Study in Microtonality and Lutherie – one

All sorts of developments in music theory distinctively originated from different cultures all over our planet. This can be the most wonderful thing, but cultural traditions tend to be limited in technological influence and limiting to aesthetic variety. Musical harmony is something more involved than melody, and it takes a bit of science to develop an understanding of how it works. I find that, in most cultures, the capabilities of harmony are typically limited, and sometimes harmed, by various theoretical and compositional past-times: the integration of temperament, theoretical emphasis on melody, a lack of harmonic theory all-together, and aesthetic conditioning.

Timbre is the textural quality of a sound, and its existence is most easily explained in terms of the harmonic series – the spectrum of all ratios built on positive integers (1:2:3:4:5:6:7, etc.). In the spectrum of audible frequencies, harmonics are equidistant, separated by the same frequency value. Thus, the frequency of a fundamental pitch, the first harmonic, can be used as a multiplier to find the frequencies of other harmonics.

The central component of an instrument is usually a harmonic oscillator, something that can project multiple frequencies simultaneously, i.e. a string or a column of air. The physical character of an instrument’s body affects the volume of the different harmonic frequencies created by these oscillations. The variations in volume are responsible for the differences one acknowledges between timbres.

Say two pitches each contain the same pattern of harmonics above them. If the frequencies of the two pitches are distanced by a ratio that results in the overlap of harmonics of each pitch, then the two pitches reinforce each other and sound comfortable with one another. This is the origin of harmony. To better understand harmony, one would next learn how comfortable or uncomfortable certain ratios feel, and the emotional characters of their relationships. Then one would work to understand the comfortability of larger groups of notes.
Let’s say that the ratio between two frequencies is identified as 2:1, a relationship called the octave. In this scenario, all of the odd harmonics of the lower pitch form a harmonic series identical to that of the higher pitch. The harmonic overlap within the octave is incredibly reinforcing and it is probably for this strength that the human ear evolved upon a 2:1 ratio. Our brains understand two pitches separated by an octave as having almost identical characters. Accordingly, every pitch within the bounds of an octave has a character almost identical to those of pitches within higher octaves. This concept of octave equivalence can be considered from biological, scientific, and mathematical perspectives as having inherent truths. Even so, compositional theory need not yield to it; although sound itself must comply to nature, the process by which sound is intentionally designed is free to one’s will. The same goes for the other ratios in the harmonic series.
A scale is a pattern of notes and most cultures have their own traditional scale. Most scales are octave equivalent and are built within a single octave. The scale of the modern West was standardized about one hundred years ago, however variations of it had been the norm for a few hundred years. It is built on a pattern of twelve tones in an octave. They are logarithmically separated as to grant the ear the perception that the distance between each pitch is equivalent. It is, however, the ratios between the pitches that the ear understands as distance. Remember that pitches separated by an octave have the same character, but only displaced. Consider that an octave series is derived exponentially (100 hz, 200 hz, 400 hz, etc. – or 100 x 2^n), not linearly. In this, the human ear’s perception of exponential relationships can be described as linear. From this knowledge and a little math, you could derive that a scale built on twelve “equidistant” tones must be built on a ratio of the twelfth-root of two (to one). It can also be derived that the ear does not understand the harmonic series as behaving linearly, but rather as a curve.

The system in which one spaces notes equally is called equal temperament. This is because, in the court of equidistance, one must shy away from some pure ratios in the harmonic series. This is, in essence, an act of temperament – the establishment of a purer ratio at the cost of another. One can alter the distribution of ratio purity by the selection of the number of tones in the temperament. Westerners primarily chose twelve tones because the lowest ratios of the harmonic series, which are considered the most consonant, are very accurately expressed. However, any multiple of twelve will yield the same twelve ratios, plus however many more. In fact many non-western cultures have built their musical systems on twenty-four equally spaced tones.

The notion of equally tempering pitches is not only a little trivial, but based on an assumption that the human ear has a taste for scales bookended by the octave. Some non-western musics, like Javanese gamelan, intentionally misapproximate the 2:1 ratio to create “shimming,” a distortion that occurs when two similar pitches try to synchronize with one another. In just-intonation, one uses only the pure ratios of the harmonic series, which treats octaves differently than tonal tuning systems do. However, this has a few disappointing qualities: it is very difficult to perform without building instruments; the majority of justly intonated music has been composed with midi-coding, which, when played out of a speaker, defeats the quality of pure acoustics; composing only with pure intervals may ignore beautiful pitch choices.

In my experience, musicians do not often discuss the relationship between harmony and timbre. From my descriptions above, I’m sure that most readers can draw parallels between the two. Timbre is, in fact, in itself, a harmony. This is a large part of what I study, and it is unfortunate that it is considered a specialized approach to music composition and theory. The other bulk of what I study is the usage of pitches that are not in the twelve-tone Western scale. The Parisian Spectral School of composition made a motion towards composing based on considerations of sound spectra. Spectralists frequently use just-intonation, considering how the harmonic series applies to particular instruments in specific environments. Despite their contributions to theory, the students of the Spectral School write music of the Spectral genre, which I don’t find particularly appealing. I am, however, very influenced by the composer Horaţiu Rădulescu, who studied with the Spectralists. He used many different non-traditional and non-twelve-tone tuning systems (also called microtonal tunings) to make harmonic funnels which each touch on a different fundamental pitch. Rădulescu would exploit small sections of these funnels by allowing them to touch in special ways.
All of this information demonstrates that there is a lot to understand about the the role of mathematics in music. It is not necessarily intuitive to compose with all of this in mind. Unfortunately, it is also difficult to study microtonal music without an instrument designed to play the specific frequencies needed for a particular work. I am a guitarist and, for my summer project, I am building a guitar that will allow me to perform and dabble in whatever harmonic designs I please, and I will post samples and demonstrations on this blog. Unfortunately it won’t seem very special until the very last steps, when I install an apparatus for changing the fret-boards of the instrument. This is in preparation for the other half of my project, in which I will compose some pieces with this instrument in hand.
I leave you with an image of a fret-board prototype that I made a year ago. I refretted the instrument itself, so the fret-board can’t be changed without serious woodwork. It is based on a justly intonated 12-tone scale, modulated into five keys, and slightly tempered for performance – too complicated and, literally, too difficult to grasp for a first design. When I build the new fret-boards, I am going to begin with equal temperaments, which are easier to build and much easier to play.


Microtonal Fretboard

I will also leave you with the work that got me into music and microtonality when I was younger: 

and the man who is, by far, my king:

Expect a post on my progress soon.

-Jean-Paul Wallace


  1. Here’s a website where you can hear samples from that Toby Twining Album – for those who don’t want to buy it: