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Mathematical analysis or statistical analysis requires that something be measured, so it's not surprising that as applied to the arts it is able to first find a foothold in literature. Books, essays, any sort of written output, present a numerical collection, containing all kinds of primary data. Letters, words, sentences and so on, can be counted and basic statistics can be gathered: How often is a given word used What is the distribution of word length What is the distribution of sentence length etc. Do this and you are guaranteed to get numbers. What is not guaranteed, and hence is surprising, is that out of these numbers, patterns emerge, both on the scale of society, as well as the individual. In the former we see the patterns that seem to be intrinsic to any form of communication, and in the latter, we seem to be able to distill aspects of the idiosyncratic patterns of usage that form the basis of a person's writing style. The idea that mathematics might be useful for determining authorship is usually attributed to the 19th-century British mathematician Augustus de Morgan. De Morgan was in many respects ahead of his time, especially as regards to what appears to have been a broad-based investigation of the power and possibility of the formalization of thought. An interest in the formal or quantifiable aspects of creative work easily fits into this program. As recorded in his wife's memoirs, de Morgan wrote a letter to a friend in 1854 that states, "It has always run in my head that a little expenditure of money would settle questions of authorship" by determining if the writings of "the latter do not deal in longer words than the writings of the former." It must have been the case that de Morgan never had the extra pocket change to pursue this thought, as it was several years later in 1886 that the first published account of a mathematical analysis of literature appears, written by American physicist Thomas Mendenhall. The subject is Shakespeare, a favored focus of the question of attribution. Mendenhall tries to distinguish between Francis Bacon and William Shakespeare according to their relative use of four-letter words. It's not a success, but nevertheless, a discipline is born. It finds a name when 10 years later, Wincenty Lutoslawski looks at 500 numerical attributes in each of Plato's dialogues in order to reconstruct the order in which they were produced, working with the basic philosophy that works that are close mathematically should be close temporally. He called his methodology "stylometry." Modern trends in stylometry pull from the full bag of tricks of artificial intelligence and advanced statistical analysis. Some approaches focus on aspects of predictability-using the empirical likelihood that one word is followed by another. These ideas were first proposed in the early 1900s by the Russian mathematician A. A. Markov who used them to construct a very simple model for the cadences found in Pushkin's poem "Eugene Onegin." Today, so-called "Markov chains" are among the most commonly used tools in the mathematical modeler's workbelt and can be found identifying patterns in all sorts of places, ranging from genetic to financial data. Word frequencies used in one way or another remain the heart and soul of literary stylometry. Some of the most successful techniques focus on usage statistics of "function words." As opposed to "content words," these include pronouns, conjunctions, and prepositions, which generally carry very little non-contextual meaning and serve instead as grammatical connective glue. Literary style, it appears, resides in the degree to which we choose "that" rather than "which" or "however" as opposed to "nevertheless." The starting material for many a stylometric analysis begins by first isolating and recording the frequencies of the favorite function words among works of known authorship and then, in one way or another, considering the degree to which the frequency pattern in a contested work is statistically similar to the patterns in the secure works. Among other examples, function word usage has been used to distinguish between the writings of Alexander Hamilton and James Madison as well as to pinpoint authorship in the Wizard of Oz series. One of the most striking results in the field is the discovery that there are certain patterns of usage that seem to be simply intrinsic to the act of communication. In 1949, using a corpus of a range of works by a number of different authors, Harvard linguist George Zipf discovered a remarkable empirical fact, known today as "Zipf's law." It states that the result of multiplying a given word frequency by its "rank" (the most frequent word has rank one, the second most frequent is rank two, and so on) was approximately the same over all the words in the corpus. Zipf published his analysis in an amazing book titled Human Behavior and the Principle of Least Effort, which derives its name from the basic argument that predicts the Zipfian discovery: Imagine any author as a chance-driven machine in which at each step a coin is tossed-if it lands heads up, then a previously used word is chosen at random, while if it lands tails up, then a new word is written down. This is a "rich get richer" sort of model in which the more a word is used in the past, the more likely it is that it will attract more use in the future. In fact, Zipf finds similar relations (called "power laws"-a familiar distribution in the complex systems world) among all sorts of ranked lists, ranging from sizes of towns (here you would consider the product of the population of the town with its rank) to income distributions (where it also goes by the name of "Pareto's law"). Words can be counted-that's the main reason that stylometry came first to literature. But, other art forms have natural numbers too. Given the success ofliterary stylometry and the empirical ubiquity of Zipfian behavior, it's something of a surprise that it was not until just a few years ago that a broad analysis of that other great symbolic language that is musical composition was undertaken. Charleston College computer scientist Bill Manaris led a small group of researchers that counted note usage over a range of composers and works, and discovered a basic power law structure. Using that as the foundation of his analysis, he and his colleagues were able to derive a collection of statistical features from musical scores that successfully allow an automatic classification of musical works from jazz, classical, and rock n' roll.


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