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three excellent ideas polansky Seattle, March 2009 1.Tempo Melody (Cowell/Beyer) “ If we wish to use tempo as melody, we have but to establish the tempo value of various tones, and change them as the piece progresses…

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three excellent ideas

three excellent ideas


Seattle, March 2009


“If we wish to use tempo as melody, we have but to establish the tempo value of various tones, and change them as the piece progresses…

This use of consecutive changes of tempo is of course no new thing; it is the mathematical ratio between tempi that has not been systematized. It can be observed in current practice that when the relations between successive tempi do not follow a simple ratio, but are accidental and arbitrary, the result is felt to be rhythmically rough. Conversely, I am convinced that further investigation would confirm my own experiments, which show that when successive tempi…. Give an impression of smoothness, [the] ratio … is demonstrably simple.”

Cowell: tempo melody

NMR, p. 92

beyer first clarinet suite 4th movement excerpt first 8 phrases
Beyer, first clarinet suite, 4th movement, (excerpt) first 8 phrases

Frog Peak edition, edited by Daniel Goode and Dennis Bathory-Kitsz

beyer second clarinet suite 4th movement excerpt first 6 phrases

Beyer, second clarinet suite, 4th movement, (excerpt) first 6 phrases

Frog Peak edition, edited by Marguerite Boland


Examples of Beyer’s use of tempo melodiesClarinet Suites 1 and 1b, 4th mvmts.

Performed by Daniel Goode (1)

Craig Hill (1b)

From Johanna Beyer: Sticky Melodies, New World Records


4th Movement, 2nd clarinet suite (Beyer)

Note names

Line Tempo Scalar Tempo Ratio Note Name

1 1/1 --- C

2 3/2 3/2 G

3 3/2 9/4 D

4 1/3 3/4 G

5 3/2 9/8 D

6 3/2 27/16 A

7 3/2 81/32 E

8 2/1 81/16 E

9 3/4 243/64 B

10 2/1243/32 B

11 3/2729/64 F#

12 2/1729/32 F#

(cycle of 5ths, “mixolydian” mode)

4 th movement 2 nd clarinet suite beyer interval sequence
4th Movement, 2nd clarinet suite (Beyer)Interval sequence

Line Tempo Scalar Interval Name Pitch (octave reduced)

1 1/1 C

2 3/2 P5th G

3 3/2 P5th D

4 1/3 P5th (down) G

5 3/2 P5th D

6 3/2 P5th A

7 3/2 P5th E

8 2/1 Octave E

9 3/4 P4th (down) B

references tempo melodies beyer
References (tempo melodies/Beyer)

Boland, Marguerite. 2007. “Experimentation and Process in the Music of Johanna Beyer.” VivaVoce No. 86, Journal of the Internationaler Arbeitskreis Frau and Musik (in German). (English version on Polansky, website)

— 2007a. (ed.) Suite for Clarinet Ib. Annotated performance edition. Frog Peak/Johanna Beyer Project #17. Series editor: Polansky, Larry. Hanover: Frog Peak Music (A Composers’ Collective)

Cowell, Henry. 1930. New Musical Resources. Republished by Something Else Press, 1969.

Goode, Daniel. 2007. Suite for Clarinet I. Annotated performance edition. Frog Peak/Johanna Beyer Project #16. Kitsz, Dennis, copyist and co-editor.

Kennedy, J. and Polansky, L. 1996. “’Total Eclipse’: The Music of Johanna Magdalena Beyer: An Introduction and Preliminary Annotated Checklist.” Musical Quarterly. 80/4.

Polansky, Larry. 2008. “’Sticky Melodies’”: The Choral and Chamber Music of Johanna Magdalena Beyer.” Liner notes to Sticky Melodies, New World CD 80678.

— “Johanna Magdalena beyer: miscellaneous materials”:

1930. Seeger, Charles. Tradition and Experiment in the New Music, in Studies in Musicology II, (ed.) Ann Pescatello, University of California Press. Berkeley, CA. 1994.

Thanks to Marguerite Boland and Amy Beal for help with these ideas

2 dissonant counterpoint and statistical feedback cowell seeger tenney ames

2. Dissonant counterpoint and statistical feedback(Cowell/Seeger/Tenney/Ames)

melodic dissonant counterpoint
melodic dissonant counterpoint

“Carl Ruggles has developed a process for himself in writing melodies for polyphonic purposes which embodies a new principle and is more purely contrapuntal than a consideration of harmonic intervals. He finds that if the same note is repeated in a melody before enough notes have intervened to remove the impression of the original note, there is a sense of tautology, because the melody should have proceeded to a fresh note instead of to a note already in the consciousness of the listener. Therefore Ruggles writes at least seven or eight different notes in a melody before allowing himself to repeat the same note, even in the octave.”

Henry Cowell, NMR, pp. 41-42

“Avoid repetition of any tone until at least six progressions have been made.”

Seeger, Manual of Dissonant Counterpoint.” p. 174.

tenney on the evolution of carl ruggles melodic style
Tenney on the evolution of Carl Ruggles’ melodic style

“I believe that what he was primarily concerned with was freshness — newness, maximal variety of pitch-content — and the sustaining of a high degree of atonal or atonical (but nevertheless harmonic) tension.”

James Tenney, 1997. “The Chronological Evolution of Carl Ruggles’ Melodic Style”


Tenney: Chronological graphs of ALSD, ALSCin Ruggles’ works(average length of different pitch classes; average length of consonant pitch strings)

“Consider for a moment what is meant by the incredibly high values for ALSD reached in Sun Treader, the Evocations, and Organum. It is, in each case, almost 9, which means that at every moment in the process of composing these melodic lines there were only four pitch-classes remaining to chose from for the next tone — and not even all of these four would necessarily satisfy certain other conditions, such as the desire for dissonant relations in close proximity. Very severe constraints indeed, for a music which sounds so free!” (Tenney)

statistical feedback charles ames
Statistical Feedback(Charles Ames)

“Along with backtracking, statistical feedback is probably the most pervasive technique used by my composing programs. As contrasted with random procedures which seek to create unpredictability or lack of pattern, statistical feedback actively seeks to bring a population of elements into conformity with a prescribed distribution. The basic trick is to maintain statistics describing how much each option has been used in the past and to bias the decisions in favor of those options which currently fall farthest short of their ideal representation”

  • Charles Ames “Tutorial on Automated Composition.”
uh oh


  • limited frame size
  • probability vs. statistics
  • colored local distributions
  • “odd” strings
  • method, not result
tenney dissonant counterpoint melody algorithm incorporating statistical feedback simplest version
Tenney, dissonant counterpoint (melody) algorithm (incorporating statistical feedback) simplest version

1. Take N elements and associated probabilities pn

2. Using a pseudo-random number generator, pick an element

3. Set the selected element’s probability to zero (or some very low value)

4. Increment all other probabilities by some uniform or weighted amount

5. Pick again

tenney algorithm probability progressions 2
Tenney algorithmprobability progressions (2)

Thanks to Kimo Johnson for his collaboration on these graphs

different probability increment functionsTenney dissonation algorithmautocorrelation over n trials

Original, linear function (no curvature)

Strong positive curvature

Linear function with MAX =1

Very strong positive curvature

Thanks to Mike Winter and Alexander Barnett for collaboration in this work.

example of tenney s use of the melodic algorithm seegersong 2 solo flute
Example of Tenney’s use of the melodic algorithmSeegersong #2solo flute

Performed by Margaret Lancaster

Forthcoming on her solo CD, Have Faith!, New World Records

selected references melodic dissonant counterpoint statistical feedback
Selected References: Melodic dissonant counterpoint/statistical feedback

Charles Ames

1996. “Thresholds of Confidence: An Analysis of Statistical Methods for Composition: Part 2: Applications,” Leonardo Music Journal 6.

1995. Thresholds of Confidence: An Analysis of Statistical Methods for Composition, Part 1: Theory.” Leonardo Music Journal 5.

1993. “How to Level a Driver Sequence,” Leonardo Music Journal 3.

1992. “Catalog of Sequence Generators,” Leonardo Music Journal 2.

1991. “Catalog of Statistical Distributions: Techniques for Transforming Random, Determinate and Chaotic Sequences,” Leonardo Music Journal 1.

1990. “Statistics and Compositional Balance,” Perspectives of New Music, 28/1.

1987a. “Automated Composition in Retrospect,” Leonardo, 20/2.

1987b. “Tutorial on Automated Composition,” Proceedings of the ICMC. International Computer Music Association. Urbana, Illinois.

1986. “Two Pieces for Amplified Guitar,”Interface: Journal of New Music Research, 15/1.

James Tenney

1977. “The Chronological Development of Carl Ruggles' Melodic Style,” Perspectives of New Music, 16/1.

Larry Polansky, Mike Winter (with Alexander Barnett).

In progress.“A Few More Words About James Tenney: Dissonant Counterpoint and Statistical Feedback.” Paper given NYC (2007) and Phoenix (2006)

Michael Casey

1992. “HS: A Symbolic Programming Language for Computer Assisted Composition.” M.A Thesis, Dartmouth College.

scales in arion s leap
“scales” in Arion’s Leap

Ya chengs

3-part chord with the intervals 7/6 and 4/3, tuned as A-C-D (4/3, 14/9, 16/9), transposed up 25 /24, 16/15, 6/5, down 25/24.

Troubadour Harp

Adds Eb (50/27), Bb (25/18), G (32/27) to the total fabric

Metal Strung Harp


tyvarb(B’rey’sheet) (in the beginning ... ) (Cantillation Study #1)(1985; revised 1987, 1989)for voice and live computerJody Diamond, voiceLarry Polansky and Phil Burk, live computer systemsfrom The Theory of Impossible MelodyNew World Records, 2009(reissue of Artifact CD, #4, 1991)

references freestyle tuning paratactical intonation

References: freestyle tuning, paratactical intonation

Lou Harrison

1971. Music Primer. New York, C.F. Peters

1974. A Phrase for Arion’s Leap. Score published in Ear (West) #1. Recording on Tellus #14: Just Intonation (1986).

Leta Miller and Fredric Lieberman

1998. Lou Harrison: Composing a World. Oxford University Press.

Larry Polansky

1987. “Item: Lou Harrison as a Speculative Theorist.” In A Lou Harrison Reader. Santa Fe: Soundings Press, edited by Peter Garland.

1987. “Paratactical Tuning: An Agenda for the Future Use of Computers in Experimental Intonation,” Computer Music Journal, 11(1).

1987. “HMSL Intonation Environment,” 1/1: The Journal of the Just Intonation Nework, 3:1.

Ezra Sims

1987. “Letter to CMJ”, Computer Music Journal, 11/4

James Tenney

1987. “About ‘Changes: Sixty-four Studies for Six Harps’,” Perspectives of New Music, 25:1/2

Harold Wagge

1985. “The Intelligent Keyboard,” 1/1: The Journal of the Just Intonation Nework, 1:4.