Forms as algorithms — The case of the “normal form”

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1. Forms as algorithms — The case of the “normal form” Eamonn Bell Columbia University epb2125@columbia.edu www.columbia.edu/~epb2125 April 28, 2018 (Buffalo State College)
3. Overview 0. This introduction 1. Working definition(s) of algorithm and their ubiquity 2. Resemblances between textbook descriptions of normal form and computer code 3. Making use of the rhetoric of pseudocode, authors “algorithmize” the notion of normal form
4. (1.) Algorithms everywhere? Kitchin [D]ozens of key sets of algorithms are shaping everyday practices and tasks, including those that perform search, secure encrypted exchange, recommendation, pattern recognition, data compression, auto-correction, routing, predicting, profiling, simulation and optimisation.1 1Rob Kitchin, “Thinking Critically About and Researching Algorithms,” Information, Communication & Society 20, no. 1 (January 2, 2017): 14–29, doi:10.1080/1369118X.2016.1154087, p. 15. Kitchin’s laundry list is inspired by the nine algorithms described in John MacCormick, Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today’s Computers (Princeton, N.J.: Princeton University Press, 2013). MacCormick’s account is a more popular survey. 2Ed Finn, What Algorithms Want: Imagination in the Age of Computing (Cambridge, MA: MIT Press, 2017). p. 15.
5. (1.) Algorithms everywhere? Kitchin [D]ozens of key sets of algorithms are shaping everyday practices and tasks, including those that perform search, secure encrypted exchange, recommendation, pattern recognition, data compression, auto-correction, routing, predicting, profiling, simulation and optimisation.1 Finn Algorithms are everywhere. They already dominate the stock market, compose music, drive cars, write news articles, and author long mathematical proofs—and their powers of creative authorship are just beginning to take shape.2 1Rob Kitchin, “Thinking Critically About and Researching Algorithms,” Information, Communication & Society 20, no. 1 (January 2, 2017): 14–29, doi:10.1080/1369118X.2016.1154087, p. 15. Kitchin’s laundry list is inspired by the nine algorithms described in John MacCormick, Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today’s Computers (Princeton, N.J.: Princeton University Press, 2013). MacCormick’s account is a more popular survey. 2Ed Finn, What Algorithms Want: Imagination in the Age of Computing (Cambridge, MA: MIT Press, 2017). p. 15.
6. (1.) What is an “algorithm” (I) Diakopolous a series of steps undertaken in order to solve a particular problem or accomplish a defined outcome.3 3Nicholas Diakopoulos, “Algorithmic Accountability Reporting: On the Investigation of Black Boxes.” A Tow/Knight Brief (Tow Center for Digital Journalism, December 2013), p. 3 4Webster’s Third New International Dictionary, Unabridged, s.v. “algorithm,” accessed April 21, 2018, http://unabridged.merriam-webster.com.
7. (1.) What is an “algorithm” (I) Diakopolous a series of steps undertaken in order to solve a particular problem or accomplish a defined outcome.3 Merriam-Webster a procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an operation; broadly: a step-by-step procedure for solving a problem or accomplishing some end especially by a computer.4 3Nicholas Diakopoulos, “Algorithmic Accountability Reporting: On the Investigation of Black Boxes.” A Tow/Knight Brief (Tow Center for Digital Journalism, December 2013), p. 3 4Webster’s Third New International Dictionary, Unabridged, s.v. “algorithm,” accessed April 21, 2018, http://unabridged.merriam-webster.com.
8. (1.) What is an “algorithm” (II) Sedgewick (“pragmatist”) a method for solving a problem5 Finn (example of “computationalist”) [Algorithms] do not merely describe cultural processes with more or less accuracy: those proceses are themselves [held to be] computational machines that can be mathematically duplicated.6 5Sedgewick 6Finn, What Algorithms Want, p. 22.
9. (2.) Motivating “normal form” (Straus) A pitch-class set can be presented musically in a variety of ways. Conversely, many different musical figures can represent the same pitch-class set. If we want to be able to recognize a pitch-class set no matter how it is presented in the music, it will be helpful to put it into a simple, compact, easily grasped form called the normal form. The normal form—the most compressed way of writing a pitch-class set—makes it easy to see the essential attributes of a set and to compare it to other sets.7 7Joseph Nathan Straus, Introduction to Post-Tonal Theory (Upper Saddle River, N.J.: Prentice Hall, 2005), p. 35
10. (2.) Definition of “normal form” (Straus) 1. Excluding doublings, write the pitch classes as though they were a scale[.] […] 2. Choose the ordering that has the smallest interval from first to last[.] […] 3. If there is a tie […] choose the ordering that is most clustered away from the top. […] If there is still a tie, compare the intervals between the first and second-to-last notes[,] […] and so on. 4. If [this process] still results in a tie, then choose the ordering beginning with the pitch class represented by the smallest integer.8 8ibid., p. 36
11. (2.) Definition of “normal form” (Straus) 1. Excluding doublings, write the pitch classes as though they were a scale[.] […] 2. Choose the ordering that has the smallest interval from first to last[.] […] 3. If there is a tie […] choose the ordering that is most clustered away from the top. […] If there is still a tie, compare the intervals between the first and second-to-last notes[,] […] and so on. 4. If [this process] still results in a tie, then choose the ordering beginning with the pitch class represented by the smallest integer.8 (2.) Computer code-like characteristics 1. Ordered sequence of imperative statements 8ibid., p. 36
12. (2.) Definition of “normal form” (Straus) 1. Excluding doublings, write the pitch classes as though they were a scale[.] […] 2. Choose the ordering that has the smallest interval from first to last[.] […] 3. If there is a tie […] choose the ordering that is most clustered away from the top. […] If there is still a tie, compare the intervals between the first and second-to-last notes[,] […] and so on. 4. If [this process] still results in a tie, then choose the ordering beginning with the pitch class represented by the smallest integer.8 (2.) Computer code-like characteristics 1. Ordered sequence of imperative statements 2. Use of “if…then” (control flow) 8ibid., p. 36
13. (2.) Definition of “normal form” (Straus) 1. Excluding doublings, write the pitch classes as though they were a scale[.] […] 2. Choose the ordering that has the smallest interval from first to last[.] […] 3. If there is a tie […] choose the ordering that is most clustered away from the top. […] If there is still a tie, compare the intervals between the first and second-to-last notes[,] […] and so on. 4. If [this process] still results in a tie, then choose the ordering beginning with the pitch class represented by the smallest integer.8 (2.) Computer code-like characteristics 1. Ordered sequence of imperative statements 2. Use of “if…then” (control flow) 3. Generality 8ibid., p. 36
14. (2.) Definition of “normal form” (Morris, 1991) Rahn/Morris Algorithm: Normal-Form Representative Definition: Span (sub-K) of an ordered set X: () = − 0 where ≤ # − 1 1. is a member of some SC . Write pcset as an ordered set in ascending numerical order. Call this . 2. Let = # − 1. 3. Construct the set consisting of all rotations of and 0 . If contains duplicate members, prune all duplicates. 4. Find for each member of . 5. Find the smallest value of from members of . Call it . 6. Delete all members of with greater than . 7. If has only one member, go to step 11. 8. = − 1. 9. If = 0, find the member of with the lowest first pc, delete all other members of , and go to step 11. 10. Go to step 4. 11. Transpose the remaining member of to begin pc 0. The result is the normal-form representative of SC and can be found on both the Rahn and Morris set-class tables.9 9Robert Morris, Class Notes for Atonal Music Theory (Hanover, NH: Frog Peak Music, 1991)
15. (3.) Concluding remarks
16. Select Bibliography I Bernard, Jonathan W. “Chord, Collection, and Set in Twentieth-Century Theory.” In Music Theory in Concept and Practice, edited by James M. Baker, David W. Beach, and Jonathan W Bernard. Rochester, NY: University Rochester Press, 1997. Diakopoulos, Nicholas. “Algorithmic Accountability Reporting: On the Investigation of Black Boxes.” A Tow/Knight Brief. Tow Center for Digital Journalism, December 2013. Finn, Ed. What Algorithms Want: Imagination in the Age of Computing. Cambridge, MA: MIT Press, 2017. Kitchin, Rob. “Thinking Critically About and Researching Algorithms.” Information, Communication & Society 20, no. 1 (January 2, 2017): 14–29. doi:10.1080/1369118X.2016.1154087. MacCormick, John. Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today’s Computers. Princeton, N.J.: Princeton University Press, 2013. Morris, Robert. Class Notes for Atonal Music Theory. Hanover, NH: Frog Peak Music, 1991. Schuijer, Michiel. Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. Rochester, NY: University of Rochester Press, 2008. Straus, Joseph Nathan. Introduction to Post-Tonal Theory. Upper Saddle River, N.J.: Prentice Hall, 2005.