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Claudia Doppioslash
Mind Contortionist and Bit Enchantress.
Mind Contortionist and Bit Enchantress.


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A group in mathematics is an object that measures symmetry in the same way that a number measures quantity.  As the recent paper explains, groups can be used to cast light on neo-Riemannian theory, which is a topic in music theory that gives some insight into progressions of major and minor triad chords.

Neo-Riemannian theory is not named after the famous 19th century German mathematician Bernhard Riemann, but rather after the German music theorist Hugo Riemann, who lived somewhat later.  The three basic operations in the theory, known as P, R and L, transform triad chords into other triad chords as follows.

The P transformation exchanges a major triad with the parallel minor triad: for example, it exchanges a C major triad (C-E-G) with a C minor triad (C-E flat-G).

The R transformation exchanges a major triad with the relative minor triad: for example, it exchanges a C major triad (C-E-G) with an A minor triad (A-C-E).

The L transformation exchanges a triad for its leading-tone exchange: for example, it exchanges a C major triad (C-E-G) with an E minor triad (E-G-B).

The animation shows the notes of the scale on a torus.  The red lines (respectively, green lines, blue lines) connect notes that are a major third (respectively, minor third, perfect fifth) apart.  Each triangle represents a major or minor triad.  Reflecting a triangle about its red (respectively, green, blue) boundary line transforms the associated triad by R (respectively, L, P. )  For example, reflecting a C-E-G triangle in its green boundary transforms it to E-G-B.

The picture of the triangular grid also shows the torus diagram, but unwrapped and displayed as a repeating pattern on an infinite plane.

Two other important transformations in the theory are transposition and inversion.  Here, "transposition" has its musical rather than mathematical meaning, of shifting everything by a fixed number of semitones in a fixed direction.  It may be helpful to describe "inversion" by means of an example.  If one starts with the triad C-E-G and applies an inversion centred at E, we replace the note 3 semitones above E (i.e., G) with a note 3 semitones below (i.e., C sharp) and we replace the note 4 semitones below (C) with a note 4 semitones above (G sharp).  So the result of inverting C-E-G about E is to obtain C sharp-E-G sharp.

It is convenient to index the notes of the scale by the number of semitones that they lie above C.  So C is denoted by 0, D by 2, E by 4, F by 5, and so on.  These numbers are considered modulo 12, so that -10, 2 and 14 all denote the note D.  We denote by T_k the operation of transposition upwards by k semitones, and we denote by I_k the operation of inversion about the note k.  So, for example, T_2 would send C-E-G to D-F sharp-A, and I_4 would send C-E-G to C sharp-E-G sharp.

The paper mentioned above gives mathematical formulae for the operations P, L and R in terms of the operations I_k.  The right hand hexagon in Figure 1 shows the result of applying the P, L and R transformations to a triad.  (Reading clockwise from top left, we have the chords C minor, C major, E minor, E major, G sharp minor, A flat major.)

A mathematically important feature of these transformations is that P, L and R commute with the transformations T_k and I_k.  This means, for example, that the result of applying P and then T_3 to a triad is the same as the result of applying T_3 and then P.  More remarkably, the only permutations of the set of 24 major and minor triads that commute with all of P, L and R are the transformations T_k and I_k.  Conversely, the only permutations that commute with all of the T_k and I_k are the ones that can be obtained as combinations of the P, L and R.

When one of the P, L or R transformations is applied, the input and output triad have two tones in common.  One somewhat annoying feature of the transformations is that the two common tones are exchanged by the operation.  One of the things the paper does is to consider modified versions, P', L' and R' that keep the two common tones in the same relative position.  These are shown in the left hand hexagon in Figure 1.  For example, the C major chord (0, 4, 7) is sent to (11, 7, 4) by L, but to (11, 4, 7) by L'.  This follows the principle of vocal parsimony: moving the notes in each chord by the minimal possible distance.  In this case, we only need move one note in C-E-G: moving the C down one semitone to B.

The paper shows that the group generated by the modified operations P', L' and R' is the dihedral group of order 24, which is the same as the symmetries of a regular 12-sided shape.  There are also some more technical results about group theory in the paper, like Theorem 3.2, which the authors argue gives a theoretical justification for a particular Schoenberg string quartet.

Other related material on this topic includes the recent paper which adapts the theory to deal with pentachords.  +John Baez also posted about this topic last year, and his post may be found at

#mathematics #sciencesunday #musictheory  
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The geometry of music revealed!  The red lines connect notes that are a major third apart.  The green lines connect notes that are a minor third apart. The blue lines connect notes that are a perfect fifth apart.

Each triangle is a chord with three notes, called a triad.  These are the most basic chords in Western music.   There are two kinds:

A major triad sounds happy.  The major triads are the triangles whose edges go red-green-blue as you go around clockwise.

A minor triad sounds sad.  The minor triads are the triangles whose edges go green-red-blue as you go around clockwise.

This pattern is called a tone net, and this one was created by David W. Bulger.  There's a lot more to say about it, and you can read more in this Wikipedia article:

and this great post by Richard Green:

The symmetry group of this tone net is important in music theory, and if you read these you'll know why!
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I've been chewing on this problem for many years: deriving optics from the Yoneda lemma. It turns out profunctor lenses and prisms are related to Tambara modules.
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Digital tools help revitalize rare languages #linguistics
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Red sprites

Far above a thunderstorm in the English Channel, red sprites are dancing in the upper atmosphere.

You can't usually see them from the ground - they happen 50 to 90 kilometers up. People usually photograph them from satellites or high-flying planes. But this particular bunch was videotaped from a distant mountain range in France by Stephane Vetter, on May 28th.

Sprites are quite different from lightning. They're not electric discharges moving through hot plasma. They involve cold plasma - more like a fluorescent light.

They're quite mysterious. People with high speed cameras have found that a sprite consists of balls of cold plasma, 10 to 100 meters across, shooting downward at speeds up to 10% the speed of light... followed a few milliseconds later by a separate set of upward moving balls!

Sprites usually happen shortly after a lightning bolt. And about 1 millisecond before a sprite, people often see a sprite halo: a faint pancake-shaped burst of light approximately 50 kilometres across 10 kilometres thick.

Don't mix up sprites and ELVES - those are something else, for another day:

You also shouldn't confuse sprites with terrestrial gamma-ray flashes. Those are also associated to thunderstorms, but they actually involve antimatter:

A lot of weird stuff is happening up there!

The photo is from here:

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AudioSet: A sound vocabulary and dataset

We are excited to announce the release of AudioSet, a comprehensive ontology of over 600 sound classes and a dataset of over 2 million 10-second YouTube clips annotated with sound labels. The ontology is specified as a hierarchical graph of event categories, covering a wide range of human and animal sounds, musical instruments and genres, and common everyday environmental sounds.

By releasing AudioSet, we hope to provide a common, realistic-scale evaluation task for audio event detection, as well as a starting point for a comprehensive vocabulary of sound events. Explore more at
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Reduction semantics! Evaluation contexts! Abstract machines! Redex! #lang! And so much more goodness! Come discover the +Racket world: ideal for not only PhD students but also advanced undergraduates. We're finalizing the funding; for now we're just soliciting interest: Sign up for notifications. Please forward to relevant student groups.
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