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July 23, 2009

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, {C,E,G}\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: CEC E, ECE C, EGE G, GEG E, CCC C, EEE E, GGG G, and CGC G. The fourth GCG 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 12Z_{12}, where CC corresponds to 00, GG corresponds to 11, DD to 22, and so on. The monoid of affine transformations Z 12Z 12Z_{12} \rightarrow Z_{12} which preserve {C,E,G}\lbrace C,E,G\rbrace as a set is called the triadic monoid. Recall that an affine transformation (i.e. xmx+bx \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 CGC 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 CGC G to a consonant subinterval of {C,E,G}\lbrace C,E,G\rbrace.

Thomas calls the triadic monoid TT and considers monoid actions of TT on sets. The category of such forms a topos, since a TT-action on a set SS is the same as a presheaf on T opT^{op} which takes the unique object of T opT^{op} to SS. (Here we view the monoid T opT^{op} as a one object category). In other words, the category of TT-actions is a presheaf topos. The subobject classifier consists of all left ideals in TT. Thomas showed us how to read off the left ideals from the Cayley graph of TT, 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 TT-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 12Z_{12}, where CC corresponds to 00, CC\sharp corresponds to 11, DD to 22, and so on. A scale is simply a subset of Z 12Z_{12}. (Actually, one does not need the ambient chromatic scale, just the Pythagorean fifth. But we use Z 12Z_{12} to simplify this discussion.) As an example of a scale as a subset of Z 12Z_{12}, we have the CC-major scale {0,2,4,5,7,9,11}\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 CC-major scale, or any major scale, is 2-2-1-2-2-2-1. So if we write aa for 2 and bb for 1, we get the word aabaaaba a b a a a b. Thus from a musical scale with scale steps of two sizes, we get a word in the alphabet {a,b}\lbrace a, b\rbrace. By studying these words one can study scales. For example, note that the bb’s are distributed as evenly as possible amongst the aa’s if we consider aabaaaba 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 12Z_{12}. But where does the major scale begin? A reasonable place to begin the CC-major scale {0,2,4,5,7,9,11}\lbrace 0,2,4,5,7,9,11 \rbrace may seem to be 0 because we often start with 0, but our choice of CC 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 aabaaaba 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 abaaabaa b a a a b a, and is commonly found in Renaissance music. The Lydian mode is also noteworthy, it has scale step pattern aaabaaba 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 {a,b}\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 {x,y}\lbrace x,y \rbrace as follows. Take a mode of the CC-major scale, then rewrite it by starting on the note F=5F=5 in the same register, and going up by a perfect fifth 77 or down by a perfect fourth 55. 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 xx for up by a perfect fifth, and writing yy for down by a perfect fourth. For the Lydian mode for example, we obtain the word xyxyxyyx y x y x y y.

Next we need the monoid St 0St_0 of special Sturmian morphisms. These are certain monoid homomorphisms {a,b} *{a,b} *\lbrace a,b \rbrace^* \rightarrow \lbrace a,b \rbrace^* where {a,b} *\lbrace a,b \rbrace^* is the free monoid on the set {a,b}\lbrace a,b \rbrace. The monoid St 0St_0 is generated by G,G˜,D,D˜G,\tilde{G},D,\tilde{D}. On generators, these are

G:aa,babG: a \mapsto a, \; b \mapsto a b

G˜:aa,bba\tilde{G}: a \mapsto a, \; b \mapsto b a

D:aba,bbD: a \mapsto b a, \; b \mapsto b

D˜:aab,bb.\tilde{D}: a \mapsto a b, \; b \mapsto b.

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

Except for the Locrian baabaaab a a b a a a, every mode can be written in the form F(ab)F(a b) for an appropriate special Sturmian morphism FF, so we get canonical factorizations of these modes as F(a)|F(b)F(a)|F(b). Except for the fifth-fourth folded Mixolydian yyxyxyxy y x y x y x, every fifth-fourth folded mode can be written in the form F(xy)F\prime(x y) for an appropriate special Sturmian morphism FF\prime, so we get canonical factorizations of these folded modes as F(x)|F(y)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 pp and qq be relatively prime positive integers. The Christoffel word of slope pq\frac{p}{q} and length n=p+qn=p+q is by definition the lower discretization of the line y=pqxy=\frac{p}{q}x, where aa means to move to the right by one and bb means to move up by one. For example, for p=1p=1 and and q=2q=2 we get the Christoffel word aaba a b. For p=2p=2 and q=5q=5, we get the Christoffel word aaabaaba a a b a a b, which is the Lydian word. The Christoffel dual to the Christoffel word ww of slope pq\frac{p}{q} is the Christoffel word w *w^* of slope p *q *\frac{p^*}{q^*}, where p *p^* and q *q^* are the multiplicative inverses of pp and qq modulo n=p+qn=p+q. So for example, the Christoffel dual to aabaab is xxyx x y and the Christoffel dual to the Lydian word aaabaabaaabaab is xyxyxyy x y x y x y y, precisely its fifth-fourth folding! Clampitt-Domínguez-Noll extend the Christoffel duality operator ww *w \mapsto w^* to a bijection for each Christoffel word ww called plain adjoint:

() :{conjugates of w}{conjugates of w *}.(-)^\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 rr be a conjugate of a Christoffel word. Then the following are equivalent.
1) r=F(ab)r=F(ab) where FF is a composite of GG’s and DD’s (no twiddles)
2) The dividing note of rr and its plain adjoint r r^\square is the same.

So, if ww is the Lydian word aaabaaba a a b a a b, then its conjugates are precisely the 7 modes, its dual w *w^* is xyxyxyyx y x y x y y, and the conjugates of w *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(ab)F(a b) with FF consisting only of GG’s and DD’s is the Ionian mode GGD(ab)=aabaaabG 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 ww, 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.

Posted at July 23, 2009 8:10 AM UTC

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7 Comments & 1 Trackback

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.

Posted by: Charles G Waldman on July 24, 2009 11:08 PM | Permalink | Reply to this

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

I wonder if the rise of interest in symmetry in mathematics somehow went along with the rise of equal temperament. Symmetry is a wonderful thing, but our quest for symmetry often makes us treat different things as effectively ‘the same’, and some people dislike equal temperament for how it deprives different keys of their distinctive ‘personalities’. (While others, like you, dislike it because they enjoy the sound of a nice fraction.)

The application of topos theory seems to assume you have group (or monoid, or groupoid, or category) of symmetries acting on your musical system, but maybe there’s hope here for non-equal-tempered systems, because in this game you’re allowed to make up subtle symmetries that don’t really map chords to chords that sound ‘the same’ in any obvious sense. The transformation

xax+bx \mapsto a x + b

sends major triads to minor triads when a=1a = -1, after all, and it’s even weirder when (say) a=5a = 5.

A really comprehensive mathematical theory of music should, however, also take actual frequencies and frequency ratios into account, not just a group acting on some set of abstract ‘pitch classes’.

Posted by: John Baez on July 25, 2009 9:13 AM | Permalink | Reply to this

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

Noll’s work can be extended to other tunings by recognizing that 12-tet is just one point on the syntonic temperament’s tuning continuum.

Posted by: Jim Plamondon on September 17, 2009 9:14 PM | Permalink | Reply to this

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

It might also be interesting to repeat Noll’s analysis using a 53-note equal-tempered scale - see e.g. this article on Wikipedia. Not only does this give you a much closer approximation to the Pythagorean intervals, but also since 53 is prime, all of the affine maps x -> ax+b (for a!=0) are all invertible over Z/53.19-step equal-temperament is interesting in this way as well, although as I wrote above, I regard these as more-or-less successful approximations to the “true” harmonic intervals (derived from overtone series).

Posted by: Charles G Waldman on July 26, 2009 1:30 AM | Permalink | Reply to this

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

I’ve been continuing to explore some of the questions raised by Noll’s paper. I still have more work to do here, but I wanted to squeeze out a posting before any more time goes by. I hope there is still some interest in this thread.

I’ve written some small programs to examine the behavior of different chords under the affine transformations. By the way, Python is a very nice language for doing this, you can write things like

 if map(affine(a,b), T).isSubset(T):
    stab[T].append((a,b))

Also, the Python Numeric library makes it very easy to synthesize sounds via sums-of-sines, and with a suitable audio output module you can play them over a speaker so as to actually hear what these transformations sound like. (I can make my code available if anybody wants to see it, I could also supply some sound files for your listening pleasure).

I’ve been playing around more with these transformations, and I’m coming up with more questions than answers. But it’s a lot of fun.

It’s also been very helpful to read Padraic Bartlett’s paper “Triads and Topos Theory”

I’ve also been reading the book “How Equal Temperament Ruined Harmony” by Ross Duffin, which is confirming some of my anti-ET biases. (He’s not an advocate of just intonation (JI) tuning, but seems to prefer some blending of sixth-comma meantone and ‘expressive intonation’ - I’m still reading this book). The main theme is that the standard story that musicians unanimously agreed at around Bach’s time that ET was massively superior just isn’t true… and Bach’s “Well Tempered Clavier” was not in fact written for ET. (WT!=ET). I found this quote from Donald Tovey, which appeared in the 1929 Encyclopaedia Britannica article on Harmony:

“No true harmonic ideas are based on equal temperament”

As I said earlier, I have some reservations of my own about ET, and I also have reservations about other aspects of Noll’s paper - for example, the switching, apparently at convenience, between semitone encoding and circle-of-fifths encoding. To my thinking, a major triad is best expressed in numbers as 4:5:6 (frequency ratios), but leaving that aside, should it be encoded as 0-4-7 (semitones) or 0-4-1 (fifths)? When looking at the behavior of the triad under Noll’s affine transformations, the choice of coding makes a difference. Perhaps one can argue that the 0-4-1 is more invariant - when passing to a 19 or 53 note scale, the circle-of-fifths encoding stays the same, whereas this is clearly not the case with (generalized) semitone encoding.

But does anybody really think of a major third as being composed of 4 stacked fifths? In that case, wouldn’t a major second (2 stacked fifths) be more fundamental? (Instad of four fifths, I think of it as 5/4, pun intended… these little numerological coincidences keep popping up, like the way the 7th scale degree approximates the 7th harmonic - there’s (apparently) no reason for it, but it’s amusing).

If you stack up 5 ET fifths (and transpose down by octaves as needed, we’re always working “modulo octaves”), you get a ration of 1.25992, rather than the ideal 1.25 (a difference of 13.7 cents). But if you do this with pure fifths (3/2) you get the “Pythagorean third” which is 81/64 or 1.265626 (7.8 cents sharp from ET, 21.5 cents sharp from JI), and this interval doesn’t sound very good at all. So, it’s not convincing to me that the major third should be though of as being obtained by going 4 steps around the circle of fifths. The major triad is more easily explained as the successive terms 4:5:6 occurring within the harmonic series.

Note that if you play an E-major chord on a guitar and double the root note on a bass, you are practically playing the harmonic series. If you make the chord into a dominant 7 (jazz it up!) you are playing 1:2:3:4:5:6:7 - the first seven terms of the harmonic series. (7th harmonic is pretty close but not exactly equal to the flatted 7th scale degree - perhaps this explains the ubiquity of the dominant 7th chord in jazz).

I also have reservations about the perceptual/musical meaning of this “neo-Riemannian” approach, ie. affine transformations mod 12, especially w.r.t. the circle-of-fifths encoding. I’ve played some of these chords and moved them around through affine transformations, and it’s not convincing to me that these transformations are related to the way we hear music.

Leaving all these reservations aside, I’m continuing to explore Noll’s ideas, because I still find them very interesting.

I’ve looked at every 3-note subset of the 12-tone scale, and computed their stabilizers (i.e. the set of affine transformations that map them onto a subset of themselves). The sizes of the stabilizers of three-note sets range from 4 to 36. You always have at least 4: the identity, and the 3 affine transformations 0x+b which collapse the 3-note set onto one of its elements. The major triad has a stabilizer of order 8.

When I first started thinking about this, I thought the structures with the largest stabilizers should be special, but these groups of notes do not sound particularly harmonious, they have a sort of unsettled sound. For example: 0,4,8 which is C-E-G# in circle-of-fifths encoding. The other sets with large stabilizers are 1-5-9 (G-B-D#), 2-6-10 (D-F#-A#) and 3-7-11 (A-C#-F) - these are of course the same chord with different roots - root,major 3rd, minor 6th (or sharp 5th if you prefer) - a subdivision of the octave into equal parts, so it makes sense that these are highly symmetric - but I’m not sure about the musical meaning of this fact.

Thinking about it more, there’s sort of an argument in favor of the mid-sized stabilizers. You could just play all 12 notes at once, and this would be stabilized by everything - stabilizer of order 144. But this is very dissonant.

Thinking about geometry, and characterizing geometrical shapes in terms of their symmetry groups (i.e. Klein’s Erlangen program), a sphere in R3 has the largest stabilizer - all of SO(3) - but it’s not the most interesting geometrical object in R3. Neither are completely asymmetrical shapes, which have minimal stabilizers. The things we find most beautiful/interesting are shapes like dodecahedra where the stabilizer group lies somewhere between {I} and SO(3)

In fact Noll’s characterization of the major triad is not that its stabilizer is particularly large, but that the stabilizer relates to the triad in a particularly special way. He constructs a mapping between the stabilizer and the set of intervals occurring within the triad (including all inversions of the triad). Similarly, there is a 2-1 mapping from the symmetry group of the dodecahedron to the set of its edges - given a “reference edge”, that can be mapped to any other edge (with a possible flip, reversing the edge) and that completely determines the rigid motion on the rest of the dodecahedron.

However, Noll’s mapping between the stabilizer and the intervals seems somewhat ad-hoc. One might agree with Bartlett’s assement that Noll’s condition “seems rather stretched” (note, Bartlett goes on to find other justifications for Noll’s characterization of the major triad).

As I stated in a previous post, I’m interested in generalizing this to 19- and 53-note equal-tempered scales, which offer better approximations to the 3/2 fifth than the 12-tone scale. Although this may sound exotic, there’s a a lot of historical precedence - Isaac Newton considered the 53-note scale, but it was far from new in Newton’s time: it goes back as far as Jing Fang (78-37BC) who calculated that (3/2) 53(3/2)^53 is very nearly 2 312^31: i.e. that 53 JI fifths is almost the same as 31 octaves. Of course, 31 octave is far outside the range of human hearing! See wikipedia for more about this, and also this article for information on the 19-note ET scale, which goes back to at least the 16th century - some 19-note keyboard instruments have even been built.

Note that in the 53-note ET scale, there are 2 different approximations available to the major third. It could be composed of 4 fifths, as in the 12-tone system, giving a ratio of 1.26542555968 (407 cents), but it could also be 45 fifths: 1.24898375884 (384.9 cents). In 12-note ET, the major third is 400 cents, but in JI, it is 386.3 cents, so the 0-45-1 encoding may be considered better (it sounds better to me, but I like my M3’s on the narrow side).

The thing is, once you pass to 19 or 53-tone ET, none of the stabilizers are very interesting! The largest ones are of order 6 - as in the 12-tone case, they always contain the identity, the 3 mappings that collapse everything to one of the given notes (0x+b for b in T), and sometimes one or two other self-maps, but this does not seem to generate a particularly interesting monoid/topos. And, choosing the note combinations with stabilizers of order 5 or 6 does not seem to pick out particularly harmonious chords. Maybe 12 really is special? Or maybe Noll’s condition is just a numerical coincidence. (It’s also somewhat interesting that in 12-note ET, the mapping from semitone encoding to circle-of-fifths is its own inverse, which is not true in 19- or 53-note ET).

While researching this topic, I came across Euler’s writings on music theory, which are much more physics-motivated - he’s considering the beating of one wave against another, and the arrival time of “blows”. There’s an interesting paper on Euler and music here

Euler defines a function he calls “suavitatis gradus” (degree of sweetness, however the way he defines it it should really be degree of non-sweetness - higher numbers are more dissonant).

It’s defined as follows: for an integer N, write out its prime factorization as p i k i\product {p_i}^{k_i}. Then SG(N) is defined as 1+k i(p i1)1 + \sum {k_i}(p_i-1).

Next, for a ratio M/N (reduced to lowest terms), Euler defines SG(M/N)SG(M/N) as SG(MN)SG(MN) - this seems a bit odd but it works out that the unison scores 1, the octave 2, two octave or a 3:1 ratio (tritave) scores 3, the perfect fifth scores 4 - intestingly a major 6th (4/3) is rated as more consant (5) than the major 3rd (5/4) which has an SG of 7. This function definition may seem a bit weird and arbitrary, but the cited article gives some justification. There are also some interesting visualizations of this function here and here.

Interestingly, using Euler’s function, the “exotic” ratio 11/8 (which I’ve been using in my own musical experiments) has an SG of 14, which makes it more sonant than the Pythagorean third 81/64, with SG=15. And 11/8 sounds pretty to me.

Now, there’s some difficulty applying this to ET, where all the ratios are irrational numbers! It’s not clear how to compute the SG of the major third in ET: 635/504 is a good approximation to ET M3 (399.9994 cents) but has a high SG according to Euler’s formula - 144. An approximation which is almost as good is 63/50 (400.1085 cents) with SG=20, so perhaps the thing to do is to minimize SG among ‘reasonably close’ approximations… (of course this leaves the question of what is ‘reasonably close’) Doing some computation with all approximations with denominator < 1000, there seems to be a cluster of very good approximations with SG values around 20, so I take this to be the answer… you only get lower than 20 by using a pretty far-off approximation.

Given that SG(5/4)=7SG(5/4)=7 and SG(2 1/3)~20SG(2^{1/3}) ~ 20 (by my method), this would support the arguement that the M3 in ET is just too wide and doesn’t sound as good as it does in JI.

Of course perhaps the concept of SG is completely flawed?

Another way of deciding which intervals are sonant/dissonant is the concept of “limit”, where you only allow ratios where the numerator and denominator (“odd limit”) or the largest prime factor of the numerator and denominator (“prime limit”) are constrained to lie below some fixed value - thus ruling out things like my 11/8. I found this quote from Euler, who apparently was quoting a letter of Leibniz to Christian Goldbach:

“In music we do not go beyond the number 5, similar to those people, who also in arithmetic have not advanced beyond the ternary, and in that German phrase about primitive man of which it is the origin: They cannot count over 3.”

The 20-th century American composer Harry Partch devised a 43-note scale which was in fact 11-limit JI - i.e. all ratios with factors <= 11, thus incorporating my 11/8 ratio, and built his own instruments as well. (I’ve got to listen to more of his music, I’ve only heard a little of it). Perhaps as our culture progresses and we get used to more complex harmonies, this “limit” increases. In pre-Renassaince music the most important interval was the 5th (3/2), and it was a step forward to start emphasizing the third (5/4). This is can be seen as a move from ‘3-limit’ to ‘5-limit’. And perhaps jazz harmony represents a move toward ‘7-limit’.

Getting back to Noll and topoi - there’s one more topic I alluded to earlier: Seventh chords and jazz! Looking at 4-note sets, the Dom7 (most important chord in jazz) has a 12-element stabilizer, and the M7 (classical 7th chord) has an 8-element stabilizer. The sizes of the stabilizers go from 5(minimal) to 48. The intervals present in the Dom7 chord are: unison,M3,m3,4,5,tritone,2,m7,m6,M6 - there are 10 of them, less than the size of the stabilizer, so I’m not sure if the “Noll condition” can be met here. I’m still working on this - I need to go through all 495 possible 4-note combinations, calculate how the note-sets relate to their stabilizers, and most importantly listen to all of them! I’m having trouble finding enough time to pursue all my interests, which is certainly preferable to the inverse situation…

Posted by: Charles G Waldman on August 6, 2009 7:34 AM | Permalink | Reply to this

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

In response to Charles G Waldman’s lengthy post:
1. The way to generalize Noll’s results beyond 12-tet is by recognition that 12-tet, 19-tet, 31-tet, and a host of other tonally-relevant tunings are just points on the tuning continuum of the syntonic temperament.

2. The way to generalize Noll’s results beyond the Harmonic Series (and, indeed, beyond the syntonic temperament) is to recognize that the source of consonance is not ratios of small whole numbers – which is a special case, applying only to harmonic timbres – but rather the alignment of a tuning’s notes with a timbre’s partials. By electronically aligning a timbre’s partials to align with the current tuning’s notes in real time, new musical effects, such as dynamic tonality, are made possible.

I submit that:
- Dynamic tonality offers the biggest expansion of the framework of tonality since jazz demonstrated the emotionally-affective use of the 7th partial a centuy ago, and
- Dynamic tonality ends what Schoenberg described as the “crisis of tonality” by expanding tonality’s framework to include new emotionally-affective sources of tension and release.

It should be noted that Thomas Noll was an advisor to Andy Milne, the lead author of the papers in which the above discoveries were documented.

Posted by: Jim Plamondon on September 17, 2009 9:37 PM | Permalink | Reply to this

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

None of these links work. Maybe you can just type them in like

http://www.example.com/

if you can't fix the markup?

[I fixed them. Jim’s writing a http://www.example.com/. It needs to be a href=”http://www.example.com/” in angled brackets. - DC]

Posted by: Toby Bartels on September 17, 2009 10:33 PM | Permalink | Reply to this
Read the post A Look at the Mathematical Origins of Western Musical Scales
Weblog: The n-Category Café
Excerpt: See how the rational numbers 2 and 3/2 gave birth to the Western musical scale.
Tracked: February 26, 2010 12:26 AM

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