Wakeman - 1 In considering the composers of the early twentieth century, Bartok and Stravinsky are two names that come to the top of the list as significant innovators. There are many similarities between them — they both enjoyed success in Europe, both were outstanding pianists, both utilized folk music in various ways in their compositions, both explored alternate approaches to harmony, rhythm, and pitch organization, and both eventually relocated to the United States, although under very different circumstances. However, their approaches to octatonic materials were very different, from both a theoretical and an aesthetic perspective. To lay a bit of ground work, it is important to understand that the octatonic collection, as referred to in this study, is a symmetrical collection of pitch classes featuring alternating half and whole step intervals. The octatonic collection is limited to only three possible unique transpositions shown in example 1. It is typically presented in most texts as an ordered collection, but it is often not used as an ordered collection.
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The tritone is a restless interval, classed as a dissonance in Western music from the early Middle Ages through to the end of the common practice period. This interval was frequently avoided in medieval ecclesiastical singing because of its dissonant quality. However, stories that singers were excommunicated or otherwise punished by the Church for invoking this interval are likely fanciful.
At any rate, avoidance of the interval for musical reasons has a long history, stretching back to the parallel organum of the Musica Enchiriadis. In all these expressions, including the commonly cited "mi contra fa est diabolus in musica", the "mi" and "fa" refer to notes from two adjacent hexachords. For instance, in the tritone B—F, B would be "mi", that is the third scale degree in the "hard" hexachord beginning on G, while F would be "fa", that is the fourth scale degree in the "natural" hexachord beginning on C.
Later, with the rise of the Baroque and Classical music era, composers accepted the tritone, but used it in a specific, controlled way—notably through the principle of the tension-release mechanism of the tonal system. In that system which is the fundamental musical grammar of Baroque and Classical music , the tritone is one of the defining intervals of the dominant-seventh chord and two tritones separated by a minor third give the fully diminished seventh chord its characteristic sound.
In minor, the diminished triad comprising two minor thirds, which together add up to a tritone appears on the second scale degree—and thus features prominently in the progression iio—V—i. Often, the inversion iio6 is used to move the tritone to the inner voices as this allows for stepwise motion in the bass to the dominant root. In three-part counterpoint, free use of the diminished triad in first inversion is permitted, as this eliminates the tritone relation to the bass.
Wagner, Prelude to Act 2 of Siegfried. Debussy, String Quartet, 2nd movement, bars — The tritone was also exploited heavily in that period as an interval of modulation for its ability to evoke a strong reaction by moving quickly to distantly related keys.
According to Dave Moskowitz , p. Since the perfect 11th i. Also in jazz harmony, the tritone is both part of the dominant chord and its substitute dominant also known as the sub V chord. Because they share the same tritone, they are possible substitutes for one another. This is known as a tritone substitution.
The tritone substitution is one of the most common chord and improvisation devices in jazz. In the theory of harmony it is known that a diminished interval needs to be resolved inwards, and an augmented interval outwards. However, if one plays a just diminished fifth that is perfectly in tune, for example, there is no wish to resolve it to a major third. Just the opposite—aurally one wants to enlarge it to a minor sixth. The opposite holds true for the just augmented fourth.
One then notices that the just augmented fourth of It is no wonder that, following the ear, we want to resolve both downwards. The ear only desires the tritone to be resolved upwards when it is bigger than the middle of the octave. Therefore the opposite is the case with the just diminished fifth of
String Quartet No.5, Sz.102 (Bartók, Béla)
Nomenclature[ edit ] In St. Petersburg at the turn of the 20th century, this scale had become so familiar in the circle of composers around Nikolai Rimsky-Korsakov that it was referred to as the Korsakovian scale Корсаковская гамма Taruskin , As early as , the Russian theorist Boleslav Yavorsky described this collection of pitches as the diminished mode уменьшённый лад , because of the stable way the diminished fifth functions in it Taruskin , —13, citing Yavorsky In more recent Russian theory, the term octatonic is not used. Instead, this scale is placed among other symmetrical modes total 11 under its historical name Rimsky-Korsakov scale, or Rimsky-Korsakov mode Kholopov , 30; Kholopov ,
Project (analysis): Bartok, Diminished Fifth (1940)
The tritone can be used to avoid traditional tonality: According to a complex but widely used naming conventionsix of them are classified as augmented fourthsand the other six as diminished fifths. Diminished Fifth, for piano Mikrokosmos Vol. Melody against double notes. In other meantone tuning systems, besides tone equal temperament, A4 and d5 are distinct intervals because neither is exactly half an octave.