John Rahn's book Basic Atonal Theory gives a systematic, organized and well thought out approach to the study and appreciation of atonal (including serial) music. The reader is provided with the essential basics of the twelve pitch-class system by covering the fundamental mathematical principles associated with this analytic method. Rahn takes us through the pitch-class system one step at a time, constantly providing the reader with practical exercises and relating what has been taught to musical examples. It is perfect for composers, performers or listeners who have a good background in tonal music and wish to acquire a better understanding of atonal music.
 Rahn starts by saying "To analyse music is to find a good way to hear it and to communicate that way of hearing it to other people." Indeed, the pitch class-system lives up to Rahn's expectations of a good analytic model. The labelling of the twelve pitches using numbers 0 through 11 ensures that the equality of the twelve tones we hear in atonal music is also represented visually within its symbolic notation. (The equality between the pitches we here are notated as evenly and equally spaced.) In fact, Rahn's approach to pitch-class analysis is for the most part very musical.
 Rahn speaks of what he calls "tonal filters" vs. "atonal filters." He believes that most Classical Western musicians hear music through "tonal filters" because of cultural and social conditioning-- the constant playing of and listening to tonal music. Our musical ears naturally become accustomed over time to hearing music in a tonal way. Rahn suggests that when people are first exposed to atonal music they attempt to hear this music using their "tonal filters." It is Rahn's aim to help the listener develop "atonal filters" through which to hear a piece of atonal music. Certainly his "musical " approach to pitch-class analysis aims to do just that.
 Rahn's "atonal" ear training approach is evident from the start. His analysis of Webern's op.21 concentrates on inviting the listener to hear the "thema" without the aid of a score (and confusingly without the aid of which movement to listen to as well). In fact, he insists we listen to this thema over and over again. Only after the student thoroughly uses his/her ears are they allowed a peak at the score. As well, Rahn creates a variety of small, creative, illustrative diagrams for the analysis of this piece and later musical examples within the book. The clarity within these diagrams are often invaluable and expose the reader to various options for thinking about and sketching these ideas creatively and efficiently.
 Rahn clearly sees both music and analysis as hinging on principles of organization. This is not hard to imagine in light of his clearly organized approach to teaching pitch-class analysis. Rahn Proceeds to take us step by step through the foundation of pitch-class analysis which he calls "The Integer Model of Pitch." It is here that the reader first becomes emersed in the ‘mathematical' component of the analytical method. Sometimes I felt Rahn indulging in the mathematics of the pitch-class theory. Detailed mathematical definitions, for example, may have seemed necessary to Rahn for the validity of the analysis but sometimes over complicated issues. Robert D. Morris in Music Theory Spectrum (1982) says "the presentation provides an excellent model for the student of music theory due to its professional organization, precision, and apt use of current mathematical style." However, as a student of music theory, I honestly found myself simplifying Rahn's "current mathematical style" into a more practical understanding of the basic principles. The mathematics, after all, is rather simplistic and I often doubted whether it really needed such stylistic mathematical proof. I myself do not see how specific mathematical terminology or phrases he occasionally throws in such as "mod 12" will help me analyse an atonal piece using pitch-class theory more effectively. Occasionally I feel Rahn confuses matters rather than keeping them simple and direct. (For example, why should he change ip(x,y) to ip(b,c)?) On a more positive note, I thought the practical pitch-class analysis exercises as well as the "Answers to Selected Exercises" section was a great idea as it allows the student to work more securely independently. The appendices reviewing mathematical notations, pitch-class analysis principles and set tables are also handy.
 Overall, when Rahn is "musical" he is very musical. His philosophies clearly and logically support his musical intuitions and musical model of analysis. When Rahn is "mathematical" he is very mathematical. He seems to leave no stone unturned and gives us more than we technically need. It is interesting to see how he joins the musical and mathematical aspects of music in a quasi reconciliation. What Rahn forgot to mention was that "atonal filters" apply not only to our ears but also to our eyes in the way we visualize and symbolize notation and to our minds in the way we think about music in general.
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