A theory of musical features is presented. The theory emphasizes how a given work is distinguished from other works in some musical corpus. The theory is illustrated by evaluating the principal motive proposed by Allen Forte in his analysis of Brahms's opus 51, no. 1. Forte's "alpha" interval-class set is shown to be unable to distinguish quartet No. 1 from other quartets by Brahms. On the basis of the theory, several refinements are made to Forte's alpha feature. Only when the prime form of the interval-class pattern is joined with a long-short-long rhythm does the resulting feature become distinctive of the work. Perceptually-pertinent properties are shown to assist in assembling a feature definition.
Imagine that you were robbed by someone and were later asked by police to provide a description of your assailant. Suppose you began your description by noting that the robber had a nose, two eyes, a mouth, and two ears. These "facts" would undoubtedly be "true," but the police would be rightly dismayed by your description for the simple reason that the facts fail to distinguish your assilant from a world of potential suspects.
When characterizing a musical work it is important, not simply to make accurate or truthful observations about its organization, it is also important to identify those characteristics that set the work apart from other musical works. In this paper, I propose to argue that music analyses can occasionally fall prey to such empty descriptive language. Like the description of our hypothetical robber, many otherwise truthful observations simply fail to be informative.
The paper is divided into two parts. Part One begins with a general theoretical discussion concerning the notion of a "feature." In the course of this discussion a number of properties of features will be distinguished and a set of corresponding terms defined. In Part Two, the concepts developed in Part One will be illustrated by referring to Allen Forte's analysis of the first movement of Brahms's string quartet, opus 51, no. 1.  In particular, the analysis will focus on what Forte identifies as the principal motivic feature of this work. Using the theoretical concepts related to features, several refinements will be made to Forte's motivic description. In effect, a `nose' will transformed into an `aquiline nose;' `two eyes' will be transformed into `close-set eyes.' A systematic analytic path will lead us to a motivic description that closely resembles a traditional diatonic motive as the principal intratextual feature of this work.
At the outset, it is important to reassure readers about the goals of this essay. My essay raises no criticisms whatsoever about set theory or the application of set theory to tonal music.  Indeed, set theory has made some admirable strides in providing greater clarity in feature descriptions. Nor should the paper be regarded as an attack on Dr. Forte.  Forte's analysis simply provides a convenient illustration of a problem that is common in music analysis. Rather, my intention is to demonstrate that grievous problems of descriptive language are evident even in the work of exceptionally observant theorists -- and to show that such problems are (at least in principle) avoidable.
In keeping with a set-theoretical approach, my analytic points here will stress the musical foreground. Since set-based analytic methods have difficulty with background features (without recourse to Schenkerian or other models), my own analytic points will also leave background characterizations to other analytic methods. My focus on foreground features is merely a matter of convience and of avoiding undue length. As the reader will readily understand, the inferential analytical approach employed here is as pertinent to the comparative evaluation of proposed background, middleground, and process-related features as it is to foreground motivic features.
Broadly-speaking, two closely-linked classes of features can be distinguished: intratextual and intertextual. Intertextual features arise from relationships between works, whereas intratextual features arise within the context of the work itself. At a minimum, we might expect a proposed feature to be present in an artifact or class of artifacts. If it is claimed that X is a feature of work Y, then we would naturally expect X to be located, or evident in Y. Sometimes features may be notable by their absence, however. For example, the absence of harmonic thirds and sixths in a tonal-period work would be something of an aberration. Since, the absence of something commonplace may itself be noteworthy, we might refer to this property as negative presence. Note that negative presence presupposes the existence of a normative repertoire or established set of expectations. That is, one can't recognize that something is missing unless one has a sense of what is normally present.
As in the case of positive presences, negative presences may be either intertextual or intratextual. An intertextual negative presence arises when the accumulated evidence suggests that a work belongs to a certain class of works, yet fails to exhibit a property that is otherwise assumed to be essential to membership in that class or group. By contrast, an intratextual negative presence arises within a work when the work itself establishes the expectation of some event or phenomenon that nevertheless remains unrealized. Since the thwarting of an expectation may quickly become a cliché, intratextual features are constantly being transformed into intertextual norms. 
Note that even in the case of positive presences, a proposed feature is rarely directly represented in an artistic artifact. For example, the pitches of a score do not directly encode the melodic intervals. The concept of "melodic interval" relies on the assumption of an underlying "voice" or "part" and deciphering voicing sometimes entails remarkably sophisticated interpretations. On what basis, then, can one defend the assumption of voice? Those theorists who have contemplated such matters typically rely on one of two appeals. One might appeal to notational conventions such as the use of separate staves or differentiation via stem direction. A more common appeal is to the perceptual experiences that affirm the subjective phenomenon of `musical line' and hence of `melodic interval.' We often assume that the notational conventions of stem directions and independent staves are straightforward reflections of a common psychological experience. That is, theorists rightly consider melodic intervals to be implied by individual successive pitches, and rightly assume that such intervals may be readily derived in most circumstances. 
In fact, few features can be identified in the "raw data" of an artifact. Like melodic intervals, most features are deciphered, interpreted or derived from the raw data. Of course even the "raw data" are interpreted. What makes the data seem ontologically "raw" is that the interpretations are comparatively stable and uncontentious. We can continue to use such terms as "raw data" as a linguistic convenience, although the term should be understood as a short-hand for interpretations that are less contentious -- and the term should never be seen as a foundation that is immune to further conceptual analysis or challenge.
Of course derived presences may be much more indirect than is the case for melodic intervals. Derived presences may include such concepts as deceptive cadences, prolongations, syncopations, suspensions, ritornellos, and other phenomena. Often, these derived concepts are more notable and compelling features than directly notated pitches or durations.
Note that there are an infinite number of possible derived presences, and not all such derivatives can have the same analytic status. For example, one might rigorously define the property fuddle to be the semitone pitch distance between the second-last notes occurring in odd numbered measures whose stem directions extend upward. Implicit in any theory is the idea that certain properties (such as melodic intervals) are not reified derivative concepts, and that, unlike the concept of fuddle, they represent some organizing principle that has influenced either the construction of the work, or its reception, or both. In the case of conventional tonal theory, the notion of a diatonic step is an example of such a central concept. In the case of set theory, the notion of an interval-class is an example of such a derivative concept.
In discussing above how theorists might justify the traditional voice-assumption underlying the concept of the melodic interval, two different forms of support were cited. One was based on subjective perceptual experience, and the other was based on notational practice. It is important to understand that these are just two of innumerable forms of appeals. One need not refer, either to perception or to the notation, in order to justify a useful descriptive concept. Consider, by way of example, the notion of something being "idiomatic." In works written for trumpet, it has been observed that difficult finger/valve combinations are systematically less common in works written by trumpet virtuosi than in trumpet works written by non-virtuosi.  In most circumstances, it is only the experienced trumpet player who is able to apprehend the specific properties that distinguish an idiomatic arrangement from an unidiomatic arrangement. That is, a trumpet player may experience a passage according to such categories as "easy to finger" or "difficult to finger." Moreoever, these conceptual and descriptive categories can be defended, even if listeners are unable to perceive them, even if non-trumpet performers fail to experience them, and even if theorists are oblivious to their existence.
Descriptive languages are often unique to particular communities. Usually, the language is largely given a priori, but there are always opportunities to expand the descriptive conceptual vocabulary. A composer might choose to create a work whose organization is intimately linked to the fuddle concept. If this were the case, the organization would have salience only for readers of this article -- and even then, only if the reader has access to the notated score. This simply illustrates that useful descriptive concepts can spring into existence within a given community, or culture.
When we describe features, we are at liberty to choose the descriptive language, although the language must contain concepts whose assumptions have some degree of support. In describing a robber to the police, an optician might focus on the robber's brand of eye-glasses, a fashion designer might focus on features of the robber's clothing, and a dialect expert might focus on the robber's manner of speech. However, within each of these descriptive domains, it is still possible to give either fruitful or empty descriptions. The optician's recognition a common brand of sun-glasses, the fashion designer's notice of poor color-coordination, and the dialect expert's animated description of the robber's impolite vocabulary may be of little use in distinguishing our robber from from anyone else.
The mere presence of some element or property does not necessarily make it a good feature. A good feature must in some ways draw attention to itself. It must be notable, or what might be dubbed salient. Of course there are innumerable ways by which something may draw attention -- some of which we will discuss later. One of the most venerable properties is the simple prevalence of an event; that is, how frequently a pattern recurs.
Apart from such intratextual factors, there are also intertextual factors that can contribute to salience. A single statement of "B-A-C-H" or the opening notes of Dies Irae can be sufficient to establish the notability of the event. Intertextual factors may be either intentional or unintentional on the part of the artist. Intentional factors may include quotation, allusion, parody and model -- concepts that have received some lucid theoretical attention.  To these may be added the concept of evocation where a passage unintentionally reminds the listener of a similar passage in another work. (Since evocation is defined as unintentional, the clearest examples occur where a musical passage reminds us of a similar passage in a work created much later by another composer.)
To summarize, we may define salience as a heightened attention that can arise due to either intratextual factors (such as phenomenal accent) or intertextual factors (such as quotation).
In identifying a feature, we must always be cognizant of the question "feature of what?" Features may be characteristic of a work, of a movement, of a composer, of a style, of tonal music in general, and so forth. What constitutes a feature depends on the scope of our gaze. For example, a non-Western listener may deem the sound of the pianoforte to be a significant feature of Western music -- without distinguishing any further divisions. To the music analyst, the features of principal interest have been those that characterize individual works, or sections thereof. At the supra-opus level, features of interest may include stylistic features, and the three basic lines of Schenkerian analysis, etc.
All scholarly disciplines are founded on some type of descriptive enterprise. Features provide the essential building-blocks for such descriptions. Each discipline establishes its own criteria for good description, however, some evaluative criteria are nearly universal and possibly tautological. Perhaps the most important property of a description is the degree to which it distinguishes the objects of our attention from other (possibly like) objects. If the goal of an analytic description is to convey what is unique or distinctive of a given object or class of objects, then good features must embody or define some of that distinctiveness.
Another important property of feature descriptions is the economy of the means of expression. When asked for directions by a passing motorist, the success of our description depends not only on the distinctiveness of the landmarks we mention, but also -- since human memory is fallible -- on the brevity of our description. Of course, in giving directions to our motorist, we might ask if they are already familiar with certain landmarks. In inquiring about their existing knowledge, we are, in effect, determining whether intertextual references will have any value in enhancing the description. Whether or not our motorist has knowledge of the pertinent geography, a good description will identify a succinct set of distinctive features. 
In the absence of distinctiveness, many otherwise truthful characterizations are without merit due to the excessive breadth of the description. In concrete terms, we may define distinctiveness as the property of "relative salience" -- that is, where a feature is more characteristic of the artifact being described compared with other artifacts or phenomena. Note that the property of distinctiveness is necessarily comparative.
Once again, what constitutes a distinctive feature depends on the scope of our gaze. A "nose" is a feature of faces in general, but the presence of a nose cannot be a feature of a particular face. It is the `stubby nose' or the `pixie nose' that may be a feature of some given face -- but only because of the implied comparison with other noses.
All of the foregoing attributes or properties of a feature are attempts to approach the central issue related to a feature: its importance, eminence or significance. By their very nature, the concepts of importance and significance are open-ended. A single, apparently trivial detail may be the source of a striking story. Significance is always contextual, and there is no telling the historical, social, personal, formal or other domains that establish the context for some event -- however apparently minor or trivial. This means that it is impossible, in principle, to produce an exhaustive inventory of the significant features of some work.
Although we cannot describe all the potentially significant features of a work, or even the most significant features of a work, this does not preclude the possibility of identifying some features that are significant. My claim here is that, salient distinctive features will always bear some degree of significance in a work -- even though other features may exist that can claim to be of greater significance.
By way of summary, the feature-related properties we have discussed are reviewed in Table 1. In general, a good feature may be defined as a succinctly described characteristic that is salient, distinctive, and significant. That is, a good feature is something that attracts our attention in some way, that distinguishes an object or class of objects from other like objects, and that is worthy of interest.Table 1
|presence||existent within an artifact|
|negative presence||absent from an artifact (although expected)|
|salience||noticeability of an event|
|distinctiveness||greater salience compared to occurrences in other artifacts|
|significance||notable, important, worthy of attention|
|Some Intratextual factors contributing to salience:|
|prevalence||noticeable because it recurs frequently|
|accent||noticeable because it is stressed (e.g., dynamic, agogic ...)|
|recency||noticeable because it is last|
|primacy||noticeable because it is first|
|mnemonic||noticeable because it is easily remembered|
|Some Intertextual factors contributing to salience:|
|evocation||unintended reminder of similar passage in another artifact|
|quotation||intended exact quotation from another artifact|
|allusion||intended indirect reference to another artifact|
|parody||intended exact or indirect reference, intended to spurn|
|model||intended or unintended borrowing of a structural framework|
There is no question that the alpha pattern is present in opus 51, no. 1. The pattern occurs in the opening three notes of the first violin and appears numerous times throughout the movement. According to our analytic taxonomy, we need to consider the degree to which occurrences of this pattern are salient and distinctive.
Of the various factors contributing to salience, consider first the simple property of prevalence. Out of 7,045 interval-class melodic diads found in this movement, the alpha pattern is the most common interval-class pattern, occurring 352 times. (The prime form alone occurs 136 times making it the 13th most common two-interval pattern.) 
However, since scale-like movements of tones and semitones are ubiquitous in tonal music, we might well expect the alpha pattern to be common in most musical works. Returning to the police station, suppose that we remain unconvinced by the police officer's protestations that saying our assailant "has a nose" is utterly vacuous. The officer might pull out a college yearbook and ask us to count the number of photographs where a person is without a nose. Having counted a large number of people, each with a nose, we might be more convinced that our description does indeed fail to be distinctive. Such a statistical demonstration might seem crude or unwarranted, but it serves the important purpose of illustrating the shortcomings of our description.
In order to determine whether the alpha pattern is distinctive of the first movement of opus 51, no. 1, we need to compare this movement with other musical works. But what other works would provide an appropriate comparison? If we compared Brahms's opus 51 to (say) a Brazilian samba,  then the origins of any observed differences between the two pieces would be difficult to interpret. Perhaps the differences would arise due to different historical periods, or different nationalities of the composers, or different instrumentation, or different styles. In other words, the observed differences may be manifestations of different classes of compositions (such as genres) rather than differences in the works themselves.
In order to minimize these interpretive problems it is preferable to select comparison works that are as similar as possible to the work we are attempting to describe. A good sample of music for such a comparison is all of Brahms's remaining string quartet movements. Brahms claimed to have written over 20 string quartets in his youth -- works which he later destroyed. Of his mature works, Brahms published three string quartets: opus 51, no. 1; opus 51, no. 2; and opus 67. These works were written over a period of about a decade. By choosing this repertoire, we can be confident in dismissing claims that any differences we may observe are due to different composers, nationalities, genres, styles, instrumentation, etc. That is, any observable differences are more likely attributable to the different characters of the works. Of course the string quartet movements are not exactly "matched" -- there remain differences. The various movements exhibit different tempi, moods, and forms. Rather than comparing the first movement of Quartet No. 1 with the other movements of the same work, in some ways, a better comparison might be to compare the first movements from all three quartets (No. 1: Allegro (3/2 meter), No. 2: Allegro no troppo (2/2 meter), and No. 3: Vivace (6/8 meter)). The different meters and the different keys mean that we cannot dismiss the possibility that any observed differences can be attributed to these factors.
Table 2 tabulates the number of occurrences of each of the alpha patterns for each of the first movements of Brahms's three string quartets. Separate results are shown for all four set variants of Forte's interval-class motive.Table 2
|pattern||No. 1||No. 2||No. 3|
|352 (5.00%)||421 (6.37%)||594 (7.92%)|
|of||(7045)||(6612)||(7498) two-interval instances|
The results of Table 2 show that the alpha interval-class pattern is prevalent in all three quartet movements -- not just in the first movement of quartet No. 1. In fact, the alpha interval-class pattern is proportionally more common in each of the other string quartet movements than it is in quartet No. 1. This implies (but does not prove) that the alpha interval-class is a musical commonplace -- at least in the case of Brahms, but probably for tonal music in general. In summary, we can say that the alpha interval-class pattern is present and prevalent in quartet No. 1, but, on the basis of this single salience factor, we cannot say that alpha is distinctive of quartet No. 1.
As we noted earlier, a good feature must draw attention to itself in some way. Apart from simple repetition (prevalence), there are innumerable other ways to draw attention. If we trust Forte's intuition, we must conclude that more than mere prevalence contributed to Forte's selection of the alpha motive. We need to look for other possible salience-enhancing properties. One place to begin is by noting that Forte selected the initial three notes of the work. In memory and learning tasks, psychologists have established that the first visual or auditory items in a sequence are more salient than other items -- a phenomenon dubbed primacy. (Similarly, people are better at recalling and learning items toward the end of a sequence -- a phenomenon known as recency.)
We might also note that Forte chose the upper-most voice, rather than, say, the initial notes in the viola part. We can only guess as to the source of Forte's intuition, but it is suggestive that this choice also accords with experimental research showing that listeners find outer voices, especially the upper-most voice, more perceptual salient. While we are considering these salience-enhancing treatments, lets consider some other common ones. On the most superficial level, salience can be enhanced by increasing the amount of acoustical energy (e.g., loudness and/or duration) that attends some event. All types of foreground accent (dynamic, agogic, melodic, etc.) can contribute to salience.
Perhaps it is the case that alpha tends to occur in foreground or melody parts rather than in accompaniment contexts. Motivic statements might tend to be presented via unison or octave doublings, or appear in outer-voice parts. Various forms of accent may be applied; statements may coincide with metrically strong positions. The feature might be isolated from preceding and ensuing material; it might tend to appear at the beginning and ending of the work, or at the beginnings and ends of phrases.
Tables 3a-f show the results of six exploratory analyses that attempt to determine the extent to which Forte's alpha patterns are linked with various contextual treatments that may be expected to enhance salience. Table 3a shows the number of instances of the alpha patterns that involve pitch-class doublings (such as unison or octave statements). Table 3b shows the number of instances of the alpha patterns that follow a rest (implying perceptual primacy). Table 3c shows the number of instances of the alpha patterns that precede a rest (implying perceptual recency). Table 3d shows the number of instances of the alpha patterns that coincide with the beginning of a phrase or slur mark (also implying primacy). Table 3e shows the number of instances of the alpha patterns that begin on the strongest metric position (down-beat) in a measure. Table 3f shows the number of instances of the alpha patterns that occur in outer voices ('cello and first violin). In each case, the results for opus 51, no. 1 are contrasted with similar tallies for the first movements of Brahms's other two string quartets.Table 3a
|pattern||No. 1||No. 2||No. 3|
|+2,+1||11 (34th)||6 (39th)||12 (30th)|
|-2,-1||0||1 (95th)||32 (7th)|
|-1,-2||2 (85th)||3 (64th)||31 (10th)|
|+1,+2||9 (42nd)||5 (51st)||11 (32nd)|
|22 (1.72%)||15 (1.83%)||86 (7.26%)|
|of||(1282)||(820)||(1184) pitch-class-doubled instances|
|of||(120)||(112)||(108) unique pitch-class-doubled interval patterns|
|pattern||No. 1||No. 2||No. 3|
|+2,+1||24 (2nd)||9 (9th)||17 (3rd)|
|-2,-1||13 (9th)||11 (8th)||7 (10th)|
|-1,-2||4 (20th)||15 (3rd)||11 (4th)|
|+1,+2||1 (58th)||19 (1st)||10 (6th)|
|42 (11.83%)||54 (15.84%)||45 (17.24%)|
|of||(355)||(341)||(261) rest-linked instances|
|of||(75)||(88)||(71) unique rest-linked interval patterns|
|pattern||No. 1||No. 2||No. 3|
|+2,+1||18 (3rd)||14 (4th)||11 (1st)|
|-2,-1||4 (24th)||18 (2nd)||9 (3rd)|
|-1,-2||2 (36th)||14 (4th)||5 (10th)|
|+1,+2||3 (29th)||10 (7th)||1 (55th)|
|27 (7.56%)||56 (16.23%)||26 (9.89%)|
|of||(357)||(345)||(263) rest-linked instances|
|of||(78)||(102)||(85) unique rest-linked interval patterns|
|pattern||No. 1||No. 2||No. 3|
|+2,+1||27 (2nd)||35 (2nd)||22 (13th)|
|-2,-1||30 (1st)||12 (16th)||2 (79th)|
|-1,-2||4 (42nd)||4 (63rd)||36 (6th)|
|+1,+2||13 (14th)||41 (1st)||10 (26th)|
|74 (11.24%)||92 (9.27%)||70 (6.66%)|
|of||(658)||(992)||(1053) slur/phrase-linked instances|
|of||(118)||(220)||(138) unique slur/phrase-linked interval patterns|
|pattern||No. 1||No. 2||No. 3|
|+2,+1||42 (2nd)||18 (9th)||7 (38th)|
|-2,-1||17 (9th)||12 (20th)||29 (9th)|
|-1,-2||6 (24th)||9 (28th)||23 (12th)|
|+1,+2||12 (13th)||6 (39th)||17 (19th)|
|77 (9.55%)||45 (4.53%)||76 (7.20%)|
|of||(806)||(994)||(1056) downbeat-linked instances|
|of||(118)||(165)||(158) unique downbeat-linked interval patterns|
|pattern||No. 1||No. 2||No. 3|
|+2,+1||79 (5th)||35 (12th)||71 (12th)|
|-2,-1||60 (7th)||50 (6th)||110 (6th)|
|-1,-2||33 (20th)||49 (7th)||86 (11th)|
|+1,+2||50 (9th)||42 (8th)||63 (13th)|
|222 (6.68%)||176 (5.50%)||330 (8.53%)|
|of||(3321)||(3200)||(3868) outer-most pitch interval-diad instances|
|of||(341)||(385)||(327) unique outer-most pitch interval-diad patterns|
Each table entry identifies the number of instances found, followed (in parentheses) by the rank position of that pattern in a list of all patterns in the movement -- ordered by frequency of occurrence. Summary statistics for each of the three quartet movements are provided at the bottom of each table. Specifically, the total number of alpha-related patterns are tabulated, followed (in parentheses) by the percent occurrence. The total numbers of pertinent measures are also identified followed by measures of the total number of unique patterns conforming to the defined context.
Consider first the summary statistics in each table showing the totals for all four interval patterns. In particular, compare the percentage occurrences for all three quartet movements. Tables 3a, 3b, 3c and 3f do not show any comparative coincidence between occurrences of alpha and pitch-class doubling, adjacent rest boundaries, or outer-voice positions. For example, Table 3a shows that only 1.72% of the alpha interval-class occurrences involved pitch-class doublings, whereas quartets No. 2 and No. 3 exhibited 1.83% and 7.26% respectively. However, Tables 3d and 3e appear to show a heightened occurrence of the alpha interval-class motive coinciding with slur or phrase onsets and beginning in down-beat metric positions -- compared to the other two quartets.
The results shown in Tables 3a-f are more telling if we focus separately on the four individual interval patterns subsumed by Forte's alpha interval-class pattern, Tables 4a-d recast the data in Tables 3a-f comparing the results for each of the string quartet movements for each of the four interval patterns: (a) the prime interval-pattern (+2,+1), (b) the inversion (-2,-1), (c) the retrograde (-1,-2), and (d) the retrograde inversion (+1,+2). Only the prime form (+2,+1) in Table 4a shows a significant increased presence compared with the comparison movements.  With the exception of the pitch-class doubling context, the (+2,+1) interval-pattern is linked with each of the examined salience-enhancing contexts much more often than for the other quartets. That is, the (+2,+1) interval pattern is more likely to be preceded or followed by a rest, is more apt to coincide with the beginning of a slur or phrase mark, is more apt to begin a measure, and is more apt to appear in an outer voice.Table 4a
|No. 1||No. 2||No. 3|
|Table 3a||11/1282 (0.9%)||6/820 (0.7%)||12/1184 (1.0%)|
|Table 3b||24/355 (6.8%)||9/341 (2.6%)||7/261 (2.7%)|
|Table 3c||18/357 (5.0%)||14/345 (4.1%)||11/263 (4.2%)|
|Table 3d||27/658 (4.1%)||35/992 (3.5%)||22/1053 (2.1%)|
|Table 3e||42/806 (5.2%)||18/994 (1.8%)||7/1056 (0.7%)|
|Table 3f||79/3321 (2.4%)||35/3200 (1.1%)||71/3868 (1.8%)|
|201/6779 (3.0%)||117/6753 (1.7%)||130/7422 (1.8%)|
|No. 1||No. 2||No. 3|
|Table 3a||0/1282 (0.0%)||1/820 (0.1%)||32/1184 (2.7%)|
|Table 3b||13/355 (3.7%)||11/341 (3.2%)||7/261 (2.7%)|
|Table 3c||4/357 (1.1%)||18/345 (5.2%)||9/263 (3.4%)|
|Table 3d||30/658 (4.6%)||12/1053 (1.1%)||2/1053 (0.2%)|
|Table 3e||17/806 (2.1%)||12/994 (1.2%)||29/1056 (2.7%)|
|Table 3f||60/3321 (1.8%)||50/3200 (1.6%)||110/3868 (2.8%)|
|124/6779 (1.8%)||104/6753 (1.5%)||189/7422 (2.5%)|
|No. 1||No. 2||No. 3|
|Table 3a||2/1282 (0.2%)||3/820 (0.4%)||31/1184 (2.6%)|
|Table 3b||4/355 (1.1%)||15/341 (4.4%)||11/261 (4.2%)|
|Table 3c||2/357 (0.6%)||14/345 (4.1%)||5/263 (1.9%)|
|Table 3d||4/658 (0.6%)||4/1053 (0.4%)||36/1053 (3.4%)|
|Table 3e||6/806 (0.7%)||9/994 (0.9%)||23/1056 (2.2%)|
|Table 3f||33/3321 (1.0%)||49/3200 (1.5%)||86/3868 (2.2%)|
|51/6779 (0.7%)||94/6753 (1.4%)||192/7422 (2.6%)|
|No. 1||No. 2||No. 3|
|Table 3a||9/1282 (0.7%)||5/820 (0.6%)||11/1184 (9.3%)|
|Table 3b||1/355 (0.3%)||19/341 (5.6%)||10/261 (3.8%)|
|Table 3c||3/357 (0.8%)||10/345 (2.9%)||1/263 (0.4%)|
|Table 3d||13/658 (2.0%)||41/1053 (3.9%)||10/1053 (0.9%)|
|Table 3e||12/806 (1.5%)||6/994 (0.6%)||17/1056 (1.6%)|
|Table 3f||50/3321 (1.5%)||42/3200 (1.3%)||63/3868 (1.6%)|
|88/6779 (1.3%)||123/6753 (1.8%)||112/7422 (1.5%)|
Tables 4a-d suggest that it is not the alpha interval-class pattern that is distinctive of the first movement of Brahms's opus 51, no. 1; rather the results suggest that it is the interval-specific form (+2,+1) that is distinctive. This is not to suggest that motivic inversions (for example) don't appear in the movement. (Salient inverted statements clearly appear in measures 92-105.) It is only to say that the inversion is a rare variation of a more basic feature. It is appropriate then that Forte identifies the interval-specific pattern (+2,+1) as the prime form of the interval-class set -- and uses this term in preference to the proper normal form (1,2). However, the retrograde, inversion, and retrograde inversion forms lack distinctiveness in this movement.
If we wish to improve upon Forte's alpha feature, we need to consider how the feature description can be modified or extended such that it will truly become clearly distinctive rather than merely prevalent. There are a number of candidate ideas for modifying the feature description. One of the most important candidates is to consider the relationship between pitch and duration. For example, the three notes of Forte's alpha motive seem to be linked to a long-short-long rhythm. This suggests that we re-analyse the work with regard to patterns consisting of both interval diad and relative duration. We might begin by first analyzing the long/short duration patterns in the three first movements in order to see if the long-short-long pattern is present, prevalent, and distinctive of opus 51, no. 1. All of the rhythmic-diad patterns are shown in Table 5.Table 5
|Duration Pattern||-------------||Brahms Quartets||------------|
|No. 1||No. 2||No. 3|
|68 (1.18%)||152 (3.07%)||99 (1.67%)|
|1322 (22.94%)||1090 (22.05%)||925 (15.59%)|
|610 (10.59%)||1169 (23.64%)||1206 (20.33%)|
In Table 5, the absolute duration of the notes has been of no concern; rather, a duration is deemed to be long or short depending upon the length of the preceding note. Thus the patterns half-quarter-whole and half-eighth-quarter would both be deemed long-short-long rhythms despite the fact that the quarter-duration is deemed `short' in the first pattern and `long' in the second pattern. Of course, this is only one of many possible approaches to characterizing rhythmic patterns.
Table 5 indeed appears to suggest that the long-short-long pattern (i.e. "shorter-longer") is more prevalent in the first movement of string quartet No. 1 than in the first movements of Brahms's two other quartets. The pattern appears 600 times out of 5,762 contiguous 3-note instances in quartet No. 1 (i.e. 10.4% of all instances). The corresponding percentages for quartets Nos. 2 and 3 are 8.1% and 5.0% respectively. We now need to consider possible relationships between this potential rhythmic feature and the alpha interval pattern.
Tables 6a-c show the durational contexts of the alpha interval patterns for each of the first movements of the three Brahms string quartets. Some patterns (such as longer-longer and shorter-shorter) are notably rare. The important question is whether the alpha patterns in quartet No. 1 are more strongly linked to the shorter-longer (i.e. long-short-long) rhythmic contour. The answer is a resounding yes. But once again, the answer must be qualified to say that only the interval-specific pattern (+2,+1) is strongly linked to the shorter-longer durational pattern. The remaining interval-specific patterns display no more rhythmic linkage than is found in the other quartets.Table 6a
|shorter longer||44 (47.8%)||7 (10.9%)||4 (9.8%)||3 (5.2%)|
|shorter longer||7 (14.0%)||25 (26.6%)||6 (8.1%)||3 (5.3%)|
|shorter longer||25 (22.9%)||34 (16.9%)||25 (14.5%)||31 (31.6%)|
An association or link between two parameters may be either unidirecitonal or bi-directional. For example, if we find that people with X-colored hair tend to have Y-colored eyes, it does not necessarily follow that people with Y-colored eyes tend to have X-colored hair. When two characteristics are reciprocally linked they form a bi-direction association. Bi-directional associations are especially noteworthy since they imply a common source or origin for both characteristics or parameters. The preceding tables (6a-c) established only that the (+2,+1) interval pattern has a strong tendency to occur in a long-short-long (shorter-longer) durational context. We should also determine whether shorter-longer durational patterns tend to coincide with the (+2,+1) interval context.
Table 7 shows the dozen most prevalent interval patterns exhibiting the shorter-longer durations in the first movement of quartet No. 1. As can be seen, the interval-specific (+2,+1) pattern is ranked first. The other set variants for alpha are not shown in Table 7 due to their especially low rankings -- 27th (-2,-1), 42nd (-1,-2), and 58th (+1,+2). In summary, Tables 6a and 7 show that the bond between the interval pattern (+2,+1) and the long-short-long rhythm is both strong and bi-directional. This linkage suggests that the interval and rhythm attributes share a common origin and are inextricable components of a single pattern. This finding suggests that it would be inappropriate to define the interval feature independent of its rhythmic properties. By contrast, none of the other set variants show any such rhythmic linkage. For example, a separate analysis of the retrograde forms of alpha showed no correlation with retrograde forms of the rhythm.Table 7
|rank||semitone interval pattern||# of instances||percent|
Further inspection of Table 7 reveals that the dozen or so most prevalent patterns show a marked predominance for ascending intervals. This bias toward ascending pitch sequences is confirmed in Table 8. Table 8 tabulates the predominance of all up-up, up-down, down-up, and down-down pitch contours in opus 51, no. 1. Of 553 interval diads (pitch triads), nearly 40 percent are solely upward in their contours -- twice as many occurrences as the down-down contour. This suggests that the specific interval sizes may be less important in our feature definition than we might think. The right-most column of Table 8 shows the number of occurrences of the various contours -- where all of the alpha interval-class occurrences have been excluded. As can be seen, the up-up pattern continues to predominate. In other words, the up-up pattern remains prevalent in this movement independent of the alpha interval patterns.Table 8
|Contours||alpha included||alpha excluded|
|up-up||198 (37.1%)||151 (31.8%)|
|down-down||101 (18.9%)||90 (18.9%)|
|up-down||106 (19.9%)||106 (22.3%)|
|down-up||128 (24.0%)||128 (26.9%)|
Returning to Table 7, we can see that the second- and third-ranked patterns outline the minor third interval -- an interval that Forte also highlights in his analysis. Unfortunately, Forte's discussion of the relationship between alpha and the ascending minor third leaves the omission of the second pitch as a mysterious compositional act. By contrast, when viewed in the rhythmic-metric context, the elimination of the second pitch is far less enigmatic. Given the long-short-long contexts of the minor-third occurrences, it is clear that the `middle' note of (+2,+1) has been treated as a dispensable "unaccented passing tone" -- a status that befits a note that we now know is interposed between two notes of longer duration and which occurs in a weaker metric position than either of its neighbors.
The sixth- and eighth-ranked patterns in Table 7 show what can only be interpreted as upward rising major and minor triads, whereas the seventh-ranked pattern corresponds to clear statements of the upward rising motive in the major mode (i.e., measures 232 to 260). However, due to the rejection of diatonic intervals in existing set theory, the (+2,+2) interval pattern must be defined as a separate set (Forte's epsilon motive, see Example 2).
With regard to diatonic intervals, a further analysis found that all of the instances of the prime form of alpha occuring in the first movement of quartet No. 1, are spelled as an ascending major second followed by an ascending minor second. By contrast, nearly ten percent of the alpha retrograde instances are spelled enharmonically as diminished thirds followed by minor seconds.
In light of the above analyses, we can now summarize the state of our alternative motivic description of the principal foreground feature in the first movement of Brahms's opus 51, no. 1. The feature consists of pitches in an upward-rising sequence, linked to a long-short-long rhythm, generally starting a phrase or slur, often following a rest, beginning in a strong metric position, and most likely to occur in an outer voice. (See Example 3.)Example 3.
The feature bears more than a superficial resemblance to the opening statement in the first violin. In traditional tonal theory, this feature would have been informally labelled `the principal motive.' All of the above analysis has merely formalized the evidence in support of this informal intuition.
It is important not to draw the wrong conclusion from the foregoing analysis. One might suppose that the analysis reinforces three old criticisms of set theory, especially in the analysis of tonal music:
In the case of Brahms's opus 51, no. 1, the above three criticisms are indeed warranted. But they are not warranted because rhythm is inherently important, diatonic intervals are privileged, or set theory exhibits a bias toward certain set variants. There is no telling how any given work may distinguish itself: rhythm may or may not be important, diatonic intervals may or may not be important, and set variants may or may not be important. In our analysis, these properties became important simply because they help to define a feature description that distinguishes the object under consideration from other similar objects.
In presenting the above analysis, there are a number of caveats and disclaimers that must be made explicit. First, it is important to recognize that the above analysis dealt only with intratextural elements and did not consider possible intertextural features to opus 51, no. 1. It is always possible that intertextural properties overshadow the intra-textural elements. As noted earlier, a single statement of, say, "B-A-C-H" may prove to be of great significance, even though such a statement may not be prevalent. Forte proposes a plausible (and fascinating) intertextual property of this movement when he links the C minor/Eb minor key relations of the first and second subjects to those of the opening movement of Beethoven's opus 13 piano sonata (Pathétique). This observation is consistent with the view that Beethoven's work provided a model for Brahms's writing.
Second, only a handful of salience-enhancing properties were explored in the foregoing analysis. Specifically, we examined prevalence, metric position, voice position, primacy, and recency-linked properties. In the specific case of metric and primacy properties, there are alternative ways of measuring such properties -- other than those used in this paper. Other accent types (such as melodic accent), and mnemonic properties were not explored in the above analysis. In addition, other ways of characterizing features -- such as scale-degree, diatonic interval (rather than semitone intervals), or relative interval size (e.g. step/leap) -- were not explored. Rather than investigating long/short durational relations, a better rhythmic feature might arise from characterizing the precise durational proportions (e.g., the 3:1 ratio of the dotted rhythm). Any of these alternative approaches might lead to a better feature description than the one shown in Example 3.
Third, the above analysis was restricted to foreground features only. In principle, however, the same comparative methodology can be used to evaluate claims of background features. In the case of Schenkerian analysis, this approach must await some computable implementation -- if this is possible.
Fourth, the above claims are limited by the comparison group of works. The failure of the alpha pattern to distinguish Quartet No. 1 from Brahms's other string quartets does not mean it is not a distinctive feature of some larger group of objects. For example, the original alpha pattern might distinguish Brahms's string quartets from works by other composers, or other works by Brahms. Or it may be that the alpha pattern is characteristic of Brahms in general, or of Western tonal music compared to other types of music.
Once again it bears emphasizing that the motivic feature shown in Example 3 is not claimed to be the "best" intratextural feature in opus 51, no. 1. In principle, it is impossible to determine the best feature, since there may always be some analytic perspective that provides a more penetrating characterization. My claim is merely that our refined feature is better than Forte's original alpha motive.
This paper has presented an analytic method for refining and evaluating feature descriptions. The method is inherently comparative -- that is, it clarifies features by placing a work in the context of a musical corpus. The method highlights what makes one work different from another.
The method allows different musical parameters to be integrated within a unified feature description. For example, the method allows the analyst to resolve whether or not particular articulation marks are integral components of a motive.
In applying the method to Brahms's opus 51, no. 1, the above partial re-analysis leads to a number of conclusions:
First, it was found that although Forte's alpha interval-class pattern is prevalent (occurs frequently) in opus 51, no. 1, comparison with other string quartet movements by Brahms indicated that the pattern is not distinctive. Distinctiveness arose only when this pattern was linked to contexts deemed a priori to further increase their salience. This observation has a number of repercussions. It suggests that the concept of salience is not a mere fiction. It also suggests that our operationally-defined estimates of salience (such as phrase onset or outer-voice position) can be useful indices of salience. In our analysis, salience was measured in terms of phrase-related primacy and recency, rest-induced grouping, unison doubling, outer-voice position, and metric position. All of these factors have been shown experimentally to enhance the perceptibility of musical events. That is, at least some forms of salience arise from known perceptual phenomena. In short, our analysis provides additional evidence that perceptual factors are often important in musical analysis.
Second, we showed that it is not the alpha interval-class pattern that is the principal feature of this movement, but a particular form of the pattern. Although the concept of interval-class identity may be important in certain kinds of music, there is no indication in this movement by Brahms's that interval-class patterns play any significant role.
Third, we showed that the pitch-interval feature was inextricably linked to a particular rhythmic context -- namely, long-short-long. Specifically, it was demonstrated that this link is bi-directional: the pitch contour is most commonly associated with the rhythm, and instances of the rhythm are most commonly associated with the pitch contour. In extending set theory to tonal works, theorists ought to develop methods for characterizing pitch-rhythm interdependencies.
Finally, it should be noted that the concept of "distinctiveness" is simply an introduction to music analysis of one of the most important concepts in contemporary scientific method -- namely the idea of the "disproof of the null hypothesis." The basic idea is that whenever an observation is made, scholars should endeavor to adjudicate its pertinence by asking the question "How likely would it be that this observation might arise merely by chance?" In experimental methodology, it is the so-called "control group" that plays a critical role in establishing the likelihood of observing something by chance. In the foregoing analysis, we have shown that examining a comparison group (in this case the opening movements of Brahms's other two string quartets), permitted us to advance more readily in identifying a distinctive feature of opus 51, no. 1. 
 Forte's analysis of this work was first published in 1983 and subsequently reprinted in 1987. "Motivic design and structural level in the first movement of Brahms's String Quartet in C minor." Musical Quarterly, Vol. 69, No. 4 (1983 Fall) pp. 471-502. Reprinted in: Michael Musgrave (editor) Brahms 2; Biographical, Documentary and Analytic Studies. Cambridge: Cambridge University Press, 1987; pp. 165-196.
 See Forte's The Structure of Atonal Music. New Haven: Yale University Press, 1973; also "Bartok's `Serial' Composition." Musical Quarterly, Vol. 46, No. 2 (1960) pp. 233-245.
 On the contrary, I regard Forte's goal of increased rigor in analytic tasks as a goal worthy of the flattery of imitation.
 Further discussion regarding this point may be found in Leonard Meyer's Emotion and Meaning in Music Chicago: Chicago University Press, 1956.
 The assumption that lines-of-sound are psychological "real" rather than "reified" is supported by a wealth of perceptual research. As theorists are well aware, not all pitch successions evoke intervals. For an extensive review of the pertinent perceptual evidence see Albert Bregman, Auditory Scene Analysis, MIT Press, 1990.
 This observation is chronicled in detail in David Huron and Jonathon Berec, "A Method for characterizing instrumental idiomaticism: A case study of the B-flat valve trumpet." MS.
 See, for example, the discussion of quotation and allusion in Kenneth Hull, Brahms the Allusive: Extra-compositional reference in the instrumental music of Johannes Brahms. PhD dissertation, Princeton University, 1989.
 There are other possible definitions of a good description, although I am not aware of any in the field of music theory. The definition proposed here merely echoes the widespread notion in theory (promoted by Schoenberg) that music and musical descriptions ought to seek an economy of expression. It is noteworthy that this definition of good description is analogous to the concept of efficiency in technical disciplines.
 An introduction to the historical background of this work may be found in Michael Musgrave and Robert Pascall, "The String Quartets Op. 51 No. 1 in C minor and No. 2 in A minor; a preface." In Michael Musgrave (editor) Brahms 2; Biographical, Documentary and Analytic Studies. Cambridge: Cambridge University Press, 1987; pp. 137-143.
 All of the ensuing measurements were carried out using the Humdrum Toolkit software. All repeats were expanded in the electronic scores prior to processing. See David Huron, Unix Software Tools for Music Research; The Humdrum Toolkit Reference Manual. Menlo Park, CA: Center for Computer Assisted Research in the Humanities, 1995.
 A ludicrous comparison to be sure, but one that well illustrates the point.
 The term "significant" is used here in the formal statistical sense of the word. Pooling the data for the two control movements, a chi-square analysis for the ratios of expected to actual instances produces the following results: prime form of alpha (X=56.47; df=1; p<<0.001 significant); inverted form (X=1.82; df=1; p=0.18, not significant); retrograde form (X=53.39; df=1; p<<0.001, significant absence); retrograde inverted form (X=5.22; p=0.02, significant absence).
 Forte's motivic description would appear to have only one advantage over the feature developed in this paper and shown in Example 3: namely, the alpha motive is more succinct. However, since the alpha interval-class motive is not distinctive of the work in question, the feature description must be regarded as too brief.
 Readers familiar with Forte's paper may have noted that our analysis has focused exclusively on Forte's alpha motives without mentioning the innumerable other subsidiary patterns discussed in his analysis. The reason should now be clear. Many of Forte's other motivic sets suffer from the same problems, and other sets appear to be attempts to patch-up the short-comings of the alpha pattern.
 This research was undertaken while the author was visiting scholar at the Center for Computer Assisted Research in the Humanities, Stanford University. The author is grateful to the Center's Director, Dr. Walter Hewlett, for providing critical feedback and advice.