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


Scott Stambler said...

well written!

Anonymous said...

It is my opinion that there are a lot of personal preferences, as well as common misperceptions being represented as scientific fact in this article.

There was a japanese researcher who discovered the world's best superconductor before the people who patented it, but he didn't bother to test the substance because he was told by authorities that it was a certain color that never produces good superconductors. And so he missed out on the discovery of a lifetime.

Likewise, a lot of people miss 99.999% of the vast unexplored continents of microtonality because they read textbooks and psychoticacoustical studies done with 7 participants, all of whom are graduate music students with 4 years of ear training.

I recommend that everyone forget everything that is being said or taught or thought about microtonality and just start writing music outside of 12. Don't limit yourselves to octaves, or one sort of interval, or whatever, just try everything you can think of, no matter what color it is, no matter what the self-styled experts tell you, and find out for yourself what works through the amazing phenomenon and truth of first person get your hands dirty experience. None of the theory people write music anyway, so no one will lose out on anything by ignoring 100% of what they write about in their prestigious journals.

Daniel Thompson said...

Thank you, Mr. Scott, for your comments. You exposed a bias in my article that I have no choice but
to admit. This article was limited to tuning systems based on octaves, but I did not mean to imply that you have to base music on octaves.

I feel that it is useful to use the abstract notion of precisely tuned octaves to demonstrate the
different properties of "purely" tuned intervals as opposed to tempered intervals. I believe that
octaves are very important to both the history and theory of music and that they are a powerful force,
but not absolute.

I am also willing to admit that the importance of the octave may be
somewhat exaggerated. Certainly, some of the studies that you have mentioned are suspect and may have
been misinterpreted. It is also likely that the numerical fascination that many theorists have with small whole number ratios may have contributed to an over emphasis on the importance of the octave and, possibly, a biased interpretation of the music of other cultures.

I have experimented with non octave music and have been impressed with the possibilities. Since a type of conflict arises in systems of just intonation, I feel it is also worthwhile to explore intervals that aren't purely tuned. This could include tempered devisions of the octave or music that disposes of the octave entirely. I am actually working on an article about non octave music. I am having some trouble because I feel that I have a lot more to learn about the subject. I also plan on spending more time on
my non octave compositions after I finish some other projects.

Thanks again for your comments. I didn't intend to present personal preferences or common misperceptions as scientific fact. Please feel free to contact me personally or leave another comment if you feel there are other problems with my
article that I haven't addressed.