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Notes from the Margin Harmonic Graphs and Complexity in Tonal Music

Volume XII • 2016

Calder Morton-Ferguson (University of Toronto)

In the 1920s, Arnold Schoenberg developed the twelve-tone system, a way of composing music which was amenable to analysis using ideas from set theory, algebra and combinatorics. [2] This analysis informed musical understanding of the structure of atonal music, a style which shifts the focus from major and minor harmonies to sequences of individual notes. Although tonal music constitutes most of the music we hear every day, it hasn’t received the same sort of extensive combinatorial attention as twelve-tone music. In tonal music, harmonies and the relationships between them are the central structural units of a piece. Our goal is to develop a meaningful system to describe the relationships in such music. In a model of tonal harmony, we would require, rather than a sequence of notes, a set of harmonies and a set of relationships between them, since the harmonic structure of a piece of music is determined not just by the choice of harmonies, but by their order and their functional relationship to one another.

Given this focus on harmonic relationships, tonal music can naturally be modeled by a graph in which the Calder Morton-Ferguson nodes represent harmonies and the edges represent I’m not yet seasoned enough the relationships between between. The harmonies to decide which type of math represented by the nodes could be the twelve major I like best, but I knows that I like playing and composing for chords, the twenty-four major and minor chords, or the piano as well as listening even all possible pairs or triads of notes. Once a choice to music of all sorts -- usually of nodes is fixed, the flexibility and power in such a without regard to the shape model lies in the choice of edges. of its harmonic graph. Harmonies can relate to one another in many ways. For example, in Western music theory, C major and G

major are very closely related. Because they are a fifth apart, they sound naturally connected and are often used in succession. Continuing to compare chords using fifths, C major and B major are harmonically dissimilar: we need to step up a fifth five times to get from a C major chord to a B major chord. In contrast, we might compare chords using chromatic steps and say that two chords are similar if their root notes are close together on a keyboard. In this sense, C major and B major are very similar, as a C major chord is just one chromatic step up from a B major chord. Given these differences, how should we choose the connections between harmonies, and thus the edges in our graph? The answer to this question depends on many factors, including the time period in which the music was composed, the composer’s style, and the individual characteristics of the piece. Unlike in atonal music, where the relationship between notes is fixed by their order, the relationship between harmonies in tonal music reflects these creative and cultural factors. After a few examples, we will introduce a notion of complexity, which will allow us to compare different pieces of music through appropriate choices of the


Preamble

by Kyle MacDonald (McMaster University)

I have spilled food on at least two of the authors featured in this edition of the Margin.

Editor Kyle MacDonald

McMaster University

Editor at Large Kseniya Garaschuk

University of British Columbia

All four of them attended and gave talks at the Canadian Undergraduate Mathematics Conference (CUMC), held at the University of Victoria this July. It was my first time at CUMC, so I got to see what the admin notes in the Margin have been talking about for the last two years. At meals or during coffee breaks between talks and activities, I spent time pestering speakers and asking them to submit an article to the Margin. In addition to making a few friends and over-excitedly spilling muffin crumbs on them when they tried to explain Lie algebras to me, I fulfilled my editorial mission: in the weeks following the conference, several people sent me their writing, some of which you are now holding. Lena, Luke, and Alexis based their articles directly on the talks they gave at CUMC, while Calder gave a talk on integer sequences and sent me an article about something completely different. To the best of my knowledge, he and Lena both wrote about musical applications of graph theory without ever having met or discussed the topic. Mathematics is a small world, and CUMC apparently smaller still. The next iteration of CUMC will be held in Montreal in July 2017. It will be hosted by Université de Montréal, with collaboration and support from several other universities in the city. The chair of the organizing committee is in fact an author in this issue. If you ever need to explain Lie algebras to a little prince on an asteroid, his article should prove invaluable. I personally look forward to seeing how his enthusiasm for mathematics shapes next summer’s conference. More information will available in due time at studc.math.ca. Speaking of things yet to come, my time as editor-in-chief is coming to an end. If you enjoy reading Notes from the Margin and think that you could improve something about it, I invite you to apply to take over. You won’t get paid, but you will learn about obscure LaTeX packages, which is almost as good. More importantly, you will get to help mathematics students share ideas, entertain each other, and develop as mathematical writers. Applications will be available shortly at studc.math.ca. If you have any questions about the role, please feel free to send them to student-editor@cms.math.ca. Cover photo credit: Bruno Pinho edges in our graph. This choice of edges will capture some of the differences between the cultural and creative factors in the music. A simple piece such as Twinkle Twinkle Little Star might only ever shift between major chords a fifth apart: say, from C to F to C to G. In a graph of the piece, edges would only be present between chords a fifth apart. Now we take a look at a more complex passage by Claude Debussy:

Taking only major chords as our harmonies1, we see that Debussy moves from A major to E major – going up a fifth – and then from E major to C major – going

Figure 1

2

1 and not the beautiful-but-too-complex-for-this-example suspensions and 9ths in the passage.

down a major third. This last transition is less common in tonal music than transitions up or down a fifth – we do not see it, remember, in Twinkle Twinkle Little Star. Since Twinkle Twinkle only uses one type of transition while Debussy’s passage uses two, Debussy’s way of connecting major chords is more complex. Modeling the major chords and their relationships in the context of each piece yields the graphs shown in Figure 1. In both graphs, the nodes are the twelve major chords. In the first graph, only chords a fifth apart are connected, while in the second graph, chords a fifth or a major third apart are connected. We could construct a graph for a single piece, an individual composer’s style, a time period in music history, or even a school of tonal harmony: if a certain transition occurs in a piece or style, then an edge between nodes is placed in the graph. Once these edges and nodes are in place, the question of how “close” one harmony is to another in the context of a certain passage of music can be precisely defined. Let be a function which takes a pair of nodes , and returns the nodes’ geodesic : a positive


integer giving the shortest possible distance between the two nodes by traveling along edges in the usual graph-theoretic way. For example, in the Twinkle while in the Twinkle graph above, Arabesque graph, . It is an exercise to show that this geodesic indeed defines a metric on harmonies. Convinced of its existence, we use this metric to formalize our notion of complexity.

Although our previous examples used the set of major chords as our choice of , the study of complexity becomes even more interesting when we choose larger sets of harmonies. Measuring and comparing the geodesic complexity of passages taken from some of tonal music’s masterpieces over various sets of harmonies yields some surprising visually intriguing results.

For a passage of tonal music and a set of harmonies we define the harmonic graph of with respect to to be the graph with a node for each element in . Every time that a harmony occurs immediately before or after some harmony , an edge is drawn in the graph connecting the nodes corresponding to and . If any other pairs of harmonies are equal to the original pair up to transposition2, then an edge is placed between any of these pairs as well.

If is the section shown below of Beethoven’s is the set of all twentyPathétique sonata and four major and minor chords, we get the following harmonic graph and geodesic complexity:

For example, since Debussy connects a C chord to an E, we must also place edges between all other node pairs which are separated by a major third, since all major chords are equal up to transposition. This ensures that when a harmonic relationship is identified, the relationship is respected rather than just the individual harmonies involved the first time that the relationship occurs. We say that a harmonic graph is valid when, for any two nodes and , there exists some path by which we can travel from to .

The opening passage in Bach’s famous Chaconne from his 2nd partita for solo violin has some interesting properties: with respect to the seven chords of the harmonic minor diatonic scale, the passage is as geodesically complex as possible, as shown below. This means that this short passage of music manages to embody all possible types of connections between the diatonic chords in the harmonic minor scale. When is the set of all major and minor chords, the graph is no longer complete.

Now, for the passage , we define the geodesic complexity of the passage as

where is the metric defined by the geodesics in the harmonic graph of with respect to . We say that the closer the geodesic complexity is to , the more harmonically complex the passage. Notice that if all harmonies in are connected to one another in , then the harmonic graph of is the complete graph . with nodes, and we get The word “complexity” in our term “geodesic complexity” can be misleading. This usage relies on the natural but subjective notion that passages featuring close harmonic connections are complex, and vice versa.

Harmonic graphs do not mark the first appearance of graph theory in the mathematics of music (for further examples, see [1]); however, they serve as a unique way to visualize and measure harmonic relationships. Through harmonic graphs and geodesic complexity, we can formally model and study the historical, cultural, and stylistic subtleties which make tonal music such a rich and fascinating form of art.

References [1] Duncan, Andrew. 1991. “Combinatorial Music Theory.” Journal of the Audio Engineering Society} 39, no. 10: 784. [2] Morris, Robert. 2007. “Mathematics and the TwelveTone System: Past, Present, and Future.” Perspectives of New Music 45, no. 2 (Summer): 76-107.

2 That is, musical transposition: shifting all notes and chords up or down by the same number of semitones, rather than a permutation of two elements of a set. The corresponding mathematical symmetry is a translation. 3


Le Petit Prince et l’algèbre Alexis Langlois-Rémillard (Université de Montréal)

Après un vol interplanétaire raté, vous atterrissez sur une tout petite planète peuplée d’un seul petit être qui vous regarde avec de grands yeux. Vous vous enlevez de la tête l’impression de déjà-vu et tentez de vous réveiller jusqu’à ce qu’il vous demande : Alexis Langlois-Rémillard Écrire des mathématiques, c’est raconter une histoire sur des personnages parfois familiers évoluant dans une contrée toujours changeante.

« Dessine-moi une algèbre! » Sans faire ni une ni deux, vous sortez un papier et un crayon et vous exécutez :

moi, ce sont simplement des moitiés de diagrammes.

Soient K, A, + : A ⇥ A ! A, · : A ⇥ A ! A, ⇤ : K ⇥ A ! A respectivement un corps, un

L’unique(4, 4)-lien :

ensemble et trois opérations. Si la donnée (A, +, ·) forme un anneau et que la donnée (A, +, ⇤) forme un espace vectoriel et que la condition ↵ ⇤ (a · b) = (↵ ⇤ a) · b = a · (↵ ⇤ b) tient pour tout ↵ 2 K, a, b 2 A alors la donnée (A, +, ·, ⇤) est une algèbre sur K. Votre mine bien fière en pâtit lorsque le petit garçon vous redemande : « Dessine-moi une algèbre! » Vous vous dites qu’il n’a probablement pas apprécié le faite de prendre K comme un corps et vous apprêtez à modifier votre définition avant qu’il ne vous arrête à nouveau par sa question : « Dessine-moi une algèbre! » Vous pourriez rester bien coi, mais il s’avère que vous connaissez juste ce qu’il faut : l’algèbre des n-diagrammes! Vous lui dessinez ces quelques dessins : (Figure 1) Le petit garçon s’en va, bien satisfait. Vous venez de lui dessiner l’algèbre des 3-diagrammes paramétrisée par sur les complexes. Un peu plus formellement, un n-diagramme est un objet composé de deux lignes sur lesquelles sont disposés équitablement 2n points reliés entre eux de façon à ce que jamais deux lignes ne se croisent. Les cinq 3-diagrammes sont dessinés plus haut.

Diagram 1

Soient n un entier et 2 C . L’algèbre des n-diagrammes Dn est le C-espace vectoriel des sommes formelles des n-diagrammes auquel est

adjoint l’opération · : Dn ⇥ Dn ! Dn définie par concaténation des n-diagrammes avec la multiplication par le paramètre lorsqu’il y a une boucle fermée. L’algèbre Dn est une algèbre associative, unifère, non-commutative et de dimension finie. Ici, sommes formelles veut simplement dire les sommes à coefficients dans C des n-diagrammes vu comme vecteurs de la base de l’espace vectoriel.

Diagram 2

4

Une famille de modules de l’algèbre Dn est définie par les (n, k)-liens. Un (n, k)-lien est un objet formé d’une ligne sur laquelle est déposée n points; de ces n points, k ont un défaut, une ligne qui n’est pas reliée et le reste des points sont reliés entre eux. Entre vous et

; les(4, 2)-liens :

;

et les (4, 0)-liens : L’action de Dn est encore une fois la concaténation avec la multiplication par lorsqu’il y a une boucle fermée. (Figure 2) La dimensiondel’algèbre Dn est le n-nombre de 2n 1 Catalan, n+1 n . En effet, il y a une bijection entre les éléments de l’ensemble des n-diagrammes et à l’ensemble des marches entières de qui ne traversent pas la diagonale. Ces marches sont comptées par le n-nombre de Catalan. La bijection est explicitée dans le dessin qui suit :

« Quel est l’intérêt de cette algèbre? » me direz-vous. Il réside dans le fait que cette algèbre est isomorphique à l’algèbre de Temperley-Lieb, une algèbre très utile en physique statistique.

T L( ) = h1, e1 , . . . , en : e2i = ei , ei ei±1 ei = ei , ei ej = ej ei |i − j| > 2i

Comme les représentations de l’algèbre de n -diagrammes sont amusantes à calculer et qu’elles sont intéressantes pour Temperley-Lieb, c’est le meilleur des deux mondes. Bien entendu, ceci n’est qu’un bref aperçu de la richesse de la théorie sous-jacente à cette structure, mais j’ose espérer que si vous rencontrez mon petit Prince, vous saurez lui dessiner une algèbre!


Graphs, groups, and music theory Lena Ruiz (University of Victoria)

Western classical music organizes musical pitches into twelve notes. The smallest permissible difference in pitch is a semitone, and one octave equals twelve semitones. An increase of an octave corresponds to a doubling in frequency, and notes differing by an octave are considered equivalent. A note is therefore an equivalence class of pitches wherein two pitches are considered equivalent when the quotient of their frequencies is some integer power of 2. Equal tempered tuning is a system in which the frequencies of notes a semitone apart are always in a fixed ratio. Enharmonic equivalence refers to a practice of identifying notes purely by their frequency class. All examples in this paper assume both equal tempered tuning and enharmonic equivalence. The twelve notes correspond bijectively to elements of the group . This correspondence allows many concepts in music theory to be formalized in terms of modular arithmetic. For example, transpositions and inversions of melodic motifs are ubiquitous in the classical repertoire. A transposition raises or lowers every note in a melody by the same number of semitones, while an inversion turns a rising melody into a falling one and vice versa. Musically, these operations form pleasing patterns. Mathematically, they form a group isomorphic to the dihedral group D12 , which is the group generated by the rotations and reflections of a regular 12-gon. Let the transposition tj by the transposition factor j be the function sending each note x to the note x + j, for each j in . Each transposition is equivalent to a rotation of a 12-gon whose vertices are consecutive notes. Let the inversion ik be the function sending every note x to the note k x, for any k in Z12. Each of these inversions is a reflection of the same 12-gon about some axis. Each transposition and inversion is distinct and there are 12 of each, so the set of transposition and inversion functions is the group of all symmetries of this 12-gon, where the group operation is composition of functions. Several other examples involve triads, which in Western music are usually major or minor and form the basic structure of most chords. We need a definition of triads that permits mathematical manipulation. Let a triad (r, s), r 2 Z12, s 2 {−1, 1}, represent the triple of notes (r, r + 3.5 + .5s, r + 7) . This way, (r, s) is a triad with root $r$ which is major when s is +1 and minor when s is -1. Let a triadic transformation be any member of the group of 24! permutations of all triads.

An interesting group of triadic transformations is the PLR group, generated by the operations P , L, and R. Let P : (r, s) 7! (r, −s) , sending triads to their parallel majors and minors: that is, exchanging a C major triad with C minor. Let L : (r, s) 7! (r + 4s, −s) . This is known as a leading tone exchange, and would for example exchange C major and E minor. Finally, let R : (r, s) 7! (r − 3s, −s) , which sends triads to their relative majors and minors. For example, R would send a C major triad to an A minor. Then PLR is generated by P , L, and R.

Lena Ruiz The beauty of math is that it is strong, unbreakable and dependable, yet lies at the core of even the most fanciful art.

The group PLR is isomorphic to D12. The permutations R, RL, RLR , ..., (RL) 12 are all distinct, as are L, LR , LRL, ..., (LR) . Since L and R are both involutions and (RL)12 R = (LR)12 L = e , L and R generate a subgroup of PLR isomorphic to D12. Furthermore, P = RLRLRLR, so PLR is in fact generated by L and R. 12

The P , L and R transformations each alter a single note by one or two semitones, and therefore lend themselves well to parsimonious voice leading for choral music. These transformations also provide a metric on triads where the harmonic distance between two triads is the minimum number of P , L and R functions necessary to transform one into the other.

Hugo Riemann (1849-1919) Neo-Riemannian music theory is named after him, not Bernhard (no known relation).

This is the distance between the triads in Waller’s Torus, a toroidal graph whose vertices are triads and whose edges connect vertices at harmonic distance 1. Its dual is the Tonnetz torus, whose vertices are notes and whose edges connect notes both present in some common triad. We could superimpose these two tori in such a way as to place each note vertex in the Tonnetz torus in the region bounded by vertices in the Waller’s torus that correspond to triads containing that note. We could then add edges between each note vertex and all the triads bounding that note’s region. A random walk in the resulting graph would then produce a composition with a smooth harmonic structure and a conservative melody (that is, the melody would contain no dissonant leaps). An example of a computergenerated walk in this graph can be found at [1]. We could also construct a weighted digraph of chords with edges weighted according to the frequency with which one chord tends to move to another within 5


This was the first Tonnetz, drawn by Leonhard Euler in 1739.

musical tradition. A randomly generated walk on such a graph should produce a composition with local harmonic structure that sounds “average” in the musical context from which the edge weights were taken. Such a composition can be found at [2].

IRCAM (the Institute for Research and Coordination in Acoustics/Music) to contain three Hamilton cycles.

The Tonnetz graph was originally an infinite nonrepeating lattice because it didn’t use equal tempered tuning or enharmonic equivalence. An infinite number of nodes were needed for each note, and the nodes associated with a single note represented different pitches. The modern, finite, toroidal versions of the Tonnetz and Waller’s tori and the decision diagram are all Hamiltonian. (Recall that a graph is called Hamiltonian if it contains a closed loop that passes through each vertex exactly once). While there is no well-known example of a composition that unintentionally used a Hamilton cycle in Waller’s torus in the melody, measures 143-176 of the second movement of Beethoven’s 9th Symphony contain 19 vertices of one. The piece at [3] has been deliberately composed by Moreno Andreatta and Gilles Baroin from

[1] Online Sequencer, sequence 279863. Available at: https://onlinesequencer.net/279863

References

[2] Online Sequencer, sequence 279962. Available at: https://onlinesequencer.net/279962 [3] Gabriele D’Annunzio (lyrics), Moreno Andreatta (music and vocals), Gilles Baroin (graphics). “Aprile: 2D and 4D visualisation of Hamiltonian path.” 2013. Available at: https://www.youtube.com/ watch?v=oazDu9t\_DTk [4] Alissa S. Crans, Thomas M. Fiore and Ramon Satyendra, Musical Actions of Dihedral Groups, The American Mathematical Monthly, Vol. 116, No. 6 (Jun. - Jul., 2009), Mathematical Association of America, pp. 479-495.

The Student Committee of the Canadian Mathematical Society (CMS Studc) is looking for enthusiastic, hard-working students interested in joining the committee. In particular, we welcome applications for the position of Co-Chair, a two-year term beginning in July 2017, and for the position of editor-in-chief of Notes from the Margin, also a two-year term, to begin as soon as possible. Joining Studc is an excellent opportunity to contribute to the Canadian mathematics community. More information about the committee’s activities can be found on our website (https://studc. math.ca) and through social media (https://www.facebook.com/studcCMS). If you have any questions about the committee, its activities, or member responsibilities, please do not hesitate to contact Aaron Berk and Aram Dermenjian (chair-studc@cms.math.ca), the current Studc CoChairs. Questions about the position of editor-in-chief can also be directed to Kyle MacDonald (student-editor@cms.math.ca), the current editor-in-chief. Applications for the position of Co-Chair are due March 31st, 2016. Preference for both positions shall be given to bilingual applicants. Le Comité étudiant.e de la Société mathématique du Canada (Studc SMC) est à la recherche d’étudiant.e.s dynamiques, intéressés à joindre le comité. Plus précisément, nous recherchons des candidat.e.s pour le poste de coprésident.e, un mandat de deux ans qui commence à juillet 2017, et pour le poste de rédacteur en chef de Notes from the Margin, aussi un mandat de deux ans, qui commence aussitôt que possible. Joindre Studc est l’occasion parfaite de contribuer à la communauté mathématique du Canada. Plus d’informations sur les activités du comité étudiant peuvent être trouvées sur notre site web (https://studc.math.ca) et notre page de Facebook (https://www.facebook.com/ studcCMS). Si vous avez des questions sur le comité, ses activités ou les responsabilités de ses membres, veuillez contacter Aaron Berk et Aram Dermenjian (chair-studc@cms.math.ca), les coprésidents actuels de Studc. Si vous avez des questions plus précisément sur le poste de rédacteur en chef, veuillez contacter Kyle MacDonald (student-editor@cms.math.ca), le rédacteur en chef actuel. Les candidatures pour le doivent être envoyées au comité au plus tard le 31 March 2016. Les applicants bilingues seront préférés pour les deux postes.

6


Simple is Elegant Luke Polson (University of Victoria)

A consecutive side-length Pythagorean triple (CPT) is a triple of natural numbers where and . Geometrically, and are the “consecutive side lengths” of a right angle triangle and is the hypotenuse. Consider the first five triples:

How do we produce (20, 21, 29) given (3, 4, 5), or (119, 120, 169) given (20, 21, 29)? In general, how do we produce a “greater” triple given a “smaller” triple? We will first present a recursive formula, which will allow us to move from one triple to another, then use it to obtain a simple solution: a function such that where is the shortest side length of a CPT. From this, and can be easily calculated.

Similarly, we find that We now have three equations:

.

Thus

Luke Polson “If you can’t explain it simply, you don’t understand it well enough.” – attributed in various forms to various mathematicians, physicists, and intellectuals throughout the centuries

We know that the first triple is given by , so we rewrite the matrix equation as:

Consider the recurrence relation Assume that is a CPT. We prove by is also a CPT, induction that where and . To do this, we need to show that are natural numbers. Since is a CPT, is a natural number, and by substitution into the recurrence relation, is thus the sum of three natural is a natural number itself. , so is also a natural number. We finally need to show that is a natural number. By substitution,

This matrix can be diagonalized, leading to the equation

where

We note that numbers, so

By our matrix equation we determine a closed-form solution:

Again, by the inductive hypothesis, is a natural number, so the same holds for . Having determined a recursive formula, we proceed to a closed-form solution. We examine

and

in greater detail. First,

7


The Distractions Page A Coast to Coast Christmas By Tyrone Ghaswala

Tyrone Ghaswala I find entering grades excruciatingly boring. To combat this, I try to find anagrams of student’s names. Sally Sixpack? Pascal’s Kylix! Hey, it’s better than just entering grades without anagrams.

13 nary tarantulas 12 noxious units 11 yellow trolls 10 arrogant bulls 9 boring cats 8 milky bowties 7 naïve bylaws 6 newborn lions 5 non salamis 4 ornamental noses 3 pleated elephants 2 QR codes 1 smelly kitten

Solution : Something’s Amiss The key here is to notice that the clues on the left hand side match up to words on the right hand side, by changing one letter from the answer. Here are the answers and their corresponding words: Ten cent coins – Dimes – Denim, Went around in circles – Swirled – Riddles, Put into a bank account – Deposit – Stomped, Cleverly invented – Innovated – Deviation, Monarch or rule – King – Sink, Tender and nostalgic – Sentimental – Enlistments. Once you have these, the next step is to work out which letters changed. Reading from top to bottom, the letters that changed from the actual answers are SWINGA, and the letters that changed from the given words are NDAMISS. Putting these together you get swing and miss, which is the answer.

You want to keep informed of the latest events and activities for math students? An easy solution: follow the CMS student committee in the social media. Search for CMS Studc on Facebook, Twitter and Google+. Vous désirez demeurer au fait des tous derniers événements et activités destinés aux étudiants en mathématiques? Une solution facile: suivez le comité étudiant de la SMC sur les réseaux sociaux. Cherchez CMS Studc sur Facebook, Twitter et Google+.

Notes from the Margin is a semi-annual publication produced by the Canadian Mathematical Society Student Committee (Studc). The Margin strives to publish mathematical content of interest to students, including research articles, profiles, opinions, editorials, letters, announcements, etc. We invite submissions in both English and French. For further information, please visit studc.math.ca; otherwise, you can contact the Editor at student-editor@cms.math.ca.

Visit us at http://studc.math.ca 8

© Canadian Mathematical Society 2016. All rights reserved.

Notes from the Margin est une publication semi-annuelle produite par le comité étudiant de la Société mathématique du Canada (Studc). La revue tend à publier un contenu mathématique intéressant pour les étudiants tels que des articles de recherche, des profiles, des opinions, des éditoriaux, des lettres, des annonces, etc. Nous vous invitons à faire vos soumissions en anglais et en français. Pour de plus amples informations, veuillez visiter studc.math.ca; ou encore, vous pouvez contacter le rédacteur en chef à  student-editor@cms.math.ca.

Notes from the Margin Vol. XII  

Fall/Automne 2016

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