4.0 Endnote, with Peregrinations[In classical style, Part 4 was to be an effervescent finale connecting Leibniz' Ars Combinatoria, Pascal's assistance to an inveterate gambler, the contents of Haydn's library, and 18th century parlor games. An alternate model might have been Wittgenstein's Tractatus, with its final toss of labored calculation into the void. (Asked whether Wittgenstein's numbering system should be consulted, PZ said "NO! I have hated that book my entire life," but then added, "well, maybe..."). What follows is a collection of PZ's draft fragments, together with materials he planned to use, and some commentary. – Craig Pepples, Sept 2017] 4.1 The Milieu"Outsiders have enquired, with a persistence verging on personality, and with a recklessness scarcely distinguishable from insanity, to whom we are to attribute the first grand conception of the work." (Lewis Carroll, The New Belfrey of Christ Church, Oxford, Dover ed., 1872) Some questions now arise. We have seen what perhaps appears to be, and may actually be, a systematic progression through a list of ratios. Of course there are gaps in the progression, but were someone to systematically continue such counting for the majority of Haydn, would these gaps be filled in? And if yes, given the mass of works in question, would that be expected, due only to happenstance, or would such a finding indicate a deeper mathematical underpinning? I can hear many "musicians" saying that we are making far too much of the arithmetic or mathematical aspect of these compositions; that music is not numbers (or at least it was not until the dreaded 20th Century); that we leave no room for inspiration; intuition; art; creativity – all those concepts that are hurled about in order to provide an excuse for vibrato without thought. Let me use a fairly simplistic simile to clarify the distinction between, and the complementarianism of, these two supposedly opposing views. Consider how composing is like poker: one is dealt a "hand", and yes, the potential combinatorics (or ratios, if you prefer) of that "hand" have much to say about the future possibilities; but balancing this, skill, nerve, savvy, and intuition oftentimes allow a player to win a large pot thru bluff alone; and the corollary, by lack of nerve and intuition, a player can lose very badly despite an incredibly strong "hand". So it is with composition. Regardless of the combinatorics, the proof of the pudding is in how composers play the "hand"; and performers who (despite unbounded belief that a composition exists solely as a means for them to express themselves) did not actually compose the work in question, would profit by considering the ratios inherent in the musical material they wish to play. 4.11 The NumbersHard as this may be to accept, I also posit that "mathematics", during H's time, had far more KNOWN personal proximity and resonance, and was a more integral, and familar-familial part of music and music-making, than they are they today (no matter the general rending of garments in regards the serialists et al).1 As evidence for this heresy allow me to begin with Leibniz, who stated that "Musica est exercitium arithmeticae occultum nescientis se numerare animi", which has been translated as "The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic".2 From at least the Quadrivium onwards, music had been part and parcel with math and geometry3; and this habit of thought was very much alive in the 18th century. Rameau's 1722 Treatise cannot be read without some concept of the math behind the harmonic series. Nor can one study intonation (in the 18th or any century) without a knowledge of ratios. 4.2 Speculations on originsGranted, we will probably never know by what specific (mathematical? proportional?) mechanism H obtained the observed variety in his Ms, but we can ask from whence such a variety arose, and base our answer on (highly) relevant circumstance. 4.21 M FormLet us begin by disabusing ourselves of a (probably) fairly common misconception i.e. just because the overall M structure is quite formalized does not necessarily imply that the internal schemes are equally iron-clad. A minuet is not a sonnet, which MUST have 14 lines, with a fixed number of syllables per line, usually in a fixed meter (often honored in the breach), and with a fixed rhyme scheme.4 The M internally is a far freer form. In addition, because the initial segment was frequently recalled as the end of the second segment, the 10-part AABBCCDDAB form that we normally associate with the M is often, in reality, a 13-part (!) AABaBaCCDDABa;5 the desire and/or need to insert or avoid (as the case may be) the lower-case "aaa" could itself account for some portion of the structural variety that we have seen. Furthermore, were we to do numeric counts of first or last movements of the H quartets, I doubt anyone would be surprised to find similar vagaries in the number-counts; so do not be misled by the outward corset. This was a time of huge musical ferment; as H himself stated, the composers themselves were inventing musical style as their work proceeded. In addition, there was, in mathematics, an entirely new endeavor, which was to have a surprising connection to musical composition. 4.22 GamingAntoine Gombaud, Chevalier de Méré (1607-1684), an avid gambler, hoped for wealth from the gaming tables. He tried mathematical approaches with disastrous results and, determined to find why his strategy had failed, consulted his friend Blaise Pascal. Pascal's work on this problem initiated an important correspondence with fellow mathematician Pierre de Fermat (of the famous and only recently solved last theorem). Their letters laid the foundation for much of probability theory, especially as applied to dice games. By 1657, Christiaan Huygens published the first book on probability, De Ratiociniis in Ludo Aleae. And, curiously enough, in 1650, Athanasius Kircher had published his Musurgia universalis, in which is included an arca musarythmica, a device by which a non-musician could compose a piece of four-part music using prearranged musical fragments inscribed in wands arranged in columns inside a box. Somewhat later, Leibniz, pre-eminent philosopher of the day, continued to espouse both the connection between mathematics and music and the idea of applying mathematics to the composition of music. All of this, probably in combination with other influences (including Die Anfansgründe des Generalbasses, nach mathematischer Lehr-art abgehandelt und vermittelst einer hierzu erfundenen Maschine aufs deutlichste vorgetragen, 1739)6 produced a cottage industry built upon the use of dice to "create music" as a parlor entertainment or game; an entertainment accorded the title of "Ars Combinatoria".7 4.23 M GamesJohann Philipp Kirnberger published the first such dice game in Der allezeit fertige Polonoisen- und Menuettencomponist8 in 1757. While in his introduction, Kirnberger admits he did not invent the concept, the gist of this and all subsequent publications of this ilk (and there were myriads through the early 19th century), is that a batch of pre-composed measures are assigned ID numbers; a matrix then arranges the ID # in a way that guarantees more or less smooth transitions from whatever choice is made to whatever next choice is made. One then throws dice to determine which of the pre-composed 1st measures will actually be used for the version being created; throwing the dice again determines which pre-composed 2nd measure will be used; and the process repeats until the "composition" is complete.& Be aware that the people who created, or were credited with creating, these games, were not just the runts-of-the-litter. In addition to Kirnberger, the list included CPE Bach (Einfall, einen doppelten Contrapunct in der Octave von sechs Tacten zu machen, ohne die Regeln davon zu wissen, or "A method for making six bars of double counterpoint at the octave without knowing the rules", 1758, which appeared on p. 167 in Ferdinand Marpurg's Historisch-Kritische Beyträge zur Aufnahme der Music, part 3, a book owned by one Joseph Haydn). Speaking of books owned by H, his library also contained a Cabala per Comporre Menuetti which is purportedly no longer in existence today; and 33 years after Kirnberger, H may have written/is credited with writing his own Gioco filarmonico o sia maniera facile per comporre un infinito numero de minuetti e trio anche senza sapere il contrapunto, ("Philarmonic Joke, or the Art of Composing an Infinite Number of Minuets Without the Least Knowledge of Counterpoint", Luigi Marescalchi, Naples, 1790).9 The salient point is, that H had, in his possession, the CPE Bach game, as well as the Cabala, and probably knew of the Kirnberger, all of which date from when H was approximately 25 years old, i.e. a time when a bright young lad and composer (which H indubitably was) is at least expected to be open to new ideas; the idea of probability, perhaps even "randomness", might have influenced him, to some extent, for the remainder of his compositional life-span. How might this connect to the variety we have observed in our charts above? H was probably aware that an 8-bar times 8-line game matrix could generate millions of versions of an opening section, but these millions of versions would all be 8 bars in length. They would not vary as did H's 1st "A" (and subsequent) sections. Where, then, is the connection? 4.24 Point and CounterpointTo my mind, it is not such a leap from the idea of using dice to create randomly ordered equal-measure Ms, to conceiving that a throw of the dice could as well vary the number of total measures. I am not going to put money on the idea that H actually used dice to determine successive lengths of the sections of his Ms, but perhaps exposure to the idea of random generation within the fixed framework of a dice game matrix gave rise to a freer approach to overall phrase-length succession. There could of course be a far less fantastical explanation for the observed variety – i.e. almost every composer must have known of these games, and would have associated them with mass- production and overall sameness. Perhaps it is precisely the sameness of these games which spurred the Haydns of this world to, with malice aforethought, go out of their way to create Ms that were CLEARLY not from the same mold as the games. In short, it could be since all the Ars Combinatoria Ms were (within one version) all of the same length, and could be generated by every Tom, Dick and Harry, H (et al.) HAD NO CHOICE but to concentrate on changing segment lengths to demonstrate some compositional ingenuity and/or originality. Furthermore, this desire to produce unique forms may have been combined with knowledge of prime numbers, Fibonacci series, and Pascal's Triangle, which, leavened with the understanding of the combinatorics of the game, may have also affected how the truly composed Ms were put together. 4.25 InharmoniousnessThere is yet another aspect whereby the Ars Combinatoria Ms may have influenced the "art" Ms – i.e. we have noticed much ambiguity in the composed Ms, especially as regards harmonic implications, and/or establishment and resolution, as well as how to partition phrases. Could this be a result of the problems of the Ars Combinatoria matrices? i.e. one can not generate an M table and retain much clarity as to where a tonic, or a modulation, will be established, since to pin down such a locus means that all possibilities must conform, else one will have some very unsubtle progression within the multiple Ms. Therefore, to guarantee smooth transitions, one must avoid being overly definitive, and that practice could have carried over to the "art" Ms. On the other hand, the Ars Combinatoria games were not without a downside, as the idea (in the public 's consciousness) that Ms are dull and stupid things (as opposed to the metier on which one learned how to compose) may have originated from dice games hawked with the claim that anyone could write Ms without the slightest knowledge of music. Whatever the effect of the M dice games, there is no denying their existence; the fact that they were commonplace; that Haydn knew of them and was credited with producing one. All that taken together must have in some way influenced the general environment of M composition. What the Ars Combinatoria does not explain is how there came to be so few repeated bar-count schemes in the H Ms, or what method H used to guarantee such variety. 4.3 The ReconstructionAlso -- and this CAN NOT be said firmly enough; what I am proposing above in terms of how to "interpret" the Ms is not " deconstruction"; this is not influenced in the least by a Derrida, or even a Foucault. What I am doing is more in the way of "reconstruction" or revivification. After all, the bar-number-counts are the bar-number-counts. I did not invent those, or their variety. The baby-step (and that is what it is) I have taken is to accept that data, and to posit: (a) perhaps the variety we have seen can be explained by the fact that the standard M was so "common" (that various people even created random Menuet tables of the type we have seen); therefore H, perhaps to show off, or otherwise differentiate his "real" Ms, went out of his way to be as varied as possible. (b) if there is such immense variety in the global H M perhaps there should also be the same in the individual local sections of the M. Also note the issue of covariance between "A" & "B" sections. i.e. the usual form of the M implies a repeat of the "A" section as part of (the terminus of) the "B" section, so that the totality, with repeats = A,A:||B(A),B(A):||C,C:||D,D:||A,B(A). Were "A" to always be repeated in entirety within "B", the opportunity to vary the measure counts of "B" would be restricted to whatever new material is introduced. However, H is astonishingly free in his recall of "A". Sometimes "A" is present in full within "B"; but very frequently "A" is truncated; or broken into component parts which are almost collaged. This shows a healthy playfulness with compositional method, at variance with the received view of the closed master-pupil apprenticeship from earlier generations and the later cult of the inspired genius. 4.4 Additional Gaulimaufrey[PZ planned to refer to parts of an email from Anton Vishio of July 31, 2005, which references an article:]
4.41 And Nonsequitur[The end of PZ's draft includes the following quote from the National Review of Sept 18, 2005, "Remembering Close Reagan Aide Peter Hannaford":]
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