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.

The Best, Little-Known Harmony Books

In his preface to Traditional Harmony, Paul Hindemith wrote, “No one is really satisfied with what he learned long ago in his harmony study, which in general has so pitifully little influence on the practical musical accomplishments that have to be learned in the early years.  So one buys the latest harmony book, as one has bought others before it, in order finally to make up for what one has missed[.]”  In my own teaching experience, I tend to agree with Hindemith’s observation.  I have witnessed competent student performers, including pianists who play thousands of chord progressions from the classical canon daily, experience puzzling difficulty when asked to harmonize chorales, or improvise classical chord progressions.  Though there are many reasons for harmonic deficiencies--I need not name them--harmonic perception is essential for supreme musicianship.  The books I’m about to recommend are clear, concise guides to “tradtional” harmony.  Most of them are not used in classroom settings today; however, they are written by experts, and actual composers, who made use of the same harmonic materials and principles discussed within their pages throughout their lives.  Because of the age of these texts, Roman numerals used to label chords are all capitalized, but this should not confuse students with adequate knowledge of major and minor scales.  Without further ado, here are some of the best, little-known harmony books:

1. Applied Harmony by George A. Wedge

American organist George A. Wedge provides a clear, methodical guide to harmony, beginning with the basics.  Exercises are progressive in nature, and even give students the opportunity to practice instrumental harmonization in more than four parts.  (Many basic harmony books limit writing to four parts, rarely discussing scoring for polyphonic instruments like the piano.)  Wedge’s second volume, Applied Harmony Book II: Chromatic Harmony, continues beyond diatonicism, offering insightful and immediately useful techniques for writing in a chromatic, tonal idiom.

2. Harmonic Practice by Roger Sessions

This text consists mostly of exercises composed to introduce students to harmonic principles one step at a time.  Sessions’ own commentary, words from a capable composer, is amusing and encouraging from the start.  No matter what a student’s level of experince with harmony, this book will fill in any theoretical gaps.  I am convinced that any student who works through this book entirely will gain a profound, intimate knowledge of harmony.   

3. Guide to the Practical Study of Harmony 

     by Peter Ilyitch Tchaikovsky

Written by a master composer and harmonist, this book is a gem.  Well-organized with abundant examples, the text begins with the basics, progressing to more advanced techniques.  Sequences, modulation, free voice-leading, melodic harmonization, “free” prelude writing, and harmonic deviations are covered in this informative, historic find.  

4. Practical Manual of Harmony by Nikolai Rimsky-Korsakov

Another book by a master composer, the text is abundant with harmonic examples and useful tips.  Topics of the last few chapters introduce chorale harmonization, enharmonic modulation, and sudden modulation.  The back also contains exercises for use with material in corresponding chapters.  With the curious mentioning of the “harmonic major mode” on pages 5 and 6, this book is another invaluable find for  music theory and history lovers.

5. Figured Harmony at the Keyboard by Reginald Owen Morris

While it is often assumed that the focus of figured harmony courses is on learning to realize figured bass at a keyboard instrument, such knowledge of figured bass, acquired by the hands in conjunction with the head, enables students to digest harmonic concepts more rapidly.  Realizing figured bass is, essentially, harmonization in action, and this book by composer R. O. Morris introduces key harmonic concepts and techniques immediately applicable to composition, improvisation, and completing harmony exercises in other books.  While I am convinced great benefit is derived from playing through this book, it can also be worked through, with staff paper in hand, like a traditional harmony text.  

6. Continuo Playing According to Handel: His Figured Bass Exercises 

     edited by David Ledbetter

The great composer’s figured bass exercises introduce students to harmonic concepts Handel, himself, deemed exceptionally important.  The exercises, including those in fugue, aim to instruct students about composition, too.  Filled with musical wisdom, and immediately applicable, this book is a “must-have” harmonic resource for all serious composers and performers.  Mentioning that it is a text with great historic value, too, is obvious.

With the abundance of harmony books available today, give the six above some serious atention.  If there is a gap in your harmonic training, one of them is sure to uncover and remedy it!

So, You Want to Write for Organ?

As an organist, I’m asked questions about how to effectively write for the instrument.  Many times, competent composers have shown me finalized movements, or long excerpts of compositions, hoping there will be little that requires change.  Though these pieces may be extremely effective, were they intended for the piano, they often require major revision to be as effective on the organ.  Even a composer as experienced as Arnold Schoenberg needed the help of Carl Weinrich, a professional organist, to create a playable edition of his Variations on a Recitative for Organ.  (Schoenberg’s original version, literally unplayable as written, is still available from Belmont Music Publishers for organists daring enough to create their own editions!)

But what makes writing for the organ so difficult?  

The organ is not a piano.  This is something many musicians, because they’ve never sat at an organ console, tend to overlook.  That said, most of the problems that arise in organ writing are purely technical.  There’s no sustain pedal, so a passage like the following, dependent upon the harmonic accumulation of notes, is not possible:

If the pedals are not allowed to assist, or are absent or occupied, only select notes from the above passage will be able to be sustained.  This compromises the intended effect of the gradually-acquired, full harmony, and gives the passage an entirely different character.  The selection below is one of many possible realizations:  

   

Passages involving seamless legato leaps, very well executed by use of the piano’s sustain pedal, will sound choppy on the organ, no matter how quickly the organist hops from chord to chord.  The same is true of extensive octave passages in either hand.  To make this apparent, all a composer needs to do is attempt to play a Beethoven piano sonata on the organ; the differences between the two instruments, including the number of keys per keyboard, will be made embarrassingly clear.

Unlike the piano, dynamics on the organ cannot be achieved by altering one’s touch.  Unless the registration of a manual (keyboard) is changed in real time, intended loud and soft passages played on it will sound at the same dynamic level.  For this reason, dynamic contrasts are frequently achieved by switching manuals.     

Unlike a piano, where sustained notes naturally decrescendo according to the order in and velocity with which they are struck, notes on the organ do not.  Therefore, dissonant chords derived from the accumulation of notes will sound more fearsome on the organ than on the piano.   Though organ literature is ripe with dissonances, it is worth understanding that a passage like the following will sound much less dissonant on the piano:

While it is true that some organs have expression pedals designed to adjust the instrument’s volume, these pedals impact all held notes on a particular manual.  It is also worth noting that the availability and use of such expression pedals varies from organ to organ.  Some players, because of instrumental limitations, cannot even execute the mildest crescendi and decrescendi, let alone excessive dynamic swells.  If such expression is desired, it is worth remembering that these pedals are operated by the feet.  Therefore, assigning a pedal note to each foot while notating an expressive crescendo is not wise.  Though there are instances in contemporary scores where this can be done with great awkwardness, many composers today seem to be fixated on “extended techniques” for their own sake, thus making pieces more difficult than necessary.  With a complex instrument like the organ, one that has so many timbral possibilities at hand, it is advised that such exploration be commenced only after thorough knowledge of what the instrument can achieve is gained.

My friend, the late Daniel Pinkham, once remarked that composers don’t realize that the majority of performance mishaps occur because of pedal writing.  I heartily agree.  Examining the organ works of masters like Bach and Buxtehude, notice how the manual parts become easier as the pedals become more difficult.  Also, note how frequently the pedals rest during a given work.  Too many composers think that, because the pedal is given its own staff, a continuously-active pedal part is necessary; it is not, and simplifying the demands of the pedal will often improve the performance of many new works.

Take your body’s balance into consideration, too.  Attempt to play some of your own passages with both hands and feet.  See if you can easily balance on the organ bench.  Whatever is awkward, sending you jolting forward or backward, is not ideal.  Charles Ives, an organist keenly aware of such issues, wrote below a difficult pedal passage in his fifth variation from Variations on America, “This passage was often played by the pedals while the left hand hung on to the bench.”  

While this blog post did not cover registration, the organ equivalent of orchestration, registral matters are often of secondary concern when technical difficulties stemming from unidiomatic writing are paramount.  Taking the time to sit at an organ and attempting to play with hands and feet together is the best education for any composer who wishes to write for it.  And while you’re at the console, refer to Sandra Soderlund’s A Guide to the Pipe Organ for Composers and Others.  (Daniel Pinkham used to refer this guide to composers.)  Inside this comprehensive, little book, you’ll find all the basics a composer needs to know to start writing effectively for the instrument.  When in doubt, ask an organist.  Better yet, let an organist direct you to an empty organ bench!  Happy pedaling!

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!

Dulcimer Days

When recording music for the Drury Drama Team’s  production of The Diviners, a play by Jim Leonard, Jr., I used an Appalachian dulcimer to play the melody of Amazing Grace.  A long-time fan of Appalachian music, this was the first time I  had the opportunity to get my hands on a beautiful mountain dulcimer.  The instrument belongs to the zither family of musical instruments, meaning that the strings do not extend beyond the sounding box; the body of a mountain dulcimer is as long as the fingerboard.  (Compare this with a guitar and you’ll see the difference.)  

Me and My Appalachian Dulcimer

With a little research, I discovered many prominent musicians have also used mountain dulcimers on recordings, Joni Mitchell and Cyndi Lauper among them.  (Lauper studied the Appalachian dulcimer seriously with David Schnaufer, and featured it in her albums A Night to Remember, Sisters of Avalon, and The Body Acoustic.)  After recording my part for the Drama Team’s production, I became even more taken with the instrument.  I continue to play it, and hope to showcase it at a future date during a “Mountain Mass” at my church.  If you’re curious to hear some Appalachian music, I recommend checking out a truly fantastic, historical recording by Butch Baldassari and David Schnaufer: Appalachian Mandolin and Dulcimer.  (This is the last album David Schnaufer recorded before he passed away in 2006.)  The album, packed with traditional toe-tapping tunes, is simply delightful.  It truly captures the spirit of Appalachia.  The duo’s artistic mastery is evident from the first track, Ground Hog, and their expressivity makes a tune like Wayfaring Stranger even more deeply moving.  I know my exploration of the Appalachian dulcimer is still only in its infant stages, but I’ve certainly been enjoying it!  If you want a good introduction to Appalachian dulcimer music, this album is a sure bet.  I was quite pleased, indeed.

Text-to-Music: a Medieval Approach

Although Medieval theorists documented various techniques used to compose chant melodies, a very curious method is discussed by John “Cotton” (ca. 1100) and Guido d’Arrezo (995-1050). Both theorists proposed a method of composition that uncovers a text’s inherent pitch possibilities. One can then fashion a melody out of these possibilities, shaping the text into a melody. This is a bit similar to some of today’s websites that attempt to “translate” text into music. (Visit http://www.p22.com/musicfont for one example.) Unlike the online example, the Medieval system for composing monophonic settings of chant utilized only the vowels (a e i o u) of a text to determine pitch possibilities. This one-to-one correspondence (vowel equals pitch) thus strengthened the bond between text and music. It provided a framework in which a composer could make choices, depending on taste, that would aid the process of composition and unite text with music more cogently. For a simple example of this system, what follows is a short summary. Any five sequential notes belonging to a given mode (scale) are written out, and above them are written the five vowels a e i o u.  Each syllable of the chant is assigned a note based on the vowel it is linked with.  For instance, if we use the following order:

       a   e   i   o  u

      D  E  F  G  A

and one of the words in our chant is Dominum, it would be chanted as follows: the syllable Do would be given a G pitch, mi an F pitch, and num an A, producing the melody G-F-A.  For even more pitch choices, an entire scale can be used, and more than one pitch choice can be assigned to a vowel. For instance, the following is possible:

a   e    i   o   u   a   e   i 

G  A   B  C  D  E   F   G

i   o    u   a   e   i   o   u

Thus, the floodgates are opened and the total number of pitch possibilities per vowel is no longer limited to a meager few! Chants composed with such structural emphasis on vowels belong to the realm of “spectral composition” where the structure of the chant is governed both by melodic contour as well as sonic (vowel) opposition. Try this for yourself! You’re bound to come up with some interesting tunes!