Pandel Collaros
Bethany College
210 Steinman Hall
Bethany, WV   26032

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.

Preliminary Definitions

Two terms are defined here.  The first, pitch-class transition (PCT), is a melodic unit represented by two different pitch classes.  For example, A-C is a PCT; B-B is not.  For purposes of this discussion, a melodic unit that consists of two pitch classes that are enharmonically equivalent is not a PCT:  for example, B-A is not a PCT.  This discussion uses the concept of PCT to contextualize the more abstract notion of a movement from one "scale-step" or "scale-degree" to another.  It does this by referring for the most part to actual pitch classes rather than to labels such as "tonic," "supertonic," "mediant," and the like.

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.

Tonic Paradigm for Pitch-Class Transitions

The Tchaikovsky melody in Example 1 is in D major.  A paradigm that in a broad sense accounts for many of the PCTs of many melodies in D major can be constructed on the basis of several examples and discussions found in the music theory literature.1  The simplest version of this tonic paradigm is shown in Example 2.

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:
  1. Pitch classes that are not members of the tonic triad move to the nearest members of the tonic triad, except that
  2. The tonic pitch class exerts a stronger pull on the supertonic than does the mediant pitch class, as indicated by an arrow drawn from E to D instead of from E to F.
Recent literature has shed light on these two principles from the perspective of music cognition.  The first of these principles corresponds with what Steve Larson calls "magnetism,"  "the tendency of an unstable note to move to the nearest stable pitch"; the second corresponds with Larson's definition of "gravity," "the tendency of an unstable note to descend" (Larson 1997, 102).  These musical forces of magnetism and gravity operate within the "basic space," a concept described by Fred Lerdahl (1988, 319-21), and that is equivalent in the present discussion to the tonal hierarchy known as the key of D major.2

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.

Beyond the Tonic Paradigm

The rest of this discussion deals with modification and expansion of the tonic paradigm prototype to produce analytic tools that I call pitch-class transitions paradigms and pitch-class transitions quotients.  These tools help to explicate the transitions of PC strings more precisely than can the unmodified tonic paradigm of Example 2.  In this article, use of these tools is referred to as the analytic procedure.

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 .

Underlying Theory

PCTs may be roughly analogous to words or at least to phonemes; this is an assumption that underlies the analytic procedure described herein.  If words have meanings and phonemes have the ability to convey distinctions in meaning, PCTs may be similarly important in conveying meaning in PC strings.

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.

Pitch-Class String

The PC string is the analytic object.  By definition, immediate repetition of a pitch class, or the immediate following of one pitch class by its enharmonic equivalent, is eliminated from the PC string.  The purpose of constructing a PC string is to separate information that is used in the analytic procedure (PCTs) from information that is not used in the analytic procedure (register, duration, timbre, dynamics, and articulation for example).

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.

Pitch-Class Frequency Analysis

The next step in the analytic procedure is the creation of a pitch-class frequency analysis (PC frequency analysis) which permits the analyst to make fine distinctions among pitch classes.  These distinctions are based on a directly measurable attribute:  the frequency of a pitch class' occurrence in a PC string.  The cognitive importance of the frequency distribution of elements such as pitch class is discussed at length in Carol Krumhansl's Cognitive Foundations of Musical Pitch.  In this and several other passages Krumhansl discusses the importance of the distribution of pitch classes in the internalization of tonal hierarchies:
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).8
The 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.

Pitch-Class Transitions Table

The next step in the analytic procedure is the creation of a pitch-class transitions table (PCT table).10  A PCT table lists all the different transitions in a PC string, the number of occurrences of each transition, and the frequency rank assigned to each transition.  This information is used in subsequent steps of the procedure.  A PCT table for the Tchaikovsky string is shown in Figure 3.

Figure 3.  PCT table for the Tchaikovsky PC string

Frequency rank

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 transition.

Pitch-Class Transitions Paradigms

The PCT table and the PC frequency analysis provide the information needed to construct the pitch-class transitions paradigms (PCT paradigms), which are synchronic representations of the transitions in a PC string.  A different paradigm is constructed for the PCTs in each frequency rank listed in the PCT table.  Each paradigm is labeled accordingly.  A  first-rank paradigm represents the most frequent transitions.  A second-rank paradigm represents the next most frequent transitions, and so on.  It is important to remember that rank refers to the frequencies of PC transitions (PCTs), not to the frequencies of occurrence for individual pitch classes.

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.

Pitch-Class Transitions Quotient

The pitch-class-transitions  quotient is a measure of how closely the distributions of PCTs as a whole conform to the most frequent transitions in a string.  As such, it is a mathematical index of the overall distributions in the string.  The quotient can be calculated from the information in the PCT table.  First, the frequency of every PCT, (y1, y2, . . . , yn), is multiplied by its associated frequency rank, (y1 + 1 - y1, y1 + 1 - y2, . . . , y1 + 1 - yn); y1 is the designation given to the largest y value.  Next, the sum of these products is divided by the number of transitions in the string, (y1 + y2 + . . . + yn).  Finally, this preliminary quotient is divided by the lowest frequency rank, or y1 + 1 - yn, where yn  refers to the frequency of occurrence of the least frequently occurring PCT.   The formula is shown in Figure 4.

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:

  1. The closer the PCT quotient is to 0, the more closely the PCTs of the string conform to the higher-rank PCT paradigms, "higher-rank" referring to those paradigms which contain the greatest frequencies of PCTs.
  2. The closer the PCT quotient is to 1, the more the string is characterized by a few common transitions amidst a great variety of rare transitions.
  3. There is an exception to hypotheses 1 and 2:  if the PCT quotient is 1, the string consists of a perfectly equal distribution of PCTs.
The PCT quotient appears to function similarly to the concept of relative entropy found in information theory, but does so without recourse to such sophisticated mathematical theory.  Norman Dale Hessert writes:
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.

Combination Pitch-Class Transitions Paradigm

A combination PCT paradigm combines differently ranked paradigms of a PC string into a concise representation for various reasons, one of which is the clarification of the most characteristic voice leading in the string.  The Tchaikovsky string again will be used as an example.

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, E-D, A-F, 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:

  1. A finer differentiation among pitch classes according to frequency of occurrence rather than according to a prior notion of stability associated with the tonic triad,
  2. The inclusion of only the most frequent transitions found in the string under consideration, and
  3. A reduction in the number of pitch classes considered:  from eight in the tonic paradigm to only five in the combination paradigm (a direct result of factor #2).
It should be mentioned that other combination paradigms can be constructed based on different analytic goals.  For example, the fifth- through eighth-rank paradigms of the Tchaikovsky string could be combined to construct a concise paradigm of the rarest (and therefore, contributing to expressivity as discussed below) PCTs in the string.  Another example would be to combine all the ranks into a single paradigm that represents all the transitions; however, one must exercise care that the resultant graphic representation does not obfuscate more than it clarifies.  In any case, such efforts will be left for another time.


The development of quantitatively descriptive paradigms for a PC string can be problematic if one assigns special statuses to particular pitch classes without examining their precise roles in the string under consideration.  Establishing such biases, such as through the invocation of conventional tonal hierarchies, may result in paradigms that are not sufficiently descriptive of the string to which they are applied.  An example is the tonic paradigm which has clearly demonstrated pedagogical applications as evidenced in the literature, but is not appropriate as an analytic tool of any precision as was demonstrated here.  Even so, the tonic paradigm has inspired the development of a new tool, the PCT paradigm.

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.


1. The tonic paradigm is synthesized from various sources.  These include Zuckerkandl 1956, 35-36; Drabkin 1980, 325; Harder 1985, 19; Aldwell and Schachter 1989, 8-9; Kostka and Payne 1995, 79;  and Gauldin 1997, 35.  The paradigm in Example 2 most closely resembles that found in Harder.
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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 procedure.
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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 discussion.
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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 1968, 187).
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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 1980.
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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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