Béla Bartók's Mikrokosmos is a six-volume set of 153 piano pieces arranged in order of increasing technical and musical difficulty. Since its publication in 1940, Mikrokosmos has been used to introduce novice pianists to aspects of technique and expression. It has also served as a source of models for aspiring composers. Music theorists have used Mikrokosmos pieces to illustrate techniques and materials of 20th-century composition.

With the advent of pitch class set theory, theorists now have a rigorous and systematic method for studying pitch organization in Mikrokosmos and other 20th-century masterworks. But set theory can be highly abstract and difficult for novices to comprehend and apply. Before describing my software in detail, I would like to reflect briefly on the difficulty of that task

In his book Women, Fire and Dangerous Things, George Lakoff (1987) uses the term "generative category" to denote a type of category characterized by at least one central member, the generator, plus a set of rules. The rules take the generator as input and yield the entire category as output. In the theory of unordered pc sets, the generator is a Tn/TnI-type or prime form, and the rules are the operations of pitch class transposition (Tn), or inversion followed by transposition (TnI).

Typicality effects are often observed with generative categories. It is common for novices to regard the generator as more central to a category than the instances which it spawns. Robert Morris acknowledged this when he remarked that novices often equate an instance of a set class with the set-type (prime form) that represents the class, or they may equate a specific realization of a set with the set itself. This is especially true in set theory where metonymy is often the guiding principle for coining names.

Lakoff explains this tendency during a summary of research on basic-level categorization when he notes that "of all the possible names for something in a category hierarchy, a particular name, at a particular level of categorization, ‘has a superior status’" (1987, p. 31). These basic-level names seem to correlate with nonlinguistic actions, and they are usually short. In addition,

Lakoff goes on to note that basic-level categorization depends upon experiential aspects of human psychology: gestalt perception, mental imagery, motor activities, social function, and memory. Superordinate and subordinate categories have less psychological centrality; they seem more like "achievements of the imagination" (Brown, 1965) and, hence, are more abstract.

So how does this work apply to music theory pedagogy? I would submit that notes are basic-level categories for most novice musicians, at least for those who have been taught to read standard musical notation. Notes can be perceived visually and aurally as discrete objects; for a beginning performer each note requires a specific set of motor activities and often a specific mental image as well. Later, during the course of their theoretical studies, novices must develop a more abstract conception of music. We, their instructors, try to lead them to the point where interval types, scale types, chord types, and tonal contexts become familiar enough to assume the status of basic-level categories. Any experienced teacher of music theory knows that this is hard work; especially for students who resist musical mind training. For them, music is something that one does, not something that one thinks about.

Another set of challenges appears when our students study 20th-century art music from the viewpoint of recent music theory. At this stage, they must add more superordinate categories to their mental framework. Some of these categories, such as pitch class and interval class, really aren't new, they're just new terms for ideas that should already be familiar. If we were to compare a typical student's mental framework to a house with basic-level categories as its first story, then these new categories might constitute the second or third story. But we rarely stop there; we continue to add categorical levels such as ordered sets, unordered sets, set classes, and set types. As we do so, we move further away from basic-level categories and deeper into the realm of abstraction.

In his review of Michael Friedman's Ear Training for Twentieth-Century Music, William Thomson (1993) noted that pitch classes are inherently intangible things. Strictly speaking, we can't see or hear a pitch class; we can only see or hear instances of pitch classes. Thomson contends, therefore, that pitch classes are first-order abstractions. And if that is true, then ordered pc sets are second-order, unordered sets are third-order, pc set classes, or set-types, are fourth-order, and the various genera of pc set-types are fifth-order.

These abstract categories are truly achievements of music theorists' imaginations. They enable us to ignore surface-level distinctions and concentrate instead upon aspects of equivalence and similarity that lie beneath the surface. In doing so, however, we lose contact with notes, chords, and motives--musical objects that can be heard, seen, and expressed in action and movement. But on the other hand, if we concentrate solely upon such primary objects, we can easily overlook significant higher-level complexes such as intervals and interval classes, harmony, tonality, meter, rhythmic grouping, and form.

If higher level categories are essential for a sophisticated conception of music (and I believe they are), then an instructor's task is to nurture their development and utilization. To do so, one must provide an environment in which novices can perceive, remember, and compare numerous instances of a class or category. The teacher must also develop the student's mental apparatus for conceiving and naming those instances at various categorical levels. For the more instances or realizations students encounter, the more mature their conception of given categorical level becomes. As students invoke various levels of categorization and name their instances, they become more facile at navigating freely between categorical levels.

Mikrokosmik Sets is a suite of hypermedia documents that facilitate exploration of Bartók's Mikrokosmos from the viewpoint of pitch-class set theory. It is designed to expedite the task of building a framework of music-theoretical categories, connecting the musically concrete to the abstract. The software is currently available in two formats: as a suite of Hypercard stacks, and as a World Wide Web site. Each format supports three audio CD recordings of the complete Mikrokosmos. A concise description of both versions is given below.

The entire suite is normally accessed from Mikrokosmik Sets, the launching pad for other stacks in the suite. Each of those stacks contains buttons that provide links to other relevant stacks at appropriate moments.

Present Sets! provides an introduction to basic concepts, terms, and symbols of set theory using examples from Mikrokosmos. Topics addressed include integer representation of pitch, pitch class, name class, and octave, and of the corresponding intervals between such elements, transposition and inversion of ordered sets, normalization of unordered sets, interval cycles, pc set-types (classes) interval class vectors, and the diatonic and pentatonic collections including their modes and subsets. Several cards provide interactive features and MIDI playback.

Concordance of Set-Types expedites the task of learning to recognize and hear instances of interval classes and pc set classes in their myriad forms of realization. The stack provides more than 160 examples of pc set classes in Mikrokosmos. These examples can be searched by Forte name, Tn/TnI-type (prime form), piece number, and piece title (Figure 1). Users can also search for instances of Russom's referential scale collections (RSCs) and the various subsets of those collections. Each example card provides information about the set class including a clockface representation that can be transposed and/or inverted to match the specific set under consideration (Figure 2). Buttons are also provided to play the relevant track from an audio compact disc. Links to related example cards and to RSC cards are available from popup buttons. A notepad button opens a scrolling field into which the user can type notes or answers to questions that are posed on each example card.

Figure 1. Search Options card for Concordance (Hypercard version)

PC Set Genera illustrates various schemes that theorists have devised for classifying and comparing pc set types. Using buttons, the user can generate and display instances of Russom's eight referential scale collections (RSCs) on interval cycles 1/11, 5/7, 7/5. Buttons are also provided to generate and play MIDI realizations of these collections.

Octatonic Set illustrates various properties of the octatonic set-type. These include various ways to partition an octatonic collection as well as symmetrical features, interval class content, and ordered interval patterns of such collections. Several cards provide buttons that trigger MIDI examples.

Diminished Fifth provides an animated analysis that shows how two [0,2,3,5] tetrachords and the octatonic collection that results from their union are transposed during the course of the piece (Mikrokosmos, IV/101). The tetrachords are displayed in conventional notation and clockface representation. The transposition process can be viewed and heard in successive animated stages or synoptically. Subsequent cards enable exploration of the number of common pcs between adjacent octatonic collections.

Chromatic Invention (1) illustrates how melodic lines are transposed and inverted in both the pitch and pitch-class domains. After a brief introduction to the topic of imitation by transposition and transposed inversion, the user explores the relationship between dux and comes for each point of imitation as well as the bridge passage.

Figure 2. Example card for Concordance (Hypercard version)

The Web version of Mikrokosmik Sets provides the Concordance in a frame-based format as well as additional documents that provide a brief history of the genesis of Mikrokosmos and illustrate various musical styles and topics. It also enables playback of relevant pieces from a compact disc in the user's CD-ROM drive using Voyager's CDLink software.

The Web version provides historical background concerning the Genesis of Mikrokosmos and its place in Bartók’s output. Styles and Topics exemplified in Mikrokosmos are illustrated in an easily accessed in a frame-based format. Each example can be played from any of three audio CDs in the user’s CD-ROM drive using Voyager’s CDLink software.

A frame format is also used in the Concordance of Set Types. The various pc set-types are listed by cardinality in a scrolling frame to the left. Each list item links to an example card that provides basic facts about that set-type along with a clockface representation that cannot be transposed or inverted. The set is also shown in standard musical notation with its registration and voicing preserved. Comments and questions are provided as in the Hypercard version, but a notepad is not provided. Finally, buttons are provided for playing the piece from an audio CD and for navigating forward and backward.

Efforts are currently underway to convert animated analyses from the Hypercard version to QuickTime movies. Two analyses of DiminishedFifth which track the transposition of an octatonic collection throughout the piece are now available. Others will be added in the near future.

The new electronic technology offers music theory pedagogues the opportunity to develop computational environments in which novices can explore basic musical concepts in new and interesting ways. Such environments are more conducive to the development of mature understanding than traditional settings in which students must often juggle a textbook, a musical score, and a recording. The software described above is based on the assumption that the user has access to the six-volume score of Mikrokosmos and to one of three audio CD recordings of the complete work. It is not intended to replace those primary documents but rather to provide easy access to relevant verbal commentary, analytical examples, graphic images, and the corresponding musical sound. In this multimodal environment students can more easily establish connections between the familiar and the novel. In doing so, they can expand their network of music-theoretical knowledge, develop their analytical technique, and become familiar with a 20th-century masterpiece.