### Thomas Noll’s Talks at Chicago on Mathematical Music Theory

#### Posted by John Baez

*guest post by Thomas Fiore*

Thomas Noll’s visit to Chicago in June was amazing! His two talks were filled to the brim with mathemusical insights and musical examples. Soon after the talk, I met him again at the International Conference for the Society of Mathematics and Computation in Music, where we heard many other exciting talks on mathematics and music. The field is growing quickly, I was delighted by the huge turnout and work presented.

I’ll report on just a couple of topics in Thomas’ talks in Chicago, he said way more than I can write here. Some topics were mentioned in the thread The Mathematics of Music at Chicago, begun by John back in May.

His first talk, The Triad as Place and Action: a Transformational Perspective on Stability, was about his article The Topos of Triads, its recent extensions, and the PLR-group. The idea is to relate Carl Stumpf’s conception of the major triad to stability. In Stumpf’s understanding of the triad, $\lbrace C,E,G\rbrace$ is viewed not as three notes, but instead as a concord of consonances, i.e., as its 8 consonant ordered subintervals: $C E$, $E C$, $E G$, $G E$, $C C$, $E E$, $G G$, and $C G$. The fourth $G C$ is not considered consonant, and so does not appear in this list. But of course, the fourth sounds consonant. Here is Thomas’ explanation:

Carl Stumpf, in his paper “Konsonanz und Konkordanz”, intends to build a bridge between his tone psychological investigations and Hugo Riemann’s music-theoretical ideas. Music theory considers the primary fourth in counterpoint as dissonant. Stumpf, however, situates the concept of consonance in perception and regards the fourth as a consonance. His main concern is to overcome the traditional critique of the failure to acoustically explain the minor triad. The entire paper is a kind of substitute for the unwritten volumes 3 and 4 of his Tonpsychologie. In order to be precise, I should say, that I borrow Stumpf’s idea of concordance, but I do not follow him with respect to the meaning of ‘consonance’. I use the music-theoretical meaning rather than the perceptual one.”

To mathematize Carl Stumpf’s conception of the triad as a concord of consonances, we consider the circle-of-fifths
encoding of the 12 pitch classes. This is the bijection between the
pitch classes and $Z_{12}$, where $C$ corresponds to $0$, $G$
corresponds to $1$, $D$ to $2$, and so on. The monoid of affine
transformations $Z_{12} \rightarrow Z_{12}$ which preserve $\lbrace
C,E,G\rbrace$ as a set is called the **triadic monoid**. Recall that an
affine transformation (i.e. $x \mapsto m x + b$) is completely
determined by its values at 0 and 1. One can check that the triadic
monoid consists of precisely those 8 affine transformations which
map $C G$ to the 8 consonant intervals listed above. In short, the
major triad is a concord of consonances in the sense that its
monoid-action stabilizer consists of those affine transformations
mapping the perfect fifth $C G$ to a consonant subinterval of
$\lbrace C,E,G\rbrace$.

Thomas calls the triadic monoid $T$ and considers monoid actions of $T$ on sets. The category of such forms a topos, since a $T$-action on a set $S$ is the same as a presheaf on $T^{op}$ which takes the unique object of $T^{op}$ to $S$. (Here we view the monoid $T^{op}$ as a one object category). In other words, the category of $T$-actions is a presheaf topos. The subobject classifier consists of all left ideals in $T$. Thomas showed us how to read off the left ideals from the Cayley graph of $T$, and presented lots of pictures.

Now comes perhaps the most amazing part: Thomas uses this topos to analyze Scriabin’s Etude Number 3, Opus 65. Essentially, he partitions the piece into harmonic cells, considers a $T$-action in each cell, and tabulates the characteristic functions for subactions in each cell. From this data he draws several musical conclusions: the two tones with the lowest truth value are missing entirely from the piece, the paradigmatic tritone oscillation of the piece has the largest non-true truth value, and the stretched triad is always locally present. There is more, but you′ll have to see his article for that. Another place to read more is the paper Triads and Topos Theory by Padraic Bartlett. Back in 2007, he wrote an exposition of Thomas’ paper as an REU project with me.

Thomas’ second talk, Diatonic and Tetractys Modes as Instances of Christoffel Duality, exemplifies the newfound kinship between the theory of musical scales and algebraic combinatorics on words. Here is one point of contact between scales and words. For this discussion, we use the semi-tone encoding of pitch classes. This is the bijection between the pitch classes and $Z_{12}$, where $C$ corresponds to $0$, $C\sharp$ corresponds to $1$, $D$ to $2$, and so on. A scale is simply a subset of $Z_{12}$. (Actually, one does not need the ambient chromatic scale, just the Pythagorean fifth. But we use $Z_{12}$ to simplify this discussion.) As an example of a scale as a subset of $Z_{12}$, we have the $C$-major scale $\lbrace 0,2,4,5,7,9,11 \rbrace$. Notice that consecutive scale steps come in intervals of 2 or 1. The scale step pattern for the $C$-major scale, or any major scale, is 2-2-1-2-2-2-1. So if we write $a$ for 2 and $b$ for 1, we get the word $a a b a a a b$. Thus from a musical scale with scale steps of two sizes, we get a word in the alphabet $\lbrace a, b\rbrace$. By studying these words one can study scales. For example, note that the $b$’s are distributed as evenly as possible amongst the $a$’s if we consider $a a b a a a b$ as a cyclic word. This property of maximal evenness has been studied by Jack Douthett, John Clough, Richard Krantz, and others, and is key in the study of scales.

So I mentioned that a scale is simply a subset of $Z_{12}$. But where does the major scale begin? A reasonable place to begin the $C$-major scale $\lbrace 0,2,4,5,7,9,11 \rbrace$ may seem to be 0 because we often start with 0, but our choice of $C$ as 0 in the semi-tone encoding was completely arbitrary. So that’s no explanation. If we consider all seven possibilities, we obtain the diatonic modes, each of which has its own scale step pattern (we don’t view words as cyclic words anymore).

I would like to highlight a couple of the diatonic modes. The Ionian mode has scale step pattern $a a b a a a b$ and is typically called the major scale. This mode has been declared the winner by history. The Dorian mode has scale step pattern $a b a a a b a$, and is commonly found in Renaissance music. The Lydian mode is also noteworthy, it has scale step pattern $a a a b a a b$. Notice that the step scale patterns for the seven modes are all the rotations of the Lydian mode, that is, the entire conjugacy class of the Lydian word in the free monoid on $\lbrace a, b\rbrace$ is precisely the set of modal words.

But again, where does the major scale begin, that is, which mode is the right one? Thomas Noll answered this question in his talk. In joint work with David Clampitt and Manuel Domínguez, Plain and Twisted Adjoints of Well-Formed Words, he mathematically characterizes the Ionian mode amongst all seven modes. It is a mathemusical justification for the historical winner. Their characterization, called the Divider Incidence Theorem, is based on the fifth-fourth folding of the modes and the Clampitt-Domínguez-Noll extension of Christoffel duality called plain adjoint. It builds on work of Valérie Berthé, Aldo de Luca, and Christophe Reutenauer.

The fifth-fourth folding associates to a modal word a new word in the alphabet $\lbrace x,y \rbrace$ as follows. Take a mode of the $C$-major scale, then rewrite it by starting on the note $F=5$ in the same register, and going up by a perfect fifth $7$ or down by a perfect fourth $5$. We go up or down according to which one remains in the same register as the chosen mode. (It is natural to try to remain in the same register and ignore pitch class equivalence, since our ears our sensitive to register.) We get a scale step pattern for this fifth-fourth folding by writing $x$ for up by a perfect fifth, and writing $y$ for down by a perfect fourth. For the Lydian mode for example, we obtain the word $x y x y x y y$.

Next we need the monoid $St_0$ of **special Sturmian
morphisms**. These are certain monoid homomorphisms $\lbrace a,b
\rbrace^* \rightarrow \lbrace a,b \rbrace^*$ where $\lbrace a,b
\rbrace^*$ is the free monoid on the set $\lbrace a,b \rbrace$. The
monoid $St_0$ is generated by $G,\tilde{G},D,\tilde{D}$. On
generators, these are

$G: a \mapsto a, \; b \mapsto a b$

$\tilde{G}: a \mapsto a, \; b \mapsto b a$

$D: a \mapsto b a, \; b \mapsto b$

$\tilde{D}: a \mapsto a b, \; b \mapsto b.$

We can similarly apply special Sturmian morphisms
to words in the alphabet $\lbrace x,y \rbrace$, just change $a$ to
$x$ and $b$ to $y$ in the definition above. If a word $r$ can be
written as $r=F(a b)$ for some special Sturmian morphism $F$, we have
a canonical factorization $r=F(a)|F(b)$, and the vertical line is
called the **divider**.

Except for the Locrian $b a a b a a a$, every mode can be written in the form $F(a b)$ for an appropriate special Sturmian morphism $F$, so we get canonical factorizations of these modes as $F(a)|F(b)$. Except for the fifth-fourth folded Mixolydian $y y x y x y x$, every fifth-fourth folded mode can be written in the form $F\prime(x y)$ for an appropriate special Sturmian morphism $F\prime$, so we get canonical factorizations of these folded modes as $F\prime(x)|F\prime(y)$. The divider for the Locrian mode and the fifth-fourth folded Mixolydian are obtained by other means. Now, Clampitt–Domínguez–Noll observe that the Ionian mode is the only mode which has the same note under its divider as under the divider of its fifth-fourth folding! So the Ionian mode is quite special in this regard.

To state the Divider Incidence Theorem, we only need
Christoffel words, their duals, and plain adjoints. Let $p$ and $q$
be relatively prime positive integers. The **Christoffel word** of slope
$\frac{p}{q}$ and length $n=p+q$ is by definition the lower
discretization of the line $y=\frac{p}{q}x$, where $a$ means to move
to the right by one and $b$ means to move up by one. For example,
for $p=1$ and and $q=2$ we get the Christoffel word $a a b$. For $p=2$
and $q=5$, we get the Christoffel word $a a a b a a b$, which is the
Lydian word. The **Christoffel dual** to the Christoffel word $w$ of
slope $\frac{p}{q}$ is the Christoffel word $w^*$ of slope
$\frac{p^*}{q^*}$, where $p^*$ and $q^*$ are the multiplicative
inverses of $p$ and $q$ modulo $n=p+q$. So for example, the
Christoffel dual to $aab$ is $x x y$ and the Christoffel dual to the
Lydian word $aaabaab$ is $x y x y x y y$, precisely its fifth-fourth
folding! Clampitt-Domínguez-Noll extend the Christoffel duality
operator $w \mapsto w^*$ to a bijection for each Christoffel word
$w$ called **plain adjoint**:

$(-)^\square : \lbrace \text{conjugates of } w \rbrace \rightarrow \lbrace \text{conjugates of } w^*\rbrace.$

The Clamitt-Domínguez-Noll Divider Incidence Theorem now reads:

**Theorem.** Let $r$ be a conjugate of a
Christoffel word. Then the following are equivalent.

1)
$r=F(ab)$ where $F$ is a composite of $G$’s and $D$’s (no twiddles)

2) The dividing note of $r$ and its plain adjoint $r^\square$ is the same.

So, if $w$ is the Lydian word $a a a b a a b$, then its conjugates are precisely the 7 modes, its dual $w^*$ is $x y x y x y y$, and the conjugates of $w^*$ are the fifth-fourth foldings of the seven modes. The plain adjoint operation assigns to a mode its fifth-fourth folding. We observe that the only mode of the form $F(a b)$ with $F$ consisting only of $G$’s and $D$’s is the Ionian mode $G G D(a b)=a a b a a a b$. And the only mode with divider incidence is the Ionian mode.

So, the Ionian mode is distinguished amongst all modes by this theorem, and the theorem works for any Christoffel word $w$, such as the tetractys or the pentatonic! In the overview article Sturmian Sequences and Morphisms: A Music-Theoretical Application, Thomas Noll lists several other equivalent characterizations of the Ionian mode.

Thomas has placed the slides from the two talks on his webpage with lots of illustrations.

## Re: Thomas Noll’s Talks at Chicago on Mathematical Music Theory

I was at the first of these talks and it was indeed very interesting. I promised at the time to post a review/summary, which, alas, never got written… (I also skipped the second talk, because I had to get back to my day job, working on grid computing for the LHC). Tom Fiore’s post here is much better than anything I could have possibly written. I do have a couple of small comments to add.

The mathematics involved was very interesting and it was really nice to see an example of a “small” topos with a non-trivial subobject classifier which one could easily wrap one’s head around. If nothing else, this was a lovely “toy model” for topos theory - with a nice example of a Lawvere-Tierney topology.

The musical example, a “recomposed” version of Scraibin’s Opus 65, was a nice way to conclude the talk, tying up the musical theory with a real example.

I had one reservation about this approach, however, and that is that it seems to require a complete ‘buy-in’ to the 12-tone, equal-temperament tuning system. The starting point seems to be that you can map a 5th to any other interval by using a mapping of the form ax+b (in Z/12). Of course, if you’re going to be analyzing Scriabin, this is completely appropriate. However (and I realize I risk sounding like a crank by saying this!) I regard the ET tuning system as a bit of a hack, my physical intuition tells me that scales are based on the harmonic series, and I really do prefer the sound of a 5/4 just-intonation major 3rd to the 1.2599:1 ratio you get in ET. I guess that’s why my bass teacher was always telling me to play my 3rds nice and wide! Left to my own devices I play them a little flat compared to the piano, but that’s what sounds right to me. (In my own experimental musical compositions, I sometimes use exotic integer ratios like 11/8, which I like the sound of).

Anyhow, this is not meant at all as a critique of Noll, whose work I think is first-rate and fascinating, and completely appropriate for analyzing 12-tone music. It just didn’t line up with my own Pythagorean leanings, where frequencies and integer frequency ratios are the foundation of musical harmony, not affine transformations in Z/12. I haven’t had time to investigate this, but I was curious whether Noll’s topos-theoretic approach can be applied outside of the 12-tone equal temperament tuning system.