Practicality in Dmitri Tymoczko's Graphs Today's music industry is made up, for the most part, by performers, composers and nonmusicians who deal with managerial aspects of what the rest of them require. Music theorists have the joy of discovery what the others seems to overlook, not that it is any consequence of laziness or oblivion but rather that their area of expertise doesn't apply the insistent demand for the music's existence. I draw a thin line between composers and theorists partly because of how they are perceived by the rest of the industry, most obviously by their output, not in the amount but who their output reaches. A composer's finished product can be heard by anyone just as a theorists product can be examined by anyone. Of course, the aural element is much more palatable to a wider audience than understanding a gamut of foreign equations and terminology. "..it is possible that with extensive training ordinary listeners can sensitize themselves to the sequences’ structure.." Dmitri Tymoczko is one of those individuals who has dug deep to discover new formations. Being able to defending what composers have given us is an integral part of composing and performing. Music of the future is influenced by studying music of the past. In Tymoczko's research the details go far to justify his ideas but to often these ideas seem to be exaggerated. I'm in no intellectual position to brush off Tymoczko's findings on grounds that there are too many exceptions or that details don't matter. My hope is that by applying these tactics to some familiar pieces in my standard repertoire I can better understand the context of where his constraints and graphs can improve my understanding of a musical genre, set of works or (at the very least) an individual piece of music. As Tymoczko leads us through a rediscovery of what is already known he first presents a powerful statement whose only purpose is to completely confuse us, then he has the opportunity to slowly rebuilding the concept. In our state of ignorance examples are pulled from many different breeds, such as his reference of a Clementi sonata and some Debussy when detailing his OPTIC system. We are in no position to slow our rebuilding process by questioning the significance of his methods and how they relate to the entire collection of those composers' music. The examples compliment his current argument well but may prove unable to stand up to their claims when more is to be proved. We are taken through a maze of models and instances of the OPTIC application and its importance of simplifying chords we encounter. The pitch-class philosophy is not a complex item to iterate. The lengths he goes to spell out the different ways to interpret distance seems wasteful. Luckily, if in the earlier chapters of his book I have missed the boat captained by Dmitri at least I have his pardon. "..it is possible to understand the gist of later chapters even while remaining somewhat fuzzy about the technical material in Chapters 2–4.." Pitch Space vs. Pitch-Class
Voice Leading is taken to a whole new level by Tymoczko. Again, the process remains the same while our understanding of it is overshadowed by the elaborate attempt at exposing new ideas. Trained musicians can already see these chordal relationships that the OPTIC system represents. Our theory 101 courses conveyed voice leading well enough to trust Bach, Clementi, Haydn and Mozart. As we witness in the following century we can see a slow progression of creativity and risk taking of 19th century composers. We had music that foreshadowed Wagner such as the experimentation by Beethoven in his late career, Weber and Liszt. Tymoczko's voice leading theories are so forgiving in that they allow a melody to justify any harmonic progression. It's hard to conceptualize a major chord being anything more than major. Tymoczko lays out ever relationship between standard chords in terms of semi-tones and then retraces his steps claiming that more is to be seen in between the notes. The pitch-class system ultimately removes one problem we encounter when condensing a chord for the purpose of identifying it; which pitch will be the constant as the other voices are moved to an appropriate octave within a 12 semi-tone space near the constant note. Larger gaps within the newly arranged chord can be quickly eliminated by repositioning notes, one at a time, from one end of the chord to the other until the smallest number of semitones exists. The same can be done to the neighboring chords to see if the voice-leading is effective. "Thus when we say that an object is a major chord, we are neglecting an enormous number of musical details, leaving behind something that is very abstract—an ordered sequence of clockwise distances around the pitch- class circle.” The phrase "chord progression" seems to be equivalent to cursing when Tymoczko uses it. His portrayal is that we don't see the movement in the voices from chord to chord. His pitch-class wheel does present a new visual for following a particular voice but forces us to completely turn off our ears. Musicians using the pitch-class system (that has already been set place by previous theorists, like Howard Hansen) know to use it only for brief isolation of a voices path. Tymoczko's pitch space graph (or line) is less invasive to our sensitivity toward voice motion. Here there is an equally as simply layout with a voice's path easy to distinct from its colleagues. His wording is so bold that you'd think he is defending a new standard for notation. We've always been able to move from chord to chord in multiple ways, is he really the first one to point out the shortest path? No one wishes to use a pitch system that forces us to ignore voice leading and settle for a "tunnel vision" approach to chord analysis. In the case of most other theories there is no need to leave out a reference to pitch on some sort of vertical gauge unless eliminating doubling on the staff proves to be far too cumbersome. Spiral piano But back to the OPTIC system… Tonal music illustrates a state of aural homeostasis; adhere to the guidelines and you'll fit right in. The procedures explained in A Geometry of Music are quite broad. So often Tymoczko walks us through a chord to chord modification showing our current staff notation side-by-side to the graphs of the newly introduced pitch space and pitchclass systems and then halts the discussion before relating any convincing element of science. This book is all about using the math to justify voice-leading paths but there is a significant lack of mathematical equations. The application of a combination of permutation, inversions and the others operations can create endless potential in harmonic progression but the details
are missing that should include specific equations for creating the desired movement. Tymoczko anticipated public connection to his book would be much improved if a companion manual containing these equations existed. It is unclear where the math and science are involved in the OPTIC symmetries. Tymoczko insinuates that voice leading is incorrectly taught in regard to the size of the voiceleading. In a search for “reasonable” voice-leading we naturally consider the distance between all of the voice motion and a consistent path for each voice to follow. Section 2.7 of his book encourages having a field day with voice crossing. More limitation should be enforced on this topic for the sake of following voices as we analyze multiple voices together and for our ear to capture the motion. Where possible, only two voices leaping distances greater than a major third and any voice crossing does not appear but a single instance per stationary chord as to not disrupt an attentive listener (“stationary” is referring to a chord that is blocked or vertical as opposed to arpeggios in the harmonic structure). For instance, Tymoczko could recommend for his readers to reserve voice leaping by more than a third to one distinct voice that, more or less, takes the role as a melody and the other voice being the bass (assuming the style requires the lowest voice to provide the root of the chord for the majority of the time with the exception of inversions). Here Tymoczko has an opportunity to demonstrate efficient voiceleading on his pitch-class “axis of symmetry” and alongside it a staff containing a single clef and the notes of the chord condensed within one octave. It seems that in a constant search for a suitable voice leading the first step should be discovering the absolute smallest path that all of the voices could follow collectively and deviating from it for the voices that require greater leaps. The smallest series of voice leading represents the constant that a science experiment would contain and remains in the shadow of the chord as reference. A topic that refuses to permeate my skull is the issue of dividing the octave evenly. Tymoczko’s tables poorly assist his in delivering his argument. Even and nearly even divisions distract from his ultimate point of why chords sound good. As found on the Princeton music course website, Tymoczko’s 4th handout very basically explains the theory of how the octave is divided evenly: “The mathematical reason for this is slightly complicated. It’s related to the fact that, when we think in terms of fundamental frequencies, the perfect fifth and the major triad divide the octave exactly evenly: the note 330 Hz (E4) divides the octave between 220 Hz (A3) and 440 Hz (A4) into two equal (110-Hz-sized) parts. (Note that E4 is a perfect fifth above A4.) Similarly, the A major triad 330 Hz (E4), 440 Hz (A4), and 550 Hz (Cs5) divides the octave between 330 Hz (E4) and 660 Hz (E5) into three equal (110-Hz-sized pieces). It turns out that when we go from fundamental frequencies to ordinary note labels, we transform perfectly even divisions into nearly even divisions.” It’s good to know he is able to lay it out in a simple manner for his students. Actually, it is comical how in his book chapter three seems to unfold smoothly leaving some readers completely bewildered and feeling ignorant and then warns his freshmen class how intellectual one must be to understand such theories but coddles them with an easy explanation of Hz division. The problems start when this theory is tested on an equal
tempered instrument with twelve pitches and then looking at how the octave is divided on the staff. By comparing wave length and our diatonic scale system we are deceived. In fact, when understood through this process, augmented chords become the nearly even triads and major triads are perfectly even. This makes more sense on paper before applying it to an instrument. In the three dimensional chord theory, this replaces the augmented cubes with major cubes. By dividing the scale like Tymoczko we have not created greater or fewer possibilities for harmonic progression but just changed where the central axis appears. He prefers the method in his book because it applies so fairly to the Brahms Piano Quartet he presents. If my suggestions were explored the outer limits of the three dimensional tiles would be well suited for the baroque and classical styles. If major chords act as the focal point then this three-dimensional space would translate easily into representing movement between scales (more appropriately modes). As we move from the cube that represents the C major scale one note is altered. There are six options closely related to the scale and then three for the one following. If working with baroque or classical styles few scales will be utilized but, as Tymoczko explains, Brahms, Shostakovich and Debussy will use many exotic scales. No matter the way the octave is divided, chords with doubled notes will still be found on the walls of the diagram, strongly suggesting that the piece of music at hand is not tonal. If a composer were applying this system to a new piece of music written in the late romantic style he would find it useful in finding desired voice leading on a chord-to-chord basis or at most the motion over a small phrase. I appreciate math’s presence in music and I use it as often as needed. The circle of fifths is an invaluable tool for training and reading music by referencing the distance from one note to the next rather than resetting the brain for every change. The significance of using small distances for voice leading and harmonic progression is reawakened by the in-depth application of Tymoczko’s symmetries. By bewildering and then convincing the reader that these findings are original encourages more credit than is due. Pitch space and three-dimensional diagrams assist in the rediscovery of what we already have understood. With the help of other theorists such as Julian Hook
not enough math. Excuses it as science. Julian Hook, Roger Scruton Original but exaggerated
Review of Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford University Press, 2011) Julian Hook http://www.mtosmt.org/issues/mto.11.17.3/mto.11.17.3.hook.html Dmitri Tymoczko -Handouts 4 and 20 http://dmitri.mycpanel.princeton.edu/files/pdfs/MUS105handouts.pdf Scruton The Space of Music 2 review of The Geometry of Music http://www.rogerscruton.com/work-in-progress/19-books/69-the-space-of-music.html
