![Undertone series on C](https://upload.wikimedia.org/wikipedia/commons/f/f2/Undertone_series_on_C.png)
''Otonality''
and ''utonality''
are terms introduced by
Harry Partch
Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...
to describe
chords
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ( ...
whose
pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
es are the
harmonics or
subharmonics of a given fixed
tone (
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
), respectively. For example: , , ,... or , , ,....
Definition
An otonality
is a collection of pitches which can be expressed in
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s, expressing their relationship to the fixed tone, that have equal
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s and consecutive
numerator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s. For example,
,
, and
(
just
Just or JUST may refer to:
__NOTOC__ People
* Just (surname)
* Just (given name)
Arts and entertainment
* ''Just'', a 1998 album by Dave Lindholm
* "Just" (song), a song by Radiohead
* "Just", a song from the album ''Lost and Found'' by Mudvayne ...
major chord
In music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the " rudiments", that are needed to understan ...
) form an otonality because they can be written as , , . This in turn can be written as an extended ratio 4:5:6. Every otonality is therefore composed of members of a
harmonic series. Similarly, the ratios of a utonality share the same numerator and have consecutive denominators.
,
,
, and () form a utonality, sometimes written as , or as . Every utonality is therefore composed of members of a
subharmonic series
In music, the undertone series or subharmonic series is a sequence of Musical note, notes that results from inversion (music), inverting the intervals of the harmonic series (music), overtone series. While overtones naturally occur with the phys ...
. This term is used extensively by Harry Partch in ''Genesis of a Music''.
An otonality corresponds to an
arithmetic series
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
of
frequencies, or lengths of a
vibrating string
A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating ...
.
Brass instruments naturally produce otonalities, and indeed otonalities are inherent in the harmonics of a single fundamental tone.
Tuva
Tuva (; russian: Тува́) or Tyva ( tyv, Тыва), officially the Republic of Tuva (russian: Респу́блика Тыва́, r=Respublika Tyva, p=rʲɪˈspublʲɪkə tɨˈva; tyv, Тыва Республика, translit=Tyva Respublika ...
n
Khoomei singers produce otonalities with their vocal tracts.
Utonality
is the opposite, corresponding to a subharmonic series of frequencies, or an arithmetic series of
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
s (the
inverse of frequency). The ''arithmetical proportion'' "may be considered as a demonstration of utonality ('minor tonality')."
If otonality and utonality are defined broadly, every just intonation chord is both an otonality and a utonality. For example, the minor triad in root position is made up of the 10th, 12th and 15th harmonics, and , and meets the definition of otonal. A better, narrower definition requires that the harmonic (or subharmonic) series members be adjacent. Thus 4:5:6 is an otonality, but 10:12:15 is not. (Alternate voicings of 4:5:6, such as 5:6:8, 3:4:5:6, etc. would presumably also be otonalities.) Under this definition, only a few chord types qualify as otonalities or utonalities. The only otonality triads are the
major triad
In music theory, a major chord is a chord that has a root, a major third, and a perfect fifth. When a chord comprises only these three notes, it is called a major triad. For example, the major triad built on C, called a C major triad, has pitch ...
4:5:6 and the
diminished triad
In music theory, a diminished triad (also known as the minor flatted fifth) is a triad consisting of two minor thirds above the root. It is a minor triad with a lowered ( flattened) fifth. When using chord symbols, it may be indicated by the s ...
5:6:7. The only such tetrad is the
dominant seventh
In music theory, a dominant seventh chord, or major minor seventh chord, is a seventh chord, usually built on the fifth degree of the major scale, and composed of a root, major third, perfect fifth, and minor seventh. Thus it is a major triad t ...
tetrad 4:5:6:7.
Microtonalists have extended the concept of otonal and utonal to apply to all just intonation chords. A chord is otonal if its
odd limit increases on being
melodically inverted, utonal if its odd limit decreases, and ambitonal if its odd limit is unchanged. Melodic inversion is not inversion in the usual sense, in which C–E–G becomes E–G–C or G–C–E. Instead, C–E–G is turned upside down to become C–A–F. A chord's odd limit is the largest of the odd limits of each of the numbers in the chord's extended ratio. For example, the major triad in close position is 4:5:6. These three numbers have odd limits of 1, 5 and 3 respectively. The largest of the three is 5, thus the chord has an odd limit of 5. Its melodic inverse 10:12:15 has an odd limit of 15, which is greater, therefore the major triad is otonal. A chord's odd limit is independent of its voicing, so alternate voicings such as 5:6:8, 3:4:5:6, etc. are also otonal.
All otonalities are otonal, but not all otonal chords are otonalities. Likewise, all utonalities are a subset of utonal chords.
The
major ninth
In music, a ninth is a compound interval consisting of an octave plus a second.
Like the second, the interval of a ninth is classified as a dissonance in common practice tonality. Since a ninth is an octave larger than a second, its ...
chord 8:10:12:15:18 is also otonal. Examples of ambitonal chords are the
major sixth
In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions (see Interval number for more details), and the major sixth is one of two commonly occurring sixths. It is qualified as ''major ...
chord (12:15:18:20) and the
major seventh
In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. It is qualified as ''major'' because it i ...
chord (8:10:12:15). Ambitonal chords often can be reasonably interpreted as either major or minor. For example, C
M6, in certain contexts or voicings, can be interpreted as Am
7.
Relationship to standard Western music theory
Partch said that his 1931 coinage of "otonality" and "utonality" was "hastened" by having read
Henry Cowell's discussion of undertones in ''New Musical Resources'' (1930).
The 5-
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
otonality is simply a just major chord, and the 5-limit utonality is a just
minor chord
In music theory, a minor chord is a chord that has a root, a minor third, and a perfect fifth. When a chord comprises only these three notes, it is called a minor triad. For example, the minor triad built on C, called a C minor triad, has pit ...
. 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
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...
and a
perfect fifth, a 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
Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. M ...
,
augmented sixth chord
In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, was further developed in the Baroque, and became a distinctive part of the musi ...
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
7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14 ...
otonality, called the
tetrad
Tetrad ('group of 4') or tetrade may refer to:
* Tetrad (area), an area 2 km x 2 km square
* Tetrad (astronomy), four total lunar eclipses within two years
* Tetrad (chromosomal formation)
* Tetrad (general relativity), or frame field
** Tetra ...
. This chord might be, for example, A-C-E-G
. 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
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 bass note. It is so named as it is ...
, 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.
Whereas
5-limit chords associate otonal with major and utonal with minor,
7-limit
7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14 ...
chords that don't use 5 as a prime factor reverse this association. For example, 6:7:9 is otonal but minor, and 14:18:21 is utonal but major.
Consonance
Though Partch presents otonality and utonality as being equal and symmetric concepts, when played on most physical instruments an otonality sounds much more
consonant
In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wi ...
than a similar utonality, due to the presence of the
missing fundamental phenomenon. In an otonality, all of the notes are elements of the same
harmonic series, so they tend to partially activate the presence of a "virtual" fundamental as though they were harmonics of a single complex pitch. Utonal chords, while containing the same dyads and roughness as otonal chords, do not tend to activate this phenomenon as strongly. There are more details in Partch's work.
Use
![Mystic chord on C overtone scale](https://upload.wikimedia.org/wikipedia/commons/f/ff/Mystic_chord_on_C_overtone_scale.png)
Partch used otonal and utonal chords in his music.
Ben Johnston often uses the otonal as an expanded tonic chord: 4:5:6:7:11:13 (C:E:G:B:F:A) and bases the opening of the third movement of his String Quartet No. 10 on this thirteen-limit Otonality on C. The
mystic chord
In music, the mystic chord or Prometheus chord is a six-note synthetic chord and its associated scale, or pitch collection; which loosely serves as the harmonic and melodic basis for some of the later pieces by Russian composer Alexander Scriabi ...
has been theorized as being derived from harmonics 8 through 14 without 12: 8:9:10:11:13:14 (C:D:E:F:A:B), and as harmonics 7 through 13: 7:8:9:10:(11:)12:13 (C:D:E:F:(G:)A:B); both otonal.
Yuri Landman
Yuri Landman (born 1 February 1973) is a Dutch inventor of musical instruments and musician who has made several experimental electric string instruments for a number of artists including Lee Ranaldo of Sonic Youth, Liars, Jad Fair of Half Japan ...
published a microtonal diagram that compares series of otonal and utonal scales with
12TET
Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resulti ...
and the
harmonic series.
[http://www.hypercustom.nl/utonaldiagram.jpg ] He applies this system for just
transposition with a set of electric microtonal
koto
Koto may refer to:
* Koto (band), an Italian synth pop group
* Koto (instrument), a Japanese musical instrument
* Koto (kana), a ligature of two Japanese katakana
* Koto (traditional clothing), a traditional dress made by Afro-Surinamese women
* K ...
s.
See also
*
Scale of harmonics
The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale.Yin, Wei. ''Z ...
*
Tonality flux ''Tonality flux'' is Harry Partch's termPartch, Harry (1949). ''Genesis of a Music'', p.188-190. Da Capo Press . for the kinds of subtle harmonic changes that can occur in a microtonal context from notes moving from one chord to another by tiny in ...
*
Tonality diamond
In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthol ...
References
External links
Otonality and ADO systemat ''96-EDO''
Utonality and EDL systemat ''96-EDO''
{{DEFAULTSORT:Otonality And Utonality
Harmony
Harry Partch
Pitch (music)
Musical tuning