Utonality and Otonality in Just Intonation
In the realm of music theory and microtonalism, the concepts of otonality and utonality provide a mathematical framework for understanding how chords are constructed based on the physics of sound. While traditional harmony often relies on equal temperament, these concepts are rooted in just intonation—a system of tuning where frequencies are related by whole-number ratios.
These terms were used extensively by composer Harry Partch in his seminal work, Genesis of a Music, to describe the relationship between pitches and their underlying series.
Key Facts
- Otonality is based on the harmonic series (overtones) and uses ratios with equal denominators.
- Utonality is the inversion of otonality, based on the subharmonic series (undertones) and uses ratios with equal numerators.
- Otonalities correspond to an arithmetic series of frequencies, while utonalities correspond to an arithmetic series of wavelengths.
- Ambitonal chords are those whose odd limit remains unchanged when melodically inverted.
- Harry Partch used the term Monophony to describe a system of just intervals derived from a single starting pitch.
Otonality: The Harmonic Foundation
An otonality is a collection of pitches expressed as ratios with equal denominators and consecutive numerators. For instance, a just major chord can be represented by the ratios 4/4, 5/4, and 6/4, which simplifies to the extended ratio 4:5:6. Because these ratios are derived from the harmonic series (the sequence of overtones produced by a fundamental tone), otonalities are naturally found in the physics of sound.
Practical examples of otonalities occur in nature and performance: brass instruments naturally produce them, and Tuvan Khoomei singers utilize their vocal tracts to create these harmonic structures. Mathematically, an otonality corresponds to an arithmetic series of frequencies, such as 110 Hz, 220 Hz, 330 Hz, and 440 Hz.
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Utonality: The Subharmonic Mirror
A utonality is the inversion of an otonality. Instead of building intervals upward from a root, a utonality is formed by building the same interval sequence downward. This mirrors the subharmonic series (also known as the undertone series).
In a utonality, the ratios share the same numerator but have consecutive denominators. For example, the ratios 7/4, 7/5, 7/6, and 7/7 (1/1) form a utonality, often written as 7/7:6:5:4. While otonality relates to frequency, utonality corresponds to an arithmetic series of wavelengths (the inverse of frequency), such as 1 foot, 2 feet, 3 feet, and 4 feet.
Defining Otonal and Utonal Chords
There is a distinction between a strict "utonality/otonality" and a chord that is "utonal/otonal." A narrow definition requires the members of the harmonic or subharmonic series to be adjacent. Under this strict rule, only a few chords qualify:
- Otonality Triads: Major triad (4:5:6) and diminished triad (5:6:7).
- Otonality Tetrad: Dominant seventh tetrad (4:5:6:7).
Microtonalists have expanded these definitions using the concept of the odd limit—the largest odd number found in the prime factorization of the numbers in a chord's extended ratio. To determine if a chord is otonal or utonal, it is subjected to melodic inversion (turning the intervals upside down, such as C–E–G becoming C–A♭–F).
- Otonal: The odd limit increases after melodic inversion. (e.g., a major triad 4:5:6 has an odd limit of 5; its inverse 10:12:15 has an odd limit of 15).
- Utonal: The odd limit decreases after melodic inversion.
- Ambitonal: The odd limit remains unchanged.
Examples of ambitonal chords include the major sixth chord (12:15:18:20) and the major seventh chord (8:10:12:15). These chords are often ambiguous and can be interpreted as either major or minor depending on the context.
Summary Table of Tonalities
| Feature | Otonality | Utonality |
|---|---|---|
| Basis | Harmonic Series (Overtones) | Subharmonic Series (Undertones) |
| Ratio Structure | Equal denominators, consecutive numerators | Equal numerators, consecutive denominators |
| Physical Property | Arithmetic series of frequencies | Arithmetic series of wavelengths |
| Example Ratio | 4:5:6 (Major Triad) | 7/7:6:5:4 |
| Melodic Inversion | Odd limit increases | Odd limit decreases |
Frequently Asked Questions
What is the difference between an otonality and an otonal chord?
An otonality is a strict subset of otonal chords. To be an otonality, the harmonic series members must be adjacent (e.g., 4:5:6). An otonal chord is defined more broadly: any chord whose odd limit increases upon melodic inversion is considered otonal, regardless of whether the members are adjacent.
What is an ambitonal chord?
An ambitonal chord is a chord whose odd limit remains the same after melodic inversion. Because of this symmetry, ambitonal chords, such as the major seventh chord, can often be interpreted as either major or minor.
How does melodic inversion differ from standard chord inversion?
Standard inversion involves moving the root note to a different position in the chord (e.g., C–E–G becoming E–G–C). Melodic inversion involves flipping the intervals upside down; for example, a major triad (C–E–G) becomes a minor-like structure (C–A♭–F).
Who developed these specific terms?
The terms utonality and otonality were used extensively by the composer Harry Partch in his work Genesis of a Music as part of his exploration of just intonation and Monophony.