1. hobbies and interests
2. guitar

The Geometry of Musical Chords
Eric Thul Dr. Godfried Toussaint, Professor Final Project for Pattern Recognition COMP­644
December 4, 2006
1 Introduction
Hi! Welcome (that's me on the right). Today we are going to talk about a little music, a little math, and a little geometry. But first, consider the following.
Imagine if you will: You are a world­
renowned music composer. You've written hundreds of scores, and the newswire has named you the hottest composer since Johann Sebastian Bach. How does that feel? Must feel quite good. Quite good indeed...
So, now that I have a candid interview with the best composer since Bach, there is one question weighing on everyone's mind. And that is, "How do you do it?" Computer Scientists, Mathematicians, Psychologists, Musicologists, and even my next door neighbour want to know. How is it that you create your compositions?
Perhaps this isn't a fair question. In fact, you probably cannot give a scientific answer. But that's okay. Let's explore this question together. Let's try to find out what might be going in that mind of yours when you sit down to compose a magnificent piece of music. However, I suppose that I should mention there is one catch. And that is, we are going to have fun! Yes, indeed taking a journey to your musical mind will undoubtably be hard work. But it will pay off. Maybe we will even get to compose a little music together.
Are you in?
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Yes, I am totally in! No... and by no I mean yes Good, I am glad we are together on this. Before we begin, let us consider one caveat: Scientists currently (December 4, 2006) do not know how people compose music. This is a problem still waiting to be solved. However, there are a few strong theories about what might be going on inside a composer's head. How he or she may visualize the relationship within the music and see where they want to go within a musical piece.
We will be exploring one such theory. This theory was proposed in a article called, "The Geometry of Musical Chords" [1] by Dmitri Tymoczko of the Department of Music at Princeton University.
2 Music
Let us begin our jounery with a little music­­ music theory that is. Now, I know that you are the world's top composer; however, we should start from the very beginning. At the beginning there are two main concepts in music theory which are important to us. 1. Harmony: Can be thought of in the simple case as two people each singing a note, where the notes may be the same or different [11, p.55]. Figure 2.1: Example harmony with two people singing at the same time. 2. Counterpoint: Can be thought of in the simple case as a duet between two people singing, literally meaning note against note [11, p.A14]. Figure 2.2: Example counterpoint between two people singing. These definitions of harmony and counterpoint may not be very detailed, but consider the sub­sections below which elaborate on these two important concepts.
2.1 Harmony
The definition of harmony that we've already seen captures the main idea. Sounding two notes together pertains to harmony. However, we do not have to limit ourselves to just two notes. For example, we can sound three, four, five, etc. Typically, Western music practices using harmony using two, three, or four notes.
I realize that your skill in harmony must be of the highest level, but if you want to get a feel for harmony, run this simple java applet (and make sure your speakers are set to "on"!). All you have to do is click on the notes you want to hear together. You have to hold down Ctrl to select more than one note. Once you press the button, the notes will all sound together.
If you tested out the applet, then you probably found combinations of notes that sound "good" and ones which sounded "bad". Let us call the ones which sounded good (or more familiar) consonant, and those which sounded bad (or less familiar) dissonant. These are musical terms which classify groups of notes. For example, try the notes C,E,G. I bet it sounds pretty good, right? Well, that is what is known as consonance. If you were to try C,C#/Db,F, then it might not sound as familiar to your as the previous example. This is a dissonant (very dissonant) chord.
Another important aspect of harmony is the idea of chords. We've been discussing "groups of notes", but really, the musical term for this is a chord. There are a few types of chords we are concerned with [11, p.9­10, 55].
Chord Type
Description
Dyad
A dyad is the example of two people singing at the same time. It consists of just two notes.
Triad (trichord)
A triad can be thought of as three people singing at the same time. This consists of three notes.
Tetrachord
We simple just add another voice. So in a tetrachord, there are four notes playing at the same time.
Example
Table 2.1.1: Chord types For each type listed above there are many properties which can be described about them. However, for our purposes we are only concerned with the distance between each note in the chord. This means give the chord C,E,G, we want to know how far C is from G. In order to find this out, we just have to count. Consider the twelve notes in order as on a piano.
Figure 2.1.1: Notes on the piano To find out the distance between C and G we can starting counting the semitones from C to G. This gives us 7 semitones. But what happens if you want the distance from A to D. Well, once we hit B we just wrap around to C since if you were to look at this sequence in the middle of the piano, you'll notice it goes back to C again after B. This will give us 5 semitones. This cyclic sequence can be realized on a circle. We can see that counting clockwise around the circle gives the distance between notes. Keep the following figure in mind because this will become very important.
Figure 2.1.2: Pitch circle So my composer friend, I hope that this discussion on the basics of harmony has been helpful.We now are going to move to counterpoint which incorporates ideas from harmony.
2.2 Counterpoint
Recall from the previous section about harmony that we were concerned with a single chord and the distance between the notes. This can be thought of as a vertical distance. In counterpoint, we are concerned with having more than one chord. For example, having two chords played sequentially. Thus, in counterpoint we are concerned with the horizontal distance between chords. Let's see how this works.
Figure 2.2.1: Previous counterpoint example In the simple example above, we see there are four chords that each have two notes. Each chord has a high note and a low now. Just for the record, in the example we see two different clefs at the beginning. The one on the top roughly looks like a G whereas the one on the botton roughly looks like an F (I know it's a stretch, but that is how they get their nicknames: G­clef (treble) and F­clef (bass)). All we need to know is that in the example, the notes are lower in the bass cleff than in the treble cleff.
Let's give names to these notes. The notes in the bass clef are called the bass voice. The notes in the treble cleff are called the soprano voice. To measure horizontal distance between these two chords we find the distance between each bass voice and each soprano voice. The following diagrams shows this process for the first transition. On the left we see the transition from G to E in the soprano voice. Note that since we are going from a higher pitch to a lower pitch, we go counter­clockwise around the circle. On the left we see the transition from C to G in the bass voice. Note that since we are going from a lower pitch to a higher pitch, we go clockwise around the circle.
Figure 2.2.2: Horizontal distance in the soprano voice counting around the circle counter­clockwise from G to E is 3 semitones. Figure 2.2.3: Horizontal distance in the bass voice counting around the circle clockwise from C to G is 7 semitones. From this example, you can see that we can then calculate the distance between each note on each voice transition.
Figure 2.2.4: Annotated counter point example where the distance between the notes are labeled in semitones above for the soprano voice and below for the bass voice. If you want to hear some counter point in action, check out this simple java applet which randomly creates a melody from a given bass line. Well, I know this must be a lot to absorb at once. But hey, you are the composer, so I hope so far so good. If the counterpoint between dyads (chords of two notes) made sense, we can easily expand the idea to triads or even tetra chords. This process adds more voices to the score. And also adds complexity since more considerations come into play. For example, we now have to look at four (in the tetrachord case) notes to harmonize and also the flow of four notes between chords. Consider the following as an example of what voice leading with tetrachords might look like. And by the way, I am not claiming the following is an award­winning score. I just put it together for an example.
Figure 2.2.5: Counterpoint example with four voices. From bottom to top we can name the voices: bass, tenor, alto, soprano [11, p.85]. Considering the four voices, we must take care at the choices of each transition. This was also present in the example of two voices, but with four voices, the idea stands out. This presents an important topic which is the basis for our discussion here today: voice leading. We can define voice leading to be the interaction of the notes in each voice and between each voice [11, p.85]. In other words, when we go from chord to chord, the notes in the bass voice must be carefully chosen and the same goes for the other voices. But not only are the transitions considered, but also the harmonies that each chords forms are considered. In general for Western music, the bass voice along with the tenor and alto represent the underlying harmony, whereas the soprano voice has more melodic freedom [11, p.85].
This idea of voice leading is what drives the curiosity of how a composer's mind works. Since a composor must factor in all of these variables, a new idea presented by Dmitri Tymoczko [1] incorporates mathematics and geometry to visualize what might be going on. However, to describe what is going on, a bit of mathematics must first be introduced. But don't be afriad, I mean, you are a great composor, so don't you have a flair for math? [10] Okay Okay, I won't make any assumptions. Let's ease into the subject.
3 Mathematics
Let's start the discussion very basic. But I want you to keep in mind the previous section about music, since that is how we are going to learn the necessary math.
Through this discussion, the basics will be introduced, then theory about pitch classes, and finally voice leading will be discussed. These sections are very important because they form a basis for the geometric construction that is to be presented. So, please check them out because they will greatly help you in your future reading.
3.1 Basics
Recall from the previous section our circle of notes (as seen in the last section). Here we have twelve notes. We made the relationship to keys on a piano by saying that if you travel clockwise around the circle (to the right on a piano) then you will eventually hit your starting point again. As you play each note the tone (frequency) that you hear gets higher. Once you hit the starting note the tone will be twice as high. Thus the frequency will be two times the first note played.
Let's take an example of this. Pick note C. If you start at C on our circle of notes and go all the way clockwise around, we hit C again. This is the same note, but just an octave higher. The octave is a musical term to describe this relationship. We first played C then played C again but an octave higher. Consider the following picture.
Figure 3.1.1: Frequency of a note at octave 1 Figure 3.1.2: Frequency of a note at octave 2 We can see that there are twice as many periods on the right than there are on the left. The result of this is a note which sounds an octave higher. The reason we are reviewing this is because all the notes on a piano have some frequency. And we can represent these in a more simple manner by chaning the scale. Consider the following equation [1, p.72], where f is the frequency some note on the piano and p is the pitch number. Just to mention, this is the standard used by MIDI to represent pitch values.
p = 69 + 12log2(f/440) Using the change of base rule [15], we can convert this into a function that you can plug into any modern calculator. Just in case you wanted to test this out on your own.
p = 69 + 12ln(f/440)/ln(2) In the above equations, we are scaling the pitch frequencies to a new logarithmic range. The idea for the equation is based on the frequency of 440, which is the note A on the piano above middle C, where middle C is in the fourth octave. As a matter of fact, the frequency 440 is ISO 16 [16] at the International Organization for Standardization If you're interested, you can hear the note being played.
The following table shows in the fourth octave (starting at C4, the values for all of the pitch frequencies in this new representation. Note that rounding took place to get an integer value because the frequencies are not exact. They are from here.
Pitch Frequency (Hz) Integer Value p
C
261.8
60
C#/Db 277.246
61
D
293.74
62
D#/Eb 311.28
63
E
64
329.868
F
349.241
65
F#/Gb 370.238
66
G
392.176
67
G#/Ab 415.477
68
A
440.348
69
A#/Bb 466.475
70
B
494.017
71
From the table, we see that we get values in the range [60,71] at increments of 1 per semitone. This is very convenient because it yields sets of 12 values per octave. Note that we start at octave zero, note C which has integer value 0. So, we only need this set, {0,1,2,3,4,5,6,7,8,9,10,11} to specify our pitches for octave zero.
With this idea in mind, we can go one step further and represent all the pitches of all the octaves using the set of twelve integers above. We can do this by taking any number in the set and adding 12 * octave. It is important to mention that this logarithmic scale actually goes lower than the notes on the piano, since we can have octave = 0. So, if we wanted to get C in the fourth octave (on the piano) we have to multiply by octave = 5. The idea is just that the notes on the piano are a subset of this huge integer range.
3.2 Pitch Classes
Now that we have established that we can generate any note using {0,1,2,3,4,5,6,7,8,9,10,11} plus 12 * octave, we can say something a little more formal.
Consider again the the set of pitches, Pitches = {0,1,2,3,4,5,6,7,8,9,10,11}. How can we define a set to contain all of the pitches of any octave. For example, we want a set containing 0,12,24,36,etc... which represents the pitch C in any octave. Well, we can use the following set definition where we can pick p to be any pitch.
PitchClass = {p + 12k such that k is an integer >= 0} The above is a pitch class. This makes it convenient generally refer to pitches and even caculate the distance between pitches. If you're wondering how this distance is calculated then consider the equation below [2, p.1].
Pitch Distance = |q­p|, where p,q are pitches
The above equation calculated the integer absolute value of the difference between the two pitches p and q. For example, take pitch p=9, note D and take pitch q=2, note A. We can apply the equation: |2­9| = |­7| = 7. This is the distance between the two pitches. Now, consider the following equation to calculate the distance between two given pitch classes [2, p.1].
Pitch Class Distance = ||a­b||12Z , where a,b are pitch classes The notation here might seem a bit strange, but let me explain. First we have to recall that our pitches can be seen on a circle, such that going clockwise around the circle once will land on the same note, but at a higher octave. And going around counter­clockwise once goes back down an octave. Let's just remind ourselves with a picture.
Figure 3.2.1: The mapping between the integer notes and our pitches. Looking at this picture, we want to find the smaller distance either going clockwise or counter­
clockwise to get from one pitch class to the other. For example, if we look at the pitch class a=2 and b=10, we see that the distance to travel around the circle is 3 going counter clockwise and 9 going clockwise. So, 3 would be our pitch class distance.
3.3 Voice Leading
Now that we are comfortable with calculating the distance between one pitch and one pitch class, we can in fact calculate the distance between multiple pitches or multiple pitch classes. When we consider multiple pitches/pitch classes, we are essentially considering chords.
Recall the idea of a chord, or just a set of pitches. For example, P={C,E,G} is a set of notes. Also, consider another set of pitches, say, Q={D,E,A}. We can define a voice leading to be the following. Fix some ordering of σp=(E,G,C) and fix some ordering of σq=(D,E,A). A voice leading is a transition between these two orderings. So we have (E,G,C) → (D,E,A). Notice that we can have many different voice leadings between these chords. But for now, we just picked one.
In order to determine the size of this voice leading, we can use our distance equations listed above. Now, notice that I just gave a set of pitches without any indication of octave. This is important. What I should have said, to be precise is that I am referring to pitch classes, which we talked about before. So, the actual octave doesn't play a role. We can safely ignore the octave since we are concerned with efficient voice leadings, meaning minimal distance from chord to chord. This implies that we pick the best octave to achieve this goal of minimality [3, p.7] ­­I mean, would you pick octave 8 to go to a note in octave 2? ..I didn't think so. That would just unnecessarily give huge distance.
Okay. So now that we have that out of the way, let's again consider our voice leading of pitch classes. (E,G,C) → (D,E,A). We know that since these are pitch classes we have to use the second equation for distance. So, let's apply the equation and generate the following table to summarize results.
Chord (E,G,C)
Chord (D,E,A)
Pitch Class Distance
E
D
2
G
E
3
C
A
3
We can also view this on the circle. Consider the following representation which shows the same chords and distances as in the table above. Note that each pitch is color coded by voice. So a red circle on the left is the same voice as the red circle on the right.
Figure 3.3.1: The chord EGC Figure 3.3.2: The chord DEA As a result from either interpretation, we have what is called a displacement multiset [2, p.2]. The displacement multiset is just a set with duplicate entries allowed which contains the distances from the voice leading. So, from the above we have the set {2,3,3}. When we are comparing different possible voice leadings between sets of pitch classes, we use the displacement multiset. This gives us the intervals between the pitch classes, independent from transposing (rotating all the pitches in the chord by the same amount in the same direction) or inverting (flipping the circle about the centered y axis, but leaving the labels in place) a voice in the chords. So, by using the displacement multiset, we have a method for comparing different possible voice leadings to find the best one.
In order to find the most efficient voice leading, we must take into consider three main ideas. The first pertains to voice crossing (to be explained). The second pertains to whether each pitch class set has the same cardinality or not. And the third is with regard to element uniqueness in the pitch class sets.
Let us quickly take care of the pitch class set uniqueness. Let us defining each pitch class set to be a multiset. So, we are allowed to have duplicate pitch classes in our multi­sets [2, p.2]. Now consider the following two subsections for our other considerations.
3.3.1 Consideration A: Voice Crossings
We can think of voice crossing simply as when two voices cross. So, recall that we have four standard voices: bass, tenor, alto, and soprano. Typically the bass sings the low range, the tenor sings in the low­
mid range, the alto sings in the mid­high range, and the soprano sings in the high range.
When there is no voice crossing, the voices never stray from this model. So, the bass will never sing higher than the soprano, and the soprano will never sing lower than the bass. This might be a rediculous example, but it gets the point across. More often voice crossing may occur between the tenor and alto. But with no voice crossing, the tenor will never sing above the alto and the alto will never sing below tenor. Consider the following example of two chords: one without voice crossing (left) and one with voice crossing (right). The colored notes show each voice, blue=bass, green=tenor, orange=alto, red=soprano.
Figure 3.3.1.1: Voice leading with no crossings Figure 3.3.1.2: Voice leading with the orange and green voices crossing Now that we know what voice crossing is, when finding the most efficient voice leading all voice leadings with can be ignored. Voice leadings without voice crossings always produce a smaller voice leading size [2, p.4]. Before the theorem and proof idea is shown, we must first review an important constraint when comparing voice leadings.
The displacement multi­set we've been working with satisfies the distribution constraint [1, p.1]. So, we know that in the displacement multi­set, there are properties which it assumes. Let's take the example we had before of {2,3,3}. Consider the following properties [17].
1. Reflexivity, by picking any element from the set, the element is less than or equal to itself. 2. Transitivity, by picking any three elements x,y,z from the set, if x ≤ y and y ≤ z then x ≤ z. 3. Totality, by picking any two elements x and y, we see that x ≤ y or y ≤ x. With these properties, the distribution contraint has two requirements [2, p.2].
1. That given the sum, k of all the elements in a displacement multiset D, then the following holds: {k,0,0,...,0n} ≥ {d1,...,dn} ≥ {d1/n,...,dn/n}. This means that a more even multi­set with the sum k is potentially smaller than an uneven multiset with sum k 2. If any element in the multi­set is increased, then the total number of elements in the multi­set is not decreased. If a displacement multi­set satisfies both requirements then it strictly satisfies the distribution contraint.
Here is the theorem and proof sketch for this claim [2, p.4].
THEOREM 1 (Tymoczko 2006). Let P and Q be any two n­note multisets of pitches, and let our method of comparin voice leadings be a total preorder satisfying the distribution constraint. Then there will exist a minimal bijective voice leading from P to Q that is crossing free. If the total preorder strictly satisfies the distribution constraint, then every minimal bijective voice leading from P to Q will be crossing free. PROOF IDEA. There are two directions which are shown. The first direction is that if the method chosen to compare voice leadings obeys the distribution contraint, then there is a bijective voice leading between between any two chords that is crossing free. The second direction is that if the chosen method for comparing voice leadings violates the distribution contraint then some crossed voice leading will be smaller than the uncrossed version of the voice leading [2, p.4­5]. The full proof is a bit long, so it wasn't included here. However, the main idea exploits the properties of the distribution contraint. This shows that given a voice leading from one chord to another, swaping the voices in one chord does not make the voice leading size any smaller. This is an important result since it rids us of having to deal with voice crossing possibilities in the search for an efficient voice leading. Given this, we can now discuss the possibility of having multi­sets of pitches with different cardinalities.
3.3.2 Consideration B: Bijective and Non­Bijective Pitch Class Sets
To clarify notation, a bijection between pitch class sets means that we have two multi­sets with an equal number of elements. For example there is a bijection between, {C,E,G} and {D,F,A}. We can create a one­to­one mapping between the elements. A non­bijection would be if the mult­sets did not have the same number of elements. So the multi­set with the smaller number of elements would map to two or more elements of the larger multi­set. Consider {C,G} and D,F,A, there is not a one­to­one mapping between them; someone has to share!
3.3.2.1 Function on Bijective Chords
Now, before we can draw any conclusions from this, let us explore what we know about the simple case: a bijection of a pitch class set with itself. For example, given the pitch class set {C,E,G}, what are the possible voice leadings to {C,E,G}.
Let us define two functions: transposition and inversion. Consider each one separately below.
•
Transposition is denoted Tx where x is the number of semitones to transpose. For example, when we transpose an ordered set of pitch classes (a chord), we add x to each element of the chord. Consider the chord A=(C,E,G). We can transpose A by say, four semitones. We have the following.
T4(A)=(T4(C),T4(E),T4(G)) =(0+4,4+4,7+4)=(4,8,11)=(E,G#,B) We can say that these chords are related via transposition. And transposition is exactly the same as a rotation of the notes around the circle. We can see this below. Figure 3.3.2.1.1: Chord (C,E,G) •
Figure 3.3.2.1.2: After rotation by four units clockwise, (E,G#,B) Inversion is denoted Ix where the function returns the values which send the input to zero; it returns the inverse. Since we are dealing with a cyclic group remember that any negative number is moving counter­clockwise on the circle. For example, ­1=11, ­2=10, etc.... So, for the inverse, let us consider the example of A=(C,E,G) Ix(A)=(Ix(C),Ix(E),Ix(G)) =(0,­4,­7)=(0,8,5)=(C,G#,F) We can say that these chords are inverses of each other. This also presents an interesting geometric perspective where we have the mirror image. Consider the images below. Figure 3.3.2.1.3: Chord (C,E,G) Figure 3.3.2.1.4: After inversion, (C,G#,F) There is an observation which can be drawn given a chord A and some function F over that chord. If there is a permutation of F(A) which yields a chord similar to A, then there is an efficient voice leading between those chords [2, p.8].
Consider the following example where A=(C,E,G). If we take the inverse, we get A'=(C,G#,F). Now we can permute the elements to arrive at σ(A')=(C,F,G#). This yields a displacement multi­set of {0,1,1}. Where in [2, p.8­11] there is not a clear definition of resemblence, this voice leading appears quite efficient. So (C,E,G) → (C,F,G#) is perhaps an efficient voice leading.
We can draw two conclusions from this. The first is that if the chord has pitch classes which are all very close to one another, like (C,C#,B) then there will only be an efficient voice leading when the applied function returns a chord which is close to trivial. The second conclusion is that if the pitch classes divide the octave evenly, which means we try to maximize the minimum distance between each pitch class. For example, there are 4 semitones between each pitch class for {C,E,G#}. Consider the following image.
Figure 3.3.2.1.5: Maximally Even Chord (C,E,G#) From this, we can arrive at an efficient voice leading by transposition where we transpose by the size of the chord. In this case, by 4. If we do this, then we will arrive at (E,G#,C), which is the same by permutation.
3.3.2.2 Minimal Bijective Voice Leading
From the case discussed above, we can prove that the lower bound on the a voice leading between transposition is from the chords which evenly divide the octave. For example, the {C,E,G#} multi­set described previously. This follows from the statement, "Thus the perfectly even chord... has the smallest possible minimal bijective voice leading to all of its transpositions" [2,p.11]. As the chords become less even then the voice leading among the transpositions tends to increase. Consider the following theorem which presents this idea more formally.
THEOREM 2 (Tymoczko 2006). Let A be any multi­set of cardinality n, and let our method of comparing voice leadings be a total preorder satidfying the distribution contraint. For all x, the minimal bijective voice leading between A and Tx(A) can be no smaller than the minimal bijective voice leading between E and Tx(E), where E divides pitch­class space into n equal parts. PROOF IDEA. The main idea behind this proof uses the fact from the previous section that with a maximally even chord that divdes the octave, we will have a voice leading size of zero. Then by taking advantage of the distribution constraint, we can find that the sum of the displacement multi­set will be at most a constant which can be picked to be as small as possible for the even chord. Then, considering a chord not maximally even, the lower bound on the smallest sum of the displacement multi­set works out to be at least that constant picked. From the theorem, we see that the chords which evenly divide the octave present the most effiecient voice leadings among the transpositionally related chords.
4 Geometry
Ah, glad to see that you're still with me. You might be wondering: isn't the title of this project "The Geometry of Musical Chords"? You are correct. However, we weren't ready to jump into the geometry... until now. Here we consider the geometric aspects of what we've been talking about all along, and hope to shed some light on how we can visualize what might be going on in that composer mind of yours.
In a sense, we will use geometric spaces to navigate paths through music! And this is what Tymoczko [1] describes to be how composers may write music. That they see this path through a musical geometric space and navigate it. Along the way they discover interesting harmonies and choose paths which yield efficient voice leading. So this is where all of what we've been talking about fits together. From the very basics of music theory we mentioned, to the idea of what efficient voice leading is and how to measure it, the geometry is what ties everything together. In this music­geometric space, we have chords which sit in this space and connect to one another depending on their boice leading. Also, chords that evenly divide the octave will be in the center of this space.
Now, this might sound complex, but we are going to start simple and use a lot of pictures! So, let us begin with the one dimensional chord geometry. This can be seen as a straight line, or if you prefer, we can use a circle. Consider the following discussion on the one dimentional chord geometry.
4.1 One Dimension
You may have noticed that even if it wasn't explicitly state, we've been looking at a few geometric aspects of music. Recall the figure of our pitch­class circle. This is what is known to be a pitch­class space [1, p.72]. However, when we are just dealing with pitches, then we can view them as points on a line that extends in both directions [9]. These are the one dimensional cases of our musical space.
Just to refresh our memories, let us take a look at the pitch space and pitch­class space in the figures below.
Figure 4.1.1: Example of the pitch space on a line, the number in the point is the octave. Figure 4.1.2: Example of the pitch­class space on a circle. Now, you might be wondering how this space is navigated. Well, the basic form of motion is by step around the circle, or on the line. And just for ease of discussion, let's consider only the pitch­class space. This is because the circle is easier to visually represent motion through the space. This motion is linear, so it is just going from one point around the circle to the other. If we are just talking about one voice, then that voice can move around from pitch­class to pitch­class generating a melodic line. Consider the following animated figure which shows the first few measures in the first verse of Bob Dylan's Blowin' In the Wind.
Figure 4.1.3: The opening verse of Bob Dylan's Blowin' In the Wind, note that this is from Sheetmusicplus.com We can also view more than one voice moving in this space. For example, Dmitri Tymoczko created a movie which shows a chord progression of Chopin on the circle. Consider the following movie below. Note that I've included it locally for speed, but the orginal files can be found here
Figure 4.1.4: This shows Chopin's music on the circle. This was created by Dmitri Tymoczko. Just to explore this further, consider the following harmonic plan of Beethovan's Violin Sonata Op. 24 ("Spring") second movement, measures 37­54. The following image was generated from an example by Julian Hook [7, p.49].
Figure 4.1.5: The harmonic plan of Beethovan's Violin Sonata, Op.24. This was made from an example by Julian Hook [7, p49].
From all of these images and movies, we can conclude that the overal movement is fairly simple. There are no huge leaps around the circle whether we are looking at a single melodic line, or ever chords that are rotating around the circle. This is an important observation because since many people consider these musical examples to be "good", perhaps we can say that good music is related to the small size of the movement. And this is important because it is exactly what Dmitri Tymoczko is trying to find out [1]. He is curious about efficient (small) voice leading between chords.
Even though the one dimensional case is fascinating on its own, we can also look at a two dimensional space to gain a little more insight to the voice leading between chords. Consider the discussion below.
4.2 Two Dimensions
Instead of a circle, we need some geometric structure to represent two dimensions. Now, I know what you might be thinking, can we use an x­y coordinate plane? The simple answer is we cannot. And this is because our geometric space needs to have have a special property which we saw in the case of the circular pitch­class space.
This property can be thought of in the following way. Imagine yourself standing on a piece of paper (or wood if you're afraid to fall through the paper). Now, consider you have only one item, which is a compass. But the compass only has eight directions on it: N,S,E,W,NE,NW,SE,SW. Now, let's consider that you are in the very center of this space. What do you see?
Well, you see a music chord. Since this space is filled with music chords of size two. Now, since you are in the middle, you will be standing on the chord which evenly divides this space. Since we have twelve pitch classes, what two classes divide twelve evenly? The answer can be any two pitch classes, just as long as they are 6 semitones apart, like C and F#/Gb. However, let's consider C and F#/Gb the center because C is typically the starting point. Any other even chord is a transposition of (C,F#/Gb)
. Recalling this notion of transposition, let's say that when you walk East, the chord you are standing on is tranposed up by one semitone and when you walk West, the chord is transposed down by one semitone. So, if you are standing on (C,F#/Gb) and walk East on unit, you will be standing on (C#/Db,G). If you were to walk West by one unit, then you would be standing on (B,F)Let's invision what we have so far.
Note in the figure that we are just considering a set of pitch classes, so the order doesn't matter. For example, (C,F#/Gb)=(F#/Gb,C) Also, the figure uses the numeric notation of the notes, as we've seen before. The images that follow are from Dmitri Tymoczko's program used to explore these geometric spaces [4].
Figure 4.2.1: A slice of the full space to show what happens walking East or West. Now, in the picture, you might notice something odd. The use of the [9,3]. This means that when you step on this spot you are carried to the other 9 3 on far left of the picture. So, in a sense you can think of this as circular behavior. You repeat to the other side once you hit a chord in square brackets.
And as you can see since order doesnt matter we have six unique ways to evenly split the octave with two notes with (6,0) in the center.
Okay, so we have this space where we more or less have a circular representation. However, we take this one step further when considering the other compass direction we can travel. Let's dive right into the full space visualization and then go for the explanation. Consider the following space of chords size two.
Figure 4.2.2: The full pitch­class space of Dmitri Tymoczko's construction [4]. We see the familiar bracketed hot­spot which carries you to the corresponding chord. Also, notice that walking up to the top or down to the bottom you are cut off and cannot go any further. Since this is a discrete space, we have our limitations. But going left, right, or along the center diagonal brings you around to the corresponding point (those chords have a bracketed counterpart).
You might be wondering what kind of geometric space has this strange property. Well, the space can actually be mapped to a Mobius strip. This is a surface which only has one side. So, this means if you keep following the surface, like walking East or West in our diagram, you will end up back where you started. Maybe a picture will help. Consider the following Mobius strip.
Figure 4.2.3: Example of a mobius strip by Escher [18]. As you can see, we have these ants following around the path which keep walking forever. And they cannot walk to the edge since they would fall off. But let us try to map Figure 4.2.2 to this mobius strip to see if we can see what is going on.
Figure 4.2.4: Image of mapping some of the pitch class chords to the mobius strip [19]. Hopefully the above picture isn't too confusing. The big red band that is vertically on the back of the image is where the chords are glued to their counter parts in brackets. However, to do this we needed to give the picture in Figure 4.2.2 a half­twist so that they line up properly. This gives us our two dimensional pitch­class space for chords. We can view the navigation of this space similarly as in the one dimensional case, except now we can also look at aspects of voice leading.
We didn't encounter this in the voice leading section above, but there are different types of motion in voice leading. Consider the following which describes each type. Note that we have two notes in each chord. •
•
Contrary can be described as having each voice travel in the opposite direction. For example when the soprano travels to a higher note in the next chord and when the bass travels down to a lower note in the next chord we have contrary motion. Oblique can be described as one voice going up or down, while the other stays on the same note. •
For example, when the soprano travels to a higher note and the bass sings the same note in the next chord, we have oblique motion. Parallel can be described as when the voices either both travel up or both travel down together. This yields parallel motion. In the figures of our two dimensional chord space, we didnt have any lines connecting the points. Well, we can add lines which show the different types of motion between the chords. Consider the following three images which show these types of motion in our space.
Figure 4.2.5: this shows the lines which connect chords that move in contrary motion. Figure 4.2.6: this shows the paths which connect chords which move in oblique motion. Figure 4.2.7: this shows the paths which connect the chords that move in parallel motion. We can also have some fun and see how songs travel through this space. Consider the following two movies by Dmitri Tymoczko [4]. The first is Chopin moving in the 2­D space and the second is a song by Deep Purple in this space.
Figure 4.2.8: This shows Chopin's music on the mobius strip. This was created by Dmitri Tymoczko. Figure 4.2.9: This shows Deep Purple's Smoke on the Water on the mobius strip. This was created by Dmitri Tymoczko. Notice that in the examples, the chords chosen stayed around the center of the mobius strip. And I mean center when the mobius strip is cut and flattend out. Also interesting to note is that the rock song by Deep Purple stayed very close to the middle of the space. These are the chords that divide the octave most evenly remember. The Chopin musical piece also stayed close to the more even chords, but ventured further from them.
The next stop is now the three dimensional case, or in other words having chords of length three. Consider the next section's discussion.
4.3 Three Dimensions
We've seen that the mobius strip is a sufficient space to represent the properties of chords in two dimension. But how are we going to create the notion of three voices (chords of size three) in a geometric framework? The answer is very similar to what we already discussed. We don't use a mobius strip, but we still need the property of a circular space.
Consider again that we are working with the chords formed by our pitch­classes. However, they are of size three. This means we have a lot of possible chords. Here we can consider the permutations of each set of pitch­classes to be different. So, this gives us a 123 or 1728 possible chords. Our space is divided by these chords.
Recall that in the two dimentional case all we needed was a mobius strip. This mobius strip gave us a space with a line which was used to glue points together to form the Mobius strip. Recall that red strip in the figure above. This is important to note because we are working in two dimensions so we used to endpoints to make this line. And this line is considered the base of the space.
In the three dimension case we have three points which compose the base. This implies that a triangle is our new base. When we slice along the base and unwrap the space, we will have a three dimensional prism with two faces which are triangles [2, p.19]. Consider the following diagram below which shows a simple model of the space.
Figure 4.3.1: model of the three dimensional chord space [2, p.19] We can imagine where the chords lie in this space. And recall that the two triangular faces are really glued together. So this space wraps around. However, since the chords on the faces have to line up when we glue, we must twist the space before putting the faces together. Instead of doing a full half­twist as in the Mobius strip, we twist by 120 degrees. The chords in this space have similar positioning to the two dimensional case. Meaning that the chords which evenly divide the octave are in the middle (that grey line) and the chords which do not evenly divide the octave are on the edges. Consider the following example.
Figure 4.3.2: model of the three dimensional chord space [2, p.19] As you can see in the picture, as you move up or down along that dashed grey line, you get all the chords which evenly divide the octave by transposition. And the chords along the edges are those which don't divide the octave at all, namely CCC. This is the same basic ideas which was presented with the Mobius strip. We have a center of this space, and then when you reach a face, you go around the circle (since the faces are really glued together).
Let's look at the whole three dimensional space with all of the chords. This might look a bit confusing, but it shows how many choices are possible when consider music composition. Note that these images were generated from a program that Dmitri Tymoczko wrote to explore the space [4]. Note that in the figure I've highlighted one of the triangular faces in red and also drew the dashed center line.
Figure 4.3.3: model of the three dimensional chord space [4] The above figure might be a little rotated, but as you can see it is the same as figure 4.3.1 but with all the chords inside. We also can have edges to show two of the different types of voice leading that we described before: contrary and parallel. We also can see the line which connect single step voice leading. This means going up or down by a single semitone. Consider the following images depitcting these types. Note that these images show the types of voice leading between all three voices.
Figure 4.3.4: model showing contrary voice leading between three voices [4] Here we have higlighted two chords 3,4,9 = D#/Eb,E,A and 4,5,8 = E,F,G#/Ab. We can see that the first voice goes up: D#/Eb → E, the second voice goes up: E → F, but the third voice goes down: A → G#/Ab. So we have contrary motion. And since there are only two directions (ignoring oblique motion), have two voices are going to have to travel together while the other goes the opposite way.
Figure 4.3.5: model showinng parallel voice leading between three voices [4] Again consider the highlighted chords. We have 3,4,9 = D#/Eb,E,A and 4,5,10 = E,F,A#/Bb. Here we have all the voices going up: D#/Eb → E, E → F, and A → A#/Gb. Figure 4.3.6: model showing single step voice leading between three voices [4] In the single step we have two chords connected: 3,4,9 = D#/Eb,E,A and 2,4,9 = D,E,A. Here we can see that only one voice moves by a single semitone or single step.
From each of the example, we have all the lines possible to allow the freedom of true composition. As you can see, it gets complicated very quickly. So my friend the composer, what do you think of this construction? Do you navigate these geometric spaces with ease as you compose your greatest works? We don't know the true answers yet, but this notion of chord geometry provides a theory on what might be going on backstage, so to speak.
The following section presents some of the mathematics behind the geometric constructions. Continue reading if you like. Or you can jump ahead to the applet which let's you explore these spaces and compose music of your own.
4.4 Higher Dimensions
Behind all of this geometry we've been talking about is a mathematical concept which describes such spaces as the Mobius strip and the three dimensional prism with triangles for faces. The term mathematicians use to describe these spaces is orbifold.
To define an orbifold, think of what we've been doing when we construct these chord spaces. Think of the two dimensional case. We saw that we have a surface containing all the two chords of two pitch classes. And we also saw that there are points on this surface that allow you to circle back to the other side. To construct our space, we folded, twisted, and glued those points together which allowed you to circle back to the otherside. This created our chord space and is in fact the way an orbifold is created. Given a symmetric surface (our chords) and then by taking all the points which look alike we fold and twist until the identical points have been brough together (glued) [6].
The origin of the term orbifold (orbit­manifold) is due to William Thurston [5] and is a special kind of manifold. We will not get into manifolds here, but it is worth mentioning that the orifold is a kind of manifold.
The notion used for this is as follows.
Symbol
Meaning
Tn
This refers to a torus in n dimensions. You can think of a torus as a donut, or to be more specific, a torus is the product of two circles [20].
Sn
This refers to the symmetric group in n dimensions, where we have the set of permutations on the objects in S
Rn
The set of all real numbers, in n dimensions.
12Zn
The set of all positive and negative multiples of 12 in n dimensions.
Here we have a set of sets. Each element in Rn/12Zn is a set of integers. To see this, consider Rn/12Zn the case when n=1. Here we have R12Z = { {0,12,24,36,...}, {1,13,25,37,...}, ..., {11,23,35,47,...} }. You can think of these as our pitch classes.
Using the notation above, we can define an n­torus Tn to be Rn/12Zn. And we can then define our orbifold to be Tn/Sn. This works out to [Rn/12Zn]/Sn, which is really Rn/[12Zn⋊Sn]. Now, we need a special way to take the product between the groups 12Zn and Sn. We can define the product to be the semi­direct product. Without getting too much into group theory, this means that there is a group which can be formed by taking the semi­direct product between 12Zn and Sn [2, p.7]. This group retains certain properties [22].
The group Rn/[12Zn⋊Sn] is our orbifold. These orbifolds can be constructed by describing what is called a fundamental domain of 12Zn⋊Sn in Rn. What this means is that a smaller pattern can be used to describe a larger pattern [21].
This is useful since it can be used to describe the the mobius strip in the two dimensional case and the prism with triangular faces in the three dimensional case, and also describe geometric spaces up to dimension n. Consider of all the points in Rn, only the ones where the coordinates are sorted in nondecreasing order. Let them be of the form (x1,...xn). Now we further restrict these points so that xn ≤ x1 + 12 and that the sum of all the xi is greater than or equal to zero and less than or equal to twelve. By gathering all these points we will form an n dimensional prism with the base being a figure in n­1 dimensions, in this case a line. Consider the prism in two dimensions below with a line for the base (highlighted in red).
Figure 4.4.1: 2D prism from the fundamental domain construction. From the above we see something interesting. Recall that since we are working in a group modded by 12Z any negative number ­x is really 12­x. This is exactly like going counter clock­wise on our pitch­
class circle. We just start at zero on the circle and step counter­clockwise x times. So, in our space we see that the point (­6,6) is really the same as (6,6). Starting to seem familiar? It should remind you of the following figure.
Figure 4.4.2: 2D chord space If you tilt your head a little then this is what we saw earlier in the two dimensional case of chord geometries. So, we see that algebraically we can describe this space. And Dmitri Tymoczko gives a function for the twisting of the surface to match up points with thier counter­parts [2, p.7].
O(x1,...,xn) = (x2­12/n, ...,xn­12/n, x1+12­12/n) By using this function on all the points in our prism, we can find all the chords related by transposition and with the pitch classes summing to the same value. In the orbifold, this function is a rotation when the prism has odd dimension and a rotation and reflection otherwise [2, p.7].
One last note about the orbifolds. Those chords which lie at the center of the orbifold (that divide the octave evenly) are known as T­symmetrical chords. This is because by transposition, they are all related. Also we can define the chords which lie along the boundary, furthest from dividing the octave evenly. These are called P­symmetrical since up to permutation they are the same. For example CCC is a P­
symmetrical chords. Lastly, we can define those chords which are inversionally symmetrical to be I­
symmetrical chords [1, p.74].
5 Future Work
Some of the other application that this geometric representation can be used for is when dealing with rhythm. The author states that since rhythms are cyclic patterns they can also be modeled as an orbifold. Perhaps these spaces could be used to study African and other non­Western rhythms [1, p.74].
6 Applet
This section is divded into two parts. The first is for those who just can't wait to use the Geometry of Chords applet. The second part is for those who want to read a little about what the applet is about. In either case, enjoy! 6.1 Running the Applet
I made the applet load off of a separate page because it is rather big. The dimensions are 900x800. Sorry to those who have a smaller screen resolution, but I needed the extra pixels.
The following link will load the applet, and just be sure to give it a minute to load up. You should see 12 orange circles arranged in a circle along with some controls. If you dont see the 12 circles just give it a second. Also, make sure your speakers are set to on!
Click To Load Applet
As a note, for better performance, if you copy the link location above and pass it to the java command appletviewer like appletviewer copiedurl then it may run a little better.
6.2 About the Applet
The main idea behind this applet is to let you navigate the chord spaces which are proposed by Dmitri Tymoczko and which have been explain in this tutorial. The applet provides two dimensions: the first dimension as a pitch class circle and the second dimension as a pitch class grid.
The default screen that loads is the one dimensional pitch class circle. You can choose which dimension you want to compose in with the drop­down menu on the top­right of the applet. Also, there are options to pick your rhythm and instrument.
Another important option is the voice leading lines option. This is only available in the second dimension since the first dimension does not have different types of voice leading. For this you have three options: parallel, contrary, and oblique. You can chose any of them or none at all to display.
You can change the options at any time, but to add notes to My Song you have to click on the circles in the pitch class space. When you click on a pitch, it will be added to the list on the right which represents your song. You can also remove any of the pitches by clicking on the pitch in the list and then clicking the remove button.
At any time you can play the song that you're working on by clicking the play button at the bottom of the applet. The play loop will continuously play your song until you hit the stop button.
As a special remark, if you pick some notes in dimension one and then switch to compose in dimension two, you can have a song with both pitches and chords. Neat!
That is the main idea of the applet, below are some screen shots which show the applet in action.
Figure 6.2.1: Showing the initial loading of the applet in pitch class dimension one. Figure 6.2.2: Showing an example in the second dimension. Figure 6.2.3: Close up view of the options. 7 References
[1] Dmitri Tymoczko. The Geometry of Musical Chords. SCIENCE 313:72­74, 2006.
[2] Dmitri Tymoczko. Supporting Online Material for The Geometry of Musical Chords. SCIENCE 313:72­74, 2006.
[3] Dmitri Tymoczko. Scale Theory, Serial Theory, and Voice Leading. Under Review at Music Analysis. http://www.music.princeton.edu/~dmitri/scalesarrays.pdf, 2006.
[4] Dmitri Tymoczko. Chord Geometries [Online]. http://www.music.princeton.edu/~dmitri/ChordGeometries.html.
[5] William P. Thurston. The Geometry and Topology of Three­Manifolds [Chapter 13]. "http://www.msri.org/publications/books/gt3m/, 2002.
[6] Heidi Burgiel. An Introduction to Orbifolds [Online]. The Geometry Center. http://www.geom.uiuc.edu/education/math5337/Orbifolds/introduction.html, 1996.
[7] Julian Hook. Exploring Musical Space. SCIENCE 313:49­50, 2006.
[8] Richard Webb. Calculated Tones. NATURE 442:149, 2006.
[9] E.K. Seeing the Music. Humanities, Then and Now 39:1, 2006.
[10] John Baez. This Week's Finds in Mathematical Physics (Week 234). http://math.ucr.edu/home/baez/week234.html, 2006.
[11] Robert Gauldin. Harmonic Practice in Tonal Music, Second Edition. W. W. Norton & Company: New York, 2004.
[12] Criag Stuart Sapp. Pitch to Frequency Mappings. http://peabody.sapp.org/class/350.868/lab/notehz/, Accessed: 2006 Nov.
[13] Wikipedia. Chordal Space. http://en.wikipedia.org/wiki/Chordal_space, Accessed: 2006 Nov.
[14] Wikipedia. Pitch (Music). http://en.wikipedia.org/wiki/Pitch_%28music%29, Accessed: 2006 Nov.
[15] Wikipedia. Logarithm, Change of Base. http://en.wikipedia.org/wiki/Logarithm#Change_of_base, Accessed: 2006 Nov.
[16] Wikipedia. A440. http://en.wikipedia.org/wiki/A440, Accessed: 2006 Nov.
[17] Wikipedia. Total Preorder. http://en.wikipedia.org/wiki/Total_preorder, Accessed: 2006 Nov.
[18] Mobius Strip Image. http://www.math.utah.edu/~lars/mathart/artpix/escher­mobius_strip_II.jpg, Accessed: 2006 Nov.
[19] Mobius Strip Image. http://www.atsweb.neu.edu/math/cp/blog/mobius.gif, Accessed: 2006 Nov.
[20] Wikipedia. Torus#Topology. http://en.wikipedia.org/wiki/Torus#Topology, Accessed: 2006 Nov.
[21] Wikipedia. Fundamental Domain. http://en.wikipedia.org/wiki/Fundamental_domain, Accessed: 2006 Nov.
[22] Wikipedia. Semidirect Product. http://en.wikipedia.org/wiki/Semidirect_product, Accessed: 2006 Nov.
[23] Music Note on index page. http://www.giculturalsociety.org/, Accessed: 2006 Dec.
[24] South Park Game. http://images.southparkstudios.com/games/create/sp_game.swf, Accessed: 2006 Dec.