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Otonality and utonality

January 01, 1970

"Otonal" redirects here. For a Spanish word "otoñal" meaning autumnal, see es:otoñal.

Otonality[1] and utonality[2] are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone (identity[3]), respectively. For example: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,....

5-limit otonality and utonality: overtone and "undertone" series, partials 1-5 numbered Play otonalityⓘ, Play utonalityⓘ, Play major chord on Cⓘ, and Play minor chord on Fⓘ.
31-limit otonality Playⓘ
13-limit utonality Playⓘ

An Otonality is that set of pitches generated by the numerical factors (...identities)...over a numerical constant (...numerary nexus) in the denominator. Conversely, a Utonality is the inversion of an Otonality, a set of pitches with a numerical constant in the numerator over the numerical factors...in the denominator.[4]

Definition edit

G(=1/1), A(=9/8), ♮5B(=5/4), ♯11B(=11/8), D(=3/2), ♮7F(=7/4)
Otonality on G = lower line of the tonality diamond bottom left to top right.
G(=1/1), F(=16/9), 5♭E(=8/5), 11♭D(=16/11), C(=4/3), 7♭A(=8/7)
Utonality under G = lower line of the tonality diamond bottom right to top left.

A Utonality is a ...chord that is the inversion of an Otonality: it is formed by building the same interval sequence as that of an Otonality downward from the root of the chord, rather than upward. The analogy, in this case, is not to the harmonic series but to the subharmonic, or undertone series.[5]

An otonality[1] is a collection of pitches which can be expressed in ratios, expressing their relationship to the fixed tone, that have equal denominators and consecutive numerators. 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. 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. 7/4, 7/5, 7/6, and 1/1 (7/7) form a utonality, sometimes written as 1/4:5:6:7, or as 7/7:6:5:4. Every utonality is therefore composed of members of a subharmonic series. This term is used extensively by Harry Partch in Genesis of a Music.[3]

An otonality corresponds to an arithmetic series of frequencies, 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[2] is the opposite, corresponding to a subharmonic series of frequencies, or an arithmetic series of wavelengths (the inverse of frequency). The arithmetical proportion "may be considered as a demonstration of utonality ('minor tonality')."[6]

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 10/10, 12/10 and 15/10 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 4:5:6 and the diminished triad 5:6:7. The only such tetrad is the dominant seventh 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.[7] 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 chord 8:10:12:15:18 is also otonal. Examples of ambitonal chords are the major sixth chord (12:15:18:20) and the major seventh chord (8:10:12:15). Ambitonal chords often can be reasonably interpreted as either major or minor. For example, CM6, in certain contexts or voicings, can be interpreted as Am7.

Relationship to standard Western music theory edit

Main article: Harmonic dualism

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).[5]

 
Starting from the symmetrical chords, otonal chords flatten one note, while utonal chords sharpen one note.

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, a utonality is viewed as descending from what's normally considered the "fifth" of the chord,[citation needed] 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♭-G ♭[F♯] Playⓘ. 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.

Whereas 5-limit chords associate otonal with major and utonal with minor, 7-limit 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 edit

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 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.[3]

Use edit

 
Mystic chord as the 1st, 11th, 7th, 5th, 13th, and 9th harmonics (quarter tone approximation) Play 8-14ⓘ (Play 7-13ⓘ)

Partch used otonal and utonal chords in his music. Ben Johnston[8] 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.[9] The mystic chord 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 published a microtonal diagram that compares series of otonal and utonal scales with 12TET and the harmonic series.[10] He applies this system for just transposition with a set of electric microtonal kotos.

See also edit

References edit

  1. ^ a b Partch, Harry, 1901-1974 (1974). Genesis of a music : an account of a creative work, its roots and its fulfillments (Second edition, enlarged ed.). New York. p. 72. ISBN 0-306-71597-X. OCLC 624666.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  2. ^ a b Partch, Harry, 1901-1974 (1974). Genesis of a music : an account of a creative work, its roots and its fulfillments (Second edition, enlarged ed.). New York. p. 75. ISBN 0-306-71597-X. OCLC 624666.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  3. ^ a b c Partch, Harry, 1901-1974 (August 1974). Genesis of a music : an account of a creative work, its roots and its fulfillments (Second edition, enlarged ed.). New York. ISBN 0-306-71597-X. OCLC 624666.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  4. ^ Gilmore, Bob (1998). Harry Partch: A Biography, p.431, n.69. Yale. ISBN 9780300065213.
  5. ^ a b Gilmore, Bob (1998). Harry Partch: A Biography, p.68. Yale. ISBN 9780300065213.
  6. ^ Partch, Harry. Genesis of a Music, p.69. 2nd ed. Da Capo Press, 1974. ISBN 0-306-80106-X.
  7. ^ "Otonality and utonality", Xenharmonic Wiki. Opens with: "For the basic concepts, see the Wikipedia article Otonality and Utonality." Accessed: December 18, 2017.
  8. ^ Johnston, Ben. (2006). "Maximum clarity" and other writings on music. Gilmore, Bob, 1961-2015. Urbana: University of Illinois Press. ISBN 978-0-252-09157-5. OCLC 811408988.
  9. ^ Coessens, Kathleen; ed. (2017). Experimental Encounters in Music and Beyond, p.104. Leuven. ISBN 9789462701106.
  10. ^ http://www.hypercustom.nl/utonaldiagram.jpg [bare URL image file]

External links edit

  • Otonality and ADO system at 96-EDO
  • Utonality and EDL system at 96-EDO

January 01, 1970
otonality, utonality, otonal, redirects, here, spanish, word, otoñal, meaning, autumnal, otoñal, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challen. Otonal redirects here For a Spanish word otonal meaning autumnal see es otonal This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Otonality and utonality news newspapers books scholar JSTOR January 2013 Learn how and when to remove this message Otonality 1 and utonality 2 are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone identity 3 respectively For example 1 1 2 1 3 1 or 1 1 1 2 1 3 5 limit otonality and utonality overtone and undertone series partials 1 5 numbered Play otonality Play utonality Play major chord on C and Play minor chord on F 31 limit otonality Play 13 limit utonality Play An Otonality is that set of pitches generated by the numerical factors identities over a numerical constant numerary nexus in the denominator Conversely a Utonality is the inversion of an Otonality a set of pitches with a numerical constant in the numerator over the numerical factors in the denominator 4 Contents 1 Definition 2 Relationship to standard Western music theory 3 Consonance 4 Use 5 See also 6 References 7 External linksDefinition editG 1 1 A 9 8 5 B 5 4 11 B 11 8 D 3 2 7 F 7 4 Otonality on G lower line of the tonality diamond bottom left to top right G 1 1 F 16 9 5 E 8 5 11 D 16 11 C 4 3 7 A 8 7 Utonality under G lower line of the tonality diamond bottom right to top left A Utonality is a chord that is the inversion of an Otonality it is formed by building the same interval sequence as that of an Otonality downward from the root of the chord rather than upward The analogy in this case is not to the harmonic series but to the subharmonic or undertone series 5 An otonality 1 is a collection of pitches which can be expressed in ratios expressing their relationship to the fixed tone that have equal denominators and consecutive numerators 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 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 7 4 7 5 7 6 and 1 1 7 7 form a utonality sometimes written as 1 4 5 6 7 or as 7 7 6 5 4 Every utonality is therefore composed of members of a subharmonic series This term is used extensively by Harry Partch in Genesis of a Music 3 An otonality corresponds to an arithmetic series of frequencies 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 2 is the opposite corresponding to a subharmonic series of frequencies or an arithmetic series of wavelengths the inverse of frequency The arithmetical proportion may be considered as a demonstration of utonality minor tonality 6 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 10 10 12 10 and 15 10 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 4 5 6 and the diminished triad 5 6 7 The only such tetrad is the dominant seventh 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 7 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 chord 8 10 12 15 18 is also otonal Examples of ambitonal chords are the major sixth chord 12 15 18 20 and the major seventh chord 8 10 12 15 Ambitonal chords often can be reasonably interpreted as either major or minor For example CM6 in certain contexts or voicings can be interpreted as Am7 Relationship to standard Western music theory editMain article Harmonic dualism 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 5 nbsp Starting from the symmetrical chords otonal chords flatten one note while utonal chords sharpen one note 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 a utonality is viewed as descending from what s normally considered the fifth of the chord citation needed so the correspondence is not perfect This corresponds with the dualistic theory of Hugo Riemann nbsp 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 G nbsp F Play 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 Whereas 5 limit chords associate otonal with major and utonal with minor 7 limit 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 editThis section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed October 2013 Learn how and when to remove this message 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 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 3 Use edit nbsp Mystic chord as the 1st 11th 7th 5th 13th and 9th harmonics quarter tone approximation Play 8 14 Play 7 13 Partch used otonal and utonal chords in his music Ben Johnston 8 often uses the otonal as an expanded tonic chord 4 5 6 7 11 13 C E G B nbsp F A nbsp and bases the opening of the third movement of his String Quartet No 10 on this thirteen limit Otonality on C 9 The mystic chord has been theorized as being derived from harmonics 8 through 14 without 12 8 9 10 11 13 14 C D E F A nbsp B nbsp and as harmonics 7 through 13 7 8 9 10 11 12 13 C D nbsp E nbsp F nbsp G nbsp A nbsp B nbsp nbsp both otonal Yuri Landman published a microtonal diagram that compares series of otonal and utonal scales with 12TET and the harmonic series 10 He applies this system for just transposition with a set of electric microtonal kotos See also editScale of harmonics Tonality flux Tonality diamondReferences edit a b Partch Harry 1901 1974 1974 Genesis of a music an account of a creative work its roots and its fulfillments Second edition enlarged ed New York p 72 ISBN 0 306 71597 X OCLC 624666 a href Template Cite book html title Template Cite book cite book a CS1 maint location missing publisher link CS1 maint multiple names authors list link CS1 maint numeric names authors list link a b Partch Harry 1901 1974 1974 Genesis of a music an account of a creative work its roots and its fulfillments Second edition enlarged ed New York p 75 ISBN 0 306 71597 X OCLC 624666 a href Template Cite book html title Template Cite book cite book a CS1 maint location missing publisher link CS1 maint multiple names authors list link CS1 maint numeric names authors list link a b c Partch Harry 1901 1974 August 1974 Genesis of a music an account of a creative work its roots and its fulfillments Second edition enlarged ed New York ISBN 0 306 71597 X OCLC 624666 a href Template Cite book html title Template Cite book cite book a CS1 maint location missing publisher link CS1 maint multiple names authors list link CS1 maint numeric names authors list link Gilmore Bob 1998 Harry Partch A Biography p 431 n 69 Yale ISBN 9780300065213 a b Gilmore Bob 1998 Harry Partch A Biography p 68 Yale ISBN 9780300065213 Partch Harry Genesis of a Music p 69 2nd ed Da Capo Press 1974 ISBN 0 306 80106 X Otonality and utonality Xenharmonic Wiki Opens with For the basic concepts see the Wikipedia article Otonality and Utonality Accessed December 18 2017 Johnston Ben 2006 Maximum clarity and other writings on music Gilmore Bob 1961 2015 Urbana University of Illinois Press ISBN 978 0 252 09157 5 OCLC 811408988 Coessens Kathleen ed 2017 Experimental Encounters in Music and Beyond p 104 Leuven ISBN 9789462701106 http www hypercustom nl utonaldiagram jpg bare URL image file External links editOtonality and ADO system at 96 EDO Utonality and EDL system at 96 EDO Retrieved from https en wikipedia org w index php title Otonality and utonality amp oldid 1176196137, wikipedia, wiki, book, books, library,

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