Otonality and Utonality
Encyclopedia
Otonality and Utonality are terms introduced by Harry Partch
to describe chords
whose notes are the overtone
s (multiple
s) or "undertones" (divisor
s) of a given fixed tone
. For example: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,....
s, 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 instrument
s naturally produce Otonalities, and indeed Otonalities are inherent in the harmonics of a single fundamental tone. Tuva
n 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')."
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
:
In the era of meantone temperament
, augmented sixth chord
s 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.
than a similar Utonality, due to the presence of difference tone
s. 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.
Harry Partch
Harry Partch was an American composer and instrument creator. He was one of the first twentieth-century composers to work extensively and systematically with microtonal scales, writing much of his music for custom-made instruments that he built himself, tuned in 11-limit just intonation.-Early...
to describe chords
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...
whose notes are the overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...
s (multiple
Multiple
The word multiple can refer to:*Multiple , multiples of numbers*List of multiple discoveries, instances of scientists, working independently of each other, reaching similar findings...
s) or "undertones" (divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
s) of a given fixed tone
Root (chord)
In music theory, the root of a chord is the note or pitch upon which a triadic chord is built. For example, the root of the major triad C-E-G is C....
. For example: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,....
Definition
An Otonality is a collection of pitches which can be expressed in ratioRatio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s, expressing their relationship to the fixed tone, that have equal denominators. For example, 1/1
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
, 5/4
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
, and 3/2
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
(just major chord
Major chord
In music theory, a major chord is a chord having a root, a major third, and a perfect fifth. When a chord has these three notes alone, it is called a major triad...
) form an Otonality because they can be written as 4/4, 5/4, 6/4. Every Otonality is therefore part of the harmonic series
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
. Similarly, the ratios of an Utonality share the same numerator. 7/4, 7/5
Septimal tritone
The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7...
, 7/6
Septimal minor third
In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...
, and 1/1 (7/7) form an Utonality.
An Otonality corresponds to an arithmetic series of frequencies or a harmonic series
Harmonic series (mathematics)
In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
of wavelengths or lengths of a vibrating string
String instrument
A string instrument is a musical instrument that produces sound by means of vibrating strings. In the Hornbostel-Sachs scheme of musical instrument classification, used in organology, they are called chordophones...
. Brass instrument
Brass instrument
A brass instrument is a musical instrument whose sound is produced by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips...
s naturally produce Otonalities, and indeed Otonalities are inherent in the harmonics of a single fundamental tone. Tuva
Tuva
The Tyva Republic , or Tuva , is a federal subject of Russia . It lies in the geographical center of Asia, in southern Siberia. The republic borders with the Altai Republic, the Republic of Khakassia, Krasnoyarsk Krai, Irkutsk Oblast, and the Republic of Buryatia in Russia and with Mongolia to the...
n khoomei
Khoomei
Khoomei , also spelled Xoomii, Xöömei, Xöömej, Khöömei, or Höömey, refers to...
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')."
Relationship to standard Western music theory
The 5-limitLimit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
Otonality is simply a just
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
major chord
Major chord
In music theory, a major chord is a chord having a root, a major third, and a perfect fifth. When a chord has these three notes alone, it is called a major triad...
, and the 5-limit Utonality is a just minor chord
Minor chord
In music theory, a minor chord is a chord having a root, a minor third, and a perfect fifth.When a chord has these three notes alone, it is called a minor triad....
. 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
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
and a perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
, 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
Hugo Riemann
Karl Wilhelm Julius Hugo Riemann was a German music theorist.-Biography:Riemann was born at Grossmehlra, Schwarzburg-Sondershausen. He was educated in theory by Frankenberger, studied the piano with Barthel and Ratzenberger, studied law, and finally philosophy and history at Berlin and Tübingen...
:
In the era of meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
, augmented sixth chord
Augmented sixth chord
In music theory, an augmented sixth chord contains the interval of an augmented sixth above its "root" or bass tone . This chord has its origins in the Renaissance, further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.-Resolution...
s 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
Tristan chord
The Tristan chord is a chord made up of the notes F, B, D and G. More generally, it can be any chord that consists of these same intervals: augmented fourth, augmented sixth, and augmented ninth above a root...
, 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 consonantConsonance
Consonance is a stylistic device, most commonly used in poetry and songs, characterized by the repetition of the same consonant two or more times in short succession, as in "pitter patter" or in "all mammals named Sam are clammy".Consonance should not be confused with assonance, which is the...
than a similar Utonality, due to the presence of difference tone
Combination tone
A combination tone, also called a sum tone or a difference tone , can be any of at least three similar psychoacoustic phenomena. When two tones are played simultaneously, a listener can sometimes perceive an additional tone whose frequency is a sum or difference of the two frequencies...
s. In an Otonality all the difference tones are elements of the same harmonic series
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
, 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
Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
, but all its intervals are consonant within the 9 limit.