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

Take Advantage of Free Music Software

Making microtonal music with software can get pretty expensive. Fortunately, there's a large supply of free music software that might be able to fill many of your needs. Considering that you can easily spend hundreds or thousands of dollars on music software, it's well worth checking out free versions first.

There's a lot of places on the web to get free or inexpensive software, but I would recommend that you check out Hitsquad. It has a very large directory of sites that offer music software. A lot of the software isn't free, but much of it is. You can also find a lot of demo versions.

Of course, most of the software doesn't offer microtuning abilities. It can still be difficult to find what you are looking for, especially if you are into more exotic microtonal scales that require unconventional keyboard mappings. Still, it doesn't cost anything to browse the titles.

If you are thinking of spending a couple hundred dollars on some software with microtuning abilities, you will likely want to see if you can find a demo version first. It's been my experience that much of the microtonal software is so limited that it's virtually worthless for the type of music that I want to make.

You may also want to consider whether the sounds of the software match the tunings that you are interested in. It may sound like a lot of fun to use fat sounding synthesised tones with exotic and dissonant scales, but the actual result might not be pleasing to your (or any one's) ears. It can take a lot of experimenting to match sounds with scales, just another reason to try before you buy.

If there's a problem with Hitsquad, it's that it can be too much fun. It's easy to waste a lot of time downloading every title that looks interesting. I recommend that you check it out, but use some moderation so that you still have time to actually make music.

L'il Miss' Scale Oven™

Recently, I've heard a lot about the L'il Miss' Scale Oven™ microtonal software. It sounds like a great resource for microtonal musicians.

Alas, I can't comment on it personally as it's only for Mac, and I'm a PC user. I did, however, run across a glowing review of it at the Post Classic blog. You can also visit the Nonoctave.com website for ordering information.

Even if you use a PC or aren't interested in the software, it is worth visiting this site because it is a good source of information on nonoctave music, an interesting subcategory of microtonal music.

If you have used L'il Miss' Scale Oven™, please post a comment telling us what you think. If you have any comments on other Mac only tuning software, that would also be appreciated.

See also the Bohlen-Pierce Site, another good web site on nonoctave music.

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?

Scala

If you are interested in making microtonal music and use a computer, you should know something about Scala. Scala is a free program that allows you to retune certain soft synths, keyboards, and sound modules. It may be that you can use it with some hardware or software that you already own.

Learn more at the Scala Home Page

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.

Embrace the Controversy

In the past, the subject of tuning has provoked heated passion and enthusiastic debate. This is partly because music has often been regarded as more than just pleasant sound. Music was viewed as inseparable from the nature of the universe, an ideal attained in the heavens and emulated on earth. Bad music or improper tuning was a crime against nature and a danger to the fabric of society.

There was a fascination with whole numbers and the simple fractions they made. It was discovered that musical tones whose frequencies formed simple fractions like 2/1, 3/2 or 4/3 sounded especially pleasant and harmonious. Musical scales that employed these ratios were viewed as natural, a reflection of the divine order of the cosmos.

We now know that nature is far more complex than the ancient mind could have imagined. The ancient view was challenged by the Pythagorean discovery of irrational numbers. Previously, it had seemed self evident that all numbers were either whole numbers or ratios formed by whole numbers. Alas, there is an infinite quantity of irrational numbers that cannot be expressed as the ratio of two whole numbers. The ancient view of the cosmos was seriously flawed.

This "inconvenient" fact of nature would eventually influence how music was made. Purely tuned intervals based on simple ratios sound nice and so does the music based upon them, however, these simple ratios have unexpected mathematical properties that make them unsuitable for certain types of musical expression.

Purely tuned instruments have problems when they modulate from one key to another. Some keys sound good and other keys contain harsh dissonances.

What can be done to avoid this? There's a couple of possibilities. You can just live with these harsh dissonances. You can simply avoid modulations that result in these dissonant intervals. Or you could add extra purely tuned intervals to increase your freedom of modulation.

Another possibility is temperament. You can detune or temper the notes in a scale so the different keys sound either more alike or exactly alike, except for absolute pitch. Our modern system of twelve tone equal temperament allows unlimited freedom of modulation with only twelve notes to the octave. All the keys sound the same, but this system contains no purely tuned intervals, except for the octave. All the other intervals are irrational.

This is where the controversy lies. There are great practical advantages to our modern tuning system, but many people long to hear purely tuned intervals.

Some people have formed very strong opinions in favor of systems that use purely tuned intervals, perhaps viewing any sort of temperament as unnatural. Others view purely tuned intervals as impractical.

I personally don't see the need to limit myself to just one side of this question. I realise that any musical system will have some measure of tension in it because of competing musical forces. I find this to be natural and desirable. Music is dynamic and personal. Ancient theories about beauty in music may be incomplete or even wrong, but our growing understanding of music is truly exciting. This controversy about tempered verses justly tuned intervals is part of what makes music interesting.

If you make microtonal music, or are thinking of doing so, this issue is of great importance to you. You can experience for yourself the difference between tempered and purely tuned intervals and come to your own conclusions about how to express yourself musically.

I suggest you keep an open mind during your exploration. Microtonal music is full of surprises and you may find that some of your preconceived ideas will be challenged by your discoveries. I personally have gained quite a different view on tuning since I began my exploration. I have had some beliefs that I never thought of questioning until I made discoveries that caused me to think in a new way. This is part of the fun. Embrace the controversy!

See also Equal Temperament

What is Xenharmonic Music?

Xenharmonic music is a type of microtonal music that uses strange or foreign harmony. It generally does not refer to microtonal music that is similar to twelve tone equal temperament.

Linguistically, this is kind of a mess. Xenharmonic music is a useful term, but not very precise. Whether a certain tuning is xenharmonic can depend on how it's used or how it's perceived by the listener.

The term xenharmonic music may be vague, but the attitude of a composer who identifies his music as xenharmonic may be more clearly defined. Frequently, such a person will explore types of musical expression that is new and experimental.

Many traditional microtonalists look to the past for inspiration and take a very conservative approach to their selection of scales or musical styles. They may view modern twelve tone equal temperament as a deviation from pure and natural principles and long to experience the musical joys of simpler times. This is usually not xenharmonic music.

A xenharmonicist tends to go in the opposite direction, perhaps exploring nontraditional, justly tuned intervals or using scales with unusual dissonances. A truly enthusiastic xenharmonic composer will take nothing for granted. He may try music that weakens or eliminates the concept of the octave. He may use scales that feature severely detuned fifths or that use the tritone in unexpected ways.

These terms can give the false impression that these two different types of microtonal music are mutually exclusive. I prefer to view microtonal music as more of a spectrum, where some is more xenharmonic and some is more traditional. It's somewhat rare and, probably, unwise for an individual to devote all their attention to one extreme end of this spectrum.

Microtonalists are an extremely varied group of people that are lumped together in one inconveniently vague term. Many people, including myself, have spent time trying to think of better, more precise terms to describe our craft, without much success.

Maybe this is OK. We are all in the same boat, regardless of our differences. Many people are unaware that we even exist. Inconveniently vague and simplified terms may be all that the public can handle at the moment. Give them a little time time to discover us, and then precise distinctions may become more important.

I for one, am not too concerned about these differences. I have great respect for anyone who is willing to challenge the supremacy of twelve tone equal temperament. Some of our ideas may be flawed. We may make some mistakes, but in music, these mistakes can teach us as much as our successes. If we are humble we can all learn from each other.

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!

What is Microtonal Music?

Microtonal music is any music that uses a tuning other than standard twelve tone equal temperament.

Twelve tone equal temperament is the basic piano tuning that we are used to. It really works quite well, which partially explains the fact that it is the only tuning many people are aware of.

I am a restless individual, however, and am not content with the the so called norm. A lot of microtonal music is based on various historical or ethnic tunings, but I am especially intrigued by certain experimental tunings that are hardly ever used. I am currently writing a symphony in twenty-six note equal temperament, an obscure tuning with strange properties. I hope to have some links to my music soon.

See also What is Xenharmonic Music?

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