The transition from one pitch to another is perhaps the single most
defining characteristic of melody. This article presents an analytic
system that 1) focuses on pitch-class strings that are derived from
series of such transitions, and 2) can be used to characterize such strings.
First a few issues must be addressed: Preliminary Definitions
for terms used in the discussion, and an explanation of the Tonic Paradigm
for Pitch-Class Transitions.
The second term is pitch-class string (PC string). A PC
string is a diachronic representation of PCTs in a melody. In other
words, it is a melodic structure in which the elements of pitch class and
order have been isolated from other elements such as register, duration,
dynamics, and timbre. (Example 1)
Example 1. Tchaikovsky, Pathetique, I, measures 89-100, theme (top staff) and PC string (bottom staff)
In Example 1, a melody to be analyzed is represented by conventional
notation on the top staff. As will be seen, the analytic procedure
is applied to the PC string derived from this melody, as represented by
the special notation on the bottom staff. This special notation will
be discussed in the section entitled Pitch-Class String.
Example 2. Tonic paradigm for PCTs in D major
Pitch classes of the tonic triad are represented by whole notes; all other pitch classes are represented by filled-in noteheads. The tonic paradigm suggests two general voice-leading principles:
The tonic paradigm is a concise representation of the types of musical forces that Larson and Lerdahl ascribe to music of the Common Practice Period, music to which I will affix the label "tonal" in this article. Beyond its obvious pedagogical utility as a general description of voice leading in this repertoire, I believe that it can be modified for use as an analytic tool as well. As it stands, however, the tonic paradigm is not a very precise graphic representation of the PCTs in a great number of tonal melodies. The paradigm is established as a straw man here, but only for purposes of discerning how it might be adapted for linear analysis.
Consider for example that only two categories of pitch classes are represented in the tonic paradigm: tonic triad and non-tonic triad. Tonic-triad pitch classes are represented by whole notes; non-tonic triad pitch classes, by filled-in noteheads. The concision offered by such a distinction is useful as a general categorization of scale steps; but from an analytical standpoint, the lack of precision is problematic.
Because all tonic-triad pitch classes are represented by one type of symbol (), the paradigm seems to imply that they represent equal points of stability to which other pitch classes move. However, this implication is contradicted in the very paradigm that suggests it. Even though the tonic and mediant pitch classes are equidistant from the supertonic (one whole step), the supertonic pitch class moves to the tonic instead of to the mediant pitch class. The inequality in attractive force between tonic and mediant, that Larson attributes to magnetism, is not reflected by the symbology. Likewise, because all the non-tonic-triad pitch classes are represented by one type of symbol (), the implication is that they represent relatively equal points of instability, an implication that very well may not be true for many melodies in the tonal repertoire.
A second area of concern with respect to the unmodified tonic paradigm as an analytic tool is that no arrows are shown going to the filled-in noteheads, which implies that pitch classes do not move to non-tonic-triad pitch classes. Additionally, no arrows are shown coming from the whole notes, which implies that melodic activity stops with arrival at a tonic-triad pitch class--i.e., tonic-triad pitch classes are points of stability from which no melodic activity originates or continues.
In the tonic paradigm, D is approached by both E and C, which suggests a third area of concern: tonic may be emphasized unduly as a point of linear stability. Such an emphasis may be misleading: D is not the most frequent second pitch class in transitions for many melodies in D major.
A fourth area of concern is that the tonic paradigm does not allow for
multiple melodic functions: only one arrow is shown leading from
any one pitch class. This ignores transitions such as A-C
found in arpeggiations of the dominant triad, and A-D that may appear in
an anacrusic sol to do gesture. Both are common transitions
in many D major melodies, and both transitions begin with A--which points
to the possibility of multiple melodic functions for the pitch class A
in D major melodies.3 Let me reiterate, the pedagogical
utility of the concise generalizations found in the tonic paradigm is not
the issue here. The issue is the development of an analytic procedure
that takes as its point of departure the prototype of the tonic paradigm.
One of the goals of the analytic procedure is to identify and quantify
PCTs in a way that addresses a kind of musical meaning that is discussed
in the following section. Another goal of the analytic procedure
is to quantify a particular type of expressivity that is discussed at the
conclusion of this article.4 The analytic procedure produces
results that are specific to the PC string under consideration and therefore
achieves the stated goals with precision .
Harold S. Powers writes, "Individual affects, or whole classes of affect, are ascribed to musical entities like motives or tunes, or to musical features like rhythms or intervals, and these features or entities are then said to be units of discourse in a musical language of pure expression" (Powers 1980, 1).5 It is in this sense, as "units of discourse" that I wish to examine PCTs as meaningful entities in the "expression" of music. In the same article, Powers quotes the Musica Enchiriadis as an "early articulation of the language-music parallel" (49). The passage to which he refers reads, "From the coupling of tones (soni) come intervals (diastemata); from intervals, in turn, grow systems (systemata)" (Palisca and Erickson 1995, 1). It is these "couplings of tones," or PCTs if we limit our considerations to pitch class alone, that are the important links between individual pitch classes and the PC string of a melody.
However, the analogy between words or phonemes and PCTs goes only so far. The meanings of words exist somewhat independently of the sentences in which they are found. On the other hand, a clue to the meanings of PCTs (and to a type of expressivity discussed in Conclusions) lies in the frequencies of their distributions in the PC string. Ian Bent's discussion of information theory in music analysis is relevant to the issue of communicating with PCTs as opposed to communicating with words or phonemes. He writes:
Artistic "communication" is, however, different in nature from other forms of communication in that it is not primarily concerned with transmitting maximum information: it is concerned rather with transmitting structure. It therefore requires a certain degree of what information theory calls "redundancy" (Bent 1987, 100).The reader may argue that the meaning of PCTs depends very much on syntax and context, how and where particular PCTs appear in the string, as well as on frequencies of distributions. I agree, but the distribution of PCTs is an important element in determining this syntax and context. Consider how the distribution of PCTs begins to address the structural issue of why one pitch class follows another in a PC string. That is, if a string has a particular frequency distribution of transitions, the order of its pitch classes must be constrained accordingly. For example if a string contains two A-B transitions and two B-A transitions--and only those distributions of those PCTs--then the order of pitch classes must be either A-B-A-B-A or B-A-B-A-B. There are no other possibilities. This is an overly simple example. But if one experiments with reordering the PCTs in a more complex example, such as in the PC string of Example 1, one would find that recognizable motivic pitch-class segments (three pitch classes and longer) from the original string frequently appear. Therefore, the frequency distributions of PCTs, a paradigmatic feature, begins to address the overall issue of pitch-class succession, a contextual and syntactic--in another word, syntagmatic6--aspect of melody.
Another assumption upon which the analytic procedure is based is that immediate pitch-class repetition is not relevant to the concept of pitch-class transition as defined at the beginning of this article. I am concerned here with the following questions with respect to the transmittal of structure: why follow one pitch class with a different one, in what ways is this done, and what do particular PCTs mean? I do not claim to answer these questions to any satisfactory degree. I do believe, however, that the procedure outlined in this article is a good start.
A third assumption upon which the analytic procedure is based is that the frequency of a pitch class in a PC string is a defining linear characteristic of the melody from which that string is derived.7 The harmonic or vertical stability of a pitch class is a cognitive issue that is touched upon below in the section entitled Pitch-Class Frequency Analysis. Quite apart from notions of harmonic stability, the frequent occurrence of a particular pitch class in a string reflects its purely linear stability in that string. Therefore, frequency of pitch class provides a clue, if only distributionally, as to what PCTs are to be found in the string, and by extension, as to why one pitch class follows another in melody.
Based on the assumptions above, the analytic procedure is applicable to any string that can be notated as a series of pitch classes. Interesting results are expected in idioms which produce melodies that are characterized by particular transition distributions. Such idioms are based on stylistic and procedural constraints associated with a priori organizing principles such as found in tonal hierarchies or tone rows.
Therefore, to apply Bent's general principle to the task at hand, the
musical meaning of PCTs and the pitch-class successions which they comprise
have to do with the transmittal of musical structure. For the purpose
of demonstrating the analytic procedure in this article, a melody has been
chosen that is based on the underlying tonal hierarchy commonly known as
D major. The melody transmits the "fact" of D major in an idiosyncratic
way as will be shown by the analytic procedure, the elements of which now
will be addressed.
In Example 1, the PC string is represented by the notation on the bottom
staff. Noteheads that represent pitch classes of the string are aligned
with the associated notes of the melody. Because the procedure ignores
immediate pitch-class repetition, the three consecutive articulations of
the pitch-class A on the top staff in measures 92 and 93 (and again in
96 and 97) are represented by a single notehead. Notice also that
the noteheads of the string all lie within the top and bottom lines of
the staff. This simplifies tabulation in later procedures by maintaining
a visual consistency for each pitch-class representation within a concise
format (within the staff). Finally, barlines are drawn through both
staves to clarify the correspondence between the string and the conventional
notation of the melody.
Listeners appear to be very sensitive to the frequency with which the various elements and their successive combinations are employed in music. It seems probable, then, that abstract tonal and harmonic relations are learned through internalizing distributional properties characteristic of the style (Krumhansl 1990, 286).8The advantage of such a readily measurable and cognitively relevant feature provides an improvement in precision over the differentiation of pitch classes found in the tonic paradigm of Example 2. The tonic paradigm divides all pitch classes into two broad categories, the analytic limitations of which have already been discussed. The PC frequency analysis overcomes these limitations, and provides a method for constructing more relevantly precise symbols for the pitch classes represented in the pitch-class transitions paradigms which will be presented shortly.
The first thing to do in a PC frequency analysis is to count the number
of times each pitch class in the PC string occurs. The results for
the Tchaikovsky string are shown in Figure 1.
Figure 1. Count of pitch classes in the Tchaikovsky PC string
This information is then organized into the PC frequency analysis, a left-to-right graduated scalar representation of the most frequent to the least frequent pitch classes in a string. The frequency analysis for the Tchaikovsky string is shown in Figure 2.
Figure 2. PC frequency analysis for the Tchaikovsky PC string
The longer the durational symbol, the more frequent is the occurrence of a particular pitch class in the string. The longest durational symbol used in the analytic procedure is the whole note. Each symbol's durational value is halved to obtain the symbol for the next most frequent pitch class, until every pitch class in the string has been assigned a symbol.9 Vertical lines separate pitch classes into groups of pitch classes which occur the same number of times in the string. The analysis in Figure 2 shows that in this case there is only one pitch class per group, and that the most frequent pitch class is F, the next most frequent pitch class is D, and so on. For ease of reference, the pitch classes listed furthest to the left in the frequency analysis and represented by whole notes are designated first class. The pitch classes listed next furthest to the left and represented by half notes are designated second class, and so on for the remaining groups of pitch classes.
Figure 3. PCT table for the Tchaikovsky PC string
It is a simple matter to examine the Tchaikovsky string and to derive the information needed for the PCT table. The first transition in the string is F-E. Each F-E transition in the string is counted and recorded. The second transition is E-D. Each E-D transition is counted and recorded--and so on, until all the different transitions have been identified and the frequency of occurrence for each noted. The data are listed in order from most frequent transition at the top of the table (which happens to be the first transition of the string in this case), to least frequent transition at the bottom of the table.
Also shown in the table is the frequency rank of each transition.
The frequency rank of 1 is established for the most common PCT. Less
frequent transitions are reflected by frequency ranks whose ordinalities
are determined in the following way: frequency rank = y1
+ 1 - yn, where y1 = the frequency of the transition(s)
that occurs most often, and yn = the frequency of a particular
The frequencies of occurrence for individual pitch classes are represented
in the paradigms by conventional durational symbols assigned according
to the frequency analysis described above. For visual concision in
the paradigms, these symbols are placed within the range of an octave
whose high and low boundaries are defined by the tonic pitch class.
Once the paradigms for a particular string have been constructed, they
then can be organized by rank as shown in Example 3.
Example 3. PCT paradigms for the Tchaikovsky PC string
The first rank PCT paradigm is represented by the top staff as shown. In this case, the first-rank paradigm contains only one transition, F-E, which appears eight times in the PC string. The F is notated as a whole note and the E as an eighth note, according to the frequency analysis. An arrow is drawn from F to E to show the direction of the transition in the string.
The second rank paradigm contains PCTs that occur one time less than those in the first rank. It is represented by the second staff from the top and contains no transitions in this case. The third-rank paradigm contains PCTs that occur one time less than those in the second rank. It is represented by the third staff from the top in Example 3. In this case, the third-rank paradigm contains one transition, E-D, which appears six times in the string. Again, each pitch class is notated according to the durational symbol that reflects its position in the frequency analysis, and arrows are drawn to show the direction of the transition. The same procedure is used to derive the paradigms for the remaining ranks, most of which contain more than one PCT.
The paradigms of Example 3 illuminate several features of the Tchaikovsky
string. First of all, the relative frequencies of all the transitions
in the string are immediately evident in the top-to-bottom organization
of the paradigms. Secondly, the pitch classes of the tonic triad
emerge in sharp relief as represented by the longer durational symbols
of whole, half, and quarter notes. The arpeggiation of the tonic
triad is indeed a prominent surface feature in this melody and is manifested
a number of times as circled in Example 4.
Example 4. Tchaikovsky, Pathetique, I, measures 89-100, theme: surface arpeggiations of the tonic triad
Finally, the fourth-, fifth-, and sixth-class pitch classes (symbolized by eighth-, sixteenth-, and thirty-second notes in the paradigms of Example 3) suggest a second harmonic structure that is expanded in the melody: an E minor triad. This structure--a supertonic triad, or the top of a leading-tone half-diminished seventh chord--becomes apparent if the notes that represent first-, second-, and third-class pitch classes are removed from the melody, as shown in Example 5.11
Example 5. Supertonic triad pitches in the Tchaikovsky melody
In summary, Example 3 is a synchronic representation of all the pitch classes and PCTs in the Tchaikovsky PC string. The PCTs are organized to reflect how often they occur in the string. Familiar musical symbols for the pitch classes are chosen according to the PC frequency analysis shown in Figure 2. The symbols themselves (whole notes, half notes, and so forth) reflect and separate out salient melodic features: in this case the horizontalization of relevant harmonic structures.
Figure 4. Formula for calculating a PCT quotient
PCT quotient =
y1(y1 + 1 - y1) + y2(y1 + 1 - y2) + . . . + yn(y1 + 1 - yn)/ y1 + y2 + . . . + yn
y1 + 1 - yn
y = the frequency of each PCT
y1 = the greatest y value
yn = the frequency of the least frequent PCT
In essence, the PCT quotient is the divided out ratio of the average frequency rank to the lowest frequency rank of a PC string. Note that the "lowest frequency rank" contains PCTs that occur the least number of times. The lower limit of a PCT quotient approaches 0. The upper limit of a PCT quotient is 1. By comparing the quotients of different PC strings in this and other studies (Collaros 1997, 1998, work in progress), I have made the following observations:
The real value of information theory lies not in how much uncertainty there is in the individual message, but rather the amount of uncertainty in relationship to the greatest possible uncertainty, in other words, not the amount of information, but the percentage of information. This percentage of information is called the relative entropy of the message source and is expressed as the ratio of the actual entropy to the maximum entropy (Hessert 1971, 12-13).Calculation of relative entropy involves base 2 logarithms and probabilities. Calculation of the PCT quotient is simple arithmetic manipulation of PCT counts, a direct and intuitive approach to the meaning of distributions in the string in light of the three observations I have cited.
The PCT quotient for the Tchaikovsky string is .55. It will
take many more analyses of this sort to make comprehensive statements as
to what this quotient may mean from a qualitative perspective with respect
to all, most, or even many of the PC strings that exhibit this particular
quotient. Nevertheless, data from this and several other analyses
suggest some possibilities for the musical significance, especially in
terms of expressivity, of the PCT quotient in general.12
These possibilities will be presented at the conclusion of this article.
Recall the paradigms for each of the ranks of PCTs in Example 3.
The first four ranks contain more than half (twenty-nine) of all the PCTs
in the Tchaikovsky string. The first four ranks by definition also
contain the most frequent transitions, which are in this case F-E,
B-A, and D-F.
On the basis of these observations, it can be said that the transitions
in the first four ranks by and large represent the most characteristic
transitions in the string. Therefore, a paradigm may be constructed
to reflect this fact by combining the paradigms of the first four ranks,
as shown in Example 6.
Example 6. Combination PCT paradigm for the Tchaikovsky PC string
As can be seen, the combination paradigm is just as concise as the tonic paradigm of Example 2. Furthermore, by virtue of the analytic procedure that has produced it, the combination PCT paradigm as a description of PCTs in the Tchaikovsky string has three advantages over the tonic paradigm. First, all PCTs shown in the combination paradigm literally occur in the Tchaikovsky string. Second, only the PCTs that occur most frequently in the Tchaikovsky string are shown in the combination paradigm. Third, pitch classes are precisely categorized into five classes on the basis of a measurable attribute: frequency of occurrence in the Tchaikovsky string. The categories are first class (), second class (), third class (), fourth class (), and fifth class ().
The advantages of the combination paradigm bring into sharp relief the descriptive weaknesses of the tonic paradigm in the case of the Tchaikovsky string. First, the transition C-D which occurs in the tonic paradigm does not occur in the Tchaikovsky PC string. Second, the transitions shown in the tonic paradigm are not based on an examination of this one PC string, but are based on the conventional notion of a tonal hierarchy. Third, the tonic paradigm distinguishes between only two categories of pitch classes: those associated with the tonic triad, represented by ; and those that are not, represented by .
I would like to make a point with respect to one of the advantages cited above. It should be noted that several PCTs that do occur in the Tchaikovsky string are not represented in the combination PCT paradigm--simply because they are infrequent. Recall that a decision was made to construct this particular combination paradigm on the basis of most characteristic PCTs in the string. The relatively infrequent PCTs that were omitted occur one to four times each, as compared with the transitions that were included and which occur five to eight times each. On the other hand, the tonic paradigm does include one of the infrequent transitions (G-F); and, as you will recall, it includes one (C-D) that does not exist in the string at all. Furthermore, the tonic paradigm neglects several transitions that do occur very frequently in the Tchaikovsky PC string: F-E (eight times), A-F (five times), and D-F (five times). Therefore, even though infrequent transitions are not represented in the combination PCT paradigm, this loss is compensated for by improvements in clarity, precision, and concision over the tonic paradigm. These improvements are due to the following factors:
PCT paradigms arrived at through the analytic procedure described in this article are specific to the string from which they are derived. It is important to note that the procedure does not equate frequency of occurrence with harmonic stability, but operates on the following assumption: a PC string can be described more precisely by counting the frequencies of particular pitch classes and PCTs than by assuming preconceived notions of stability, instability, and resolution tendencies associated with a tonal hierarchy that does not arise necessarily from the melody being examined.13 Therefore, PCT paradigms are more likely to incorporate relevant descriptive information than is the tonic paradigm.
PCT paradigms can be employed in more than one way. As shown in Example 3, they can be quantitative representations of all the transitions within a given string. Employed in this way, they resemble the "idiolect" of Leonard Meyer who writes, "It is the goal of music theorists and style analysts to explain what the composer, performer and listener know in this tacit way. To do so, they must make explicit the nature of the constraints governing the style in question. . . ." (Meyer 1976, 11).14 PCT paradigms reflect particular strategic melodic constraints within which a composer works.
A less microscopic but more concise representation of the analytic results, and therefore more appropriately comparable to the tonic paradigm, is the combination PCT paradigm as shown in Example 6. Even though it is not as quantitatively precise as are the collective paradigms of Example 3, the combination paradigm of Example 6 reflects a more accurate approximation of the rules that govern melodic syntax in the Tchaikovsky string than does the tonic paradigm.
A broader application of the combination paradigm idea also may yield interesting results well beyond the scope of one melody. For example, if all the PC strings for a particular historical period, genre, composer, or geography could be analyzed, a general combination PCT paradigm could be derived for a targeted repertoire. Such a general combination paradigm would be equivalent to Leonard Meyer's "idiom" or even "dialect," depending upon the breadth and depth of the repertoire considered (Meyer 1976, 30).
In addition to the PCT paradigms, the PCT quotient is the other primary tool of the analytic procedure discussed in this article. As a mathematical index of transitions distributions, the quotient quantifies a specific type of expressivity the meaning of which is clarified, I hope, by the following comments.
The observations made in the section entitled Pitch-Class Transitions Quotient give rise to the following hypothesis: the greater the PCT quotient (except for strings in which all transitions occur with equal frequency), the more expressive is the string.15 This seems to be logical. A PC string characterized by a few types of frequent transitions along with a variety of rare transitions, that corresponds with a relatively high PCT quotient, is capable of more surprises as far as PCTs go--by first establishing implications, and then by diverging from the realizations of those implications.16 On the other hand, a relatively equal distribution of different transitions, that results in a relatively low PCT quotient except in the case of strictly serial strings, provides few surprises from the perspective of what pitch class follows what pitch class. It is my observation that this is the case in simple diatonic strings.17
The type of expressivity discussed here is related to concepts of information theory known as entropy and redundancy, concepts which are dealt with in depth by, among others, Norman Dale Hessert (1971) and Leonard Meyer (1957).18 Redundancy correlates with musical features that occur frequently. Entropy correlates with those features that are more rare and in the context of information theory impart more information. With respect to PCT paradigms, the higher ranks reflect redundancy of PCTs; the lower ranks reflect entropy with respect to PCTs. It is in these concepts of redundancy and entropy that the paradigms, PCT quotient, and expressivity discussed here are interrelated. These interrelationships are explicated in this article by an analytic procedure that quantifies a quality of the PC string. This quality is a particular kind of expressivity whose quantification is reflected in the PCT paradigms and quotient.
Studies I have cited suggest that the PCT quotient (.55) of the Tchaikovsky string is relatively but not excessively low.19 To say that the quotient reflects a dearth of overall expressivity in the Tchaikovsky theme is incorrect. It must be remembered that expressivity in the context of this discussion is a function of the distributions of PCTs in a string. This particular brand of expressivity reflects only one of several features such as harmony, tempo, rhythm, dynamics, and timbre that contribute to the overall expressiveness of the music.
An examination of the paradigms reveals the reasons for the relatively low quotient of the Tchaikovsky string: most PCTs are represented in the highest ranked PCT paradigms (1-4.). The rare PCTs, represented in ranks 5-8, make up less than half of all transitions in the string.20 If not already obvious to the reader, a general principle should now be made explicit: PCTs in the higher-rank paradigms establish a context in which lower-rank PCTs may contribute to the type of expressivity discussed here.
I would like to make one final point, perhaps the most important with
respect to the Tchaikovsky string and the type of expressivity I have been
discussing. It was mentioned that low PCT quotients are characteristic
of simple diatonic strings in my experience. The Tchaikovsky string
reinforces this generalization in that its most frequently occurring pitch
classes are restricted to the rather narrow gamut of the diatonic pentatonic
scale. The simplicity of the pitch-class vocabulary, along with the
scarcity of melodic half steps as reflected in the paradigms, are nevertheless
important elements in the overall musical expressiveness of this theme.
Recall that this simple melody is juxtaposed amidst a lushness of harmony
and orchestration in which dynamics swell and recede along with the contour
of the melodic line. It is my conjecture that the juxtaposition of
a "less expressive" pitch-class succession and more expressive elements
color this theme in a very distinctive way that is in part explicated by
the PCT paradigms and the PCT quotient.
2. I shall use the term "tonal hierarchy" in a way consistent with Larson's
description: "synchronic hierarchies of musical elements," the "elements"
being pitch classes in the present discussion. See Larson 1997, 113..
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3. Previously cited authors address some of these problematic issues.
See Drabkin 1980, Aldwell and Schachter 1989, and Gauldin 1997.
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4. Information theory has provided mechanisms by which to approach meaning
and expressivity in musical structures. For example, in 1948 Claude
Shannon developed a formula which measures the complexity of information
in a stochastic message source such as a PC string; see Shannon and Weaver
1964, 50-51 and the discussion of relative entropy on page 56.
The pitch-class transitions quotient, a concept similar to those explored
in Shannon and Weaver, is discussed in the present article. See also
Hessert 1971 for an overview of information theory in music analysis.
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5. Other recent work also has addressed the issue of units of musical
meaning. For example, V. Kofi Agawu writes, "if we treat the individual
note as the elementary unit, we run into the immediate problem that not
only does a single note have no meaning except in relation to others, but
also the note is, for all practical purposes, a very small unit indeed.
. . . This is surely an indication that the elementary units of music are
best defined at a level greater than the single note, and therefore that
they embody a relationship as primitive" (Agawu 1991, 16). It is
my sense that PCTs qualify as one type of "elementary unit" of musical
meaning, and as such deserve scrutiny.
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6. "Syntagmatic" refers to how the various paradigmatic units, in this
case PCTs, are assembled in time according to a syntax. Authors using
this term include V. Kofi Agawu (1991, 8-9) and Nicholas Cook (1987, 165).
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7. Note that I am referring to pitch class here, not PCT.
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8. Matt Hughes (1977) presents an alternative method of analysis based
on cumulative pitch-class durations rather than frequency of occurrence.
Because duration is beyond the scope of the present discussion, I will
focus on the frequencies of PC distributions in this step of the analytic
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9. PC strings that have a great number of different PC distributions
would seem to pose a problem in the area of symbology here. The author
has devised a method that groups similar frequencies in order to limit
the number of symbols in the analysis (Collaros, work in progress).
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10. Dean Keith Simonton (1984) has demonstrated the usefulness of tabulating
"two-note transitions" and organizing the resultant data in a meaningful
way. A two-note transition is similar to a PCT; unlike a PCT,
however, a two-note transition can include two of the same pitch class.
For example, C-C
is a two-note transition, but it is not a PCT as defined in the present
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11. Here the PC frequency analysis has uncovered two melodic structures
that recall Paul Hindemith's "degree progressions of the melody" (Hindemith
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12. Research by this author includes analyses of PC strings derived
from the melodies of Beethoven, Schubert, Brahms, Hindemith, and others.
See Collaros 1997, 1998, work in progress.
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13. Larson addresses the issue of stability in a usefully discriminating
way. He describes "musical forces" that operate within a contextual
stability that is created by prolongation. However, Larson goes beyond
the empirical description of PCTs which is the focus of the current discussion
(Larson 1997, 102-112).
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14. A passage very similar to the one quoted appears in a later writing
by Meyer as well; see Meyer 1989, 10.
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15. The author is talking of only the PC string here, which is only
one aspect of melody.
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16. The author realizes that cognition of such divergence may be retrospective.
See Meyer 1973, 111, for specific comments regarding prospective and retrospective
temporal events. Also, Eugene Narmour provides detailed discussions
of prospective and retrospective melodic structures in his books The
Analysis and Cognition of Basic Melodic Structures (1990) and The
Analysis and Cognition of Melodic Complexity (1992).
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17. For example, the phrase which begins the first theme of the first
movement of Beethoven's Symphony No. 1 (measures 13-17) yields a PCT quotient
of .48 (Collaros, work in progress). The comparatively low quotient
reflects the elegant, yet simple diatonicism of the PC string. At
first thought, some may argue that the brevity of the Beethoven string
is the reason for the low quotient. My research does not suggest
a significant correlation between length of PC string and PCT quotient;
the mathematics involved certainly do not require such a correlation.
My research does confirm that the quotient is an index of the average frequency
rank and how it compares to the most frequent transitions in a string.
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18. A more recent and very in-depth discussion of music's ability to
be expressive, without the focus on information theory, is found in Kivy
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19. Analyses by this author of non-serial melodies reveal PCT quotients
that range from a low of .48 for the Beethoven PC string cited in note
17, to a high of .95 for the string derived from the first phrase of Paul
Hindemith's Acht Stucke I (Collaros, work in progress).
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20. Recall that the PCT quotient is a measure of how closely the distributions
of PCTs as a whole conform to the most frequent transitions in a string.
See the section entitled Pitch-Class Transitions Quotient.
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Aldwell, Edward and Carl Schachter. 1989. Harmony and Voice Leading. 2nd ed. New York: Harcourt Brace Jovanovich.
Bent, Ian. 1987. Analysis. New York: W. W. Norton.
Collaros. 1997. "The Quantitative Melodic Analysis System (Quanti-MAS)."
Paper read at the 35th Annual Contemporary Music Festival at Sam Houston
------. 1998. "Pitch-Class Transitions Paradigms and Quotients in Franz Schubert's String Quartets." Paper read at the Fifth Annual Conference of the Rocky Mountain Society for Music Theory at the University of New Mexico.
------. Work in progress. Pitch-Class Transitions Paradigms.
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