Ear Training for the Tone Deaf

I suppose I could be called tone deaf, if by tone deafness you mean an inability to reproduce notes accurately after I have heard them sung or played.

This is one of two definitions of tone deafness. The other is being unable to correctly hear relative differences between notes in the first place. I think I hear the notes in music and the intervals between them just fine, since I enjoy music without reservation. But when it comes to reproducing those notes and intervals, it's hit or miss for me — mostly miss.

The latter malady "is most often caused by lack of musical training or education and not actual tone deafness," says Wikipedia. Again, check. I've had very little musical training.

One way to remedy that is through "ear training," a process of exposing the ear over and over again to various pitches, intervals, and other musical entities and asking the student to identify what he hears. I have found a web site that does that quite well, and for free. It is www.good-ear.com.

The meat-and-potatoes of it is the Ear Trainer. In addition to the beginner lessons I am currently using, it offers further lessons in intervals, chords, scales, cadences, jazz chords, note location, and perfect pitch.

The beginner lessons in intervals are simple in concept. The Ear Trainer plays a succession of two notes, and you have to decide whether they constitute a prime (i.e., a unison), a major third, a perfect fifth, or an octave. Sound easy? Try it. You may find it's a lot harder to achieve perfection — 100 right out of 100 — than you thought.

It gets even harder when you avail yourself of the option not to always use the same first note, but to change the pitch of the initial note at random. You do this by removing the check mark next to "fixed root." You can also change from the sound of an ordinary piano to that of a guitar or violin, an electric piano, or a Rhodes piano, whatever that is. There are three selectable volume levels and three available tempos.

I cheated. I went to Ricci Adams's MusicTheory.net site and popped up the ersatz keyboard window, located here. It allows you to click on any of the piano keys to hear its sound, and also to put red dots on the keys you are most interested in. Using Ear Trainer's default "fixed root" mode, I played various keys until I identified the root note — it changes each time you initialize the Ear Trainer — and then I put red dots there and on the associated major third, perfect fifth, and octave notes. When Ear Trainer played an interval that I was uncertain about, I experimented to find which red-dot key played the same second note. That was a big help.

Even so, my best score so far is 90 out of 100. It will take a lot more ear training to cure my tone deafness, I fear.


Addendum: Since I wrote the above, I have practiced some more. I managed to get a score of 99 out of 100 on the simple intervals trainer using a fixed root note, so I advanced to varying the root note. That's an order of magnitude harder for a tone-deaf person such as myself. Plus, I found that the two octaves of Ricci Adams's virtual piano were not enough to contain all the notes in the training examples, so I've moved on to Wai-man Wong's Java Piano at PianoWorld.com. It spans three full octaves. The same virtual piano keyboard is also available here.

Bach's Well-Tempered Clavier

Johann Sebastian Bach is one of the first composers you probably think of as being "classical." Bach (1685-1750) was not actually a classical composer, but Baroque. The Baroque period, with composers like Bach, preceded the classical period, that of composers like Mozart. Bach wrote polyphonic musical pieces, such as fugues, that use multiple melodic lines in counterpoint, rather than using chords per se in harmonic progressions.

Bach was among the first composers to exploit the system of tuning called equal temperament, in which the pitches of every pair of adjacent keys on a piano or organ (for example, C and C-sharp, or E♭ and E) are equidistant. Equal pitch distances allow for greater complexity of musical development — transpositions and modulations of key, satisfying harmonies, etc. — than earlier tunings did.

Earlier tunings were more concerned with making sure pitch ratios did not involve irrational numbers, which were considered inherently imperfect. Those tunings however — such as the one described by the ancient Greek philosopher Pythagoras — held back composers who longed to explore greater musical complexity.

The adjective that applies to equal temperament is "well-tempered." Bach wrote a set of keyboard pieces, The Well-Tempered Clavier, to show off the advantages of equal temperament. A "clavier" or "clavichord" was any generic keyboard instrument of Bach's time.

This site gives a wonderful introduction to The Well-Tempered Clavier and to J. S. Bach's music in general. I have just begun exploring it, but one thing I can recommend to you right away is that you position your mouse over the "Book II 2. c minor" entry in the scrolling list, then click on it. A little panel appears as the first notes of a fugue are heard. Move the mouse pointer to it and click on "Play movie!" That brings up a Macromedia Shockwave (actually, Macromedia Director) presentation concerning Bach's C-Minor fugue.

In the presentation, at the top you see the measures of the musical score roll by as the corresponding music plays. In the bottom right corner there's a timeline view of the four voices in the fugue, measure for measure. You can click to pause the music, then click again in the timeline to restart it at the clicked measure.

At the lower left is a scrollable narrative description of the fugue. In it are underlined hotlinks that, clicked on, take you to various points in the music. Some of them play the music from that point on to the end of the piece. Others just play a snippet of the music and then stop. These hotlinks help the author of the presentation make numerous key points about Bach's composition.

The timeline is interesting because it highlights crucial aspects of Bach's musical structure, such as how the theme begun in the first measure is restated by a second voice in the following measure, and then by two more voices in subsequent measures (mm. 4 and 7). Then there are reiterations and transformations of this subject or theme in (count 'em) 21 later measures, highlighted in blue on the timeline.

The theme, which lasts but a single measure, is offset by an eighth rest such that it starts after the initiation of any particular measure and laps over into the next measure. You can hear just the basic theme (the version starting in measure 1) by clicking on "subject" in the narrative.

The presentation is the best I have ever seen of a musical piece — I would love to know how to put one like it together myself — and I can hardly wait to examine the other Bach pieces having similar presentations.

Chord Geometry, Part IV

This is the fourth in a series of posts discussing the mathematical model of musical chords proposed by Dmitri Tymoczko of Princeton in his paper "The Geometry of Musical Chords," published in Science magazine. The first in the series was The Geometry of Chords. The most recent was Chord Geometry, Part III.

In that most recent installment, I went into the concept of T-symmetrical chords in some depth. T-symmetry (short for transpositional symmetry) means that a chord precisely divides the octave into equal parts. That is, the intervals between the notes of the chord, including that from the top note of the chord up to the root note one octave higher, are equal in size.

Chords that are nearly, but not precisely, T-symmetrical are much used by composers. They tend to manifest pleasing harmonic consonances, while they "can be linked to at least some of their transpositions by efficient bijective voice leadings." The latter characteristic, all-important musically, means that the separate "voices" that produce the various notes of successive chords, as those voices move independently of the other chordal voices from one chord to the next, can do so by covering relatively short (i.e., efficient) pitch distances. At bottom, that is what can give any piece of Western music its coherence.

In Chord Geometry, Part II, I explored another kind of symmetry, P- (for permutational) symmetry. Perfectly P-symmetrical chords have all their notes in the same "pitch class." For instance, a triad made of three C notes — however odd such a chord may seem — is exactly P-symmetrical. {B, C, D♭} is, on the other hand, only nearly P-symmetrical, making it and like clusters of closely spaced notes good candidates for compositions that exploit atonality and harmonic dissonance. Again, voice leadings among such chords tend to be efficient and thus musically coherent.


Tymoczko describes one more type of chordal symmetry: I-symmetry, for inversion symmetry. "A chord is inversionally symmetrical (I-symmetrical) if it is invariant under reflection in pitch-class space," he says.

To the right is a movie showing how the C major triad can be inverted to become its reflection in pitch-class space, F minor. C major is, in terms of numerical pitch classes, {0, 4, 7}. F minor is {0, 5, 8}. In the movie, the notes of the C major triad are represented as if they were "hours" on a clock face, except that there is a 0 at the top of the clock instead of a 12, and hour-numerals that are not part of the chord are not shown. When C major is inverted to F minor, the clock face is in effect flipped over by means of pivoting it around the 0-6 axis. Now the hours that are shown change from 0, 4, and 7 to 0, 5, and 8.

C major and F minor are not exactly I-symmetrical. If they were, the inversion would look just like the original. But C major and F minor qualify as being nearly I-symmetrical.

You can change the C major triad into a perfectly I-symmetrical one just by raising the middle note from pitch class 4 to pitch class 5 — E becomes E#, or F — making the chord into a C suspended triad.

Inverting C suspended yields C suspended again, so C suspended is a perfectly I-symmetrical triad.





These ideas about chord symmetry flow from musical set theory. Set theory as it applies to music, says Wikipedia, "deals with collections of pitches and pitch classes, which may be ordered or unordered, and which can be related by musical operations such as transposition, inversion, and complementation."

In the examples given above, the operation called inversion is front and center.



Here is another movie, this one of a seventh chord and its inversion. The original chord is not one with a common name. It can be called "C 0_2_4_6." The inversion has the same pattern, but now with F# as the root. You can call it "F# 6_8_10_0," since F# is of pitch class 6, where C is of pitch class 0.

"C 0_2_4_6" does not initially appear to be I-symmetrical. That's because the pivot point is 0 in the movie. Pivoting the clock face around the axis running through 0 (and also through 6) seems to give a completely different arrangement of pitches. But notice that "F# 6_8_10_0," if rotated one half turn of the clock face, looks just like "C 0_2_4_6." You do the math by adding 6 to each note in {6, 8, 10, 0}. Then, since you are using modulo-12 arithmetic, if the result is 12 or higher, subtract 12:

• (6 + 6) mod 12 = 12 mod 12 = 12 - 12 = 0
• (8 + 6) mod 12 = 14 mod 12 = 14 - 12 = 2
• (10 + 6) mod 12 = 16 mod 12 = 16 - 12 = 4
• (0 + 6) mod 12 = 6 mod 12 = 6

Mathematically, this is rotation. Musically, it is transposition. "F# 6_8_10_0" rotates/transposes to become the original chord, "C 0_2_4_6." The inversion of the original chord is identical to the original under rotation/transposition. Ergo, the original chord was I-symmetrical after all.


Below is a movie, a swatch from a longer movie of Mr. Tymoczko's, that shows why this is important:




















There are two progressions, each with two chords. The first two-chord progression is from an F# half-diminished seventh chord, {F#, A, C, E} or {6, 9, 0, 4}, to an F dominant seventh, {F, A, C, E♭} or {5, 9, 0, 3}. This example comes from Wagner's Parsifal via the Tymoczko paper.

The second two-chord progression, also cited in Tymoczko's paper, is from Debussy's Prelude à l'après-midi d'un faune. It changes from A# half-diminished seventh, {A#, C#, E, G#} or {10, 1, 4, 8}, to B♭dominant seventh, {B♭, D, F, A♭} or {10, 2, 5, 8}.

In both examples, a four-note chord that is nearly but not exactly I-symmetrical shifts to a second four-note chord that, again, is almost but not quite I-symmetrical.

In the first example, (6, 9, 0, 4) patterns with (0, 2, 5, 8), which can be thought of as its chord-structure template. (The parentheses mean the order of the numbers is significant. Curly braces would mean note order is not important.) If you add 8 to each number in (6, 9, 0, 4) — then subtract 12 if the result is 12 or higher — you get (2, 5, 8, 0). That can be reordered to (0, 2, 5, 8).

Likewise, (5, 9, 0, 3) patterns with (0, 3, 6, 8). You add 3 to each number — subtracting 12 is not necessary here — then do the proper reordering of the result: (8, 0, 3, 6) —> (0, 3, 6, 8).

It can be demonstrated that the second example likewise involves a progression between two chords whose chord-structure templates compute to (0, 2, 5, 8) and (0, 3, 6, 8), respectively.

This is why the orange ball in the diagram on the left side of the movie moves between balls labeled "0 2 5 8" and "0 3 6 8" ... in both examples!

Admittedly, it is hard to read the numbers on the various balls in the diagram; try clicking on the movie to see a larger version. The reddish ball in the center of the cluster is labeled "0 3 6 9," which means it represents all four-note chords whose chord-structure template is (0, 3, 6, 9). All (0, 3, 6, 9) chords are I-symmetrical — not just nearly so, but exactly (hence the reddish color of the ball). Here we are ignoring the fact that (0, 3, 6, 9) chords also happen to be perfectly T-symmetrical, by the way. Chords with structure (0, 2, 5, 8) and those with structure (0, 3, 6, 8) both "live right next door" to (0, 3, 6, 9) chords in Tymoczko's geometrical chord space, known as an "orbiform."

There are four flavors of (0, 2, 5, 8) chords that can be derived by minimally altering (0, 3, 6, 9), just as there are four flavors of (0, 3, 6, 8) chords ... which is why each of these chord classes is represented four times in the diagram, while the perfectly I-symmetrical (0, 3, 6, 9) appears but once. This is not tremendously important in the present context. The key thing is that the line segment between (0, 3, 6, 9) and each particular (0, 2, 5, 8) flavor is short, as is the line segment between (0, 3, 6, 9) and each particular (0, 3, 6, 8) flavor.

Accordingly, the chord progression from a (0, 2, 5, 8) chord to a (0, 3, 6, 8) chord traverses, in effect, just two of these short segments: one segment to arrive at (0, 3, 6, 9), and a second segment to arrive at (0, 3, 6, 8). Such a progression, in fact, represents an "efficient voice leading" and is therefore musically coherent.

In more practical terms, (F#, A, C, E) in the first example moves to (F, A, C, E♭) by lowering its bottom and top voices a semitone apiece while holding its second and third — or inner — voices at their original pitches. In the second example, (C#, E, G#, A#) becomes (D, F, A♭, B♭) as the lower two voices ascend one semitone apiece while the upper two voices don't change in pitch.

So the math discussed above is a roundabout way of showing that the specifics of these two actual chord progressions are equivalent in crucial ways. They both represent efficient, hence coherent, voice leadings because they both move between two chords that each live next door to the same perfectly I-symmetrical chord — namely, the one whose structure pattern is (0, 3, 6, 9).

Wagner and Debussy may not have known anything about the mathematics involved, but they clearly grasped their import.

Chord Geometry, Part III

In Chord Geometry, Part II I discussed the mathematical model of musical chords proposed by Dmitri Tymoczko of Princeton in his paper "The Geometry of Musical Chords," published in Science magazine. These remarks followed upon the introduction to the topic I made in The Geometry of Chords.

Tymoczko shows in his paper how principles of contrapuntal "voice leading" à la Bach serve also to underpin the chord-based harmonies in later Western music, right up to the dissonant, atonal compositions of the modern era. In the non-Euclidean geometric spaces Tymoczko defines to house all possible musical chords made of any number of notes of any imaginable pitch class, edge boundaries and certain other geometric loci are inhabited by chords that are perfectly symmetrical in one of three possible ways. Other chords close to these loci are nearly, but not perfectly, symmetrical. Voice leadings among such nearly symmetrical chords are favored by composers even when the music they are making is not overtly contrapuntal.

In the previous post I showed how this works for one form of (near) symmetry: permutational symmetry or P-symmetry. A 3-dimensional "orbifold" composed of all possible triads has edge boundaries. They consist of triads in which all three notes have the same pitch or pitch class. (A "pitch class" consists of a pitch and all other pitches separated from it by one or more exact octaves.) Harmonies traveling near these orbifold boundaries "bounce" off them as they progress from one nearly symmetrical triad to the next. These nearly P-symmetrical chords are characterized by their dissonance and are used in (Tymoczko says) "recent atonal composition, particularly the music of Ligeti and Lutoslawski."


What about the familiar sonorities of more consonant chordal progressions? They too involve near-symmetries of chordal geometry.

Besides P-symmetry, there are two other types of chordal symmetry, exact or near. The simpler of these two types is transpositonal symmetry, or T-symmetry. A chord that precisely divides the octave into equal parts possesses exact T-symmetry. Such a chord is the C diminished seventh. It can be represented by the ordered set (C, E♭, G♭, A) or, using Tymoczko's pitch-class designations, (0, 3, 6, 9).

Imagine these numbers in red on a clock face, with the 0 where the 12 should be. The other numbers — 1, 2, 4, 5, etc. — are in black. Now imagine rotating the clock face to bring the 3 to the top, to the spot where the 0 had been. The 6 moves to where the 3 was, the 9 to where the 6 was, and the 0 to where the 9 was. The same positions as were in red before on the clock face are still in red.

This rotation of the clock face is equivalent to subtracting 3 from (or adding -3 to) each pitch class in the original chord, using modulo-12 arithmetic in which all results are forced into the range from 0 through 11. For instance, -3 becomes 9 in modulo-12 arithmetic. Thus, (0, 3, 6, 9) becomes (-3, 0, 3, 6), which becomes (9, 0, 3, 6). Given appropriate pitch names, (9, 0, 3, 6) is equivalent to (E♭, G♭, A, C), which is sometimes spoken of as the 1st inversion of the C diminished seventh chord.

Tymoczko doesn't call this sort of transformation "inversion," though. He reserves that term for the transformation I'll discuss in the next installment. This one he calls "transposition." This is because the 1st inversion of the C diminished seventh chord is exactly the same as the E♭ diminished seventh chord when it is in root position: (E♭, G♭, A, C). Mere clock-face rotation simply transposes one chord that evenly divides the octave into another chord that also evenly divides the octave, but has a different root.

The same is true for clock-face rotations (modulo-12 additions) involving chords that do not evenly divide the octave. For example, the C major triad (C, E, G) becomes D major, or (D, F#, A), by adding 2 to each of its pitch classes. In other words, (0, 4, 7) transposes to (2, 6, 9).

Such major triads only approximately divide the octave. No clock-face rotation of a major triad can put the red numerals in the exact same positions as they were in before. This is why major triads are only nearly transposition-symmetrical, but not exactly so.

Near-symmetry makes major triads and other familiar types of chords useful in voice leading and harmonic progressions. Chords that are perfectly T-symmetrical, such as the C augmented triad, (0, 4, 8), lie at the center of their orbifold. Depending on how the orbifold is presented, this central locus can be thought of as a point or a line. All perfectly T-symmetrical chords correspond to that point or lie along that line.

Hence they are mathematically close, in that moving from one perfectly T-symmetrical chord to another can be done with maximum efficiency. "The perfectly even chord at the center of the orbifold," writes Tymoczko, "can be linked to all of its transpositions by the smallest possible bijective voice leading." ("Bijective" means that each note in the first chord leads to exactly one distinct note in the next.)

Sticking to perfectly even chords like augmented triads and diminished sevenths is not, however, good compositional strategy, since "efficient voice leadings between perfectly T-symmetrical chords are typically not independent." Voices move in parallel from one chord to the next: (0, 4, 8) transposes to (2, 6, 10), for instance. This slavish movement in parallel is considered too dull and limiting.

Once you switch to chords such as major triads that are not quite T-symmetrical, you can get quite efficient — i.e., small-step — voice leadings while preserving the independent movement of the various voices that make up the chord progression. This is how coherent progressions of common chords that are highly consonant are made.

These common chords are highly consonant because their pitch classes are harmonically related in simple ways. For example, the G pitch in the C major triad is close to 3/2 times the C pitch. (Only "close to," due to the way notes are spaced in so-called "equal temperament.")

Tymoczko's chord geometries using orbiforms express the consonance or dissonance of musical chords quite nicely in terms of how close the chords are to the central point or line of perfect T-symmetry. Chords that are consonant are close to that locus. Chords that are far away from that locus are dissonant and have to rely on near P-symmetry, as discussed in the previous installment, rather than near T-symmetry to furnish their progressions with the desired coherence.

There is also another useful type of symmetry, inversional symmetry or I-symmetry, which can apply to chords distributed not just near a central locus or edge boundary, but throughout the pitch-class space of an orbifold. "I-symmetry is exploited in both tonal and atonal music," says Tymocko. "It plays a salient role in the 19th century, particularly in the music of Schubert, Wagner, and Debussy." I'll discuss I-symmetry in the next installment.

Chord Geometry, Part II

In The Geometry of Chords I talked about the mathematics of chords as described by Princeton's Dmitri Tymoczko in his paper "The Geometry of Musical Chords," published in Science magazine. Since that post, I have downloaded his article (not free) and tried to figure out its abstruse mathematics. I don't know how fully successful my attempts have been, but I can say my understanding and appreciation of his ideas have grown.

The articles about Tymoczko's ideas which I cited in the original post seem to have given short shrift to some notions I consider key. One of these is the emphasis Tymoczko places on "voice leading," which is the basis for the contrapuntal music of Bach and others from the 18th century. "Counterpoint (or voice leading) is the technique of connecting the individual notes in a series of chords so as to form simultaneous melodies," writes Tymoczko. Composers of counterpoint aim for a system of felicitous successions of notes: one whose paths are efficient, in that changes in pitch from one note to the next in a given voice are small; one whose paths are independent, such that the separate voices do not move slavishly up and down in parallel; and one whose pitch paths ideally do not cross or intersect .

Counterpoint in (say) four voices produces a progression of four-note chords as a side effect. After the heyday of contrapuntal compositions, composers began thinking in terms of chord-based harmonies per se. There was a new set of constraints as to what sorts of harmonies were permissible. These constraints were not stated with reference to the rules of voice leading common in counterpoint. Yet, Tymoczko finds, post-Bach Western music has generally managed to "satisfy harmonic and contrapuntal constraints at once."

His fundamental purpose is accordingly to show how the two sets of constraints can both be satisfied in the various forms Western music has taken — even the most atonal, avant-garde compositions of the modern era.

Basically — at least as far I can tell — Tymoczko's math reveals that most or all "coherent" chordal harmonies, however consonant, dissonant, or atonal, partake of the same efficient, independent, ideally non-crossing voice leading that true counterpoint does.


In order to see this, you have to first construct various geometrical spaces within which all possible chords have their specific places. These spaces don't adhere to the commonsense axioms of Euclid which form the basis of everyone's (probably forgotten) high school geometry. Tymoczko's geometric spaces, in fact, loop back on themselves like a Möbius strip.

If you take a strip of paper and give it a half-twist, gluing its ends together, you have a Möbius strip. If you trace a line along its center far enough, you wind up on the "other side," exactly opposite the original starting point!

This is how Tymoczko models two-note chords, in fact. For example, take the notes C and D♭. They form two possible two-note chords, one being (C, D♭) and the other being (D♭, C) — see the first example in this movie. Though they are reversed in pitch order, they are identical in sound. They are like a certain point and the point directly opposite it on the Möbius strip.

To get from the first point to the second more expeditiously than following the center line of the strip all the way around, you can "bounce" off the edge or boundary of the strip directly over to the point immediately on the "other side" — though, as the unbroken-center-line journey proves, there is really no "other side" to a Möbius strip, and the two distinct points are the same!

This kind of geometric weirdness, when extended into an arbitrarily large number of dimensions to allow for chords with any conceivable number of notes, makes it possible to describe the various sorts of symmetries and near-symmetries that can occur in the structure of musical chords.


Tymoczko finds that not exact symmetry in chord structure but near-symmetry is the key to understanding why composers pick the chord progressions and voice leadings they do:

In many Western styles, it is desirable to find efficient, independent voice leadings between transpositionally or inversionally related chords .... A chord can participate in such progressions only if it is nearly symmetrical under transposition, permutation, or inversion ....

(C, D♭) and (D♭, C) are nearly symmetrical two-note chords, while (C, C) and (D♭, D♭) are perfectly symmetrical. (C, D♭) can also be represented as (0, 1), since in Tymoczko's math, the note C is equivalent to the "pitch class" 0 and the note D♭is 1. Similarly, (D♭, C) is (1, 0). Both are permutations of the unordered collection {0, 1} — order-independence being indicated by the use of curly braces rather than parentheses.

The idea of a "pitch class," by the way, is simply that tones whose pitches are separated by one or more octaves are essentially identical. Thus, every C in every octave on the piano keyboard is in the same pitch class.

The type of (near) symmetry in the example I just gave is what Tymoczko calls permutational symmetry or P-symmetry, since each pitch class in the first chord corresponds one-for-one with a pitch class in the second. In the most "trivial" case of P-symmetry — for example, the perfect symmetry of (C, C) leading to (C, C) — the voice leading is effectively null, and the result is, says Tymoczko, "musically inert."

Slightly less trivial near-symmetry of the permutational type — for example, (C, D♭) to (D♭, C) — is more interesting, especially when two-note chords become three-note triads such as {B, C, D♭}. Provided the triads as realized are made by three distinct voices that can be told apart by the ear — say, a flute, a clarinet, and an oboe — the resulting progression is dissonant, even atonal, but it is musically valid:

Nearly P-symmetrical chords such as {B, C, D♭} are considered to be extremely dissonant. They are well-suited to static music in which voices move by small distances within an unchanging harmony .... Such practices are characteristic of recent atonal composition, particularly the music of Ligeti and Lutoslawski. From the present perspective, these avant-garde techniques are closely related to those of traditional tonality: They exploit one of three fundamental symmetries permitting efficient, independent voice leading between transpositionally or inversionally related chords.

To get from (B, C, D♭) to, say, (C, D♭, B) involves having the voices or instruments trade off notes. The flute takes over the clarinet's note, the clarinet takes over the oboe's note, and the oboe takes over the flute's note. But the chord remains the same unordered collection, {B, C, D♭}. To capture this idea, Tymoczko has it that a voice-leading progression from (B, C, D♭) to (C, D♭, B) is reflected off the edge or boundary of his geometric space, which is called an "orbifold." As with a Möbius strip for two-note chords, this reflection off the boundary of a triadic, 3-dimensional orbifold results in the chord progression arriving right back where it started.

{C, C, C} is right on a boundary edge of the orbifold, as is {B, B, B}, as is {D♭, D♭, D♭}. Those and other same-note triads are examples of perfectly P-symmetrical chords.

{B, C, D♭}, since it contains three separate notes that are adjacent on a piano keyboard, is a "chromatic cluster," with B as its root. It is close to but not on an orbiform boundary, and so is only nearly P-symmetrical. That proximity to a boundary means the edge-reflected path from (B, C, D♭) to (C, D♭, B) is quite short. It also means that {B, C, D♭}, in any of its ordered permutations, sounds dissonant to the ear.

Taking both characteristics together, we learn why a dissonant, even atonal voice leading from (B, C, D♭) to (C, D♭, B) is said by Tymoczko to be among the "symmetries permitting efficient, independent voice leading between transpositionally or inversionally related chords." (Notice, however, that this particular voice leading is not non-crossing. The third principle of ideal counterpoint is not observed.)

In my next post, I'll take up the subject of how other near-symmetries of chord structure make for voice leadings and harmonies that are less dissonant and more dulcet to Western ears.