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INTERVAL CYCLES IN THEORY AND PRACTICE
Designing a Musical Interface for Composition and Performance
By
Daniel Leon Harrell
B. M., East Carolina University (2003)
M.M., University of Louisville (2006)
Proposal of Dissertation for the Doctorate of Musical Arts
School of Music
University of Illinois Urbana-Champaign
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 2 of 10
Abstract
This study focuses on implementing an interactive musical instrument interface for
the Apple iPad™, which constructs and manipulates interval cycles in real-time to facilitate
composition or live improvisation. In the first part of this dissertation the interval cycle and
its musical function is described. Part two discusses the implementation of the musical
interface including developing methods of visual representation, algorithmic data filtering
and specific issues of interface design.
Introduction
The impetus for this project came from my desire to understand intuitively-based
pitch organization within my own compositions. I began to find the connections I was
looking for when I was introduced to interval cycles.
Example 1. Interval cycle - IC(2 3)
Interval cycles are pitch collections that consist of a repeating pattern of intervals.
For example IC(2 3) is an interval cycle of an alternating pattern of a major second (2) and a
minor third (3). Because there is no standard method of notating interval cycles, for
convenience I will label them with the abbreviation IC and the interval pattern notated in the
number of half steps of each interval within parentheses.
IC(2 3) is an example of an interval cycle with pitches that do not repeat in each
octave. This unique feature of interval cycles makes it difficult to calculate quickly what
pitches will be in the cycle beyond a few iterations. To compound this problem, a more
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 3 of 10
complicated pattern of multiple intervals can quickly become too complex to easily manage
without a proper tool.
Statement of Problem
Although there is great potential in the organization of pitch with interval cycles,
efficiently calculating them restricts composers and performers from using them with ease.
Additionally, even if a composer has the pitches for an interval cycle at hand, the act of
composing at the piano with them is still difficult due to the layout of the conventional
keyboard. Since the piano keys are oriented in repeating patterns of twelve it is difficult to
work with interval cycles that do not align with this pattern. A keyboard that is capable of
changing its layout dynamically with the interval cycle would be ideal for eliminating this
hurdle.
Background of the Study
The use of interval cycles can be traced back as early as 1920 in a letter that Alban
Berg writes to Arnold Schoenberg in which he encloses his “Master Array of Interval
Cycles1.” This master array is a chart of the twelve possible single-interval cycles. Berg
describes this idea to Schoenberg as a “theoretical trifle” though ultimately used it as a
central method of organizing pitch in many of his works. In example 2, Berg uses IC(7) in
the bass and IC(1) in four parallel descending chromatic lines in an excerpt from Op. 2, No.
2.
1 Perle, George. "Berg's Master Array of the Interval Cycles." The Musical Quarterly 63.1 (1977): p.2
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 4 of 10
Example 2. Reduction of the opening of Berg’s Op. 2, No. 2
Howard Hansen extends the concept of interval cycles by using multiple intervals
within a cycle called a projection.2 Since some cycles, such as IC(3), exclude some pitches,
Hansen extends these cycles by inserting an arbitrary interval of the perfect fifth. This allows
any cycle to use all twelve chromatic pitches. Example 3 shows an IC(3) with the inserted
perfect fifth. This cycle would now be notated as IC(3 3 3 7).
Example 3. IC(3) with inserted perfect 5ths to use all twelve pitches
To add to the complexity of Hansen’s system he freely uses the output of an interval
cycle in its original form as a scale, with the interval cycles pitch output reordered. All of
Hanson’s interval cycles will ultimately use all twelve pitches due to the insertion of a perfect
5th. Due to this fact Hanson limits interval cycle-based pitch collections with a stopping
point. IC(7) is seen below limited to 3 pitches in both projection form and reordered scale
form.
2 Hanson, Howard. Harmonic Materials of Modern Music; Resources of the Tempered Scale. New York: Appleton-Century-Crofts, 1960. Print.
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 5 of 10
Example 4. IC(7) in projected form 1, and scale form 2.
Related to interval cycles are Olivier Messiaen’s Modes of Limited Transposition;
symmetrical scales that have less than twelve possible transpositions3. Messiaen created six
modes that meet these criteria, however it is possible to create more using interval cycles.
Any symmetrical interval cycle with pitch content that repeats in every octave will be a mode
of limited transposition. Example 5 shows that IC(1 5) will produce a mode of limited
transposition that is a subset of Messiaen's modes 2 and 7.
Example 5. (1) Mode 2, (2) Mode 7, (3) IC (1 5)
When composing with interval cycles it is also important to understand the degree to
which each interval cycle is related. To do this it is necessary to understand them within the
context of a musical space. Musical spaces organize all possible elements of the space in
order to show the relationships between each of its members. The most common type of 3 Messiaen, Olivier. Technique De Mon Langage Musical. The Technique of My Musical Language : Musical Examples. Paris: Alphonse Leduc, 1956. Print.
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 6 of 10
musical space is pitch space, commonly associated with Hugo Riemann. Example 6 shows a
portion of Reimann’s tonnetz4, a pitch space arrangement that shows all relative keys,
chromatic mediant, and dominant relationships.
Example 6. A portion of Reimann’s Tonnetz
Clifton Callendar, Ian Quinn and Dimitri Tymoczko describe multidimensional pitch
spaces5. Example 7 shows the voice leading relationships between any two possible tri-
chords. Additionally, Tymoczko uses geometric figures to visually represent the chords
placed with in the full continuum of pitch space6.
Example 7. Pitch space showing all tri-chords.
Interval cycles can be placed into an interval cycle space because they are collections
of distances between pitches. Since interval cycles can be analyzed for their interval vector
they can also be placed into an interval vector space. Since this would need to be a
4 Lerdahl, Fred. Tonal Pitch Space. New York: Oxford UP, 2001. Print. 5 Clifton Callender, Ian Quinn, Dmitri Tymoczko Generalized Voice-Leading Spaces Published 18 April 2008, Science 320, 346 (2008) 6 Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford UP, 2011. Print.
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 7 of 10
multidimensional space, algorithms would be required to calculate and map these spaces.
These spaces can be used to understanding the complex relationships between various
interval cycles.
Research Questions
The first task of this dissertation will be to construct a standard notation to represent
interval cycles in a logical manner, with both standard musical notation and alternate visual
representations. Secondly, the design of an instrument that visually shows the relationship of
pitches within the interval cycle that has logical keyboard layouts. Finally, the task of how to
manipulate the interval cycle via the interface in real-time with precision will directly
influence the usability for the end user. The core algorithms used in this program will be
used in the mapping of the interval cycle space, which will be available for the user to
compare different interval cycles.
Methods and Procedures
This study focuses on creating an instrument to produce MIDI data, rather than an
audio signal. Since the user interface of this program will be the Apple iPad™, the ultimate
goal will be a stand-alone application for the iOS platform. However, the interim step of
using the iPad as an input surface to output data to MAX/MSP or SuperCollider for
processing the pitch algorithms may be necessary due to limitations of the Apple App Store
development process. It is yet to be determined how the algorithm for computing this data
will function.
Attached to this document is a prototype schematic of the interface with sample data
output. Page 1 is a diagram of the interval cycle input interface. Page 2 shows the dynamic
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 8 of 10
keyboard interface. Page 3 shows the alternate grid keyboard interface, which is design for
efficiency on a flat surface. Page 4 demonstrates chord entry on the dynamic keyboard.
Pages 5 and 6 show two alternate views of the iPad application.
Limitations
The interface will be limited exclusively to the output of pitch information in the
form of MIDI data and is not intended for use to control other parameters of music. Also
the interface is not designed to handle any other decision-making requirements of the
composition and improvisation processes outside of the realm of pitch.
The instrument interface is designed for formulating and playing interval cycles
primarily, thus will require some method of handling pitches that are excluded from the
interval cycle. This limitation has not been accounted for yet but will be implemented in the
software.
Finally this study and instrument is limited to 12-tone equal temperament for
practical purposes, but may be extended in the future for use with alternate tunings.
Summary
The goal of this project is to create an instrument to assist in creating interval cycle-
based music. It is designed to increase efficiency in calculating and performing interval
cycles. Priority will be placed on ease of use in the interface and robustness of the algorithm
to sort interval cycles into a comprehensive interval cycle space.
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 9 of 10
References
Perle, George. "Berg's Master Array of the Interval Cycles." The Musical Quarterly 63.1 (1977):
1-30.
Hanson, Howard. Harmonic Materials of Modern Music; Resources of the Tempered Scale.
New York: Appleton-Century-Crofts, 1960. Print.
Messiaen, Olivier. Technique De Mon Langage Musical. The Technique of My Musical Language :
Musical Examples. Paris: Alphonse Leduc, 1956. Print.
Lerdahl, Fred. Tonal Pitch Space. New York: Oxford UP, 2001. Print.
Clifton Callender, Ian Quinn, Dmitri Tymoczko Generalized Voice-Leading Spaces Published 18
April 2008, Science 320, 346 (2008)
Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Common
Practice. New York: Oxford UP, 2011. Print.
Kelley, Robert Tyler., and Michael Howard Buchler. Mod-7 Transformations in Post-functional
Music. Diss. Dissertation (PhD) Florida State University, 2005., 2005. Tallahassee: Florida
State University, 2005. Print.
Lewin, David. Generalized Musical Intervals and Transformations. New Haven: Yale UP, 1987.
Print.
Lewin, David. "Transformational Techniques in Atonal and Other Music
Theories."Perspectives of New Music 1/2 (1982): 312-71. Print.
Meredith, D. (2001). MIPS: A Formal Language for the Mathematical Investigation of Pitch Systems.
Unpublished manuscript.
Pekowski, John. "A Method for Composing with Interval Cycles as Applied to an Original
Composition for Symphony Orchestra." Diss. Texas Tech University, 2007. Print.
Harrell INTERVAL CYCLES IN THEORY AND PRACTICE Page 10 of 10
References (cont.)
Quinn, Ian. "A Unified Theory of Chord Quality in Equal Temperament." Diss. University
of Rochester, 2004. Print.
Reti, Rudolph. Tonality in Modern Music. New York, NY: Collier, 1962. Print.
Travers, Aaron, and Aaron Travers. "Interval Cycles, Their Permutations and Generative
Properties in Thomas Adès' Asyla." Diss. University of Rochester, 2004. Print.
Schillinger, Joseph, Lyle Dowling, and Arnold Shaw. The Schillinger System of Musical
Composition. New York: C. Fischer, 1946. Print.
C40
C4
E4
44
E4
F#4
C4
24
E4
G4
C4
1
F#42
4
F#4
B4C4
2E4
G4
21
Step 1Step 2
Step 3Step 4
Step 5
The pitch C4 is
entered using the onscreen keyboard or from
the input of an attached
MID
I keyboard. Since there is no
interval the output is only one pitch.
Output:
[C4 ...]
The pitch E4 is entered, this
creates the first interval. B
oth pitches are side by side because this
completes the
pattern for the cycle.
Output:
[C4 E4 G
#4 C5...]
The pitch F#4 is entered causing a line to split the
cycle between the
first and second interval.
Output:
[C4 E4 F#4 A
#4...]
The pitch G4 is
entered causing a line to split the
cycle again.
Output:
[C4 E4 F#4 G
#4 C5...]
The user continues to enter pitches until the desired interval
cycle is complete. A
s m
ore pitches are added the circle is
evenly divided.
Output:
[C4 E4 F#4 G
#4 B4 D
#5 ...]
Interval cycle input interfaceAppendix 1 - p. 1
C40
C4
E4
4
CCCCCCCCCCCC
4
CE 4
G#CEG#CEG#CEG#
56
7
4
E4
F#4
C4
2
4
E4
G4
C4
1
F#42
4
F#4
A4
C4
2E4
G4
21
CE
4
F#A#CEF#A#CEF#A#
56
CE
4F#GBD#FF#A#DEF
56
CE
4F#GAC#D#EF#A#CC#
56
Step 1
Step 2
Step 3
Step 4
Step 5
Dynam
ic keyboard interface
Appendix 1 - p. 2
D#9G9
A9
A#9
F#8A#8C9
C#9
A7
C#8D#8E8
C7
E7F#7
G7
D#6G6
A6
A#6
F#5A#5C6
C#6
A4
C#5D#5E5
C4
E4F#4
G4
4
F#4
A4
C4
2E4
G4
21
Dynam
ic grid keyboard
interface
This grid keyboard layout is an alternative to the m
ore conventional layout. It has the advantage of being m
ore ergonomic on the flat
surface of the iPad.
The example show
s IC(4 2
1 2) arranged from low
est pitches on the bottom
to the highest on the top.
Also the grid is four squares
wide to align w
ith the interval cycle pattern. Each repetition of the interval cycle is the next row
upw
ard.
Appendix 1 - p. 3
4
F#4
A4
C4
2E4
G4
21
CE
4F#G
AC#D#EF#A#CC#
56
Chord entry on dynam
ic keyboard interface
Example output
The black dots represent the selected chord from
within a IC
(4 2 1 2) that will
result from the root note being pressed.
The example show
s what this chord w
ill be on each successive root pitch.
Appendix 1 - p. 4