Harmonic Space Music

by Marc Sabat (February 2020)

festival program co-curator

In his ground-breaking 1983 article “John Cage and the Theory of Harmony”, the American composer and theorist James Tenney proposes a mathematical model describing a possible ordering of “harmonic” auditory perception. He sets out a quantitative measure of how pitched sounds with various fundamentals, which may be measured along a one-dimensional continuum of vibrational frequencies from low to high, also establish other salient, multi-dimensional relations of proximity. In musical terms, these are generally described as qualities of concordance/discordance, manifesting as degrees of tuneability, applicable to successively or simultaneously sounding pitches. The building blocks of these relations are frequency ratios and the specific primes which comprise them.

When specifically referring to sound, frequency describes periodically varying oscillations of objects transferred through a medium to our auditory senses. Sounds may be considered to be mixtures of frequencies. Sounds that are perceived as bearing “pitch” are generally combining frequencies that resemble an harmonic series by implying, or including, a fundamental frequency of which they are all integer multiples. Superposition of pitched sounds is, therefore, superposition of harmonic series.

Whenever two frequencies approach unison by being almost equal, acoustic beating or amplitude modulation is perceived. Fast beating results in a sensation of discordant roughness and slow beating or unison gives a smooth, concordant percept. When two fundamental frequencies relate to each other by a proportion near to a relatively small-number ratio (like 1:2, 2:3, etc.), the respective harmonic partials align and the composite sound becomes potentially “tuneable”, i.e. the beating may be stilled by more closely establishing a ratio between fundamentals. If this ratio favours unisons lower in the harmonic series, the degree of concordance is higher.

This implies a relationship of proximity, which may be measured in terms of the lowest common partial of the two series. If the ratio, in lowest terms, is b/a, then this partial is ab. Originally proposed as a quantification of consonance by Giambattista Benedetti in a letter written to Cipriano de Rore in the 1500ʼs, Tenney modified this measure by taking its logarithm, and called the value log 2(ab) “harmonic distance”. Because the logarithm of a product is the sum of the logarithms of individual factors, harmonic distance represents a path of least resistance between two frequencies by following the most basic “harmonic relations” (absolute consonances consisting of intervals from a fundamental to one of its prime partials). The space mapped out by these intervals is what Tenney calls “harmonic space”, and it presents a generalised model extending the two dimensional lattice of triads proposed by Leonhard Euler to include higher primes.

Although harmonic space extends infinitely in infinitely many dimensions, human pitch perception is bounded in terms of both frequency range and resolution. This psychoacoustic non-linearity allows finite subsets of the space to nearly “fold” upon themselves, enabling such subsets to evoke an impression of unimpeded, cascading harmonic movement passing through various enharmonic near-equivalents.

The Extended Helmholtz-Ellis JI Pitch Notation (HEJI) is based on the principle of defining unique pairs of visual symbols (overtonal/undertonal accidentals) mapped to each prime dimension of harmonic space, extending the traditional staff notation so aggregates of symbols may distinguish intervals (transposable relations, moveable “do”) as well as notes (absolute pitch-heights, fixed “do”). The ability to parse accidental combinations determines practical limits to the notation and also provides an impulse to establish the scope of a generally applicable enharmonically viable pitch-space.

Tenney suggests that, after a century dedicated to other aspects of musical perception, perhaps it is an interesting time for musicians to creatively revisit and reimagine what “harmony” might be. He proposes an approach more closely aligned with a scientific definition of “theory”: researching quantitatively and descriptively rather than codifying habitual practices and rules specific to particular musical cultures or styles.

In founding the Berlin-based research group “Harmonic Space Orchestra”, we celebrate Tenney’s vision, exploring the idea of harmonic space both by playing his music and by creating new performance practices, compositions, and theoretical discourse.