Home > Interval (music)
3 Interval cycles
Interval cycles, "unfold a single recurrent interval in a series that closes with a return to the initial pitch class", and are notated by George Perle using the letter "C", for cycle, with an interval class integer to distinguish the interval. Thus the diminished seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0-11 to indicate the lowest pitch class in the cycle. "These interval cycles play a fundamental role in the harmonic organization of post-diatonic music and can easily be identified by naming the cycle." (Perle, 1990)
3.1 Source
4 Interval strength and root
4.1 Interval strength
David Cope suggests the concept of interval strength, in which an interval's strength is determined by its approximation to a lower, stronger, or higher, weaker, position in the harmonic series. See also: Lipps-Meyer law.
4.2 Interval roots
Hindemith and David Cope both suggest the concept of interval roots. To determine an interval's root, one locates its nearest approximation in the harmonic series. The root of a perfect fourth, then, is its top note because it is an octave of the fundamental in the hypothetical harmonic series. The bottom note of every odd diatonically numbered intervals are the roots, as are the tops of all even numbered intervals. The root of a collection of intervals or a chord is thus determined by the interval root of its strongest interval.
As to its usefulness, Cope provides the example of the final tonic chord of some popular music being traditionally analyzable as a "submediant six-five chord" ( added sixth chords by popular terminology), or a first inversion seventh chord (possibly the dominant of the mediant V/iii). According the interval root of the strongest interval of the chord (in first inversion, CEGA), the perfect fifth (C-G), is the bottom C, the tonic.
4.3 Source
- Cope, David (1997). Techniques of the Contemporary Composer, p.40-41. New York, New York: Schirmer Books. BooksEnthsiast.com.
5 Other intervals
There are also a number of intervals not found in the chromatic scale or labeled with a diatonic function which have names of their own. Many of these intervals describe small discrepancies between notes tuned according to the tuning systems used. Most of the following intervals may be described as microtones.
- A Pythagorean comma is the difference between twelve justly tuned perfect fifths and seven octaves. It is expressed by the frequency ratio 531441:524288, and is equal to 23.46 cents
- A syntonic comma is the difference between four justly tuned perfect fifths and two octaves plus a major third. It is expressed by the ratio 81:80, and is equal to 21.51 cents
- A Septimal comma is 64/63, and is the difference between the Pythagorean or 3-limit "7th" and the "harmonic 7th".
- Diesis is generally used to mean the difference between three justly tuned major thirds and one octave. It is expressed by the ratio 128:125, and is equal to 41.06 cents. However, it has been used to mean other small intervals: see diesis for details
- A schisma (also skhisma) is the difference between five octaves and eight justly tuned fifths plus one justly tuned major third. It is expressed by the ratio 32805:32768, and is equal to 1.95 cents. It is also the difference between the Pythagorean and syntonic commas.
- A schismic major third is a schisma different than a just major third, eight fifths down and five octaves up, Fb in C.
- A quarter tone is half the width of a semitone, which is half the width of a whole tone.
- A kleisma is six major thirds up, five fifths down and one octave up, or, more commonly, 225:224.
- A limma is the ratio 256:243, which is the semitone in Pythagorean tuning.
- A ditone is the pythagorean ratio 81:64, two 9:8 tones.
- Additionally, some cultures around the world have their own names for intervals found in their music. See: sargam, Bali
