INTERVAL CYCLES IN THEORY AND PRACTICE -...

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 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


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


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.


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.


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.


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


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.


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.


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

CE

4F#G

AC#D#EF#A#CC#

56

4

F#4

A4

C4

2E4

G4

21

Appendix 1 - p. 5

4

F#4

A4 C4

2E4G4

21

D#9

G9

A9

A#9

F#8

A#8

C9

C#9

A7

C#8

D#8

E8

C7

E7

F#7

G7

D#6

G6

A6

A#6

F#5

A#5

C6

C#6

A4

C#5

D#5

E5

C4

E4

F#4

G4

Appendix 1 - p. 6

INTERVAL CYCLES IN THEORY AND PRACTICE -...

Documents

Transcript of INTERVAL CYCLES IN THEORY AND PRACTICE -...