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36-418: 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. A Utonality is a ...chord that is the inversion of an Otonality: it is formed by building

72-457: 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

108-540: A microtonal diagram that compares series of otonal and utonal scales with 12TET and the harmonic series . He applies this system for just transposition with a set of electric microtonal kotos . Numerary nexus 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 . Thus the n-limit tonality diamond ("limit" here

144-572: 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, so the correspondence is not perfect. This corresponds with the dualistic theory of Hugo Riemann : 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

180-417: A ninth is an octave larger than a second, its sonority level is considered less dense. A major ninth is a compound musical interval spanning 14 semitones , or an octave plus 2 semitones. If transposed into a single octave, it becomes a major second or minor seventh. The major ninth is somewhat dissonant in sound. Some common transposing instruments sound a major ninth lower than written. These include

216-465: A set of pitch classes , where a pitch class is an equivalence class of pitches under octave equivalence. The tonality diamond is often regarded as comprising the set of consonances of the n-limit. Although originally invented by Max Friedrich Meyer , the tonality diamond is now most associated with Harry Partch ("Many theorists of just intonation consider the tonality diamond Partch's greatest contribution to microtonal theory." ). Partch arranged

252-592: 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. Partch used otonal and utonal chords in his music. Ben Johnston often uses

288-512: A single starting pitch. 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 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

324-490: A tonality diamond, such as Harry Partch 's 11-limit diamond, each ratio of a right slanting row shares a numerator and each ratio of a left slanting row shares an denominator. Each ratio of the upper left row has 7 as a denominator, while each ratio of the upper right row has 7 (or 14) as a numerator. This diamond contains three identities (1, 3, 5). This diamond contains four identities (1, 3, 5, 7). This diamond contains six identities (1, 3, 5, 7, 9, 11). Harry Partch used

360-490: Is enharmonically equivalent to an augmented octave). If transposed into a single octave, it becomes a minor second or major seventh. The minor ninth is rather dissonant in sound, and in European classical music, often appears as a suspension . Béla Bartók wrote a study in minor 9ths for piano. The fourth movement (an intermezzo ) of Robert Schumann 's Faschingsschwank aus Wien is constructed to feature prominent notes of

396-410: Is a compound musical interval spanning 15 semitones, or 3 semitones above an octave. Enharmonically equivalent to a compound minor third, if transposed into a single octave, it becomes a minor third or major sixth. See: Dominant seventh sharp ninth chord . Three types of ninth chords may be distinguished: dominant (9), major (M9), and minor (m9). They may easily be remembered as the chord quality of

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432-413: 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, CM, in certain contexts or voicings, can be interpreted as Am. Partch coined the term "Monophony" (not to be confused with monohpony ) to describe a system of just intervals deriving from

468-415: Is in the sense of odd limit, not prime limit) is an arrangement in diamond-shape of the set of rational numbers r , 1 ≤ r < 2 {\displaystyle 1\leq r<2} , such that the odd part of both the numerator and the denominator of r , when reduced to lowest terms, is less than or equal to the fixed odd number n . Equivalently, the diamond may be considered as

504-459: 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

540-405: 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')." 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

576-446: 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

612-503: Is therefore composed of members of a subharmonic series . This term is used extensively by Harry Partch in Genesis of a Music . 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

648-415: 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

684-503: The 7-limit otonality, called the tetrad . This chord might be, for example, A ♭ -C-E ♭ -G [REDACTED] ♭ [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

720-633: The melody a minor ninth above the accompaniment: Alexander Scriabin 's Piano Sonata No. 9 , 'Black Mass' is based around the interval of a minor ninth, creating an uncomfortable and harsh sound. Several of Igor Stravinsky 's works open with a striking gesture that includes the interval of a minor 9th, either as a chord: Les Noces (1923) and Threni (1958) ; or as an upward melodic leap: Capriccio for Piano and Orchestra (1929) , Symphony in Three Movements (1946) , and Movements for Piano and Orchestra (1960) . An augmented ninth

756-411: The tenor saxophone , the bass clarinet , the baritone / euphonium when written in treble clef , and the trombone when written in treble clef ( British brass band music). When baritone/euphonium or trombone parts are written in bass clef or tenor clef they sound as written. A minor ninth (m9 or -9) is a compound musical interval spanning 13 semitones, or 1 semitone above an octave (thus it

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792-404: The 11-limit tonality diamond, but flipped it 90 degrees. This diamond contains eight identities (1, 3, 5, 7, 9, 11, 13, 15). The five- and seven-limit tonality diamonds exhibit a highly regular geometry within the modulatory space , meaning all non-unison elements of the diamond are only one unit from the unison. The five-limit diamond then becomes a regular hexagon surrounding the unison, and

828-560: The Utonality the numerator is always 1 and the numerary nexus is thus also 1: 1 1 1 2 1 3 1 4 1 5 e t c . ( 4 3 ) ( 8 5 ) {\displaystyle {\begin{array}{cccccc}{\frac {1}{1}}&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&\mathrm {etc.} \\&&({\frac {4}{3}})&&({\frac {8}{5}})\end{array}}} For example, in

864-416: The elements of the tonality diamond in the shape of a rhombus , and subdivided into (n+1) /4 smaller rhombuses. Along the upper left side of the rhombus are placed the odd numbers from 1 to n, each reduced to the octave (divided by the minimum power of 2 such that 1 ≤ r < 2 {\displaystyle 1\leq r<2} ). These intervals are then arranged in ascending order. Along

900-417: The elements of the tonality diamond, with some repetition. Diagonals sloping in one direction form Otonalities and the diagonals in the other direction form Utonalities. One of Partch's instruments, the diamond marimba , is arranged according to the tonality diamond. A numerary nexus is an identity shared by two or more interval ratios in their numerator or denominator , with different identities in

936-435: The fact that the size of the diamond grows as the square of the size of the odd limit tells us that it becomes large fairly quickly. There are seven members to the 5-limit diamond, 13 to the 7-limit diamond, 19 to the 9-limit diamond, 29 to the 11-limit diamond, 41 to the 13-limit diamond, and 49 to the 15-limit diamond; these suffice for most purposes. Yuri Landman published an otonality and utonality diagram that clarifies

972-419: The integers less than n which share no common factor with n, and if d(n) denotes the size of the n-limit tonality diamond, we have the formula From this we can conclude that the rate of growth of the tonality diamond is asymptotically equal to 2 π 2 n 2 {\displaystyle {\frac {2}{\pi ^{2}}}n^{2}} . The first few values are the important ones, and

1008-417: The lower left side are placed the corresponding reciprocals, 1 to 1/n, also reduced to the octave (here, multiplied by the minimum power of 2 such that 1 ≤ r < 2 {\displaystyle 1\leq r<2} ). These are placed in descending order. At all other locations are placed the product of the diagonally upper- and lower-left intervals, reduced to the octave. This gives all

1044-571: The other. For example, in the Otonality the denominator is always 1, thus 1 is the numerary nexus: 1 1 2 1 3 1 4 1 5 1 e t c . ( 3 2 ) ( 5 4 ) {\displaystyle {\begin{array}{cccccc}{\frac {1}{1}}&{\frac {2}{1}}&{\frac {3}{1}}&{\frac {4}{1}}&{\frac {5}{1}}&\mathrm {etc.} \\&&({\frac {3}{2}})&&({\frac {5}{4}})\end{array}}} In

1080-683: The otonal as an expanded tonic chord: 4:5:6:7:11:13 (C:E:G:B [REDACTED] ♭ :F ↑ :A [REDACTED] ♭ ) and bases the opening of the third movement of his String Quartet No. 10 on this thirteen-limit Otonality on C. 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 [REDACTED] ♭ :B [REDACTED] ♭ ), and as harmonics 7 through 13: 7:8:9:10:(11:)12:13 (C:D [REDACTED] - :E [REDACTED] :F [REDACTED] ♯ :(G ↑ [REDACTED] - :)A [REDACTED] :B [REDACTED] [REDACTED] ♭ - ); both otonal. Yuri Landman published

1116-417: The relationship of Partch's tonality diamonds to the harmonic series and string lengths (as Partch also used in his Kitharas) and Landmans Moodswinger instrument. In Partch's ratios, the over number corresponds to the amount of equal divisions of a vibrating string and the under number corresponds to the which division the string length is shortened to. 5 ⁄ 4 for example is derived from dividing

Otonality and utonality - Misplaced Pages Continue

1152-744: 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. An otonality 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

1188-479: The seven-limit diamond a cuboctahedron surrounding the unison. . Further examples of lattices of diamonds ranging from the triadic to the ogdoadic diamond have been realised by Erv Wilson where each interval is given its own unique direction. Three properties of the tonality diamond and the ratios contained: For example: If φ( n ) is Euler's totient function , which gives the number of positive integers less than n and relatively prime to n, that is, it counts

1224-420: The string to 5 equal parts and shortening the length to the 4th part from the bottom. In Landmans diagram these numbers is inverted, changing the frequency ratios into string length ratios. 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

1260-450: 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

1296-497: 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. 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

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