss if it ceased to be practiced by living people.

Music is not static. Its depends on a complex interaction between multiple cultural, acoustic, and physiological factors that we, at present, don't properly understand. We can record the music, but a knowledge of the factors that lead to the music can easily be lost forever, along with an understanding of how this music was interpreted by the listener.

Different musical traditions are like different languages. Many languages are also being lost at an alarming rate. With the death of a language comes the loss of an irretrievable supply of information about the people who spoke it. In my opinion, a musical tradition is just as important as language in defining a culture.

What do you think? Is the loss of ancient musical traditions simply the price of progress? Or should more be done to preserve these traditions?

Equal Temperament

Few people give much thought to the actual notes that are used to produce music. A piano owner usually hires a professional to tune their piano. It is rare for a piano tuner to ask what kind of tuning is desired. It is assumed that the piano will be tuned to the common standard, called twelve tone equal temperament. The customer may not even be aware that there is more than one possibility.

It is amazing to me that there is so little attention given to the fundamental building blocks of music. Modern musicians may spend years studying how to arrange notes to make beautiful music, but they may give little or no thought to the notes themselves, how they are tuned and and placed within the octave.

Scientists have spent billions of dollars investigating the building blocks of matter. They perpetually seek out new elementary particles and try to discover their bizarre properties.

Its strange that this kind of curiosity that is so common among scientists is so rare among musicians.

Why is this? Why has our modern system of tuning become so ingrained that it is usually taken for granted.

First, we should consider the nature of twelve tone equal temperament. Some claim that this is an entirely arbitrary tuning that violates natural acoustic principles. This is a compelling argument because you cannot derive it simply and directly from natural acoustic principles like you can with just intonation or pythagorean tuning.

Does this mean that equal temperament is entirely arbitrary and artificial? I would say no. Equal temperament was not designed by a committee or invented by an entrepreneur.

Equal temperament grew in the rich soil of western musical thought. It developed in response to strong cultural forces, acoustic realities, and the practical needs of composers and musicians. It represents a remarkable compromise between competing interests and goals. I do not view it as arbitrary but rather as a solution that grew organically out of its environment.

This partially explains the overwhelming success of equal temperament. It is adapted very well to the way we currently make and think about music. It packs great diversity and freedom into a scale of only twelve notes. It also provides a convenient standard that allows many different instruments to play together in the same tuning.

However, I feel that some criticism is called for in regards to modern views of equal temperament. Equal temperament may be an effective compromise but it is not the only solution. There is no reason to artificially limit our musical language to a mere twelve equally spaced notes. Equal temperament cannot accommodate the full range of emotional expression that is possible in music.

Equal temperament has many good qualities, but this alone can't account for its level of success. We have become somewhat lazy and complacent. Tuning used to be the subject of passionate debate and energetic experimentation. Now, most people are content to have only one choice. The overwhelming success of twelve tone equal temperament is partly due to apathy.

Fortunately, it is becoming easier to experience alternative tunings. There's a large quantity of recordings available in historical and ethnic tunings. It's also becoming much easier to produce experimental microtonal music with computer software.

Music can be a very personal experience. We still have much to learn about how the brain and human body respond to and interpret sonic vibrations. These responses are complex and vary from person to person. Alternate tunings are a way to explore, more deeply, the relationship between music and the listener. Twelve equally spaced notes to the octave are simply not enough.