On the concept of transformation principles in music
We are about to embark on a conceptual journey through Johann Sebastian Bach’s creative mind. The Musical Offering, and the even greater Art of the Fugue, were consciously written as musical pedagogical workshops, to educate Bach’s students, and generations of future musicians, in the art of polyphonic (many-voiced) contrapuntal composition.
In the Musical Offering, Bach uses King Frederick’s "Royal theme," to create a series of ten canons. But the fun part is that they are puzzle canons -- he does not write out all of the voices. You have to solve the musical puzzles before you can perform them! 
Bach challenges you to discover one or more principles of musical change, with the aid of hints in the titles, and musical symbols. When you apply these principles of change to the canonical voice(s) provided, you are able to create one or more new canonical voices. The solution to the musical enigma is found when you add the newly created voice(s) to the given voice(s).
It is not the theme itself, but what you can do with it, how you are able to transform it, which becomes the subject of musical thought. In order to solve Bach’s musical puzzles, the student is forced to concentrate on discovering the transformation principles Bach intended to be used, and can not get lost in just performing the notes. These discovered principles are the same ones that Bach, and other great composers, used to compose their fugues, and other musical forms using polyphonic counterpoint.
Unity and diversity
Bach's creative use of these principles, in conjunction with his championing of the revolutionary new well-tempered system, which increased his freedom to explore the possibilities of musical development throughout the entire well-tempered musical domain, allowed Bach, more than any composer before him, to create compositions which presented intensive studies of a single musical idea.
--As Lyndon LaRouche has expressed it recently, in between bookends consisting of the moment of silence before the music begins, to summon the musicians' and listeners' concentration, to the moment of reflective silence after the piece is completed, a reenactment of the development of the composer's single musical idea (one idea developed through creative multiplicity) should grab the listener right from the sounding of the first note, and hold him at the edge of his seat, by building up an unceasing musical tension, until the idea is completed.
And there is something indefinable, in experiencing such a powerful performance, which is capable of expressing a quality the great conductor Wilhelm Furtwängler termed "in between the notes," that moves the listeners, sometimes to tears. And that is the real purpose of music.
Look, thus, at the transformation principles we are about to encounter, as tools to create such a masterpiece.
The musical complex domain
Discovering, and using the musical transformation principles contained in Bach's puzzle canons to perform beautiful music, enables you to experience a musical equivalent of the principles of complex numbers in science. The LaRouche Youth Movement has worked hard to try to understand this concept, developed by the German mathematician C. F. Gauss (1777-1855). Therefore, it can be helpful to see the similarity between Gauss' scientific concepts and Bach's artistic concepts.
1. As in the mathematical complex domain, there is a conflict between the “visible” universe, experienced by the senses of sight, hearing, etc., and the invisible principles lying behind the sensual universe (Also think about Plato’s “Allegory of the Cave” found in Book VII of The Republic).
In Bach’s musical universe, the sensual realm is represented by the given, “visible” music which Bach supplies. It is up to you to discover the hidden “invisible” principles of musical development, the actual subject of each canon, which Bach has chosen, even before composing the given, sensual music. These are the principles that lie behind the notes, which cause musical change. In classical music, it is not the fixed theme or melody, but such musical change, transformation, and development, which is primary.
2. In the mathematical complex domain, more than one physical principle is operative at the same time. For example, the act of “squaring” involves two, simultaneous transformation principles – extension and rotation. (To square a given line, the length is squared, and the angle is doubled. (See picture))
Similarly, in Bach’s musical universe, as you will discover, several principles of change can be applied at the same time, creating a complex musical domain.
You can also think of each principle as a physical “dimension.” Take one of Bach’s puzzle canons which employ two principles or “dimensions” you are familiar with from a previous canon. In the new canon, however, a third, new, principle is demanded – a new “dimension,” which, when applied, totally changes the “curvature” of the “two-dimensional” musical “space.” 
In the musical workshop that follows, you are encouraged to physically perform these same single, or multiple transformations with geometric figures, to better visualize the different types of musical change, and how they can be combined to create complex musical development.
As a special feature, 3D animations of these geometrical movements are provided, which will help you to explore the "harmony of the spheres" – transformation in a "spherical" musical universe.
Bach's Art of the Fugue
As stated above, Bach used the principles of musical change you are about to discover, to compose his greatest contrapuntal works. He chose themes for his fugues, for example in his Well-Tempered Klaver, based on their ability to undergo the type of musical change he wished to explore.
One of Johann Sebastian’s sons, Carl Philip Emmanuel, gave us a first-hand insight into the workings of his father’s mind – his ability to immediately see the pregnant developmental possibilities of a given musical theme, in a in a letter to Bach’s first biographer J.N. Forkel.
“When he listened to a rich and many-voiced fugue, he could soon say, after the first entries of the subjects, what contrapuntal devices it would be possible to apply, and on such occasions, when I was standing next to him, and he had voiced his surmises to me, he would joyfully nudge me when his expectations were fulfilled.” 
The Socratic Method in Music
The use of the musical principles you are about to discover, enable a composer to take his or her original musical idea, and put it into dialogue with itself. A process of musical Socratic self-reflection about the original idea, "the same," ensues, where new variations of the original musical idea are thought up, "the other," and held up to the first idea, which, then, itself, suddenly becomes becomes something "other." This experimental process leads to musical development, especially when the act of counterposing the original idea, with a variation of the idea, leads to paradoxical relationships, called dissonances, which beg to be resolved. This, then, becomes a Socratic method of production of music.
Bach’s fugues are a mirror of the playful creativity of his musical imagination. It is that creativity which you are able to joyfully discover, by taking up the challenge to solve the puzzle canons of the Musical Offering.
Note: The Musical Offering and LaRouche's Concept of "Modes"
Another line of musical investigation regarding Bach's Musical Offering, which we will not focus on here, but is important to keep in mind for further study, is the question of Bach's use and development of what Lyndon LaRouche refers to as "modes." LaRouche's concept is both more subsuming, and ambiguous, than usual use of the term “mode,” referring to the ancient Greek, or Middle Age church modes .
This author currently understands LaRouche’s expanded concept of mode, to be the musical “one” or unity of effect, that is created by the dynamical interaction between the “many” different musical “dimensions”  which the composer chooses to bring to life, each one of which having been given a special character by the composer, in order to solve the particular musical problems which her or she has chosen to be the subject of that particular composition. These challenges arise through the construction of dynamical, not mechanical, musical motion and development.
To use an analogy from the visual world of painting, the composer uses his or her creativity to form a special characteristic, or color, for each “dimension”, and a special blend consisting of the dynamical interaction between the different dimensions, which form a kind of nested quality of uniqueness – unique to that particular piece, family of similarly composed pieces, the entirety of that composer’s works, and the particular geographic area and time period in which the composer’s lived. This paradoxically creates a unique musical fingerprint, or, in this case, a “hear-print,” which can be recognized, as one recognizes who is likely to have painted a painting one has never seen before, even if the composer is experimenting with hitherto unexplored territory. It is just that particular quality of exploration, or method of setting up, and solving musical problems, which can sometimes reveal the author of the composition.
These are some of the “dimensions” of Bach’s musical universe:
1. The ambiguity, and sometimes outright conflict, arising from the musical space in which Bach wrote his masterpieces -- Bach’s complex well-tempered universe, consisting of:
a. all of the 12 major and 12 minor keys, and…
b. The old church modes, including the inherent tension of the Lydian mode (which includes the most dissonant “tritonus” – “three tones” interval of three whole steps, for example, C-F#), and…
c. Other special tonal patterns which naturally arose through the unique prosody (the rhythm and intonation of speech patterns) of different languages.
This complex multiply-connected tonal backdrop sets the scene for the development of dynamical harmonic change, not static "Rameauian"  chords, but momentary vertical relationships of simultaneously-heard independent voices -- (as if a snapshot were taken of a group of runners with fast film – a split second cross section of forward motion) through both closely related, and, with the advent of the well-tempered system, more distantly related keys and "not-exactly-keys" (a broader notion leading towards LaRouche's concept of "modes.")
In Bach’s Musical Offering, as mentioned above, the Royal Theme embodies aspects of both C major and c minor. That built-in musical paradox serves as a provocative germ for further development.
As the composition unfolds, any present moment of the piece is multiply-connected to the harmonic relationships of the past, the future, and the entire completed whole:
a. through musical memory to the previous tonal pathways,
b. through a kind of musical premonition, to the future harmonic journeys, including sometimes jarring confrontations between your expectation of which harmonies will come next, the “unheard” – what the composer “planted” in your mind to expect -- with the surprising, actual, next “heard” tones,
c. during repeats, through a paradoxical musical memory of both the past and the future)
d. and, ultimately, the “mode” of the entire, completed work.
2. The complex interaction between the different human or instrumental voices. Bach was the master of the art of creating independent, coherent, dynamical, beautiful individual voices with intense forward musical momentum, (in this case, with the added constraint that two or more of those voices, comprised either of the Royal theme or variations thereof, or a counterpoint voice, have to be in canonical relationship with each other),
The dynamical interaction between these "independent" voices, referred to as "cross-voices," which occur both consecutively, where a theme travels from one voice to another, across the voices, which the listener hears as a continuous melodic line, or;
where two or more voices are heard simultaneously through contrapuntal (point against point) interaction, forming various consonant and dissonant intervals,
In the very process of solving the problems that are thus created, Bach creates a "field" of action, where notes found in the key the piece is written in, and the keys that the piece modulates (travels from one key to another) through, but also notes not found in any of these keys, will appear, not arbitrarily, but as footprints that are impressed as Bach leads each voice from one place to the next, sometimes by an entirely new path created by solving a challenging musical paradox or problem. These musical footprints, both those found along the usual path, and those found off the path, as if left by the need to travel along a musical detour, become the evidence of a "field" of musical activity, which can not be confined within a corset comprised of fixed "keys," but, as stated above, requires a more subsuming and ambiguous notion of "mode."
3. The Pythagorean comma:
The paradoxes arising from the ambiguity of the sizes of the various intervals within the well-tempered system. This discrepancy can be heard, for example, if you sing an octave, as opposed to a series of three major thirds. You will land at a slightly different place!
This means that, especially when singing a capella (without instrumental accompaniment), or in stringed instrument ensembles, where the notes sung/played are not fixed, the musicians will slightly adjust their pitches, depending on the pitches of the other musicians, or depending on where one’s own voice line is coming from, and going toward.
For example, there will be a slight change in the pitches sung by the sopranos, if the soprano section is singing alone, together with the altos, together with the altos and tenors, or together with all four voices, depending on the tones of the others. You can think of this like a hypothetical lonely planet orbiting the sun, as opposed to it being accompanied by another big planet pulling it in its direction.
This is the challenge that Lyndon LaRouche has recently put before the LaRouche Youth Movement, to enhance their ability to perform Bach’s great motet, Jesu, meine Freude.
The various types of musical transformations we will explore in the following pedagogical workshop, then, serve as musical “agents provocateur,” or “set ups” to create just such challenging musical problems, which bring the composer, the musician, and the listener, into the uncharted territory called "modes."
This, then, is a subject for future musical pedagogical workshops, but it would be useful for you to already begin to think about this while proceeding on our current journey.
Lyndon LaRouche's comments about counterpoint to a group of young people in Seattle on May 5, 2007
To Part 3 -- Bach's Puzzle Canon Workshop
 Bach had previously written a series of puzzle canons on the theme of his Goldberg variations.
 Gauss proved the existence of the real physical, geometric principles which lie behind the misnamed “imaginary” or “impossible” numbers, previously invented to solve a certain type of equation (for example: x² + 1 = 0, x² = -1, and thus x = √-1 , which became known as x = i , because of the seemingly “impossible” existence of the square root of a negative number).
 "For, Gauss, no magnitude could be admitted, unless its principle of generation was demonstrated. For magnitudes associated with the square roots of negative numbers, that principle was the complex physical action of rotation combined with extension. Magnitudes generated by this complex action, Gauss called "complex numbers" in which each complex number denoted a quantity of combined rotational and extended action.
"The unit of action in Gauss' complex domain is a circle, which is one rotation with an extension of unit length. The number 1 signifies one complete rotation, -1 one half a rotation, √-1 one fourth a rotation, and - √ -1 three fourths a rotation. ( )
"These "shadows of shadows", as he called them, were only a visible reflection of a still higher type of action, that was independent of all visible concepts of space. These higher forms of action, although invisible, could nevertheless be brought into view as a projection onto a surface."
Bruce Director, Carl Gauss's Fundamental Theorem of Algebra, Part II, Bringing the Invisible to the Surface, May, 2002, www.schillerinstitute.org/educ/pedagogy/gauss_fund_part2.html.
 It can also be thought of as concept similar to that of the German mathematician Riemann -- a “Riemannian” space where the number of dimensions corresponds to the number of transformation principles. The addition of yet another transformation principle, totally transforms the existing musical space.
 The Bach reader, p. 277. C.P.E. Bach’s letters, in a complete English translation, give the reader an important understanding of Bach’s compositional principles, teaching method, as well as details about his life. They were the chief source for Forkel’s biography (dedicated to Baron Gottfried van Swieten, the man who introduced Bach’s music to the 25-year-old Mozart).
 In the Middle Ages, composers used a set of "modes" called Doric, Phrygian, Lydian, Myxolydian, Ionic (similar to the modern "major"), and Æolic (similar to the modern "minor"). All of the above also had Hypo- variations (except for the Ionic and Æolic) which contained the same notes, but started at a different place (for example, Hypo Dorian).
These names came from the set of "modes" used in Ancient Greece, which were the same as the church modes, but the names were shifted around (Was the tuning the same?). It is said that they were named for the region where that particular mode was used. Whether that was actually the case, is another story.
 Here, I use “dimension” in a broader sense than above. Earlier in this section, I have used “dimension” in connection with different types of transformation in time and space. Here, it also encompasses the different kinds of expressions noted in that which follows.
 Jean-Phillipe Rameau (1683-1764) French composer and stultified music theorist.