An essay on patterns in musical composition
transformations, mathematical groups, and the
nature of musical substance.

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A Belorussian translation at Webhostinggeeks is also available.

Edgar Varese once gave a definition of music as "organized sound", and this seems like a good place to start, more because of what information such a definition does not give than for what it does give. The definition is somewhat satisfying because it seems to embrace practically anything that is called music, that anyone could conceive of as music and further certainly anything that might be called music in the future.

On the other hand it is unsatisfying because there are no clues in the definition as to how music is constructed. On mulling Varese's definition over, the first question that comes to mind is a compound one: what is organization, how is the sound organized?

I will try to consider an answer to that question by starting with the most historically primitive musical notions and moving through increasing complexities to complexities of modern atonal music. Here, I mean "atonal" in its technical sense, not in it's gutter sense of harsh and "dissonant".

Undoubtedly, the most primitive musical percept is that of pulse, recognized perhaps first in our own rhythmic pulse, heart beat. Imagine an Australopithicus, on his day off, picking some object up and banging it against another; the satisfaction obtained by simply 'beating out time', in a steady repeating unit of time. This sets the stage for a pulse to music as being a fundamental parameter that can be varied: slower or faster. Nowadays we are a little more precise and sophisticated about such things and specify a pulse by giving a metronome value for an underlying note value that is tied to musical notation. For a given pulse, one has simply a sequence of unaccented and undifferentiated beats separated by equal units of time. A metronome beats a pulse, making a musical pulse mesurable and determinate.

One can overlay a foundational pulse with pulses having structural forms that give a pattern of various levels of accent, and displacements in time of pulse beats. (Synchopation/Rubato). Accents usually involve an alteration of dynamics (piano-forte, e.g.). All together, these variations of/in pulse give Rhythms.

Within modern Western musical notation this is designated by grouping the primitive pulses into bars that create a meter. For certain evolved musical forms/types the meter is prescribed: A waltz, usually is written in 3/4 meter, so that there are 3 quarter notes to a bar. A short representative table:

                  waltz               3/4
                  polonaise           6/8
                  polka               2/4
                  gigue               6/8
                  sarabande           3/4
                  chaconne            3/4
                  passacaglia         3/4
                  allemande           4/4
                  tango               2/4, 4/4, 4/8 (before 1955)

These all happen to be, or evolved from dance forms. It is the association of music with dance and also with the voice and with indigenous language that have provided given structure for many musical features.

To write music that follows words in a given language, the music must follow the accents and rhythms of that language in order to project the words meaningfully to a listener. The art of doing so is called prosody. Composers also pay attention to vowel structure of the text, with an understanding of biologically resonant pitches of vowels. In [Helmholtz 1877] a great study, "On the Sensations of Tone", with unmatched thoroughness, was done of vowel resonances and generally how sound is perceived by humans.

Though written over a hundred years ago, it is still, a Bible of the subject that covers the appropriate mathematics, biology of the ear, particulars and history of pitch standards and tempering. Every composer should at least look at it.

Languages that provide many clear open vowels (Italian, Hungarian, Spanish, Russian) are the easiest to write for. Consonantal clusters (Czech, Russian and most Slavic languages) can present a slight obstacle to the composer. Languages that present nasalized and covered vowels, along with diphthongs and triphthongs (English, French, Portuguese) require more care and restriction in setting.

In regard to language and singing, it has been said, and I think quite defensibly, that Italian (Tuscan) is the perfect (and easiest) language for mellifluous singing: all vowels are open and in abundance; there are no diphthongs, nor triphthongs. Even in Italian, however, the composer must respect the natural resonances of the vowels. If this is not done, the singer often simply changes the vowel to suit the pitch. The most dreaded of vowel sounds for singers, whether they know it or not, is the long 'e': it has the highest resonant pitch and is most likely to become unstable when sung at the top of the singer's range.

That being said of Italian, Japanese, Turkish, Kalmuk and polynesian languages also love their vowels and abhor consonants being nextdoor neighbors.

Language has been claimed to be a powerful influence on the structure and nature of what might be called a national music. Influences come from setting music in the language and then abstracting the music to that which is not sung. The rhythms of the language remain embedded in the musical thought.

On this basis one would expect different kinds of music to evolve in areas where the sound structures of language are different. In Europe, for example, Romance languages of the south have lost the inflective endings that they might have kept from Latin, while the northern Germanic, Slavic and Scandinavian languages have kept their inflective endings. One should include in the second category, Finnish, Estonian and Hungarian. The thing about these inflective endings is that in terms of stress, they are weak: rarely stressed because the word stress in these three languages fall always on the first syllable.

The music coming from inflected languages concomitantly exhibits weak cadential endings which are necessary for proper prosody in song. One can hear this distinction between say German music and Italian music. Southern European music is quite distinct in its melodic structure from northern European music.

Some of the idiomatic melodic structure in southern European music is a result of exposure and assimilation of music that comes from Turkic and Afroasiatic languages, which have an altogether different sound structure.

One might expect a kind of straightjacket from a language where the word accents are prescribed. For example, in Turkish, Hungarian, Finnish and Estonian the accent always falls on the first syllable of words. In Spanish the accent naturally falls on the penultimate syllable, with exceptions in certain constructions that then must take an explicit accent (e.g., démelo = give it to me). In English, German and Russian there is no predicting where the accent falls. Fortunately, words have different lengths, in numbers of syllables.

English is not a particularly easy language to set to music because of its many covered vowels, diphthongs and triphthongs. The development of a musical language whose rhythms and cadences fit and are derived from spoken English has been a long time coming.

For me, the prototypical American English opera is the "The Crucible" by Robert Ward, one which certainly deserves more and better performances than it gets. Other composers who have caught the rhythms of English are Samuel Barber and Ned Rorem, but none who I know of more perfectly than Ward.

While rhythm is a purely temporal aspect of music, cadence has temporal, vertical (harmonic), and horizontal (melodic) aspects. For a working definition of cadence: it is a technique, procedure or method by which one signifies the end of a musical phrase. In a more modern setting, it signifies the end of an "activity". Activities, however, do not necessarily have cadential endings. Activities can be broken, terminated abruptly, die out by a process of pitch attrition or simply melt into another activity.

To make life more complicated, or more interesting, depending on your viewpoint, the vertical and horizontal aspects of music, are not, as a rule, independent. In speaking of cadence and the horizontal aspects of music, we have entered the realm of pitch.

The natural pitching of vowels in human speech is a rather subtle and diffuse concept, especially in languages like the various dialects of Chinese, some African and Caucasian languages which distinguish meaning by tonal or pitch variation patterns applied to a syllable. In many languages, one speaks of high and low toned vowels. Though one can easily learn to distinguish changes in pitch, the actual absolute pitch as frequency is never (to my knowledge) used as semantic inflection in any human language.

The only place in nature where one hears clearly defined pitches and pitch relations is in the songs of birds. It is not hard to imagine that should birds never have existed, we would have no music beyond rhythm.

Outside my window, in Spring, I frequently hear what I call my "Beethoven Bird". It chirps quite recognizably the opening motive of Beethoven's Fifth Symphony (G G G Eb), descending the major third from G to Eb. Sometimes, it adds a trill on the Eb. I don't think Beethoven got the idea from a bird, since in truth that little motive appears in many pieces of music and is almost a musical banality.

The point of the 5th symphony is not the motive, but the variative genius of Beethoven that could make spring from it an entire symphony. That symphony, so often played that it's not really paid attention to, is an astounding solution to the fundamental problem of musical composition: how do I keep doing the same thing for half an hour, or more, and be the opposite of boring?

So, I think it a pretty good guess that the idea of song arose in human culture by imitating the songs of birds, and also by hearing that even the birds produced variations in their song. The songs of birds seemed to be not learned behaviors but rather hard wired into the bird's brain, but that is not necessarily true.

In the human species, few things come hard wired, except possibly our inclination to see, impose, and create patterns, sometimes playfully and sometimes with dire consequences.

The creation of the patterns of music, mathematics and physics is, I would assert, primarily a playful activity with no particular purpose other than the enjoyment of the activity of creation and the aesthetics of the end product. But, I digress.

The next question should be where do the pitches of music come from, and what are they anyhow?

For modern western composers, the pitches or "Urmaterie" of music might be defined as those that one finds on the keys of a [piano]. The predominating frequencies elicited by striking piano keys have a long and fairly complex history, going back at least as far as the Greek mathematician/philosopher Pythagoras [LINK] ca. (575-500 BCE), through the classical [Greek modes] of which there seemingly were 14:

        Ionian             Hypoionian
        Dorian             Hypodorian
        Phrygian           Hypophrygian
        Lydian             Hypolydian
        Mixolydian         Hypomixolydian
        Aeolian            Hypoaeolian
        Locrian            Hypolocrian

on through to 7 liturgical modes [liturgical modes] of the Roman Catholic church.

Two of these church modes survive in western music as the Major mode (Ionian) and the Minor mode (Aeolian). There is more to say about the minor mode, which has certain modifications in modern usage. There are actually two common minor modes: the melodic minor and the harmonic minor.

Why did only these modes survive?
The answer to that question lies in the particular path that music took in western culture. Western music developed a structural notion of "tonality"; in this respect it is not unique. All 14 classical Greek modes have a tonal center which is the resting or resolving note for music written in it. The liturgical modes also have tonal centers inherited from the Greek modes.

There are other modes as well: various Persian modes, Hungarian modes, Neapolitan modes, Spanish modes, these last not being restricted to 7 tones. There are various pentatonic scales with five modes each. [Colin 1973] Consider also then the multiple octave ragas Raga Asavari: RagaNet, Issue #1 [LINK] of Classical Indian music with ascending and descending aspects similar to the ascending and descending forms of our melodic minor mode.

The number of musical modes used by humans is in the hundreds. This doesn't cover the microtonal embellishments found in music of the middle east which have been said to divide the semitone, our smallest well tempered interval into 64 parts.

It seems that the average adult can discriminate 1/17 of a whole tone, while trained musicians (violinists) can discriminate as finely as 1/540 of a whole tone around concert A. Pitch discrimination ability seems to be more innate than a product of learning as does the audible range, but only to up to a point. Depending on various alterable factors, the extremes of possible perception range from about 12 Hz up to 25000 Hz. The normative range is narrower than this, sometimes much narrower. [Seashore 1938] Online, see Auditory Perception.

Why all these modes and why does this wonderful richness disappear in western (European) music?
The reason that all these modes could exist and be wanted is that music was at first essentially horizontally organized. It could maintain a tonal structure which provided an architectural wholeness to a composition either by cadential formulas of a mode or simply by droning the tonal center also called the "tonic pitch". Most of the music of the world followed the path of increasing inventiveness and complexity of horizontal structure, i.e., of phrase and rhythm.

In European culture there also developed concomitantly with the ascendency of the Roman Catholic church a definitive division between sacred music and "profane" music (that which was not sacred). The original sacred music of the church was plainsong, sung in unison, with no instruments using the liturgical modes. (In practice, Locrian mode is unstable, easily degenerating into mixolydian - but that's a story for a little later on.)

For a history of the introduction of "organum", the singing of plainchant in parallel fourths see [Grout 1973], or more briefly and online Music History. The intervals of the fourth and the fifth are more consonant in natural tempering, than in well tempering.

Although the idea of counterpoint, the playing of simultaneous melody, was in secular use, it had not previously been allowed in the church. Major changes in the arts, and the rise of universities occurred during the 11th century. Changes and upheavals that were to climax in the Renaissance, 300+ years later. The introduction of even this primitive, harmonically based counterpoint is the source of the complex counterpoint of J. S. Bach.

It is the path toward vertical structure intertwined with horizontal structure that western music has taken. This, rather than the path of increasing horizontal complexity. In western music the concept of harmony rises up as dynamic force within music that carves out its forms. This is an isolated cultural event that has not happened elsewhere, or at any other time in the history of music.

Tuning a keyboard instrument by "just temperment" confines the keys (written musical signatures) in which one can play, and therefore the amount and distance that one can modulate. [Helmholtz 1877], contains a good deal of information on the mathematics of various scale temperings, and an approach to what distinguishes consonance from dissonance between two pitches.

To my knowledge, the connections between the mathematical properties of a sound wave form and what human beings can hear as the pitch of that wave form have yet to be elucidated. A gong, for instance, produces an extremely complex sound pattern that is usually used in musical circumstances with the assumption of its pitchlessness. Yet, I remember a story of a production of Turandot, where the conductor was being driven crazy because the enormous gong that is called for, not only turned out to have a pitch, but the pitch was off key relative to the score being played by the rest of orchestra. The sound of most instruments has a certain frequency, which is weighted more heavily than the other emanating frequencies. This weighting is an easy mapping between auditory pitch and a physicomathematically defined pitch. But it does not explain the gong. A harder question is how we recognize the sound of a particular instrument.

One reasonable theory is that the [overtones] for different instruments have different weights and it is the different weights that elicit recognition. Some experiments that I remember (and can not give references to) indicated that human hearing is not so discriminating, and that the actual information as to which instrument is being played really comes from the transients of the note attacks. Should anyone reading this have answers or even remarks on the above questions, I would be more than happy to hear from them. The one possible source for such information that I know of but have not yet explored is Music and Brain Information Database (MBI) [LINK].

Dave Rusin has collected some Usenet posts on the question of why there are 12 tones [LINK] to an octave, and some others on the nature of musical sounds [LINK] In the articles by Dave Rusin and Stephen Fulling, it is explained how well tempering closes the sequence of fifths into a circle. That there are other possibilities of far greater complexity is also explored. For the CS and number theoretically inclined see Stephen Fulling's paper in which well tempering is related to the Chinese Remainder Theorem [LINK]. [Stephen's paper is in PostScript.]

That well tempering closes the [circle of fifths] and that one can play on a piano keyboard in *any* key, celebrated in the two books of Bach's "Das Wohl Temperierte Klavier", makes possible a simpler group theoretical structuring of both pitch and music than would otherwise be possible.

In an aside, the mathematical concept of [Groups of Transformations]. is defined and related to transformations applied to the keyboard in the context of well tempering.

The fact that a well tempered keyboard means that one *can* modulate from any key to any other key within a given piece, and that pieces may be written in any key, led to the development of increasingly doing so. If a thing is possible, someone is likely to do it. Thus came the advanced chromaticism of the late 19th century, seen/heard in the works of Wagner, Liszt, Reger and others, which threatened to destroy the harmonic pillar of tonality. Arnold Schoenberg, in 1924, to the rescue bringing dodecaphony and groups of transformations! [Leibowitz 1949]

The transformations of a group usually leaving a "central something" invariant.

Now, we can consider groups of transformations in a musical context.

The 12 tones of an octave are merely the alphabet of western music. We can consider groups of transformations within the intermixed syntax and semantics of the language(s) derived from them.

Every single piece of music could be considered a [language] unto itself; so we are apparently discussing simultaneously, the metasyntax and metasemantics of music, in terms of its mathematized structures. This applies, as we will see, to both tonal and atonal music.

In the little introduction to [group theory], we talked about a group of transformations related to the keyboard itself, that is, to the keyboard's structuring of the 12 tones contained within an octave. Instead of talking about the structure of what amounts to the musical alphabet, let's now talk about the linguistic structures constructed from the alphabet.

The actual process of composing music is very complicated as it involves integrating all levels of organization, where any given level can be designated (parameterized) by a time of duration; from the duration of a pitch or rest up to the time duration of the entire composition. I have never seen nor even heard of any document that purports to explain all of this. I do not think it can be written; if written, it cannot be read.

There are "style sheets" in the form of architectural standards available, e.g. sonata allegro form, fugue, rondo, etc., all with a set of large scale rules, or, you can make up your own. At the other end of the scale just above the level of pitch, is the level of phrase. This is where we can talk and work comfortably because the complexity is manageable.

Moreover, to be able to control larger structures one must be able to control the smaller structures from which the larger ones are built.

In constructing a phrase, a major consideration is that it contain a high potential for transformation. Most melodies of musical art are essentially a string of phrases. There are exceptional cases of long unbroken melodies with intensely complicated internal structure that practically defy analysis.

The ability to create such wondrous things is denied most composers. As a single example, listen to the 54 bar long melody beginning on the second page of the first movement of the sonata #3 of Chopin.

As a counterexample by the very same composer listen to and look at the score to the Etude Op 25 #10 in B minor, the so called "octave étude".

One can begin an analysis of the octave étude by noting that it is completely generated by its first two notes, giving the interval of a semitone (minor second).

Although Chopin is rarely thought of, or spoken of as a structuralist, the existence of this étude is a severe criticism of that posture. Since the étude exhibits, in A and A", primarily a simple horizontal structure, I would like to use this étude as an example of transformations before launching into further abstractions of patterns and groups.

In overall structure the étude has a sectional form given symbolically as

        A B B'-codetta transition A"-coda

        The transition has the feeling of a pianistic recitativo.
        This is a slightly more detailed A B A form of which it
        gives the illusion. As Stefan Wolpe once said "Giving the
        illusion of having done something is sometimes far more
        effective than the actual doing."

Putting aside the double octaves in both hands, the étude begins in triplets with 4/4 time signature on the pitch of the fifth degree (dominant) in B minor: (Following Chopin's harmonically correct notation let x = ##, a double sharp)

           (measure 1)                           (measure 2)
   F# E# F# G G# A  G# Fx G# A A# B  |  A# Gx A# B B# C#  B A# B C# D D#

           (measure 3)                           (measure 4)
   Cx D# E  D# E F  E E# F#  E# F# G |  F# Fx G#  Fx G# A  Fx G# A  G# A A#

           (measure 5)                           (measure 6)
   B A# B A# A G#   A G# A G# G F# G |  G F# G F# F E  D# D C# B
   D                D                   B      B       F#      F#

This covers the first six measures.

The seminal motive M is simply a descending minor second, F# F. What follows is the retrograde RM of M, F F#, and the strict inversion IM, F# G. The intervallic sequence F# G, G G#, G# A, can be seen as a classical sequence (a 3 time occurrence) of IM. From transformational manipulations of the seminal I, a new six note motive II has been constructed, which extends the pitch set {F#, F} of I. A sequence of four repetitions of motive II is presented successively raised by a whole tone, introducing a new interval of the major second.

This is transformation by intervallic expansion. The rising in whole tones gives the illusion of a diatonic structure ascending to the tonic B and contained within the highly chromatic activity. Chopin has distilled a diatonic structure from this highly chromatic activity, creating a synthesis of antitheses of harmonic ontologies.

The intervallic expansion would not necessarily be a foreshadowing but for the B and B' sections of the étude, which are of entirely different character having as its seminal interval, the major second. Notice that motive II is repeated 4 times and does not follow the classical rule that "3 times makes a sequence" (Driemal ist Gesetz.

There are three ways of justifying this, and they are complementary.
The first way: the first presentation of II is not a presentation, but a construction, and therefore doesn't really count as a presentation. The second way: The fourth appearance of II is the third transformation of II by translation by a major second, and *that* is the threefold sequence. The third way: the fourth repetition is to unprepare the listener for the radical event beginning with the 3rd measure.

So what's the radical event at measure 3?
The motive II can be seen as a coupling of two three note motives. Measure 3 introduces a transformation by fragmentation. The opening F# E# F# can be understood as merely a written out mordent embellishment of F#, thus its contraction (another transformation) makes perfect sense. By the contraction of II from 6 notes to 3 notes, an increase in urgency has been created because the rate of motivic repetition has been increased.

A sequence of derived motive III being the second half of II is now presented, and is repeated 5 times, well beyond the classical limit 3. It is rising now uniformly by minor second transpositions. In the middle of measure 4, the three note motive is not raised by the expected minor second, but repeated in situ.

This has two effects: first it is a slight pulling back of the overall ascending motion established from the start; second, the pulling back also allows the harmonic goal of the first four measures to occur on the downbeat of the 5th measure.

The pulling back strengthens the downbeat. So far, then, the goal or harmonic driving force is heard as the fundamental harmonic progression (cadence) V -> I, i.e. dominant to tonic.

The motion of measures 1-4 is ascending, the measures 5-6 are descending.

Memory shortens time intervals, so to the ear the descent balances the physical ascent. Having contracted the 6 note motive II into the 3 note motive III, implying the fragmentability of II in measures 1-4, measures 4-5 fragment and recombine the two 3 note motives in a new way. The 3 note first part is intervalically unchanged; then where the second part ascended by semitones, it now descends in semitones, being heard as an inversion. This new 6 note motive IV is presented in a 3-sequence descending in major seconds as II ascended in major seconds. In place of the possible fourth expected occurrence of the motive, the second half descending 3 note motive is repeated, hastening the descent, whose goal is the (tonic) I-chord of B minor.

With this as a start, anyone can, from the score, continue the analysis of the intricate motivic transformations that are used to generate the rest of the piece.

It is worth remarking that the flanking sections being based on the minor second have a rather stormy, savage and chromatic character. For the central sections, however, the basic motivic interval is widened to the major second, transforming the savage chromaticism of the flanking sections to a lithe diatonacism, maintaining all the while, clear connections with the motivic shapes of the flanking sections.

The middle sections present a beguilingly flowing melody that masks the severe logic of motivic transformations that give rise to it. Look at this from a reverse perspective.

The expansion of the fundamental interval is mirrored in the augmentation of the motivic material. The note progression is slowed so better to hear the diatonic nature and also to enhance the difference between the nature of the middle sections and the flanking sections.

The middle section extracts the diatonic underpinnings of the first section.

The transformations that developed historically and became solidified in contrapuntal theory at the zenith of the Baroque period of Northern Germany, with J. S. Bach, are:

    Classic Fugal transformations:
			   harmonic (usually in minor mode)
		   retrograde inversion = crab

		   diminution    - halving note durations
		   augmentation  - doubling note durations
		   transposition - usually by intervallic 4th or 5th

   The first three transformations together with an identity
   transformation form a group.  The group members are operators
   on a pitch sequence.

   The group table for combinging the contrapuntal transformations:
   (E = Identity, I = Inversion, R = Retrograde, C = Crab)

                             | E  I  R  C
                           E | E  I  R  C
                           I | I  E  C  R
                           R | R  C  E  I
                           C | C  R  I  E

   The group is commutative and every element is idempotent: X X  =  E.
   The [group] structure is then that of the Klein 4-group,
   which is composed (a direct sum) of two subgroups of order 2 with
   element pairs:

            E I   and   E R

   Diminution, (D) augmentation (A) and an identity
   (temporal scaling transformations), also form the group of order 3,

                  D A  =  E

   and are inverses of each other.  As fugal transformations, in practice,
   augmentation is multiplying the note values by 2, while diminution
   multiplies them by a factor of (1/2).  There is no reason why only 2 and
   it's multiplicative inverse need be used as scaling transformations,
   except that in Western musical notation this is most convenient.
   It is probably also most intelligeably most audible.

   Generally, one could easily, especially within the context of electronic
   music, use scaling transformations D(1/z) and A(z') with
   1 <= z <= Z, Z being some arbitrary but reasonably small number,
   like '3' or '4', and of course,

                  D(1/z) A(z)  =  E

   These will not all together easily be made into a group, but will
   have, practically speaking, a subset of the multiplicative group of
   of positive, nonzero real numbers that is not itself a group.  It
   might be useful to make it into a group by a toroidal compactification.
   If that makes sense, that's good; if not, that's ok.  I'm dropping
   the matter as being too technical for this small essay.

   The translations (up a fifth)  = U5, (down a fifth)  = D5,
                    (up a fourth) = U4, (down a fourth) = D4,
   with the identity, also form a group of order 5.  Noticing,
   however, that with octave equivalence, U4 = D5, U5 = D4, the
   group becomes another realization of the group of order 3 with

                  U5 D5  =  E

   The full group of contrapuntal transformations can then be seen
   as a group of order 8 with elements labeled 

        E I R C U5 D5 D A

   and with subgroups  E I
                       E R
                       E C
                       E D A
                       E U5 D5

Most notable works in which to experience the full spectrum of these contrapuntal transformations and vertical combinations thereof are Bach's Art of the Fugue and the fugue that serves as the last movement of Beethoven's Hammerklavier Sonata.

What makes a good fugue classical subject?
Ideally, it should establish a tonal center, provide a strong rhythmic pattern, be fragmentable, and terminate with an implied modulation to a tonal center either a fifth or fourth up. Further circumspection might be called for depending on the composer's plans for the subject's transformations and combinations thereof.

As an example of such a subject I present, in an admittedly clumsy notation, the subject of the four voiced fugue in 6/8 from the Prelude and Fugue in A minor "The Great" for organ by J.S. Bach. The subject is in two sections and is five measures long.

The first section covers the first 1.5 measures, while the remainder, all in sixteenth notes, covers the remainder of the subject, and is a modulating sequence from (I) A to (V) E.

    1/8 1/16     1/8 1/16    1/8        1/16
   |                        |           F F F |F F F        |
   |                     E  |       E    E    |      E E E  |
   |    C   C               |C             D  | D     D     |
   |      B     B      B   B|                C|         C   |
   |A         A             |   A             |   B        B|
   |                 E      |                 |             |
   |                        |                 |     G       |
      I                V        I       (III)   V of III
   (1)                      (2)               (3)

   |            |           A   |             |
   |            |               |G            |
   |            |             F#|             |
   |E E E       |         E     |   E         |
   |      D D D |D D D          |             |
   | C     C    |      C C      |             |
   |         B  | B             |             |
   |   A       A|       A       |             |
   |            |   G#          |   G         |
   |     F      |               |             |
   |            |     E         |E  <-        |
    IV^7 of III   V^7   I        V   |
                     (I=IV of V)     | second subject entrance
                          V of V       a fourth below.
   (4)          (5)             (6)

The subject begins immediately with preparation on the downbeat of the first measure being emphasized by it being an eighth note against the soon to be established 1/16 note pulse, that is initiated on the C that follows.

Within the first measure, the key of a minor is established with the pitch sequence (A-CBCAB-).

Although one sees the tonic note of C and its leading tone B, suggesting the possibility that the C major might be implied, the beginning emphasis on the opening A dissuades the ear from that academic possibility, moreover, there is no G present to associate the B as a leading tone with a V (G) of I (C).

Within the initial pitch sequence, the (CBC) can be heard at first as an embellishment on C (a mordent) so the sequence can be abstracted as (A-C-AB-) transformed to (C-A-E) in eighth notes just prior to the beginning of the second section of the subject.

Also within the first measure, a rhythmic motive is introduced, and repeated. On the repetition the phrase shape of the first occurrence is exploded from containment within a minor third (A-CBCAB-) to span an octave (B-EBEBC-).

At the beginning of the second section in the last half of measure 2, what had appeared to be an embellishment (CBC) is now understood as the generator of the activity of the second section, that suggests a kind of writing that would be typical for violin: the repeated pitches interpolated between the sequences of descending pitches. The descending pitches are also subjected to the exploding transformation in that scale like descents are immediately followed by triadal descents that, of course, outline, 7th chords so that the implied harmonic modulation sequence is effected using only a linear sequence of pitches.

Any modulation relies on the calculated ambiguity of a given chord: enter with one interpretation and leave in another. There are subtleties in the modulation above because the modulation is from A minor to E minor and it is E major that is the V of A minor.

The sleight of hand is the first appearance of G# promising a modulation to E major, but then leaving the mode ambiguous at measure 6 when the subject enters again, which it must do in E minor.

The constraints of traditional harmony can limit, and also guide sequences of allowed transformations. If these constraints are removed, the pitch deployment has more freedom and there is a wider range of transformations available in general, and also at any point in any composition.
A more general list of transformations is then:

    Motivic transformations:
	   inversion, retrograde, retrograde inversion = crab


	   diminution  -  augmentation
	   fragmentation  -  agglutination/extrapolation
	   explosion  -  implosion (using octave equivalence)
	   contraction  -  expansion (of intervals)
	   rotations in H-V plane
	   aspect abstraction
	       rhythm, pitch set, shape
	   shape shifting
	       harmonic, rhythmic
	   interpolation and its deleting reverse

The abstract musical object is defined or clarified by the transformations which it undergoes.

What is left invariant under the transformation?
In the course of transformations, one "object" may become another.

Consider again the fundamental problem of musical composition: transformation and variation that leaves some musical aspect or object invariant.

A secondary problem which is very much a matter of taste to the composer, is to maximise the relational structures and minimize the material. [Anton Webern] is a prime exponent of this taste. It is the analog of a mathematician's taste for a short elegant proof for a theorem over a longer and messier one.

The importance of temporal exposition in transformations cannot be ignored, unless the score is to be seen only and not actually heard, in which case one can question whether it is music or a graphic art form. A derogation of such graphically conceived scores by the unmusical is "Augenmusik" (Eye-music) in German.

In order for a piece of music to be understood by a presumed "hearing" listener, the transformations must normatively not be too wild; sequences of transformations cannot be so extreme, as not to be perceived rather immediately as connected.

Although, almost every composer, at times, will sacrifice this principle to the drama of the dynamic unfolding of the music.

There is a flow in time of the logic of a piece, that can be, and is disrupted by the immediate drama of a divergent and surprising event: an event which breaks the continuing flow of logic, but which is then by logicotransformational means pulled back into the normative logical flow, thus resolving the divergent to the normative. Stefan Wolpe called these divergent events "radical events", but when I think of the etymology of "radical", from the Latin "radix" (root), radical seems to me not the most propitious adjective.

Music is a logical structuring like a mathematical proof of itself, but it is not mathematics. It is more like a mathematical story told dramatically; not necessarily with high drama, but with passion. The composer must, at least ask the question:
What is the listener supposed to hear?
The questions: [What does a listener hear?] What can a listener hear? are elusive and sufficiently vexing questions that most composers, I think, ignore them, beyond the grossities that are obvious.

There is also the distinct question of what can be perceived. The retrograde transformation seems to be, for most, heard or at least hearable, but, not necessarily perceived.

In a musical forms class with [Raoul Pleskow], long ago, such a transformational relation was heard by me in a Beethoven Sonata. On announcing the relation of retrograde, Raoul asked "Do you *actually* hear that?" When I said yes, all he said was, "Yeah - only you and Bartók."

Bartók was, of course, rather fond of the retrograde transformation, and his musical pieces have a relatively high density of palindromes. There is a fine place to spend time to practice perceiving and recognizing this kind of relationship, and associated transformation.

Overlaying regimes of order by durative magnitude:
We have, thus far, been considering transformations and relationships between phrases and motives, primarily because that is easiest and simplest.

Yet, clearly, there are harmonic transformations, cross relations between activities separated both vertically and horizontally, transformations of phrase or phrase segments into vertical relations and transformations of vertical material into horizontal material.

In the last a perfect extreme example is an arpeggio; an exact transformation of a chordal or vertical structure into a phrase or horizontal structure. The ear hears the chordal reference in the phrase material. Larger patterns are created by grouping smaller patterns. The larger patterns are also subject to transformations much the same as those of the smaller patterns, just as an edifice is built from stones forming walls and other parts until the simple and classical musical architectural forms:

			 A  A'

                        A  B  A

                  A B A  C A D  A B A
are achieved.

The third happens to be a rondo form, which is clearly an elaboration/extension/expansion of the simple ABA form. The upper case letters here represent sections of different musical material, and embrace an entire piece.

But why not apply such arrangements of symmetry and variety to smaller structures below the architectural level?

           Dynamics and Architecture of the Drama: More patterns

There is a dynamical aspect of the temporal unfolding of a musical piece, beyond the observation that it has a beginning middle and end. Forces are created within and by its internal structure that motivate the unfolding.

Many listeners hear and then feel the ineluctable in fugal form. The form itself is sufficiently rigorous that once known, the directions that it can take are anticipated on the entrance of the first voice, the outcome is in a sense clear. More frequently than not, the subject is so structured, that one does not have to be told "this is a fugue"; it's obvious.

The question then arises as to how the arrangement of patterns of pitches creates these dynamical forces.

This is an easy question with a difficult answer, because the answer necessarily involves the Art of composing music, and a composer's personal solutions to this very question which he must ask himself.

At any point in composition, there are causalities linking successive activities, and yet there are no rules for these causal linkings. Different composers have different ways of creating them, and these choices are part of each composer's voice or style. The composer learns *his* art by composing, as the writer learns his by writing.

There is no other way; no one, regardless of the depth of instruction, can be taught to be a composer of music. The composer, as any artist, is called upon within, to create; he must do so ultimately with integrity, passion and conviction.

So, there are no great secrets revealed here, but rather some considerations of simple principles based on observation of our existing musical heritage.

To a great extent, I expect, these considerations have analogy and use in any artform that is temporally presented: e.g., literature, poetry, film and dance.

The basis of all more specific motivating forces seems to be manipulations of patterns of Tension and Release. So, next the question as to how tension and release are induced. A brief table of some pattern types that can be employed, with no pretensions of exhaustiveness, follows:

             Control of time (rhythm)
                     regular (repetitive)
                     irregular (usually following language stresses)
             Momentum of process (A kind of Newton's first law of music
                                  that sets up expectations.)
             Diversion or blocking of the momentum
                     stresses that pull at the underlying pulse
                     line direction
                     line speed
                     density increases and decreases
             Harmonic motion
             Harmonic rhythm

Now, I would like to talk about each of these.

The control of time by rhythm in a regular way is most frequently done by writing music that follows a dance form: the 3/4 rhythm of a waltz or mazurka, a tango, a polonaise, etc. It works, and from the viewpoint of composing effort is relatively inexpensive.
Irregular rhythms used to propel a line forward are chiefly found in music that freely follows the inflections, rhythms, stresses and semantics of language. Prime examples are operatic recitative.
A Digression:
Many people, for reasons that I cannot understand, have difficulty with the music dramas of Wagner. They're too long. (For whom?) They're too heavy. (whatever that means) Bloated orchestra. (bullshit) Poetical critics. (the poetical forms and techniques are ancient - they're supposed to be) Stravinsky, Brahms and many formalist composers have a particular animosity toward his music, because it doesn't satisfy their personal aesthetics (so what?) and so on. I don't particularly care what they either think or feel about Wagner's work, for in the control of musical time, phrasing and harmony, it is my unalterable opinion that no one has yet lived who can even approach his genius in the meaningful setting and creation of music to fit language. Only if you understand the language and know the text is this completely obvious. To hear it done with complete brilliance, I merely suggest the confrontation between Ortrude and Friedrich at the beginning of act II of Lohengrin. [The Kempe recording with Christa Ludwig and Dietrich Fischer-Dieskau.] Read the German; all the English translations that I've seen are laughably prettified, and miss completely the power of the German words, both in meaning and in sound. My favorite of all laughables is an intense line from Ortrude:

          "Ach! Wie tödlich du mich kränkst!"

   Usually rendered (and I mean rendered) as

          "You make me deathly ill."

   What Ortrude is really saying is

           "You, make me sick to death!"

The sounds of the German and the musical tones chosen by Wagner fit the emotion expressed perfectly. He wasn't an especially compassionate or even kind man, and he was a megalomaniac to boot; but so what. I mustn't get started on Wagner, or I'll never shut up. Back to pattern types of tension and release, and the forward impelling of music.

There is a kind of auditory law of momentum akin to Newton's first Law of Motion: A musical process continues (in our perception) in its motion unless acted upon by an external force. The musical process itself can't be a source of tension. The force in this case is the will of the composer. I'm sure a few examples won't offend clarity.

   First Example:

If a line or phrase rises for a sufficient length of time, the expectation that it will continue to rise is set up. Suppose the rising is at a constant rate; the expectation is then that this rate will remain constant. There is a perceived momentum of process. A tension is created in the rising, and further when the composer bends either the rising itself or the rate of rising. Since any musical parameter has a finite range, one cannot continue its rising indefinitely.

There are two ways of aborting, both of which release tension. 1) crest and descend, releasing the tension as one would with a stretched string. or 2) pull so hard and fast that the string snaps.

While the attention has been diverted by the snapping of the musical line, either another activity has taken over, or it hasn't.

In the latter case, silence becomes the new source of tension, simply because "something" is expected and nothing is given. Complicate the rising line by having a simultaneous descending line. Listen to this happening in the third movement of the Symphony #3 by Saint-Saëns.

   Second Example:

Increasing density is a source of tension building: The prelude to act I of Lohengrin opens with solo violins playing a high and open A major chord PPP in harmonics. The "grail theme" is presented. Then, there is a contrapuntal development of the theme, adding all the strings. The wood winds are added almost imperceptibly, thickening the texture, then, the horns etc. It is only at the climax a little beyond the 3/4 point, in time, of the prelude that the full orchestra is used, after which the density thins out back, in retrograde order, to the solo violins. Tension and release in the architecture of density.

   Third Example:

The Piano Sonata #7, Prokofiev, third movement is a perpetuum mobile, written in the time signature of 7/8 and shown in the score to have the 7 eighth notes divided as

                    7  =  2 + 3 + 2

So there is a prime stress on the downbeat of each measure and secondary stresses on the third and sixth eighth notes within each measure. There is a kind of nonrational aspect to this rhythmic structure that always keeps the listener slightly off balance.

The irregularity of pulse is a source of heightened tension kept up through the entire movement. Add to this the even more offbeat punctuations that Prokofiev throws in for the right hand, and you have a steady high degree of tension propelling the movement and being modulated up and down by the punctuations that constitute a melodic component.

The tension thus maintained has an explicit release of tension in the movement's ending, but I'm not telling how it's done. Figure it out.

Diversion or blocking of the momentum:
The use of rubato or syncopation is a blocking of a linear momentum which alternately creates and releases tension; see it perhaps as tension modulation.

Well Tempered [Traditional Harmony] has as it's seminal foundation the simple sequence V->I, and the concept of leading tone.

Harmonic variety is obtained by substitute sequences, and progressions which divert this cadential protosequence. In the substitutions and diversions, harmonic motion is created. For harmonic motion, which actually need no examples, I give any of the chorale settings of J.S. Bach as examples.

Motion is created like a wave motion by push and pull of harmonic tension and release.

Harmonic rhythm is created by a repeated harmonic sequence, and also by variants of equal length.

A harmonic rhythm need not be in accord with temporal rhythm. A chaconne form, usually in 3/4, is a set of variations on a harmonic sequence, which rarely if ever modulates to a different key. There the harmonic and temporal rhythms are usually in accord.

For an example where these two rhythms are not in accord (a source of tension), see the Chopin's Nocturne Op. 37 #1, in G minor. The second section in Eb Major (4/4) starts on I on the downbeat of the first measure, but then manages to avoid I on any strong beat of the measure.

Modal harmonic structures, i.e. other than major or minor are used for short periods of time by many composers. A classic and well known example is the opening of the fourth symphony of Brahms written in E major, but which actually opens in the Phrygian mode. The use of modes in a modern harmonic context, of course, has absolutely nothing to do with either the [medieval modes], and is even further removed from the use and even concept of ancient [Greek modes]. Both those concepts are completely linear and not harmonic or polyphonic.

Because the (Ionian) Major protocadence V->I is carried over into the treatment of other modal harmonies, relative to the Major (Ionian), they have harmonic weaknesses in that the V chord is usually a minor rather than a major chord.

Associated with this weakness is that the 7th degree of the modal scale is a whole tone down from the 1st degree (tonic) note, rather than a semitone. This relation weakens the function of the 7th degree in its function as "leading tone".

The Dorian, Mixolydian and Aeolian modes have minor V chords. The Lydian and Locrian have major V chords, while Phrygian actually has an incomplete diminished 7th chord as its V.

The Locrian mode is particularly difficult to treat harmonically because its I chord is an incomplete diminished 7th, consisting of two minor thirds. This does not function very well as a harmonic resting point. It is rather a harmonic chord of ambiguity.

Much early music divides the octave into 5 parts, producing various pentatonic scales. "Auld Lange Sygne" is written in a pentatonic scale as are other old Scottish and Irish songs. There is good reason to believe that ancient Egyptian music was pentatonic. Classical Chinese music is pentatonic with various modes. See [Persichetti 1961] for more details on pentatonic modes and their uses. The most used pentatonic in Western music are the black keys of the keyboard.

More modern composers have used pentatonic scales in much the same way that they use diatonic modes, for color. The most prominent examples are in Turandot by Puccini, and works by Ketelby.

Harmonic amorphics:
Scales, modes, chords, which is to say vertical and horizontal harmonic structures have harmonic function by virtue of their intervallic substructures. By making horizontal and vertical structures with homogenized intervallic substructure, the harmonic meaning or function is weakened or ambiguated. I happen to like the name "harmonic amorphics" for such things.

From a transformational viewpoint, harmonic amorphics have a group of translations that leave their substance unaltered: Translate a chromatic scale by any interval. Isolated, its structure (though perhaps not its function) is left invariant; similarly translate a whole tone scale by any number of whole tones; or translate a diminished 7th chord by any number of minor thirds, and its pitches remain the same.

This is, in fact, a useful ploy in a harmonic context. The game goes something like this: I am in some harmonic context, with a definite tonal center established. Say I'm in C major and I want to be F# major, which, harmonically speaking, is as far away as I can get from C major. There are classical ways of modulating through to F#, but I want it now.

A way is to leave the current tonal center through an harmonic amorphic, write around briefly in this harmonically ambiguous space and reemerge in F#. The alternative, if you are brave enough and your art is good enough is to pull a Beethoven. Anybody who has listened to enough Beethoven will know exactly what that means regarding Beethoven's "modulations": they are more like a divergent event, as written about, above. If you do not understand, it is a minor and expendable joke anyway.

Whole tone scales (modes) divide the octave into 6 notes; under a twelve tone system there are only two distinct whole tone scales related by a semitone.

                     C  D   E  F#  G#  A#

                     C# D#  F  G   A   B

Since the notes of whole tone scales are all equally spaced, there can be no intrinsic harmonic resting place: find some cadential formula, and it will work the same for any translation. The same property is true of the Chromatic scale.

Liszt in his later years was already writing small scale music using whole tone scales and explicit atonality. [Walker 1988], [Walker 1993], Anticipating the music of the impressionists, expressionists and atonalists of the 20th century, understanding as he did before, that new relationships and local structures demanded also new architectural structures.

An augmented fifth chord consists of and augmented 5th divided into two major thirds. The customary resolution in harmonic context is a contraction to a 5th, thus contracting one of the major 3rds to a minor 3rd, producing overall a standard triad. The augmented fifth is also a standard triad of a whole tone scale and in that context is possessed of the same tonal amorphousness as the scale itself.

The much used diminished 7th chord is composed of three successive minor 3rds, with two interlocking tritones. Harmonically, by reason of its homogeneity, it is ambiguous in function and frequently interpreted as a rootless diminished 9th chord.

For example a C diminished 9th (CEGBbDb) with C cut off the bottom is a diminished 7th chord, (EGBbDb).

The root of the chord is a major 3rd down from the E. The adding of the C disambiguates the diminished 7th. But, this is not the only possible disambiguation: choose and add a note down a major 3rd from any of its other three tones.

These also provide roots that disambiguate (EGBbDb) making it, in each case, a diminished 9th.

                Db    E     G      Bb   The vertical permutations
                Bb    Db    E      G
                G     Bb    Db     E
                E      G    Bb     C#

                C     Eb    Gb     A    Possible disambiguating roots
					forming another diminished
					7th chord

The possible disambiguating roots also form a diminished 7th chord. In terms of pitch sets there are exactly 3 diminished 7th chords: the two given above and (BDFAb).

In a context of total harmonic amorphics, anything follows anything; the trick of disambiguation at the boundaries of harmonically amorphic activities is the harmonic analog or warp/improbability drive allowing passage to distant tonal centers faster than would be possible by staying within the classical harmonic rules.

The breakdown point of harmonic structure is allowed by many musicologists to be the famous "Tristan chord" (FBD#G#) that appears in the third measure of Wagner's "Tristan und Isolde" (first performed in 1865).

The Tristan chord, coming from an established A minor (I) resolves by chromatic alterations to an E7 (V7).

Despite all the attention given the Tristan chord, one could see it simply as a distorted V of V. That Liszt had penned such harmonies ten years earlier went unnoticed for some time.

The dissolution of the tonal center, however, doesn't lie with the existence of a single subversive chord, but rather with the practice of composers of passing through tonal centers with such rapidity, through the use of harmonic amorphics (chromaticism of voice leading being one technique of amorphisation) so that the center itself is not, or barely perceived by the human ear as actually being a tonal center.

Since the tonal center was a foundation of architectural forms, these too were subverted, and the problem arose of reestablishing a sense of order and cohesiveness in large structures.

The dissolution of the tonal center, led, as one might expect, to the emergence of two opposing camps, the reactionaries and progressives. One could put Mendelssohn, Brahms, and Reger in the reactionary camp. Of the three, Reger is the most interesting to me. His use of chromatic and advanced harmonies pushed them to the limit, while the order that he imposed was that of the period of High Baroque, a prodigious example of which is his "Variations and Double Fugue and a Theme of J.S. Bach" for piano.

Reger was, of course, not the beloved of either camp. Among the progressives were Wagner, Liszt and his group of students dubbed the "Altenberg Eagles" referring to the name of Liszt's residence in Weimar. [Walker 1988] [Walker 1993] Among the eagles were Ferruccio Busoni, [Busoni 1911] Basic Repertoire List - Busoni [Link] Karl Taussig, Hans von Bülow, and the very young Julius Reubke. The eagles were the champions of "Musik der Zukunft", a phrase written in a derogatory editorial and gleefully adopted by them at the prodding by the Princess Carolyne.

The mystical Alexandr Scriabin (1872-1915)
The Scriabin Words Page [Link]
Scriabin as Synaesthete
abandoning his early Chopinesque style created a completely different kind of harmony based on chords built on fourths instead of thirds, and also based on his ecstatic and synaesthesic view of the nature of music.

Finally, in 1924 Arnold Schoenberg established a principle which instead of attempting to repair the concept of tonal center, actually made a virtue of its absence. The principle might be paraphrased by: OK, if we are not going to be able to regain a tonal center let us, by musical law, annihilate it utterly.

Tonal centers become established by returns and repetitions of the tonic note itself as well as by certain cadential formulas. If the 12 available tones are required to be democratized, then a tonal center cannot be created.

Therefore, make it a principle that one uses all 12 tones before any are allowed to be repeated. But, this is not enough since it provides a specific disorder, and does not provide an ordering principle. In sounding out the 12 tones there is always the expression of them by some permutation of them.

Suppose one picks a single permutation of the twelve tones; this then provides a recognizable piece of musical material that can be used to provide musical cohesiveness in musical composition and at the same time assures that there will be no tonal center.

This is "the method of composing with 12 tones", which we now call by the Greek neologism dodecaphony. Such a permutation was called "Reihe" by Schoenberg, which can translate as "sequence" in English. For some reason that I've never quite understood, we now say "12 tone row".

The three greats of the Viennese circle of composers to use this method were, Schoenberg, of course, Alban Berg and Anton von Webern.

   There are
              12! = 479,001,600 possible absolute dodecaphonic rows.

   ('!' indicates the mathematical "factorial function".
   The row considered modulo its 12 transpositions and therefore as an
   intervallic sequence, there are

              11! = 39,916,800

possible intervallic rows most of which are not musically interesting. Yet there is still a great deal of freedom and many possibilities within which to make music.

In [Webern 1934], Webern distilled dodecaphony to a remarkable state of minimalism, and great symmetry.

The foundation of his Concerto for Orchestra is a set of four highly symmetric rows related to each other by the same relations that generate each of the rows from a 3 note asymmetrically arched motive.

To illustrate the shapes involved, between the pitches of the row, I'll use '/' for "ascending to" and '|' for "descending to". The fundamental row is:

        B | Bb / D  /  Eb / G | F#  /  G# | E / F  |  C / C# | A

            E               C               R             I

The row partitions itself into 4 motives of 3 notes. If the first 3 note motive is taken as fundamental, then the second is the crab, the third is the retrograde, and the fourth is the inversion of E. By applying the contrapuntal operators as above, C, R and I to this row, one obtains a magic square of 3 note motives:

  (E):     B | Bb / D  /  Eb / G | F#  /  G# | E / F   |  C / C# | A

               E               C               R               I

  (R):    C# | A / Bb  |   F / F# | D  /  E | Eb / G   /  Ab / C | B

               R               I               E               C

  (C):     A / C# | C  /   F | E / G#  |  F# / G | Eb  |  D | Bb / B

               C               E               I               R

  (I):     B / C | Ab  |  G | Eb / E   |  D / F# | F   /  Bb | A / C#

               I               R               C               E

By virtue of not being invariant under any of the transformations of R, C and I, the fundamental row is transformed into distinct rows that are clearly and closely related by also being generated by the same 3 note motives.

The above magic square array also distributes four 3-note pitch sets over the four motivic shapes.

This structure involving a group of transformations acting on two distinct levels to produce rows that determine a magic square is then the foundation of a three movement concerto for orchestra of a most delicate and ethereal texture that lasts for all of about nine minutes. For Anton v. Webern, this was a very long piece.

When I first discovered this piece founded on such elegant mathematical symmetries, I wondered immediately whether there were similar structures that could be constructed that would be suitable for a work on a much smaller scale and therefore of even higher symmetry; also, could one extend the notion to provide material for larger scale works?

A little work with paper and pencil showed the answers to these questions to be very much in the affirmative, and that the two directions were related.

In the matter of higher symmetry, one can produce rows similar to Webern's fundamental row which is intervallically equivalent to its retrograde (and therefore a palindrome) or equivalent to its inversion, or equivalent to its retrograde inversion. You should work this out yourself.

In the matter of generating material for larger scale works, using either the organization of Webern's row, or the rows of higher symmetry whose second half is intervallically equivalent to the R, I, or C of the first half, one can develop a sequence of rows with the chosen symmetry, composed of three note motives where the intervals of the primary 3 note motive expand slowly by semitones while retaining the motivic shape. Thus, one can write within a dodecaphonic system with rows of high symmetry adding a process of intervallic, then motivic, and consequently [row evolution].

Problems of Dodecaphony.

There are two compositional problems that arise with use of dodecaphony. The first is what Stefan Wolpe described as a lack of "chroma", from the Greek, meaning color.

The constantly revolving and equidistributed 12 tones leads to an overall homogeneous grayness, that cannot be broken.

The second problem, especially if one follows the elegant refinements of Webern, creating patterns within patterns within ..., and taking into account the extended notions of musical line, its fracturings and abstractions that arise quite naturally in the dodecaphonic atmosphere, is that the density of interrelation and cross relations becomes so high that they can scarcely be heard.

The paradox: there is so much information being presented that it cannot be assimilated, especially by the ear that is accustomed to the idea that music is melody with accompaniment.

Even for the musically trained ear, the density of Webern's relationships coupled with the sparseness of actual material are a challenge in hearing and listening.

Composers of Webern's genius understood the requirements being placed on the listener and so composed with low material densities and aphoristically. Some pieces of Webern last less than a minute. The sparseness of music of this type is reminiscent of Gagaku, the court music of Japan.

The above problems notwithstanding, the serialization of pitch led to the serialization of other musical parameters, and a concept of Total serialism (M. Babbitt, O. Messian, K. Stockhausen). At this point, one is no longer composing music but calculating it as the output of some mathematical algorithms.

To some this was a happy and clever thing; to others it was an artistic horror, signaling the death of free will and free artistic creation. The lines of musical evolution were dividing yet again, and continue to do so.

But, I want to get at some of the organizing principles that have succeeded serialism in the rather explosive post serialistic era.

One of the responses to the perceived constrictions of total serialism was aleatorism, or music of chance, referring to the casting of a die (Alea Iacta Est), where parameters were determined "randomly" by the composer, or certain aspects of performance were left to the performer.

One can look at the latter from a positive perspective as giving the performer more freedom and therefore responsibility in the creative act, making a kind of compositional collaboration between composer and performer; the negative perspective is to view this as a simple laziness and abdication of responsibility by the composer.

Once again, the direct jump to chaos and its attendant lack of structure induced an attempt to order the chaos; and once again it is mathematics that comes to the rescue by providing the organizational tools. Iannis Xenakis, has employed mathematical techniques, some instituted on computers to chose values of musical parameters from various statistical distributions well known to mathematicians and physicists. These techniques, first published in journals, were collected into his book, "Musiques Formelles" [Xenakis 1963], thus giving structure even to the randomness of wanton aleatorism.

Along a different line, considering the above stated problems of dodecaphony composers sensed an importance of tonal center and genuine problem for the art of music in its loss. Some, as Reger did before them, used old forms to structure new material; some adopted an eclectic aesthetic understanding that tonalism, harmony and atonality were all part of western musical heritage. Two American composers who worked successfully in this atmosphere of eclecticism are Samuel Barber and Robert Ward.

In his Sonata for Piano, Op. 26, Barber writes in what is essentially classical sonata allegro form a piece in mostly Eb minor. It is in four movements: allegro energico, a scherzo marked allegro vivace e leggiero, adagio mesto, and finally a fugue marked allegro con spirito While the first and last movements are in Eb minor, the second basically in G major, the third movement is essentially atonal, and developed from a 12 pitch set exposed in the first measure by vertical pairing

                  Bb            A     
                  Eb            E    
              G        D   G#       Db
              B        F#  C        F

which is repeated in the second measure. Both halves of the measure are presented in such a way that the illusion of motivic repetition is created, but the first is really a variant of the second. The adagio is not dodecaphonic, but is clearly spiritually derived from the concept.

In Robert Ward's opera "The Crucible", at the end of act 2, there is chaconne in (6/8) (very close to being a passacaglia if one cares to make the technical distinctions), powerful both musically and dramatically (duet between John Proctor and Mary Warren) that has as it's ground bass

        G#  Fx  G#    B  Fx  G#  /   G  F#  D#   F#  E#  C#

        E  Eb  D  D  Bb  C#   C#  B#  G#  B  A  E  (G#)

The first measure establishes the key of G# minor, and introduces in its last three notes a motive (the same as the fundamental motive of Webern's Concerto, coincidentally) which then unfolds, after a leap up a major 7th, in transformations and translations in a long descending line to a novel chromatic cadence to G# minor for a repetition. The ground bass happens to pass through all 12 tones in its unfolding, but is not a dodecaphonic sequence.

The mixing of tonal and atonal idioms, that is the slipping in and out of either is accomplished using the same techniques developed in the transition era between late chromaticism and atonality, say in the works of Max Reger and perhaps more pointedly in Schoenberg's predodecaphonic "Verklaerte Nacht": which is to say using a fundamental triadalism with, chromatic (by a semitone) alterations and; added, deleted and resolved nonharmonic pitches.

Another alternative and direction appeared in the later works of [Stefan Wolpe], [Wolpe 1959], who sought ways of loosening the rigidity of dodecaphony, while maintaining a sense of structure and finding a replacement or abstraction for the concept of tonal center. The abstraction comes from the following simple observations about classical harmony.

The vertical and horizontal aspects of harmony:
Chordal (vertical) sequences imply horizontal lines; likewise horizontal lines (voices/melodies) are constructed from scalelike motions and chordal outlines, and so imply chordal sequences. This is to say that harmony has both vertical and horizontal aspects. It doesn't matter what kind of harmony.

The "meaning" of any given note within a musical composition is entirely given by its contextual function; a sort of musical version of Mach's principle in cosmological considerations. What is a chord but a set of pitches arranged in a vertical order; change the order and the harmonic function changes. In traditional harmony a triad (CEG) in root position functions differently from its first inversion (EGC) which again functions differently from the second inversion (GCE).

The same holds true in atonal composition in general, and in dodecaphonic composition more particularly. It is the variety and ingenuity of snaking the twelve tone sequence, its repetitions and transformations through the H-V plane that makes a dodecaphonic composition, by exposing the horizontal and vertical aspects and possibilities inherent in the sequence. This idea is no different from the idea that Beethoven's 5th symphony exposes just such inherent transformations and possibilities of a simple motiv (GGGEb) with which it opens.

The fundamental triads built on each degree of a scale are also vertical arrangements of subsets of the scale tones. As a generalization to atonal music, consider "the chords" to be ordered vertical arrangements of subsets of the 12 tone set.

In [Wolpe 1959], it is pointed out that in atonal music a funny thing happens with the harmonically weak octave relation (harmonic octave equivalence and the reduction to 12 tones). An interposing 7th or 9th nullifies the weak relation, and in so doing breaks the translational symmetry of octave equivalence. This is to say that one can write in such a way, in an atonal context, that octaves are not equivalent, thereby effectively obtaining more than 12 fundamental tones.

Traditionally, the tonal center is associated with a pitch and with a triad. By simply understanding the chord as representing the tonal center, the translation to atonal harmony is then also a chord. Since all arrangements of the set {C E, G} are still C major triads, the generalization to atonal harmony is that of some pitch set, not necessarily considered as a subset of a dodecaphonic sequence.

This jump having been made, it is easy to see that tonalism can be returned to atonal music, not by organizing by old rules but by organizing according to new rules where the chords of traditional harmony are replaced with pitch sets, where a pitch set center is the tonal center. General pitch sets can be exploited in any way that a traditional chord can be exploited. By doing this and by learning the lessons of symmetry from Webern, chroma can be returned in an orderly way to atonal music, dodecaphonic or not. This is saying what I understood from Wolpe in a different way. [What I call [pitch sets] are frequently called "pitch groups", or "pitch constellations" a terminology that I refrain from here for fairly obvious reasons.]

A few comments on Wolpe's article, "Thinking Twice" [Wolpe 1959]:
Every composer, in fact anyone interested in modern music should read it, at least twice. His prose is as unique and relationally dense as his music. Every sentence speaks at least a paragraph. In it, there are examples of style, of thinking and of music writing, a style that is both as elegant as Mozart or Webern, and at the same time as fiercely virtuosic as any composer I know of.

Now to some considerations of formal ways of playing with pitch sets by mathematical means.

   Permutations and combinations:
   Finite groups can be represented by permutation operations as remarked
   in [group].

   For 3 objects, there are 3! = 1*2*3 = 6 possible permutations.  These
   are all contained in the rows of two matrices.

	(1 2 3)    (1 3 2)
	(2 3 1)    (3 2 1)
	(3 1 2)    (2 1 3)

   Going down, cyclically permute the rows of the matricies, to form two
   new pairs of matricies.  Superposing the matricies of either column,
   one has a pattern for distributing 3 values of 3 different parameters:
   e.g., pitch, loudness and instrumental color.

	(1 2 3)    (1 3 2)
	(2 3 1)    (3 2 1)
	(3 1 2)    (2 1 3)

        (3 1 2)    (2 1 3)
        (1 2 3)    (1 3 2)
        (2 3 1)    (3 2 1)

        (2 3 1)    (3 2 1)
        (3 1 2)    (2 1 3)
        (1 2 3)    (1 3 2)
   These matricies (squares) do not have the double diagonal property.

   Now, consider a set of 4 objects.

   Magic squares of dimension 4 do have the double diagonal property,
   which is why they are called "magic":

   (1 2 3 4)    (1 4 3 2)    (1 3 4 2)   (1 2 4 3)   (1 4 2 3)   (1 3 2 4)
   (3 4 1 2)    (3 2 1 4)    (4 2 1 3)   (4 3 1 2)   (2 3 1 4)   (2 4 1 3)
   (4 3 2 1)    (2 3 4 1)    (2 4 3 1)   (3 4 2 1)   (3 2 4 1)   (4 2 3 1)
   (2 1 4 3)    (4 1 2 3)    (3 1 2 4)   (2 1 3 4)   (4 1 3 2)   (3 1 4 2)

   where the collection of rows totally cover all 4! = 1*2*3*4 = 24
   permutations of 4 objects.  Once again, as above, generate new squares
   by cyclically permuting the rows and generate a pattern for distributing
   4 values of 4 distinct parameters.

   It is not the summations that are really important as devices for
   musical composition, but the interlocking patterns of permutations.
   Write a 4x4 square array with a similar pattern of properties:

                          (a  c  d  b)
                          (b  d  c  a)
                          (c  a  b  d)
                          (d  b  a  c)

   so that every column, row and diagonal contains each letter of
   the set {a,b,c,d} exactly once.  This array is orthogonal, in
   a sense, to our first array just above:  Combine them as follows:

                         (1a  2c  3d  4b)
                         (3b  4d  1c  2a)
                         (4c  3a  2b  1d)
                         (2d  1b  4a  3c)

   Follow 1 through the columns.  It gets paired with each letter
   exactly once.  Do the same with 2, 3 and 4, and exactly the same
   thing happens.  If the numbers are values of some parameter, and
   the letters, values of another parameter, then the above gives
   a pattern for the horizontal and vertical display of both
   parameters simultaneously that maximizes diversity with minimal
   musical material.  For example, the numbers may label pitch sets
   while the letters label instrumental colors or motivic shapes.

   Consider this array to be only one H-V exposure of the pattern,
   generating others by rotational and flip transformations that
   form the transformation group of the square:

        E = do nothing, R1 = rotate by 90 degrees, R2, rotate by 180 degrees
        R3 = rotate by 270 degrees,
        FV = flip vertically about a central horizontal line
        FH = flip horizontally about a central vertical line
        FD = flip about the major diagonal (L->R down)
        Fd = flip about the minor diagonal (L->R up)

    which gives a transformation group of order 8 that possesses
    3 cyclic (and therefore commutative) subgroups, of orders
    4, 2 and 2.

    To add further transformations, add cyclic permutations of rows
    and columns, each being then cyclic subgroups of order 4.

All of these squares have the double diagonal property, and more stringently have the property that each row, each column and each diagonal is a permutation of the 4 symbols; within a given square, all the permutations are distinct.

The reason that these particular squares are interesting in atonal music is that they describe the pattern for deploying a set of any four distinct pitches in a horizontal and vertical event that avoids the dreaded octave cross relations. Other applications for such patterns are already indicated. This is just a basic already calculated pattern; there is no constraint necessary that it be used to construct a sequence of four chords.

In pitch deployment, the first square might take the shape

              1              4 
                 \  2  3   /
                    4  1
              3  /         \ 2 

                    /   \2
                  2        4   

   with, e.g., 1 = Bb, 2 = A, 3 = C, 4 = F.

What the actual group is, and how the event functions in context will have much to do with the shape of square presentation. Abstractly, I happen to find the above generally pleasing for the opening of a piece. Other parameters like dynamics or instrumental colors can be presented using such squares. Four of such things is a good number: enough to create interest and about right to produce a complex order that is actually audible and perceivable. The number 4 also happens to be a divisor of 12.

The number 4 turns out to be a good number for such operations from a mathematical point of view as well.

   For six objects, perhaps construct squares using the 3x3 squares, e.g.,

	(1 2 3)(4 5 6)
	(2 3 1)(5 6 4)
	(3 1 2)(6 4 5)
        (6 5 4)(3 2 1)
        (5 4 6)(2 1 3)
        (4 6 5)(1 3 2)

   A 6x6 square with double diagonal property cannot be constructed, but

        (1 2 3 4 5 6)
        (4 5 6 1 2 3)
        (5 4 1 6 3 2)
        (6 3 2 5 4 1)
        (3 6 5 2 1 4)
        (2 1 4 3 6 5)

   distributes the six tags with exactly one in each row and each column.
   Although not a square, the following 4x6 array works well for distributing
   a hexachord over four voices or parameters.

        (1 2 3 4 5 6)
        (4 5 6 1 2 3)
        (5 1 4 3 6 2)
        (3 6 5 2 1 4)

   The Magic square of dimension 16 with the double diagonal property:

	( 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16)
	( 5  6  7  8  1  2  3  4 13 14 15 16  9 10 11 12)
	( 2  1  4  3  6  5  8  7 10  9 12 11 14 13 16 15)
	( 3  4  1  2  7  8  5  6 11 12  9 10 15 16 13 14)
	(13 14 15 16  9 10 11 12  5  6  7  8  1  2  3  4)
	( 9 10 11 12 13 14 15 16  1  2  3  4  5  6  7  8)
	(10  9 12 11 14 13 16 15  2  1  4  3  6  5  8  7)
	(15 16 13 14 11 12  9 10  7  8  5  6  3  4  1  2)
	(16 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1)
	(12 11 10  9 16 15 14 13  4  3  2  1  8  7  6  5)
	( 7  8  5  6  3  4  1  2 15 16 13 14 11 12  9 10)
	(14 13 16 15 10  9 11 10  6  5  8  7  2  1  4  3)
	( 8  7  6  5  4  3  2  1 16 15 14 13 12 11 10  9)
	( 4  3  2  1  8  7  6  5 12 11 10  9 16 15 14 13)
	(11 12  9 10 15 16 13 14  3  4  1  2  7  8  5  6)
	( 6  5  8  7  2  1  4  3 14 13 16 15 10  9 12 11)

   will organize V and H relations minimizing the weak octave relations.
   With regard to pitch distributions, there are, of course only 12
   available, so we have 4 left over tags, that could be filled with
   a pitch duplication or extension, or with silence.  Rarely, however,
   are 16 separate voices and parameters employed.  Even 16 distinct
   tone colors is a bit much.

   Magic Squares - Historical Notes [Link]
    Constructing Magic Squares [Link]

The point of such arrays and groups of transformations is that they are systematic ways of generating the very possibilities that composers actually use in solving the essential problem of maintaining an essential sameness to musical material while providing, at the same time, the variations that make things different. What possibilities are used and how such transformation groups are applied and to what, is part of the composition process. The mathematics is merely a tool to extend and systematize the composer's use of his language of musical patterns. It exists, as a grammar of language exists for the poet. There is nothing intrinsically unartistic about the use of mathematics in the arts and particularly in music anymore than the use of grammar by a poet is unartistic.

The Golden Mean [Link]

Fibonacci Series and Lucas Series:

    The Fibonacci Numbers [LINK]:
   The Fibonacci series is derived from a recursion relation imposed on
   a sequence of numbers F(n), labeled by integers n = 0,1,2,3, ....
   Set F(0) = 1, F(1) = 1 and let

                   F(n+1)  :=  F(n) + F(n-1)

   which is to say, every number in the sequence is the sum of its two
   predecessors.  The infinite Fibonacci sequence then begins:

        1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 79, ...

   If different starting numbers are used, the number sequences are
   called Lucas numbers.

To learn all about Fibonacci, his numbers and their various connections with natural phenomena, the golden mean, see The Fibonacci Numbers and the Golden section [Link]. For some some musical applications see Fibonacci Gamelan Rhythms [Link].

The Geometry of Music:
One of the aspects of music that makes it amenable to and symbiotic with mathematics is that all it's parameters are readily quantifiable; since mathematics is "the science of quantity", it becomes automatically and inextricably linked to mathematical concepts.

Although I don't believe it to have actually been done, it is conceptually possible to consider musical events to be quantifiable patterns that exist in a multidimensional space whose coordinates are the totality of musical parameters. Theoretically, the space could be an infinite dimensional Hilbert space; practically it is a space of many but finite dimensions. This idea may have been part of the thinking of Iannis Xenakis [Xenakis 1963], in his development of Stochastic Music which supplies a principle of order to the notion of free aleatorism. Xenakis [Link] Xenakis' Music [Link]
Algorithmic Composition, a Definition [Link]

In his musical researches, Guerino Mazzola has explicitly approached the concept of musical geometry, in a book, apparently now difficult to find. From the little I know, I have extended his motivic transformations into the the area of Lie algebras, and Lie Groups acting as Motivic Transformations

This has been "A Brief History of Music" as a cyclic, Hegelian evolution:

         ... order -> order+invention -> chaos -> order ...

   More specifically, a linear evolution until the break down of
   the concept of tonal center:

   Linear modes -> Doubling of lines -> Organum -> Free polyphony ->
   Harmonic polyphonies -> Chromatic Harmony -> Atonalism ->

   When the tonal center disappears, the fundamental organizing
   principles of form also degenerate.  The paths foreseen by
   Liszt in his last compositions all materialize, including his
   understanding of the need to create new forms for new organizing
   principles.  The path splits into many paths, from the point of
   atonalism (usually credited to the Tristan chord of Wagner).
   The one completely individualistic path is that of Scriabin.
   It is still music that is an acquired taste.

    -> Scriabin (chords in fourths rather than thirds)
    -> Expressionism (Ravel)
    -> Impressionism (Debussy)
    -> Dodecaphony (pitch serialism) -> Webern -> Total Serialism ->

    -> "Subserialism" Wolpe
       extending and expanding on the concepts of Webern, yet diverging
       from strict serialism by balancing subsets of the twelve tones.

    -> Aleatorism -> Xenakis (Stochasticism)

The evolution is with references to mathematics as a repeatedly applied organizing principle. In the restoration of order from chaos, there is no consensus that leaves this simple diagram as one dimensional. That there are multiple schools of thought that lead in different directions is at no time more apparent than during the twentieth century which saw the toppling of tonality. In this little history, I have left out many composers who were important to the development of modern music; it is not a value judgment, merely a lack of energy.

Having opened with a broad but relatively uninformative definition of music, I should like to close with a more personal, and descriptive explanation of the process whereby that which has been defined is actually created, a quote from Igor Stravinsky and a paraphrase of a quote by Charles Wuorinen, indicating the current state of composition for composers.

"Composing, for me, is putting into an order a certain number of these sounds [i.e., the full well tempered pitch spectrum] according to certain interval-relationships. This activity leads to a search for the center upon which the series of sounds involved in my undertaking should converge. Thus, if a center is given, I shall have to find a combination that converges upon it. If, on the other hand, an as yet unoriented combination has been found, I shall have to determine the center towards which it should lead. The discovery of this center suggests to me the solution of my problem. It is thus that I satisfy my very marked taste for such a kind of musical topography."

                               - Igor Stravinsky  [Stravinsky 1942] , p. 37.

   How can you make a revolution when the last one already said
   that anything goes.
                               - Charles Wuorinen [LINK]


[Circle of Fifths] [Keyboard Octave] [Liturgical Modes] [Greek Modes] [Traditional Harmony] [Dodecaphonic Evolution]
[Mathematical Groups of Transformations] [Music as Language]
[Pitch Sets] [Problems of Post Webernian Composers]
[A Brief History of Musical Concepts]
[Lie Algebras, Groups and Motivic Analysis]
[Musical References] [Overtone Series] [Listening to Music]
[Anton Webern] [Stefan Wolpe] [Raoul Pleskow] [Music Links]


  1. Chaos, Self-Similarity, Musical Phrase and Form
  2. Finite Automata and Morphisms in Assisted Musical Composition
  3. Meaning Generation in Music Listening
  4. The Brain Opera - MIT
  5. Grame - Computer Music Laboratory
  6. Music Theory On Line No. 1

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Created: September 1997
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