Intonation and Microtonality

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About fifteen years ago, I first came across a copy of Harry Partch’s Genesis of a Music while browsing in the university library in St. John’s, Newfoundland. Picking up the autographed title by chance curiosity, I became fascinated by the mixture of archaic-looking mathematical ratios and Partch’s passionate and polemic prose, with which he lambasted the American classical music establishment of the 1940s. His radical re-analysis of musical intervals dismissed the inaccuracies of the tempered tradition and embraced the possibilities of the harmonic series’ more distant reaches. The freshness of his approach sparked my own interest in intonation and microtonality.

I would describe intonation as the art of selecting pitches, or (more accurately) pitch-”regions” along the glissando-continuum of pitch-height (following James Tenney’s description in the 1983 article “John Cage and the Theory of Harmony”). The “tolerance” or exactitude of such regions varies based on the instrument and musical style.

In this context, microtonality is an approach to pitch which acknowledges the musical possibility of this entire glissando-continuum and is not limited to the conventional twelve equal tempered pitch-classes.

Georg Friedrich Haas, in his article “Mikrotonalitäten” (1999), distinguishes four approaches to microtonal composition:

  1. music based on beating and detuning phenomena
    (i.e. Scelsi, Feldman, Lucier)
  2. music based on inharmonic equal-division systems
    (i.e. Hába, Carrillo, Wyschnegradsky; atonal music in 12-ET)
  3. music based on irrational/inharmonic structures
    (i.e. Varese, Cage, Lachenmann; random tunings, percussion spectra, multiphonics)
  4. music based on harmonic Just Intonation (JI)
    (i.e. Partch, Johnston, LaMonte Young, Tenney)

In addition, it is possible to consider the historical temperaments (Meantone and Well-Tempered) as hybrid systems combining the harmonic implications of JI with detuning. Such hybrid (“tempered” semi-harmonic) systems also include certain unconventional equal temperaments—for example 19-, 31-, 53-, 55-, and 72- ETs (among others). Especially important to contemporary composers is 72-ET (based on 1/12 tones), which has been used (among others) by Ezra Sims, Joseph Maneri, James Tenney, and in Europe by Hans Zender. This system allows for a very close approximation of many 11-limit JI ratios advocated by Harry Partch without abandoning the advantages of the conventional tempered semitones.

The increasing plurality of compositional approaches, as well as the diversity of tunings required for many traditional musics (i.e. Arabic, Chinese, European “Early Music”, Indian, Indonesian, etc.) demands a new level of refinement from instrumentalists, an ability to distinguish and reproduce different tunings. Intonation is no longer simplistically reduced to “in tune” vs. “out of tune” based on the nearest possible “consonance”. Contemporary practice demands a differentiation of various sonorities (from the simplest to the most complex), and cultivation of the ability to tune them on various instruments.

In my own approach to intonation, I am especially interested in the distinction between tunable intervals (“consonances” in an extended sense) and tunable dissonances (intervals which may be tuned through a succession of consonances). This gives me a harmonically motivated definition of intonation based on pitches which may be directly produced on instruments and sets of intervals which may be directly derived from such pitches. Subsets of tunable interval combinations can lead to different scales and tone-systems, which I generally develop based on the instruments involved in a given piece. Given a reasonable degree of tolerance for mistuning (in simultaneous sounds, +/- 2 cents), this method allows for a fairly accurate realization of many microtonal pitches because the players are always able to construct the pitches required by ear.

The first step in any microtonal intonation is an accurate method of notation. Absolute pitch-height may be notated in cents, by placing a deviation (+/-) above the appropriate accidental. This allows reference to the tempered system and to electronic tuning devices. However, as a harmonic notation it is insufficient. To accurately specify complex harmonic relations, it is necessary to know the frequency ratio of the intended interval. Following Harry Partch’s approach, one might simply write the intended ratio above the interval, for example: taking the perfect fifth G-D one would write (-3.9) for the G and (-2.0) for the D. Then above the interval, a ratio (3/2).

An alternative would be a precise just intonation notation using modified accidentals. In this case, there are two alternative approaches. In the 1950s Ben Johnston (a student of Partch) developed a tonally motivated JI pitch notation, based on the C major scale (1/1 – 9/8 – 5/4 – 4/3 – 3/2 – 5/3 – 15/8 – 2/1). Though radical in its time and significant historically, Johnston’s approach is fundamentally flawed because it is asymmetric—intervals which we have learned to accept as perfect fifths are not always so in his system (Bb-F, D-A, F#-C# are all one comma smaller than a perfect fifth with ratio 3/2).

An earlier approach to JI notation was established in the mid-1800s by Oettingen and used by Helmholtz and Riemann. It forms the basis of a notation system I have developed together with Wolfgang von Schweinitz, and which we call The Extended Helmholtz-Ellis JI Pitch Notation. It is based on the Pythagorean interpretation of the series of perfect fifths, notated with double-flats, flats, naturals, sharps, double sharps. (Cage uses this notation in his 1977 Cheap Imitation for solo violin).

For each new prime number, a comma-alteration is defined so that all ratios can be graphically notated: for example, the Syntonic comma (prime number 5 in the harmonic series) is indicated by attaching arrows to the normal accidentals. A similar approach (with different symbols) is used in the independently developed notations of Daniel Wolf and Joe Monzo. In our version, the accidental symbols include signs for tempered pitches and may also be combined with cents indications (invented by Ellis), allowing pitches to be read (both) harmonically and/or melodically.

[See illustration 1]

Using a harmonic notation with cents allows a composer to precisely analyze the intervallic structures which are inherent in string and brass instruments. Both families are based on the principle of open strings/tube lengths with harmonics/overtones. Taking the open G-string on the violin, for example, it is possible to notate each of the natural harmonics and to distinguish precisely the nodes at which these natural pitches may be realized.

[See illustration 2]

In this sense, it is possible to explicitly define and notate a harmonic approach to microtonality. On a given instrument, it is possible to arbitrarily tune each of the strings or tube lengths (scordatura/valve tuning). From each string or tube-length, it is theoretically possible to produce harmonic partials.

On the string instruments, these partials occur at nodes: when lightly pressed, the harmonic partial sounds; when pressed down to the fingerboard, the stopped node sounds. It is important to note several practicalities: (1) the partials are fairly well-tuned until 8, at which point they gradually become slightly sharper than the theoretical harmonics, at a rate of circa 4 cents per partial. This detuning due to inharmonicity varies depending on the thickness and stiffness of the string (thinner, longer, more flexible strings are more perfectly tuned). (2) The pitches indicated as nodal points are exactly accurate only if the string is very precisely pinched at the nodal point. The process of pressing down the finger and string against the fingerboard causes the pitch to rise and must be taken into account by the player and the composer when working with precisely tuned nodal pitches.

In addition to these harmonics and nodes, it is possible for a violinist to tune double-stops to open strings and to natural harmonics. As long as the double-stop is one of the intervals tunable by ear the player will be able to accurately construct many new pitches by this method.

[See illustration 3]

The composer must bear in mind the physical constraints of hand position and the reality that higher harmonics do not speak as readily, but certainly the first five partials may be used as “starting points” in this manner. It is possible to repeat this process several times to achieve complex tuned dissonances (at each stage bearing in mind the limitations of possible hand positions).

For example: Play the open D-string together with B-natural one comma down (-17.6 cents) on the A string (1st finger, pure major sixth 5/3). To the comma-lowered B-natural, tune an A-natural lowered by a comma and a septimal comma on the E string (-48.8 cents, 3rd finger, natural seventh 7/4). Then play the open A string (0 cents) together with the lowered A (-48.8 cents). This is a beating, tuned “quartertone-diminished octave” with the ratio 35/18.

On brass instruments, similar analogies are possible with a careful analysis of the valve-combinations. In this case, it is important to note the fundamentals of each valve-combination. In the case of the horn, for example, if the three valves are respectively tuned to 2/15, 1/15, and 3/15 of the open horn’s length, then the combinations will add up to produce lengths in the proportion 15:16:17:18:19:20:21. These form a descending sequence of pitches with an interval structure mirroring the harmonic series in the downward direction (sometimes called a subharmonic series). Over each of these fundamentals, the player is able to play a harmonic series of overtones (on the horn especially, the higher harmonics may be very accurately realized, even as high as the 20th partial).

[See illustration 4]

Using such instrument-based techniques, it is possible for composers to begin to investigate harmonic microtonality on acoustic instruments with confidence that musicians will readily be able to hear and reproduce these pitches. For the composer further interested in microtonality, there are a number of fine freeware tools available for investigating intervals.

Manuel Op de Coul’s “Scala” program, available for PC, Linux, and Mac OS X is one highly recommended starting point.

Freeware fonts for Sibelius and Finale, more information about the Helmholtz-Ellis Notation, as well as music examples, can be found on my website.

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Marc Sabat

Marc Sabat is a Canadian composer and violinist living in Berlin since 1999. He has written concert music for various ensembles including acoustic instruments, live computer and electronics, as well as making recorded projects involving sound and video (installation, DVD, and internet). He has recently developed The Extended Helmholtz-Ellis JI Pitch Notation and is currently teaching a course in acoustics and experimental intonation at the Universität der Künste Berlin. Sabat also performs chamber music and solo concerts and has recorded music on various labels including mode records, World Edition, and HatArt. Marc Sabat studied at the University of Toronto and the Juilliard School of Music in New York.

contact: Plainsound Music Edition, info@plainsound.org