Analyzing Hexatonic Cycles in Classical Music Theory

Scales in 20th century music include pentatonic (diatonic) scales and synthetic scales. Pentatonic scales are diatonic and include major and minor varieties. Synthetic scales cannot relate to diatonic scales and include whole tone scales (one semitone-two semitone pattern) and octatonic/hexatonic scales with symmetrical interval constructions. These symmetrical patterns—such as one semitone three semitones repeating—are especially favored by 20th century composers. The Lydian-mixolydian scale combines characteristics of both modes and contains a dominant seventh interval, making it versatile for harmonic construction.

In 12-tone equal temperament, the number 12 enables multiple symmetrical pitch collections through interval cycles. Starting from any note (labeled 0-11), repeating specific intervals generates different scales: (1) Interval cycle 1/11 creates the chromatic scale (aggregate containing all 12 pitches); (2) Interval cycle 2/10 creates the whole tone scale (six pitches, completely symmetrical); (3) Interval cycle 3/9 creates the fully diminished seventh chord (four different inversions possible); (4) Combining two diminished seventh chords creates the octatonic scale (alternating half-whole steps). These symmetrical structures create different emotional effects - the whole tone scale produces confusion because every note sounds identical relative to neighbors, while the octatonic scale offers partial resolution through its repeating pattern.

The music orients toward three referential collections: diatonic (t0, t2, t5), octatonic (t3, t6 with perfect fifths/tritones), and whole tone (t2, t4, t6 with perfect fifths/tritones). These correspond to melody types: 0-2-5-7 (diatonic), 0-2-3-5 (diatonic/octatonic), 0-3-3-6 (octatonic), 0-2-4-6 (whole tone). Collections rarely occur in pure form but function as colors or tendencies. Diatonic scales associate with human characters (young men/women) and pastoral/military modes. Octatonic scales associate with superhuman characters (old woman/diviners, elders/ancestors). Whole tone scales associate with sub-human/animalistic qualities in orgiastic dances of life or wild dances of death. These associations operate as general tendencies rather than universal rules.

This video introduces three key 20th-century scales: synthetic scales (any scale not based on traditional diatonic systems, such as Lydian dominant or Klezmer scales), the whole tone scale (a symmetrical scale with only whole steps, containing only major thirds and augmented triads, with only two possible versions that together cover all 12 notes), and the octatonic scale (an eight-note scale alternating half and whole steps, containing stacks of minor thirds that form fully diminished seventh chords, with three possible versions each combining two diminished seventh chords).

The octatonic scale is an eight-note scale with a repeating semitone-tone-tone pattern, containing eight distinct notes before repetition. It has only three transpositions (C, C#, D) because the interval pattern sums to three semitones. Classified as a scale rather than a mode, it lacks a tonic note due to its repetitive structure. The scale naturally relates to the diminished seventh chord, as two such chords a semitone apart produce all eight notes. The Petrushka chord (two major triads a major third apart) is included within the scale. The scale is enharmonically flexible, containing both sharps and flats. Clara Schumann's nocturne demonstrates early intuition of octatonic concepts. Igor Stravinsky popularized the scale in 'The Firebird' (1910), while Mozart used octatonic elements in Symphony No. 40. Messiaen classified it as his second mode of limited transposition, and Bartók employed it in Suite Op. 14. The scale's symmetrical structure creates distinctive harmonic and melodic textures that define 20th-century musical vocabulary.

The neo-Riemannian theory provides analytical tools for understanding highly chromatic Romantic-era music by defining two fundamental operations—L (lowering the fundamental by a semitone) and P (changing the mode by modifying the third)—which together generate a hexatonic cycle of six chords (three major and three minor) built upon an underlying augmented triad, allowing analysts to trace harmonic progressions in works by composers like Schubert and Brahms without assigning traditional functional roles to their acoustically ambiguous chords.

Franz Liszt employed thematic transformation—a technique for putting previous material into new guises—as a fundamental compositional method. While other composers used this technique, Liszt applied it to create the structure of entire pieces, designing material around its potential for transformation. Consequently, he rarely repeated himself exactly. This approach generated scales of Eastern Europe, symmetrical harmonic structures, and eliminated absolute music in favor of programmatic content, prompting conservative classicists like Brahms to publicly oppose this insurgent tide. Both Liszt and Wagner shared commitment to chromaticism, with Liszt transforming multi-movement forms into complex one-movement structures while Wagner worked on whole acts rather than individual numbers.

Hexatonic cycles are six-note cyclic systems in Neo-Riemannian theory that generate 24 related chords through smooth voice leading, offering an alternative analytical framework for understanding late-romantic triadic progressions; these cycles can be implemented programmatically in environments like Sonic Pi to explore their musical properties.

Franz Liszt's transcriptions and paraphrases constitute approximately half of his total output, with around 350 pieces out of an estimated 700 works. These include both literal transcriptions of orchestral and operatic works and paraphrases that develop themes from other composers into new, personal compositions. This massive body of work demonstrates his generosity and humanism, as he devoted enormous time to making other composers' works accessible to piano audiences. His relationship with Richard Wagner was complex: Liszt provided crucial financial support to Wagner, who was often ungrateful and demanding. Despite this, Liszt continued to support Wagner, viewing it as his mission to help such a genius. The relationship was complicated by Wagner's affair with Cosima, Liszt's daughter, and his subsequent marriage to her.

Franz Liszt composed La lugubre gondola I and II (S200) in response to the death of Richard Wagner, whom he deeply admired. Liszt stayed with Wagner at the Palazzo Vendramin in Venice in late 1882, where he foretold Wagner’s death and the image of his body being carried by a funeral gondola. The music transforms the barcarolle form into a tragic expression, featuring highly original harmony that evokes profound desolation. The first version, composed in 6/8 time, captures the rocking motion of a gondola and a mournful melody. Liszt later revised it into a second version in 4/4 time, intensifying the emotional impact with more powerful harmonies while preserving the gondola’s rhythm and melodic plaintiveness. He also created arrangements for violin or cello with piano. Upon learning of Wagner’s death, Liszt composed R W—Venezia, a piece marked by violent dissonance reflecting personal grief, which dismantles Wagnerian grandeur. On what would have been Wagner’s seventieth birthday, Liszt wrote the short elegy At Richard Wagner’s Grave, opening with the theme of Excelsior! from his own choral work The Bells of Strasbourg Cathedral—Wagner had used this theme to open Parsifal. The elegy concludes with a hushed reference to the bell motif from Wagner’s opera. This piece, also arranged for organ or string quartet and harp, remained unpublished for decades.

Chromatic medians are chords built a major or minor third away from the tonic that contain non-diatonic tones, creating tension commonly used in film and TV scores; they can be used by pinging between the tonic and a chromatic median, alternating between different chromatic medians, or mixing chord qualities to enhance dramatic effect.

Chromatic mediant relationships can form complete cycles. For example, you can progress through a series of chords related by chromatic mediant: D minor to F minor to Ab minor to Bb minor to Db minor to Eb minor to G minor to A minor. This creates a continuous harmonic flow that doesn't establish a clear tonal center. The octatonic scale (alternating whole and half steps) generates symmetric harmonic structures. Because the scale is symmetric, any chord built from it will appear at regular intervals throughout the scale. This creates harmonic structures where multiple chords can function as tonics, creating a sense of harmonic ambiguity. The hexatonic scale (alternating whole and augmented fourth intervals) also generates symmetric harmonic structures with similar properties. These symmetric scales create harmonic languages fundamentally different from traditional tonal harmony.

Chromatic mediant chords are non-diatonic chords created by moving a major or minor third from a starting chord to a different quality (major or minor), which film composers use to create enhanced emotional effects such as heroic, sinister, or romantic moods that cannot be achieved with standard diatonic harmony.

John Coltrane revolutionized jazz harmony with his Coltrane Changes or Coltrane Matrix, a groundbreaking system of chord substitutions that transformed jazz improvisation. Unlike traditional functional harmony based on fourth and fifth intervals, Coltrane used major third intervals to create rapid key shifts. The basic Coltrane Cycle consists of three major seventh chords moving in descending major thirds (Cmaj7 → Abmaj7 → Ebmaj7 → Cmaj7), forming a triangular pattern on the circle of fifths. This creates three distant-related tonal centers rather than a single tonic, producing a 'snake that bites its own tail' effect. The relationship between these keys is called a chromatic mediant movement, which introduces outside notes and creates unique harmonic color. This technique, while adopted by Coltrane from earlier impressionist composers, became central to modern jazz and is used in film scores and pop music like 'Under the Bridge.'

Chromatic mediant chords are chords whose roots are a major or minor third apart but do not belong to the same diatonic key; each major or minor key has six chromatic mediant chords, which were popularized in classical and romantic music for creating color and emotion, and are now widely used in modern gospel music as passing chords and in film scores for modulation, as well as in neo-soul progressions with extended chord voicings.