- Otonality and Utonality
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5-limit Otonality and Utonality: overtone and "undertone" series, partials 1-5 numbered
Play Otonality (help·info), Play Utonality (help·info), Play major chord on C (help·info), and Play minor chord on F (help·info).Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones (multiples) or "undertones" (divisors) of a given fixed tone. For example: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,....
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Definition
An Otonality is a collection of pitches which can be expressed in ratios, expressing their relationship to the fixed tone, that have equal denominators. For example, 1/1, 5/4, and 3/2 (just major chord) form an Otonality because they can be written as 4/4, 5/4, 6/4. Every Otonality is therefore part of the harmonic series. Similarly, the ratios of an Utonality share the same numerator. 7/4, 7/5, 7/6, and 1/1 (7/7) form an Utonality.
An Otonality corresponds to an arithmetic series of frequencies or a harmonic series of wavelengths or lengths of a vibrating string. Brass instruments naturally produce Otonalities, and indeed Otonalities are inherent in the harmonics of a single fundamental tone. Tuvan khoomei singers produce Otonalities with their vocal tracts.
Utonality is the opposite, corresponding to a harmonic series of frequencies or an arithmetic series of wavelengths. The arithmetical proportion "may be considered as a demonstration of Utonality ('minor tonality')."[1]
Relationship to standard Western music theory
The 5-limit Otonality is simply a just major chord, and the 5-limit Utonality is a just minor chord. Thus Otonality and Utonality can be viewed as extensions of major and minor tonality respectively. However, whereas standard music theory views a minor chord as being built up from the root with a minor third and a perfect fifth, an Utonality is viewed as descending from what's normally considered the "fifth" of the chord, so the correspondence is not perfect. This corresponds with the dualistic theory of Hugo Riemann:
Minor as upside down major.In the era of meantone temperament, augmented sixth chords of the kind known as the German sixth (or the English sixth, depending on how it resolves) were close in tuning and sound to the 7-limit Otonality, called the tetrad. This chord might be, for example, A♭-C-E♭-F♯, where the F♯ replaces G♭, which would have made it a dominant seventh. Standing alone, it has something of the sound of a dominant seventh, but considerably less dissonant. It has also been suggested that the Tristan chord, for example, F-B-D♯-G♯ can be considered a Utonality, or 7-limit utonal tetrad, which it closely approximates if the tuning is meantone, though presumably less well in the tuning of a Wagnerian orchestra.
Criticism
Though Partch presents Otonality and Utonality as being equal and symmetric concepts, when played on most physical instruments an Otonality sounds much more consonant than a similar Utonality, due to the presence of difference tones. In an Otonality all the difference tones are elements of the same harmonic series, so they reinforce the tonality, but in an Utonality the difference tones do not reinforce the existing overtone series and tend to destabilize the tonality.
It could be argued that Otonality and Utonality are equally consonant simply because they have the same sets of intervals between pairs of pitches, but in that case there are other pitch collections that must be considered. For example, the chord 9:10:12:15 is not part of any Otonality or Utonality below the 15 limit, but all its intervals are consonant within the 9 limit.
See also
Sources
- ^ Partch, Harry. Genesis of a Music, p.69. 2nd ed. Da Capo Press, 1974. ISBN 0-306-80106-X.
External links
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