Note: Octave doublings and spacings are not taken into account in the following analytic systems.
I. Additional systems.
A. Popular music (jazz) lead sheet symbols.
Chord symbols are not standardized, and students are apt to encounter variations of those given above, especially in manuscript arrangements.
B. Because of the complexity of many contemporary chord structures, neither roman numerals nor chord symbols are satisfactory for analysis. For this reason, a number of alternate systems have been devised.
These systems have in common the analysis of a group of associated pitches (for instance, a chord, scale, or melodic line) by the quality and number of its constituent intervals. In addition, the Hindemith system seeks to identify a chord root. This root will be the one note that appears as the root of the strongest interval(s) found within the chord.
1. Hanson system.
Noting the inability of more traditional analytic methods adequately to describe most twentieth-century music, Howard Hanson, in his book Harmonic Materials of Modern Music (see Bibliography for details), invented a simple system of interval labeling, given below. In this sytem (as in most interval-descriptive sys-tems), octave (registral) placement, octave doubling, spacing, and voicing are not accounted for. Octave equiv-alence and equal temperament are assumed as givens. All instances in a sonority of the pitch “D” are understood as the “Pitch-class D” and are equivalent for analytic purposes. Any collection of pitches (such as a scale, melodic line, or chord) may be analyzed using the Hanson system. Thus the sonority:
is reduced for analytic purposes to:
In this system, the symbols p, m, n, s, d, and t stand for interval classes:
p ! perfect intervals
m ! major thirds and minor sixths n ! minor thirds and major sixths s ! minor sevenths and major seconds d ! major sevenths and minor seconds t ! tritones
In the analysis of a sonority, the symbols are always given in order, with superscript numbers showing the number of a given interval class in the given sonority.
When using any interval-descriptive system, it is useful to know the number of intervals in any set of notes.
The formula n(n " l)/2 may be used to provide a quick calculation.
In the example given above, there are four pitches, so there will be six intervals [(4 # 3)/2 ! 6]. Here are two other examples:
2. Persichetti system.
Vincent Persichetti, in his book Twentieth Century Harmony (see Bibliography for details), dealt with the mat-ter of describing nonconventional sonorities via inmat-terval labeling by devising a graphic system of inmat-terval labeling. This system is similar to that of Hanson in that octave placement, octave doubling, spacing, and voic-ing are not accounted for and pitch-class equivalency is assumed. Persichetti’s system differs from that of Hanson in that intervals are grouped in four categories. Consonance includes all perfect intervals plus all thirds, both major and minor, and all sixths, both major and minor. Mild dissonance includes major seconds and minor sevenths. Sharp dissonance includes minor seconds and major sevenths. Neutral dissonance indi-cates the tritone.
= consonances (Hanson’s p, m, n)
= mild dissonances (Hanson’s s)
= sharp dissonances (Hanson’s d)
= neutral dissonances (Hanson’s t)
3. Hindemith system of root designation.
In his book The Craft of Musical Composition (see Bibliography for details), Paul Hindemith describes a method of root designation for vertical sonorities. This approach is based on the overtone series (see Part V, Unit 2) and is consistent with our understanding of root structure in tertian harmonies. The example below indicates the intervals, left to right, in increasing order of dissonance, from the octave through to the tritone.
The root of each interval is indicated. Note that the tritone, ambiguous in character, by its nature has no root.
In the sonorities below, the root of each “chord” is determined by its primacy in the structure. The pitch that is supported by the majority of intervals within a “chord,” with intervals toward the bottom of the stack pre-dominating, is understood to be its root. Thus in the first instance B is the root of the stack, as it is the root of the lowest interval, the perfect fourth. In the second structure, F is the root of the two lowest intervals, the minor second and the major third, and also the majority of intervals in the stack, and is thus the root of the
“chord.” In the third “chord,” F is the root of the majority of intervals as well as the lowest intervals, the per-fect fifth, the minor seventh, and the minor third (tenth).
FURTHER CONCEPTS FOR ANALYSIS 185
4. Allen Forte, Joel Lester, and many others have developed a method for analyzing atonal and serial music. Pitch- and interval-class theory (related to mathematical set theory) in effect also reduces a complex of notes into its constituent intervals and, much like the Hanson system, enumerates the intervals according to type. In the first example below, the numbers within the brackets indicate that there is one m2 or M7, one P4 or P5, one TT, and no other intervals.
It is beyond the scope of this book to deal with any of these systems in detail. A more detailed dis-cussion may be found in Unit 10 on pages 213–217. The standard books dealing with these systems are all listed in the Bibliography.
II. In addition to the usual items covered in analysis, with twentieth-century music, the following questions should also be considered:
A. Is the music centric? If so, how is the tonal center established?
B. What scalar material is used?
C. What harmonic materials are used? Is there a sense of harmonic progression? If so, how is it controlled?
What is the normative dissonance level? How is it established and controlled?
D. What cadential idioms and devices are employed?
E. What rhythmic and metric devices are used?
III. Suggested reading (see the Bibliography): Hanson, Hindemith, Persichetti, Forte.†
†Supplementary readings are suggested at the end of most units in Part IV; these books are listed in the Bibliography. The student will also find that reference to the various anthologies listed there, as well as to such collections as the Mikro-kosmos of Bartók, is essential.