Buffon’s needle – the probability that a needle dropped on lined paper will fall on one of the lines). The concerns of stochastic geometry can be distinguished from those of fuzzy geometry, which is the geometry of inexact measurements.
We must distinguish time series geometry from stochastic geometry in general because time series are not directly commensurate with patterns in the two-dimensional Euclidean plane. The dimensions of a time series (in this instance, pitch and time) are not measured with comparable units. In time series analysis, we can speak of the rate of change of a variable with respect to time; and, it is permissible to compare ratios between values on the different scales, because ratios cancel out the units of measurement. It is not permissible, however, to make direct comparison between values measured in different units. For example, it would not be meaningful to compute the distance between two tones in pitch and time using the Pythagorean Theorem, which would involve taking the sum of the squares of distances measured in two different units. For that reason, the subject matter of the current study is not the same as what is normally designated by the term stochastic geometry. What the two concepts have in common is that they are both concerned with randomness and geometry.
iv Vos and Troost (1989); Huron (2006), pp. 75-77. v Huron (1996); Huron (2006), pp. 85-88.
vi Generally speaking, the concepts of the climax and the dramatic arc can be traced back through
Gustav Freytag’s analysis of the five act drama to Aristotle’s tripartite theory of tragedy. In the context of Romantic music, the notion of musical form as a developing process can be interpreted as specifically Hegelian (see Schmalfeldt [2011], pp. 23-30, especially p. 269, n. 19).
vii
Mann (1943), pp. vii-xiv.
viii Jeppesen treats modes with major and minor tonics differently. 75% of Jeppesen’s cantus firmi
with minor tonics climax early, that is, in the first half. With a standard error of 12.5%, this proportion is statistically significant at the 95% level of confidence. The most common location of the melodic peak for minor tonics is in the first quarter of the melody. The distribution of peaks is more irregular for major tonics, but the most frequent area for the location of the climax is in the third quarter of the melody. In the ideal types, the affective distinction can be interpreted as follows: major tonic – bright, confident, and certain; minor tonic – dark, yielding, and uncertain. This distinction parallels what we find in Fux, although the manner in which the distinction is made differs.
ix Schenker (1910); Federhofer and Mann (1982); Snarrenberg (2005).
x See Narmour (1977, ch. 11) and Gjerdingen (1988, ch. 1 and 3) for a discussion of archetype,
schema, style form, style structure, and idiostructure. These are complex subjects, but – roughly speaking – grammar is concerned with style forms (particular features, regarded as abstract forms, such as triads) and style structures (the regular combinations of features that distinguish a style, such as cadence formulas). Rhetoric, as I am using the term, is concerned with the other, more high-level concepts, particularly idiostructures. Idiostructures – again, roughly speaking – are large, relatively freeform gestures found in individual compositions, which are as much process as they are form. Style analysis, in the sense of that which is concerned with describing or defining a musical style, is primarily concerned with style forms and style structures. This is to be distinguished from critical analysis, which is more concerned with style structures and idiostructures. In this study, we are primarily doing critical analysis, although we are also doing style analysis in the sense that we are concerned with the differences between musical styles from a critical point of view. The critical analyst, in my view, needs special techniques to help see the forest beyond the trees. As a critical analyst, I share Zbikowski’s (2002, pp. 96-134, esp. pp. 124- 26, 132-34) view that a theory of music is made up of a number of conceptual models, changing in response to circumstances, which guide understanding and reasoning, provide answers to conceptual puzzles, and which simplify reality.
xi Schenker’s description of wave motion in this place uncharacteristically bears a certain
resemblance to Kurth’s conception of musical form. See Cook (2007), pp. 263-264.
xii The concept of the Ursatz is fully formed by the first two volumes of Schenker’s Das
Meisterwerk in der Music (1925-1926), but the concept is nearly complete by the fifth issue of Der Tonville (1923). See Pastille (1990), pp. 79, 81.
xiii In the current work in progress, a melodic typology adapted to the wave paradigm has been
used to describe the distribution of arch types in the cantus firmi. (The wave typology differs from – but does not contradict – Narmour’s well-known implication-realization typology [1990, 1991, 1992, 2000]. Narmour’s melodic typology is genuinely interesting, but it serves a completely different purpose; and research on the subject is still ongoing.) In the wave typology, arches (and complementary troughs) are distinguished according to whether or not the first and last notes make a rising, level, or falling motion. They are further distinguished according to whether their top (or bottom, in the case of troughs), occurs at the mid-point, or early or late relative to the mid-point. The wave typology can be extended to include compound forms, such as
the open-low-high-close form, or the open-high-low-close form. Applying information theory to the typology, we can evaluate the unity and variety of melodic ideas in the abstract – that is, without regard to the identification of specific motives in the traditional sense. An interesting and unexpected finding – the technical details of which are to be explained more fully in a later installment – is that there is an approximate power law relationship (which appears linear in a log- log chart) between the relative frequencies and the rank orders of the basic arch types in a composite data collection from all of Fux’s cantus firmi, segregating the opening motions from the closing motions. The exponents of the power laws for the opening and closing motions differ, reflecting the greater variety of arch types in the opening motions. This power law is comparable to Zipf’s Law (Zanette, 2006).
A model of Zipf’s Law is proposed, based on the cognitive bias known as the distinction bias (Hsee and Zhang, 2004). It is hypothesized that if we always take the most preferred member (arbitrarily selecting one member where the utility is similar, even when the advantage is small, or non-existent) of a (uniformly distributed with replacement) random selection of alternatives, for a range of trial depths, the resulting distribution will approximate a power law. The exponent of the power law is related to the maximum trial depth, a larger maximum trial depth representing a more selective decision procedure.
xiv Cooke (1959, pp. 102-110) discusses examples of rising and falling melodic lines in both the
major and minor modes.
xv For a discussion of the difficulty of drawing theoretical conclusions from the overtone series,
see Lerdahl and Jackendoff (1983, 1996), pp. 290-294.
xvi
In terms that are developed in the following section, the VHF of the original Dorian cantus firmus was 0.400 for the opening motion, and 1.000 for the closing motion. The contrast in the alternative melody is theoretically nonexistent: the VHF is 0.667 in both cases. For this reason, the alternative is especially suitable to be put into retrograde motion.
The VHF belongs under the umbrella of fractal geometry. I used a somewhat more complicated fractal indicator in my study of Haydn minuets (Chesnut [1996]). The ancestor of these indicators is Hurst’s rescaled-range analysis, which Hurst used to study flows of water in the River Nile. In a parallel study, Duane (2012) uses information theory to examine the nature of melody. Fractal geometry and information theory are complementary. Each has its advantages and disadvantage, depending on the context, and what one is attempting to accomplish. Fractal geometry is concerned with spatial relationships (presumably invoking the right brain) in which