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10 Free AtonalityI. Much contemporary music is conceived in terms of the manipulation of a restricted set of intervals instead

of scale-forms, traditional materials, or serial processes. In this procedure a basic interval cell (or set) maygive rise to a whole work, in both its harmonic and linear aspects. The cell itself may be treated as a motive.A cellular approach typifies much serial music (see Part IV, Unit 11). Much music of this type is not stronglycentric and is frequently termed atonal.

II. Characteristic procedures.

A. The basic cell, or set of interval relationships, may be altered in any of the following ways:

B. The cell is usually applied to both the vertical and horizontal aspects and often accounts for most of thesounds heard in a work. The following excerpt uses the first five of the derivations in the example in IIA.

C. Many interesting chords can be built by projecting two or more different intervals in succession. In suchsonorities, note duplication at the octave or fifteenth is usually avoided. Melodic lines may be derived fromany of the resultant sonorities.

III. The transposition factor is especially relevant to interval music. Any set lacking a given interval or its inver-sion may be transposed by that interval, yielding pitches not contained in the original set.

IV. Pitch-class set analysis provides another tool for analyzing complex pitch structures or for discovering rela-tionships in the music that might not be readily apparent. Pitch-class refers collectively to all the occurrencesof a particular pitch, regardless of register. Pitches spelled enharmonically are to be considered as the samepitch. Pitch-class sets consist of from three to twelve pitch-classes. They may occur as melodic motives or ges-tures, as vertical sonorities, or as combinations, or they may be noncontiguous but related by register, metri-cal placement, or the like. Since smaller sets are easier to work with, the music is typically segmented. Thisprocess breaks longer musical figures into manageable groupings.

Sets can be further divided into subsets:

To analyze a pitch-class set, first write the selected pitches in scalar order within the octave. Consider that indealing with atonal music two basic assumptions are routinely made: 1. Enharmonic spellings are consideredto be equivalent, and spelling may be changed to make the intervals simpler. 2. Any interval and its inversionare considered to be equivalent, and therefore intervals are reduced to their smaller size. The pitch-class setis first put into normal order, as described above, and since it is also useful to consider pitch-class sets andtheir inversions to be equivalent, the set is ultimately reduced to what is termed lowest ordering or the bestnormal order. Finding the normal order may require some rotation of the pitches to assure the proper order-ing, with the smaller intervals to the left and the larger intervals to the right, and the smallest possible inter-val from first note to last. The pitches are then numbered, with the first note being the integer 0, and thefollowing notes numbered according to the number of semitones away from the first pitch. The pitch-classintegers are separated by commas, and the series of numbers is placed in brackets. The resulting integerseries becomes the label for the set: [0,1,4], [0,3,4,7], and so on. The set may then be transposed so that the firstnote is middle c; this puts the set into prime form or prime order. Allen Forte determined that there areonly 220 possible sets, and he has indexed all of these sets, providing them with convenient index numbers:[0,1,4] ! Set 3-3; [0,3,4,7] ! Set 4-17; and so on.

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The use of the commas avoids confusion when using the integers 10 and 11. Some theorists prefer to use let-ters for these integers so that the commas can be eliminated. T (or A) stands for 10, and E (or B) stands for11. Thus the set [0,2,5,8,10] can be written [0258T].

Conventional structures can also be given a set designation.

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For example, [0,2,4,6,8,10] is a whole-tone scale. Other aspects of the set can be seen by examining the inter-val vector, which is a tabulation of the total number of intervals within the set. The vector consists of six dig-its, also placed within brackets. Each digit represents an interval class. The first digit represents the numberof m2s or M7s, the second digit represents the number of M2s or m7s, the third the number of m3s or M6s,the fourth the number of M3s or m6s, the fifth the number of P4s or P5s, and the last the number of tritones.An examination of the interval vector for this set reveals that it is rich in M2s, M3s, and tritones, as we wouldexpect of a whole-tone scale, but lacking in all other intervals.

The theory of pitch-class sets is fairly detailed and complex. Those wishing to find additional materials on thistopic are referred to the Forte, Lester, Kostka, and Straus books listed in the Bibliography.

V. Suggestions for class discussion.

A. Analyze the examples in Unit 36 of Music for Analysis. Students may bring additional examples from theliterature into class.

B. Suggested reading (see the Bibliography): Cope, Dallin, Forte, Hanson, Persichetti, Ulehla, DeLone.

Exercises

1. Reduce each of the following examples to its pitch-class set, and analyze the resulting set.

2. Experiment with building massive dissonant sonorities by projecting in alternation two or three differentintervals, as suggested in IIC. Analyze the results using the Hanson system or some other method ofinterval analysis.

3. Build several chords by mirroring around a central note. Analyze according to the instructions forExercise 2.

4. Compose a brief work for piano based on the interval set GB CF . Derive both melody and harmonyfrom the set and its transpositions. Try to construct a convincing cadence.

5. Study the Bartok Mikrokosmos, no. 144: Minor Seconds, Major Sevenths (#444 in Music for Analysis). Then,compose a short work for instruments available in class using the basic cell of the example in IIA.

6. Analyze the given material, and continue in the same idiom:

a.

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b.

c.

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