The Interval Cycle

An interval cycle is a repeated pattern of an interval or set of intervals.

The most common interval cycle is the scale.

Any scale that repeats at the octave is an interval cycle. For example the major scale consists of a pattern of half and whole steps. That pattern is W,W,H,W,W,W,H (W = Whole Step, H = Halfstep). That interval pattern repeats exactly as the scale is repeated in the next octave.

Interval cycles can be less than an octave. The most common is the Whole Tone scale. Not only does it repeat at the octave, but its entire intervallic content is created by a repeated major 2nd interval.

Interval cycles can also be used that do not repeat at the octave. For example the pattern for the notes C, D, E, F creates an interval pattern of W,W,H. If you repeat this pattern by adding an additional interval at the end you can create longer strings of pitches that create scales that are smaller or larger than the octave. The result of the example (C, D, E, F, +m2) = (C, D, E, F, G, A, B, C, D, E, F#, G, A, B, C#, D, etc.)

Composers of Interest

• Toru Takemitsu
• Quigang Chen
• Olivier Messiaen
• Jennifer Higdon
• Stephen Andrew Taylor
• Gyorgy Ligeti
• Howard Hansen
• Vincent Persichetti

Research Summary: "A Unified Theory of Chord Quality in Equal Temperament" by Ian Quinn

DISSERTATION:

"A Unified Theory of Chord Quality in Equal Temperament" by Ian Quinn

ABSTRACT:

Chord quality — defined as that property held in common between the members of a pcset-class, and with respect to which pcset-classes are deemed similar by similarity relations (interpreted extensionally in the sense of Quinn 2001) — has been dealt with in the pcset-theoretic literature only on an ad hoc basis. A formal approach that generalizes and fuzzifies Clough and Douthett ’s theory of maximally even pcsets successfully models a wide range of other theorists’ intuitions about chord quality, at least insofar as their own formal models can be read as implicit statements of their intuitions. The resulting unified model, which can be interpreted alternately as (a) a fuzzy taxonomy of chords into qualitative genera, or (b) a spatial model called Q-space, has its roots in Lewin’s (1959, 2001) work on the interval function, and as such has strong implications for a unification of general theories of harmony and voice leading.

SUMMARY NOTES:

Introduction (p. 1-3)

• Q-space: The visual/spacial/mathmatical space that we as musicians place chords & sets and utilize to compare 2 chords, in order to define if the 2 chords are the same, similar, or different. The full pitch continuum and relationships of all tones to each other.
• Chapter 1 surveys the approaches of Hanson, Forte, Morris, and many other theorists. Certain well-known sonorities (e.g., the diatonic and pentatonic collections, Messiaen’s modes of limited transposition, the hexatonic scale) turn up as what Quinn calls prototypes.
• The 6 prototypes are considered the top of the Q-space (mountain tops). Chords below the prototypes that are closely related are called genera.
• Q-space consists of 'mountains' called qualitative genus that are entities characterized by prototypical sonorities and encompassing sonorities to variying degrees, according to their closeness to the prototypes.
• The closeness or distance of an arbitrary sonority to the prototypes of a qualitative genus will be described in terms of the intrageneric affinities of the genus, and abstract structural relationships among genera will be called intergeneric affinities.
• The use of Q(c, d ), where c is the number of pitch class sets in the universe, and d is the cardinality of the maximally even set prototypical of the genus.
• Chapter 3  that the theory of Q-space is a powerful starting point for future theoretical development — particularly in the direction of understanding the relationships among the abstract theories of harmony and voice leading that constitute the landmarks of recent pcset-theoretic research.

Chapter 1 (p. 4-)

• All of these theories engage what we might think of as a fivefold hierarchy of increasingly general conceptual entities engendered by a piece of typical Western art music that is being conceived harmonically: (0) the sounding music, (1) the notated
  music; (2) the pcset or chord; (3) the species; and (4) the genus. Each level of this hierarchy abstracts essential harmonic features away from more accidental features of the previous level.
• A nominalist might say that a pcset-class is nothing more than an equivalence class of pcsets under transposition and inversion, without justifying the assertion of the relationship among pcsets and pcset-classes in terms external to the theory.
• Chord quality, then, can be defined nominally — provided at least that one believes in properties — as that property that is held in common between all members of any pcset-class, and that property by which various pcset-classes are distinguished from one
  another to varying degrees. It takes its place in the hierarchy of variously essential and accidental properties that is structured by what philosophers call supervenience: Property A supervenes on Property B if and only if any change in Property A necessarily entails a change in Property B. To assert that properties of chord quality supervene on properties of harmony, which super vene on properties of “the music itself,” is to say that one can change a harmony (by transposing or inverting it) without changing its quality, but one cannot change a harmony without changing “the music itself.” It can be helpful to think of a supervenient property as an abstraction of certain aspects or facets of those properties on which it supervenes.
• The goal of the present work is to justify those implicit intuitive claims from the top down, without attempting to ground the theory in the quicksand of intuition; rather, the argument will have its foundations in the usual mathematical and nominal characterization of pcset theory, and will proceed by means of theoretical unification. (p.7)
• The locus classicus of chord quality is often taken to be the inter val-class vector; Straus, for example, obser ves that “the quality of a sonority can be roughly summarized by listing all the intervals it contains” (2000, p. 10). Howard Hanson seems to have been the first to use this principle as the basis for a complete and rigorous pcset classification system.
• Hanson's Theory (p.13), Hanson's algorithm (p. 14)
• Viewed as a complete system, Hanson’s projections have five properties that make them particularly attractive as  a set of prototypes for harmonic genera: Unique Prototype Property (UUP), Unique-Genus Property (UGP),  Intrageneric Inclusion Property (IIP), Prototype-Complementation Property (PCP), Prototype-Familiarity Property (PFP).
• each projection is what Eriksson (1986) calls a maxpoint, a chord species “containing the maximum number for its size of at least one inter val class” (p. 96). The maxpoints of ics 1 and 5 correspond one-to-one with Hanson’s projections; maxpoints of the other ics are tabulated in Figure 1.4.
• Q-space discussion begins on p.22, described as point where disparate theories converge.
• Starting at p.25 Daniel Harrison's N = 2 1/6, used to circumvent the problem of multiplying pcsets to avoid a mapping of many to one, ei. 2 * 2 = 4 and 8 * 2 = 16 (4 in MOD 12).
• p. 31 Morris’s algebraic approach: The true beginning of what Quinn is adding to the discussion on Hanson and Forte.
• Morris initially suggests a bit of fudging: if we were to decree ic 1 and ic 5 to be identical, redefining “interval
  content ” accordingly, the problem would go away, with SG(v) taking its place in the hierarchy just above SG(3).
• stopped around p.38
• (p. 71)2.4.3 Harmony and voice leading. The paradigm case of unification in a modern
  music-theoretical context is, of course, Schenkerian theor y, which grows out of the
  influential and powerful idea that harmony and voice leading are two sides of the same
  coin. The application of this idea to repertoires outside of Schenker’s restrictive canon is
  by no means limited to specifically Schenkerian approaches; Schoenberg himself, often
  characterized as Schenker’s antithesis, proclaimed famously that “ THE TWO-OR-
  MORE-DIMENSIONAL SPACE IN WHICH MUSICAL IDEAS ARE PRE-
  SENTED IS A UNIT” (Schoenberg, 1950, p. 109; emphasis original), and similar
  sentiments were common among postwar composers of the Darmstadt circle. The
  issue has recently been tackled from a pcset-theoretic perspective in several important
  articles (Roeder, 1994; Lewin, 1998; Morris, 1998; Straus, 2003). Straus begins his
  contribution with a clear statement of the theoretical problem: “ Theories of atonal
  music have traditionally been better at describing harmonies — at devising schemes
  of classification and comparison — than at showing how one harmony moves to an-
  other” (p. 305). The “schemes of classification and comparison” include the variously
  taxonomic approaches discussed in Chapter 1 in connection with the notion of chord
  quality. These are theories of chord structure, and as Straus points out later in the same
  article,