Stravinsky Verticals

In his 12-tone and serial works from 1960 onward, Igor Stravinsky (1882-1971) often employed a method of material transformation that utilized rotational arrays.  Known by several names, including what Milton Babbitt called “hexachordal transposition-rotation,” Stravinsky’s method of generating pitch-class (note) successions was actually developed almost two decades earlier by composer Ernst Krenek (1900-1991);  in 1941, it makes an appearance in Krenek’s choral work Lamentatio Jeremiae Prophetae.  The procedure is simple, and can be used for numerous compositional purposes in both 12 and non 12-tone worlds.  While Stravinsky and Krenek used sets consisting of 6 differing pitch classes (hexachords), there’s no reason why sets with more or fewer members can’t be used.  Here are the four simple steps to create your own transposed rotational arrays:

Create a series of pitch classes.
  

  

Rotate it, making the first element become the last.

Transpose the new set so that its starting pitch class is the same as that of the original set.
  

Repeat  the process for the remaining sets.
     

The total number of sets
will equal the “length” of a given set.

Compositionally, sets can be used one after the other in an ordered or unordered fashion.

Used in this manner, the frequency of a repeated pitch class (and, potentially, more that repeat), is ideal for situations requiring pitch centricity.  Choral works, for example, and more “melodic” passages easily benefit. 

Sets can be assigned to various instruments to create multiple parts.
  

Sets may even be deduced from already-present melodic fragments, or created with pitch-class repetitions.

Often dubbed “Stravinsky verticals” when applied as follows, sets can be used harmonically, too.  Instead of reading the sets left to right, read them one by one from the top down or bottom up.  They can be expressed this way as chords.  Because resulting octaves have a real intervallic function, they are permissible in such situations, even in 12-tone music.
  

Form chords by reading top to
bottom, or bottom to top. 

Single tone representation of pitch class E

Although Stravinsky liked to represent the duplicated “notes” of the first pitch class of a set by a single tone, it is also possible to compose a multi-octave representation.

Multi-octave representation of pitch class E

With a little creative thought, the possibilities are endless, no matter the idiom in which you compose.  By writing out several such transformations, and playing them over on an instrument for sonic reinforcement, you’ll soon be able to mentally “hear” all such transformational possibilities inherent in any pitch succession.  With practice, you’ll be able to fluently maneuver between sets at your disgression without relying on charts, which greatly speeds up the process of composition.  Because the ear is king, it is my conviction that the results of any musical procedure should only be used if they satisfy its dictates.  Use these techniques to best fulfill your own musical vision.

12-Tone Texts

Budding composers have often asked me which texts I recommend as introductions to 12-tone music that outline compositional techniques required to effectively compose in this idiom.  While there are many books of which I am aware, what follows are those I believe to be the most beneficial for composers who have had few encounters with this type of music.  These books, though “Reader’s Digest-like” texts for experienced composers, should enlighten young composers exploring 12-tone composition:

1. Simple Composition by Charles Wuorinen

Though opinionated and domineering, especially in the realm of aesthetics, this text is an excellent 12-tone primer.  Covering the basic series, transformations, Stravinsky verticals (hexachordal transposition-rotation), etc., Wuorinen also discusses Milton Babbitt’s time-point system, and the use of serial techniques to construct large-scale musical forms.  Though the author can be full of himself at times, take him for his tools, not his talk.   

2. Lehrbuch der Zwöfltontechnik by Herbert Eimert

This is a wonderful, musical text that demonstrates a variety of creative ways to derive 12-tone rows for numerous musical contexts.  A straightforward approach to basic 12-tone theory, this introductory text is only available in German.  (The subtle differences between American and German 12-tone “philosophy” are quite interesting.)  This little book also delves briefly into duration and dynamic rows, and permutations.    

3. Serialism by Arnold Whittal

This text provides a comprehensive survey of 12-tone and serial music from a British perspective, demonstrating numerous techniques budding composers can apply to their own compositions.  Filled with relevant musical examples and an ample bibliography, this is a text that beginners will find useful for reference.

4. Studies in Counterpoint by Ernst Krenek 

Though these studies in 12-tone counterpoint often result in music that sounds like late Krenek--similar to the way some of Hindemith’s books encourage students to write like Hindemith--these graded exercises provide composers who are new to 12-tone music with a solid contrapuntal approach for writing in this style.  In spite of the interjection of Krenek’s own stylistic ideals, the majority of the book is extremely instructional, and the exercises are worth working through.  

5. Materials and Techniques of Twentieth-Century Music by Stefan Kostka

This book, though also a survey, contains detailed sections devoted to 12-tone composition that may enlighten young composers.  Matrices and other devices are explained in detail, and the presentation of material is extremely clear.  To get the “basics”, this book is a good place to start.  It also contains various exercises and music to analyze.  (It is also written in a refreshing, objective manner.)

6. Serial Composition by Reginald Smith Brindle

This text deals with 12-tone melody, harmony, form, orchestration, etc., and outlines some generic stylistic guidelines.  Written by a composer, it presents practical material in a musical way so that its application can be grasped immediately.  Many excerpts from compositions are utilized, and a variety of creative ideas are introduced.

7. Post Tonal Theory by Joseph N. Straus

An introduction to set theory and 12-tone music, this book should be read following the others in this list.  It introduces interval vectors and other compositional ideas/classifications that may prove useful when composing 12-tone music.  Each chapter contains relevant exercises to reinforce pertinent material.  

Though there are many books that deal with 12-tone composition, the seven above provide ample practical advice for composers just beginning to explore the vast world of 12-tone music.  Happy composing!