Through second hand information on work by Guerino Mazzola (GM) I understand that GM has done motivic analysis of many pieces which range over many styles using his analytic software RUBATO In his book "The Geometry of Tone", (1986), to my knowledge, still only available in German, and which I have not had the opportunity to read; his general interest is in the geometrical aspects of music. One of the influences of this idea can be seen in Chantal Buteau MS thesis: "Motivic Topologies and Their Meaning in the Motivic Analysis of Music" [Caution: last I looked, this file has a wrong suffix; it is really a PostScript File.]
This is a further extension, since topology is an extension or generalization of geometry, that is based on a concept of "nearness" without the use of a metric.
Ever since the now seemingly simple welding together of algebra and geometry that every high school student learns, mathematics has evolved to expand and illucidate this connection, creating areas of "Algebraic Geometry", "Algebraic Topology", "K-Theory" and "Geometric Algebra". This and more can be explored at Dave Rusin's Mathematical Atlas: A Gateway to Mathematics.
Felix Klein understood a geometry as a space where a set of "geometric invariants" are preserved under the action of a group of transformations that act on the space. Ultimately, the essence of the geometry is absorbed into the group itself. I've written a very small essay on this which is easy for mathematicians, but perhaps not so easy for musicians.
Conceptually and mathematically, musical compositions are analyzed by GM in terms of motivic transformations. Once the idea of transformations is introduced, the notion of group [another small introductory essay] follows in its footsteps, these groups being intimately connected with any musical geometry.
Basically, as I understand it, GM. conceives of motivic transformations or even sectional or compositional transformations as fundamentally a transformation of the score paper, conceived perhaps as a long continuous scroll. He lays out on the score paper, arbitrarily convenient coordinate axes that musicians commonly call vertical and horizontal. The atomic units that he considers for transformations are coordinates of "motivic events" given generically at (p_k t_k), where p_k is the kth pitch and t_k is the "time" at which p_k starts. GM introduces linear transformations on such an event, or more precisely, on all the events of the motiv. The subscript 'k' distinguishes the motivic events. It is important to remember that the T_k values are start times, and not durations. Extensions of this idea by allowing extra formal dimensions for duration as well as for dynamic level for each event is certainly possible.
For a motiv, the transformations are applied to each event within the motiv. Larger structures can, of course, be considered.
For example: Apparently to Mazzola, for a motiv for any event in the motive, let the event be represented by a two dim. vector (p t). Then the transformation |-1 0 | | p | |-p | | | | | = | | | 0 1 | | t | | t |represents *an* inversion, when applied to all motivic events. [Emphasis mine] in this context, it must be understood that there is no absolute sense of inversion unless you assume that a motiv is defined by an interval sequence and not pitch sequence, i.e., assumption that "motiv" is invariant under pitch translation. (Of course, there are upper and lower practical limits on the extent of these translations.) Otherwise, to define inversion absolutely you would have to pick a horizontal line about which the reflection takes place; choosing such a line is arbitrary, even though in specific contexts like those of a tonal fugue there will be guidelines that arise from the composer's possible intentions regarding tonality.
Where, on a vertical line, thinking of a traditional piano score, you place the zero will determine a translation or vertical position for a representative of the pitch translated equivalence class that constitutes the abstract intervallic motiv.
It is important to note first that the concept of translational invariance only makes sense in an even tempered pitch evironment. That the octave is divided into 12 parts in traditional Western music is not a necessity, and you will find a very interesting discussion on this in terms of continued fractions Pianos and Continued Fractions
In presentation, the further two motivic translations are apart as pitch translations, the less similar "people will say" they are. So, apparently there are cicumstances for accepting and for rejecting the concept of motiv as intervallic and insisting that motiv be defined through pitches. I would truly only believe exclusively in definition by pitch if everyone had perfect pitch. I do not.
It seems to be true that anybody who is not insensitive to music and
has some interaction with it has a "pitch memory": if a piece of music
is known and heard repeatedly in it's written key (tonal music),
on hearing it in a wrong key they would know it was wrong.
If the first bar of Elektra were played in other than D minor I would
know it since the opening 3 notes are F D A, outlining a D minor triad.
I do not have perfect pitch: if someone where to say, sing a 440 A,
concert pitch, there isn't a chance, that I would get it right.
Perfect pitch is a "gift" that most of us do not have spontaneously,
although I think a reasonable facsimile can be trained.
People whom I have known who have perfect pitch tell me from their
experience that it is a kind of memory.
Mazzola, seems to accept and use the intervallically based concept of motiv, while still allowing that, *contextually* - since playing any motive involves choosing some translate from the equivalence class - the idea of transposition is "a motivic transformation".
Returning to the event vector, its second component "t", I see as being a time value on a horizontal line. Where you put the zero doesn't matter much if the entirety being transformed is under consideration, since where the zero is determines a time translation of the motiv, which means nothing if the motiv is all there is. There is geometrically a difference between an object's location and the coordinate system that can be used to locate it. Location is absolute, while the coordinate system is completely arbitrary. Put another way, a Euclidean plane and its mathematical concept existed millenia before Decartes layed his now famous coordinate system on it (Historically, he did not do this anyhow); other coordinate systems, such as polar can also be used and these have nothing to do with the Euclidean nature of the plane.
Mazzola again goes on to say that | 1 0 | | p | | p | | | | | = | | | 0 -1 | | t | |-t |gives the transformation of retrograde, which is of course also translated in time. "t" is a value on a time axis, and this is indeed the retrograde transformation, using a concept of time translation. The crab, that should be the result of multiplying these two commuting matrices of inversion and retrogrde is correct:
|-1 0 | | p | |-p | | | | | = | | | 0 -1 | | t | |-t | These transformations together with the 2x2 identity give a representation of the Klein 4-group, exactly as they should. E = Identity, I = Inversion, R = Retrograde, C = Crab and the abstract group multiplication table is then. | E I R C ---------------- E | E I R C I | I E C R R | R C E I C | C R I E Next, Mazzola proposes a transformation | 0 1 | | p | | t | | | | | = | | | 1 0 | | t | | p |
which geometrically exchanges the pitch displacement with time displacement. Though this might seem a litle strange, GM is talking about systematics of motivic transformations - and why not this one? It seems, without great thought on my part, to alter the pitch (hence abstract intervallic) order; in a three note capital Lambda shaped motive, the contour is basically left unchanged or inverted, while new pitches can be generated. That works for me as being legitimate, and recognizable motivic transformation. Notice that the t-p inversion can be acomplished in steps using the one parameter Lie group that it generates, so that done in small increments, the transformation sequence can be perceived (or learned to be perceived). Here again the concept of nearness of musical material and its transformation arises. The concept of "nearness" might be called the soul of the area of mathematics called topology, a soul that goes beyond metric geometry that can be done with rulers.
The retrograde transformation multiplied by the t-p reverting transformation gives | 0 -1 || 0 1 | | p | | t | | || | | | = | | | 1 0 || 1 0 | | t | |-p | | 0 -1 | | p | |-t | | | | | = | | | 1 0 | | t | | p | This transformation happens to be the "generator of rotations" in the p-t plane about the arbitrarily chosen origin, the rotations, of course being | 0 -1 | | cos theta - sin theta | exp ( theta | | ) = | | | 1 0 | | sin theta cos theta | 0 <= theta < 2 π a rotational transformation of magnitude theta given in radians,
which are the elements of the Lie group SO(2), adding, set theoretically, to the p-t inversion operator, to the rotation generator gives the Lie algebra of O(2). p-t reversion cannot be accomplished by a rotation in the p-t plane; it would require a rotation out of the p-t plane that leaves the line p=t invariant.
On the matter of Lie algebras and Lie groups, perhaps the gentlest introduction would be found at Dave Rusin's Mathematical Atlas: A Gateway to Mathematics. Another possibility is a restricted and very condensed [Appendix B] which I wrote primarily for an audience of theoretical physicists.
Mazzola introduces the "arpeggio" transformation | 0 1 | | p | | p | | | | | = | | | 1 1 | | t | | t+p |
One might be disturbed by the apparent addition of quantities of different physical dimensions and actually want to use Hz as the physical units of pitch - wrong! - frequency is logrithmically related to pitch and is not sensibly added. [To be more precise, perceived pitch while logorithmically dependent on Hz, is also dependent on loudness and the frequency spectrum of any real sound. Perceived pitch also has a rolloff at frequencies 20kHz-25kHz, and and 10Hz-20Hz. In the lower rolloff, sound becomes a tactilly felt vibration, rather than an auditory experience.] Pitch has no physical dimension, but is sensibly additive. That "t" might actually have units of time is irrelevent. This is just basically a matter of abstract visual score geometry; units of time only come into play when a tempo is set.
So what does the transformation do? Each pitch is translated by its time position. Here, where the t=0 is placed, does matter and results depending on the position of t=0, are various vertical stretching apart of the motivic intervals.
Adding these operators extends the Lie algebra to the algebra gl(2, Z) associated with the complete general linear group in two dimensions, GL(2, Z), where Z can be any field that will constitute the "scalar field" of the algebra. Realistically, Z is actually some finite Galois field that expresses the time and pitch quantizations. Idealistically Z could also be the field of real numbers; I say idealistically since this is music and there are limits to the human abilities to distinguish both pitches and times. Moreover, the parameters of of pitch and time are in practice discretized in considerations of music.
Possibly the most condensed presentation of abstract algebra in existence can be found in [Appendix J]. The purpose and path chosen there is to get to Galois fields as quickly as possible, and then use them for a specific pupose. Once again, the intended audience is theoretical physicists, for whom it may also cause a minor headache.
To the local motivic transformations expressed by gl(2, Z), or its Lie group GL(2, Z), we wish to add those transformations which simply shift a motiv in p-t space; these are usually called translations. General motivic "translations" cannot be represented by the defining matricies of gl(2, Z). These 2x2 matricies are all matricies of rotations, reflections, and stretchings. However, the inhomogeneous group that includes translations can be represented by 3x3 matrices of the form:
| 1 0 p0 | | p | | p+p0 | | | | | | | | 0 1 t0 | | t | = | t+t0 | | | | | | | | 0 0 1 | | 1 | | 1 |
Here, I have given the general form for pure translations. The 2x2 unit matrix in the upper right can be replaced with any of Mazzola's transformations, and in fact, any element of GL(2, Z). Matricies of this form must act from the left in order to produce translations in "score space" (the same as p-t space). They also multiply correctly by *left* multiplication to represent The inhomogeneous Lie group
IGL(2, Z)and implicitly, we have also the Lie algebra igl(2, Z) for this group.
The general transformation is then
| M V0 | | V | | M V + V0 | | | | | = | | | 0 1 | | 1 | | 1 | where M is any element of the general linear group GL(2, Z), V is an event in p-t space, and V0 is a translation vector in p-t space.
There is also a similar representation of the inhomogeneous group that acts by right multiplication. The general form for pure translations is then:
| 1 0 0 | | | | 0 1 0 | | | | p0 t0 1 |
So, actually following things out while sitting here, the group/algebra of transformations of motivic transformations is larger than I had originally thought.
The group IGL(2, Z) allows for the *most general* transformations in score space that includes rotations, stretchings in any direction, as well as translations. These are topological transformations (homeomorphisms) of the plane and will not preserve any specific metric or metrically derived topology. Where the fields of rationals or reals are unbounded, Galois fields close on themselves automatically then being possessed of a toroidal topology, which will be inherited by the Lie algbera as a vector space over the field.
let's consider this a little further, and just temporarily avoid some complications of Galois field stucture by allowing the field Z is actually the Real field R.
So far, the two motivic parameters considered are pitch, and start time. It would appear that there is no more room in IGL(2, R) to accomodate other parameters such as duration, loudness or perhaps "color" in the sense of the overtone spectrum that musical sounds possess. Realistically, I am ignoring intial transients and attack envelopes.
let's talk about color first, since it is by far the most complicated.
In principle, the number of overtones is infinite as a Fourier decomposition of any repeating wave form, but in practice, the actual number of overtones is finite and can suffer some absolute cutoff that is an upper bound on this finiteness. The number is actually fairly small, and one could take say 25 as a computational upper bound on the number of overtones. A genuine sense of "musical color" seems not be simply representable by a single additive paramater, but instead must be defined as a point in a 25 dimensional space, where the coordinates are the values of the Fourier coefficients of each overtone and each overtone labels an axis in the 25 dimensional space. That color should mix with the other paramaters does not seem to make any sense, yet one could easily consider separate transformations within color space in the same sense as the internal symmetries of particle physics that seem to be so divorced from spacetime.
At the same time one can consider a given finite set of colors permuted over the events in the p-t place. This also brings into play the theory of finite groups since it is a fundamental theorem of finite group theory that any finite group can be represented as permutations (transformations of) on a string of distinguished symbols. [That's not the theorem, but a fairly accurate description of it.]
Now let's return to the duration and dynamic parameters. Duration is clearly additive and one has the instinctive desire to connect this with the start time. For loudness, in computation, and taking into account the law of Weber that the perception is logorithmically related to the stimulus, an appropeiate parameter is decibels, which to the human ear is an additive parameter. The perception of pitch is physiologically connected to that of loudness, and so it makes not unreasonable theoretical sense to attach loudness to pitch.
Now for a bit of mathematical prestidigitation:
Extend the field of reals to the field of complex numbers, that can symbolically be represented by
a + b i where a and b are real numbers and i := sqrt( -1 ). Electrical engineers sometimes use 'j' instead of 'i', but in the literature of mathematics and physics, 'i' is standard notation. Algebraically then, the square i^2 = -1.Some day, maybe, I'll make a little webpage that shows the development of the "number" concept from the viewpoint of intellectual greediness. Right now, I'll just assume that this is known, or intuitively reasonable and obvious to the discerning reader.
Now, consider the complex pairs:
start-time + duration i and pitch + i loudness
The most general topological transformation (homeomorphisms) of the 4 real dimensional (2 complex dimensional) space is now simply
IGL(2, C)In words, the inhomogeneous (= translations included) general linear group (= determinant of the 2x2 matrices are not zero) acting on a 2 complex dimensional space. This is a group with 12 parameters (a Lie algebra of 12 real dimensions as a vector space).
A connection with the "internal color space" actually does exist; it is not really so internal as I suggested previously since, once again, the human perception of pitch depends also on color; in a sense, pitch is a projection from color space modified by loudness. To my knowledge, exactly what this projection is, most generally, is still unknown.
Without this intrinsic connection between pitch, loudness and color, ignoring this complexity (constraint), the group of topological transformations of C^2 becomes something like
IGL(2, C) X IGL(25, C)
The field C for the color space is not a mistake, and allows for phase differences between the overtones - trust me. :-)
This little essay on the Lie group theoretic transformations of musical motivs leaves more questions that answers, but I think some of the questions that it provokes may be more interesting than any analytical answers given.
Again, since I have not read Guerino Mazzola's works on this subject, I don't know whether he has gone as far as I have here in unifying motivic transformations into the Lie algebra/group structure, that acts locally on arbitrary motivic elements as geometric objects in t-p space, and further on the extended complex spaces.
Can human beings hear, or be taught to hear these general topological transformations of motiv? I don't know. Perhaps someone would like to find out. In doing so, please be wary of your experiments (Do they actually measure what you think the meaure?) as I am with this purely mathematical construction
What is the meaning of the Lie agebraic structure, particularly the anticommutativity of any Lie algebra?
Music has, since the time of Pythagoras been intimately associated with "Number"; number since Descartes has be connected with geometry, geometry then with groups. Those connections with number are now finding their way into understanding the complexities of music in new ways, much of which has been suggested by the musical library of the past. I know that some musical scores have been dismissed by the pejorative "Augenmusik", because of the seeming concern expressed in the score for for its geometry and appearance to the eye (implied is a lack of "musical" concerns). Personally, I understand this pejorative as a distinct lack of understanding and a lack being able to "see" music as well a s hear it. Geometry, as number, *is* at the base of the structure of music.
If one wants to "understand" musical structure, the listener is in a sense irrelevent. The mind and understanding of the work should come from the composer; yet, rarely will the composer say much *about* the work, since to him, it speaks precisely and exactly for itself.
The composer, by composing, is stretching our auditory sensibilities as he may be even stretching his own. I think it was Karlheinz Stockhausen who said that there are times in his composing that he would write what was right, what made the most sense, even if it offended his ears.
No good composer that I have had the privilege of knowing, has any idea of what any unknown listener will hear in their works - nor do they care - nor should they. They wrote what they needed to write.
Music is written for the same reason that mathematics is done; there is an internal need within the composer that demands expression; how what is expressed is perceived is of no concern to the creators of either. If there is concern, you've just met a hack composer or mathematician, more concerned with their alleged careers that with their art.
Contrary to a common idea, ART does not come the perceiver - nor should it; the perceiver comes to it - and that is exactly what is there for: it is an opportunity to expand the faculties of the perceiver. Failure to come, on the part of perceiver, is only to the loss of the perceiver, not to the artist nor to any valuation of the work, which does ultimately speak for itself.
My thanks to Dr. Ludger Hofmann-Engl and to Dr. Michael Leyton, whose stimulating discussion lead to these thoughts on Lie Alebras and Groups in musical transformations.
Email me, Bill Hammel at