Quarter-Tone Harmony/Concepts

The quarter-tone system is a musical system that splits the octave into twenty-four equally-spaced notes, rather than the typical twelve notes in the conventional system. This means that, between every adjacent pair of notes in the conventional system, there lies another note half-way between the two in the quarter-tone system: A quarter-tone is the new smallest interval that is produced, exactly half of a semitone in size. Notation The quarter-tone system requires three new accidentals to be devised. Though not standardized for quarter-tones, the most often-used system is the Stein-Zimmerman notation system: t half-sharp or semisharp, raising a note by 1 quarter-tone #t three-halves-sharp or sesquisharp, raising a note by 3 quarter-tones d half-flat or semiflat, lowering a note by 1 quarter-tone db three-halves-flat or sesquiflat, lowering a note by 3 quarter-tones A full quarter-tone chromatic scale ascending and descending is notated like so: This also produces new pairs of enharmonics: Transposition Note that the same relationships exist in the set of new quartertonal pitches as in the set of conventional pitches in the same way two identical combs interlock with one another, so passages can be transposed up or down quarter-tones like so: Intervals There are several new intervals for which there are no standardized names: Size in quarter-tonesSystem 1System 2Numerical notation1quartertonal secondsubminor second, semi-augmented unisont1, db23neutral secondd25second-thirdsupermajor second, subminor thirdt2, db37neutral thirdd39third-fourth, minor fourthsupermajor third, semi-diminished fourtht3, d411major fourthsemi-augmented fourtht413minor fifthsemi-diminished fifthd515major fifth, fifth-sixthsemi-augmented fifth, subminor sixtht5, db617neutral sixthd619sixth-seventhsupermajor sixth, subminor seventht6, db721neutral seventhd723quartertonal seventhsupermajor seventh, semi-diminished octavet7, d8 "System 1" is the system used by Wyschnegradsky in his Manual, whereas "System 2" uses more technical, modern names consistent with the traditional naming system. This book will hereafter use System 2: Subminor intervals are quarter-tone smaller than minor. Supermajor intervals are a quarter-tone larger than major. Neutral intervals are a quarter-tone smaller than major, or a quarter-tone larger than minor. Semi-augmented intervals are a quarter-tone larger than perfect, or a quarter-tone smaller than augmented. Semi-diminished intervals are a quarter-tone smaller than perfect, or a quarter-tone larger than diminished. These intervals, like traditional intervals, can be inverted, and doing so produces other quarter-tone intervals: the subminor second (1) inverts into a supermajor seventh (23), the neutral second (3) inverts into a neutral seventh (21), etc. A rule of thumb for quarter-tone interval inversions: Subminor intervals invert into supermajor intervals, and vice versa. Neutral intervals invert into other neutral intervals. Semi-augmented intervals invert into semi-diminished intervals, and vice versa. Cycles In the conventional twelve-tone system, different intervals (exclusing any larger than the octave) behave differently when repeatedly stacked in series. For example, only four intervals produce a cycle that goes through all twelve notes in the conventional system: the minor second (1 semitone) the perfect fourth (5 semitones) the perfect fifth (7 semitones) the major seventh (11 semitones) Let us now see the behavior of the new intervals: The subminor second (1 quarter-tone) produces the aforementioned quarter-tone chromatic scale. The neutral second (3 quarter-tones) splits the minor third into equal halves, producing a scale with eight equal tones in one octave:Notably, this scale consists of two interlocking diminished sevenths chords a neutral second apart. The supermajor second (5 quarter-tones) splits the perfect fourth into equal halves, producing a scale with all 24 tones in five octaves: The neutral third (7 quarter-tones) splits the perfect fifth into equal halves, producing a scale with all 24 tones in seven octaves: The supermajor third (9 quarter-tones) splits the major sixth into equal halves, producing a scale with the same eight tones as in the neutral second cycle but in three octaves: The semi-augmented fourth (11 quarter-tones) splits the major seventh into equal halves, producing a unique scale with all 24 tones in eleven octaves, not unlike a warped circle of fourths: The semi-diminished fifth (13 quarter-tones) is an inversion of the semi-augmented fourth, producing the cycle of semi-augmented fourths in reverse order. The subminor sixth (15 quarter-tones) is an inversion of the supermajor third, producing the cycle of supermajor thirds in reverse order. The neutral sixth (17 quarter-tones) is an inversion of the neutral third, producing the cycle of neutral thirds in reverse order. The supermajor sixth (19 quarter-tones) is an inversion of the subminor third, producing the cycle of subminor thirds in reverse order. The neutral seventh (21 quarter-tones) is an inversion of the semi-augmented fourth, producing the cycle of neutral seconds in reverse order. The supermajor seventh (23 quarter-tones) is an inversion of the subminor second, producing the quarter-tone chromatic scale in reverse order. Contrary Motion Interestingly, contrary motion in quarter tones starting on the unison produces all conventional intervals (labeled here with their quartertonal sizes): And contrary motion on the semi-augmented unison likewise produces all of the new quartertonal intervals: