Why is it so difficult to start up an unique band?

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thank you okay thank you very much for inviting me
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and thank you for that kind introduction I'm going to share my screen and
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um and start going okay can everyone see that
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and and you heard that okay so my talk today is called
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visualizing musical structure the problem is that musical knowledge is
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often tacit and intuitive so musicians learn their way around complex spaces of
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musical possibility without being able to articulate that understanding in words so one task for music theory is to
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make this intuitive knowledge explicit and one way we do this is by drawing
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maps that clarify the structure of the spaces that musicians work in as we will
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see these Maps can have Rich mathematical structure so one famous example of a map was drawn by John
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Coltrane in about 1967 it's visually very striking I'm not actually going to
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talk about it at all today so I'd like to you to put it out of your mind but I just bring it up as an example of a
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tremendous you know a world historically great musician who was trying to use
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some kind of visual representation to understand um musical possibilities
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so my doc is going to have four parts the first part we'll talk about the early history of musical visualization
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from about 1700 to about two thousand then I will talk about a period of
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consolidation where we developed comprehensive models of musical structure
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and then we'll talk about a third period that focused on simplifying these models and extracting actionable musical
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information from them and finally I'll give a summary of where we are now
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okay so we'll start with the history of musical visualization I think the
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earliest um ma visual model of musical structure that that I still find comprehensible
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and useful is the circle of fifths which was first proposed by the theorist David Heineken in 1711 the circle of fifths is
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a model of the 24 keys in western music so each of these pie slices here
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represents a key Cedar is German for C major a mall is a minor then we have G
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major E minor and so on around in a circle coming back to C and you will
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notice if you're a musician that the keys here are placed in a different way
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from how we would now place them Heineken has put a minor between C major
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and G major which is not how it this figure appears in contemporary textbooks
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I don't want to talk about that right now though we will come back to it instead I want to talk about some general features of this model the first
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is that locations or points represent complex musical objects in this case they represent musical Keys
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we can model these Keys as scales or collections of seven notes so we can say
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that each point on this graph represents a collection of seven notes
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the circle represents a kind of musical proximity and here it actually represents a two-fold proximity on the
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one hand there's a logical or physical proximity if we move from C major to G major which are adjacent major keys on
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the graph this requires just changing one note uh by one semitone so if you're
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playing a C major scale on the piano and you want to make it into a G Major scale all you have to do is make the minimal
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possible change on the piano so that's a kind of physical or logical proximity
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what's interesting is that the circle off also represents a kind of musical or
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cultural proximity because classical Western musicians often take short
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distance motions on this space so what's exciting here is that this figure can
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maybe explain something about cultural practice by way of a kind of pre-existing logical or geometrical
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structure I'm going to call a model of this kind of voice leading model and the these are mostly what I'll focus on
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today a voice leading for me is a mapping between one collection of notes
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and another and along with a set of paths between them okay so turn the c
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major scale into the G Major scale by raising the note F to F sharp turn the c
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major chord into the C minor chord by lowering the note E to E flat map the c
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major chord to Itself by moving C up to e e up to G and G up to C these are all voice leadings mathematically you can
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think of a chord as being a set of points on the circle and a voice leading as a set of paths on the circle that
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connects the chord at each of its endpoints so this is mostly the kind of
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model that I'm going to be talking about today all right our next geometrical model uh
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was devised about 25 years later by the musician David Kellner it's also a
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representation of the key structure in western music it's also a circle of fifths but here the major and minor keys
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are arranged in the modern format so we have major keys in this uh and minor
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keys in a concentric Circle A and C A minor and C major occupy the same
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angular position and share a key signature which is shown up here so this version of the circle of fifths is the
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modern one that you can find in contemporary textbooks almost 300 years later
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now interestingly there's a third model of key space that was devised a little less than a hundred years later by one
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of my favorite historical theorists Godfrey Weber who was just a very perceptive musician and and really had a
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lot of sort of great things to say about um many many different topics so this is
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his table of the relationship of keys and if you look at it the circle of fifths is vertical here so these capital
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letters are major Keys arranged as fifths and this figure is a Taurus
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because this F sharp major here and this g flat major there are the same on a
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piano keyboard if you take two adjacent strips you have the Kelner circle of fifths here are the minor Keys here are
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the major keys so he's embedded the circular structure of the traditional
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circle of fifths in a toroidal way he's added to the circle of fifths uh an
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extra dimension of possibility which relates minor thirds so from C major here one can move to a minor and also C
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minor why did he do that he did that because culturally in the cultural practice of music making composers often
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do move from C major to C minor and this is on the circle of fifths this is a
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fairly um this is a fairly dramatic move so here's a place where
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the intrinsic geometry seems to diverge from the actual musical practice since
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Faber was a very practical guy he just said well we're going to change the geometry so we're going to come back to
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this this is also a Taurus because you can return to your starting point by moving either
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horizontally or vertically or let's just say it appears to be a Taurus we're going to return to that question now
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moving backwards in time there's another model that's fundamentally important that was derived by the great
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mathematician Leonard Euler in 1739 he called this the net of tones the tone
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net we usually in English we call it the tone Mets using its German form this is
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a totally different model from the others I've shown you because here points represent notes so this F here
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represents the note F in the other models points represented configurations of notes complex musical objects here
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points are individual notes and chords are represented by shapes so if we wanted to represent the F major chord f
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a c we would need to do that with a triangle so an Euler's tone Nets fifths
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are represented by this diagonal line here we have fcgd a e h which is German
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for b f sharp C sharp G sharp D sharp and B which is German for B flat
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okay we have minor thirds in this direction g e c sharp d b g sharp and
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then we have major thirds in this direction so here's a more modern representation of the tone that's where
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I've regularized the three axes we have perfect fifths on the horizontal Dimension we have major thirds on this
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diagonal minor thirds on this diagonal and the simplicial structure here is
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caused by the fact that minor thirds can be built out of major thirds and perfect fits so if you want to go a minor third
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up from a to c we can go down a major third and up a perfect fifth okay so in
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some way you only need two of the axes here which is why we have this this triangular simplicial structure now this
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is a a very different model from the earlier models first because points represent notes rather than full chords
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and second because the kind of proximity that's being modeled here is an acoustic
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proximity not voice leading proximity so F and C are adjacent here not because
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they're near each other on the piano but because they make a consonance
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they sound pleasing or they sound attractive together
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they are represented by frequency ratios if we just look at the fundamental frequency of the vibrating string the
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frequency ratio is very simple it's three over two and that corresponds with this felt perception of blending or
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smoothness that musicians call consonants so this is what Euler was trying to model in the same way the
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major third foreign is also very consonant
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um and and so uh this figure is mostly used when we're dealing with problems of
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tuning and intonation essentially the job of devising a scale or a musical uh
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system is to find a collection of notes that are related to each other by nice
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consonant intervals and what this means is taking a selection from the Infinite
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Space uh represented by Euler's complete lattice so here's a selection that
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contains all 12 notes you can play on the piano keyboard here are the frequency ratios that correspond to that
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selection and the trick here is that we want to make this periodic so we want to
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say that the notes at the top are equivalent to the notes on the bottom the notes at the left are equivalent to the notes of the right and there's
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basically no way to do this while also having all the adjacent notes have the
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desired simple frequency ratio ratio is that you want so all of these lines
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represent deviations from those perfect intervals these Capital these large numbers
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represent frequency ratios that arise from perfect intervals multiplying by three over two or dividing by three over
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two in this direction multiplying by five over four in this direction here you can see that this note cannot be
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tuned so that it's a perfect fifth under this note and we have all sorts of deviations now we adjust every interval
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so that it's equally imperfect and this gives us very good perfect fifths on the
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horizontal axis and noticeably not so great uh major thirds here but we're
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used to it one interesting result from all these models is that even the seven note diatonic scale even the white notes
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cannot be tuned perfectly so here I have the I've built the scale corresponding
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to that little set segment of Euler's lattice and you'll hear that the D minor
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chord is not perfect it's a little wobbly so let me play that
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that's the wobbly one not terrible
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but but it caused a lot of consternation back in the day and there's basically no
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principled way to fix that you're always going to get what's called a wolf interval so this is the kind of problem
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that Euler set out to solve with his tone in that and you know just as a
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little mathematical aside you can compare this kind of structure to what is called field extensions in
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mathematics a field extension is a way of supplementing the rational numbers or
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the numbers in some field with an irrational so the rational numbers are represented by this hollow Q here and we
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can extend the rational numbers by adding the single irrational the square root of two and we create a new and
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bigger field of numbers that contains every rational number a and then also
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um every product of a rational number with the square root of two what's going
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on here in tuning theory is something very similar the octave is represented by a rational number the other perfect
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intervals are irrational numbers and um instead of using rational
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coefficients we use integer coefficients so basically we have like a little lattice here where you can go in the
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rational direction or in the direction of the irrational corresponding to the perfect fifth or the major third what's
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interesting about this kind of space is that it's not a continuous space so you can go a large number of fifths outward
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on the lattice and end up perceptually very close from where you started and the same is true for the major third so
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the trick is to try to find a compact region of the space anyway I suspect that that Euler was well aware of the
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mathematical analogies here between and how you're basically manipulating your rational numbers
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we're now going to Leap Forward 200 years to the next step in our story which Believe It or Not took place in
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1996. my colleague Rick Cohn who's a very very Innovative and perceptive
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music theorist he noticed that Euler's graph could be used not just to
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represent Acoustics but that it could also be used to model voice leading
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proximity and what he noticed is that if you take little steps on
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Euler's tone Nets between triangles which represent major or minor chords these little steps
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move one note by a small amount either one or two semitones
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so this thing that was designed to represent acoustic proximity also
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represents voice leading proximity or The Nearness of chords if you're playing them on an
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instrument like a piano and this was quite an amazing and revolutionary Discovery
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um and it inaugurated a huge burst of Music theoretical activity so Cohn
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published this in 1996 within two years uh you had douthit and Steinbach
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representing you know devising other graphical representations of the relations among different chords here is
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Ed Goins 1998 attempt to expand the tone Nets to a new
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um in a new direction to include sevenths here's a graph that I made
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representing voice leading relations among scales so there's Oodles of these graphs it's a little cottage industry
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where everyone could make a little lattice of their favorite musical objects and what they've left us with
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there's a wildly heterogeneous set of models each tailored to a specific musical circumstance and all of them
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looking completely different from all the others so the question is could we bring any
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order to this situation and that brings me to my second the second part of my
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talk where I'm going to be talking about the development of comprehensive models of musical structure and the attempt to
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unify all of these different models into a single um into a simulated structure
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I'm not going to model I'm not going to monitor the chat but if anyone maybe uh Monday if you're moderating if you see a
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question in chat that needs to happen now you can you have permission to interrupt but otherwise I'm going to
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plow forward and we'll deal with the chat afterwards okay okay so this part of the story starts
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with Cliff calendar uh my friend and collaborator in 2002 he described a
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two-dimensional continuous space that represented all possible three-note chord types so this is a more abstract
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graph than the ones we looked at because a point here doesn't represent a specific chord like C major or D major
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it represents every major chord it's a little more abstract and actually it's
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it's even more abstract than that this point here represents major or minor chords those are those are upside down
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versions of each other so they can sort of be thought of as the same thing nevertheless it is completely
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comprehensive in the sense that any three notes you can play not just three notes on the piano but even three notes
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that that fall between the cracks of the piano uh occupies some point on this
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graph so it is it is truly that it's the set of configurations of three notes on
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the circle modulo reflection and rotation if you want if you want the mathematical name for it but musically
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it's all the possible chords that you can possibly think of and so this really
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attracted a lot of attention because it had the kind of generality that people felt uh should be part of our discourse
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okay within a year Rick Cohn who's the one who noticed the the fact about the tone Nets he came up with a a
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tetrahedral model of four note chord types which is shown here Ian Quinn
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um Ian Quinn this is very funny he worked he didn't work with computer Graphics he worked with construction
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paper and so here is his model of four note card types this actually I've had
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these on my desk for about 20 years and you can see they're a little worse for the wear but but here are his model
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um Ian started making really important connections to mathematics uh he he
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noticed that there's some kind of quotient operation here he noticed that it's really important to include
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multiceps in these graphs multi-sets are chords that have duplicate notes like cceg and traditionally music theorists
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have ignored those but Ian realized that that they play a crucial part of the story he didn't publish anything during
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this period but he was talking to all of these people and really contributing to this investigation other important
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people here include no melkies the uh well-known math professor at Harvard and Rachel Hall a math professor here at St
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Joseph's in Philadelphia and these people are both great musicians as well as excellent
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mathematicians I would say there were three main obstacles to generalizing the
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coming up with a generalized framework for musical geometry one was purely Musical and involved understanding the
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underlying musical objects of study so this involved defining the notion of voice leading in a rigorous way for both
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chords and chord types another was learning to a or those musical assumptions and
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Concepts like cardinality equivalents that give rise to pathological geometries I talked about how musicians
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habitually did not distinguish cceg from ceg and turns out if you if you group
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those together you end up creating a giant geometrical mess so learning not
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to group those together was very difficult because it sort of went against the grain of a lot of music theoretical training
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uh another challenge was trying to find analogs to these graphs we've looked at
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that represented chords in their particular particularity rather than chord types so all these recent models
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were chord types but but really we'd like to have all the different chords in
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in different locations finally there was a purely geometrical challenge of
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understanding the geometry that resulted here what do paths mean what's the significance of the
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duplicates that seem to inevitably appear on the boundaries of these spaces how do we deal with singularities that
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act like mirrors now none of this is really hard mathematically the real challenge here was getting these two
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totally different languages the language of music and the language of math to speak to each other productively okay so
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there is this period of success um uh there's a paper I wrote called
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scale Theory serial Theory and voice leading where I think I first really Define voice leadings in a very in in
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what I would think of as the most fruitful modern way this was published later but written before there are two
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papers in science one I wrote called the geometry of musical chords one I wrote with uh Cliff calendar and Ian Quinn
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called generalized voice leading spaces and then I wrote this all down in my book my first book a geometry of Music
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which kind of is a comprehensive introduction to this new way of thinking and the framework here is is really
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pretty simple we start by situating a chord with endnotes in n-dimensional
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Cartesian space so if our chord has two notes we can put these two Notes on The X and Y axis here so a point in this
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two-dimensional space represents two notes happening at the same time one x
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axis Note One Y axis note if we want three note chords we need three dimensions here I have trumpet sax and
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trombone on the X Y and Z axes respectively so we start with euclidean
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space and a point representing you know a note on each axis if you look at the
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space that results it's periodic and has the structure of a piece of wallpaper so
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over here is a representation of um of a piece of wallpaper with um
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these different tiles where horizontally and vertically adjacent tiles are upside
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down relative to each other over here I wrote out every you know big collection
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of two note chords uh in a kind of obsessive way and this space of two note
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chords actually has the same abstract structure here it's got these four quadrants and quadrants are upside down
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relative to their horizontal and vertical neighbors okay uh then what we
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do is we glue together points in this space that are equivalent according to Conventional musical terminology or
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symmetries and what this amounts to doing is representing music on a single tile of wallpaper rather than
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inefficiently using multiple tiles so in my book I have the example of an ant
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that is walking across the wallpaper and we can represent the ant's trajectory on
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a single tile okay and so what happens here is quite interesting as the ant crosses Tiles at Point Alpha here it
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moves off the lower at the lower left edge of the tile and
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it reappears on the upper right edge of the tile right because this is the point
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that is analogous to this point over here and what that means is that this space has the structure of a Mobius
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strip its left Edge is glued to the right Edge upside down that disappearing
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off the lower left and reappearing off the upper on the upper right is the Hallmark of a kind of of a Mobius strip
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what happens over here a beta is even more interesting because as the ant crosses tiles here instead of
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reappearing on the opposite Edge it actually appears to bounce off okay so
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it looks like Point beta this upper boundary has the structure of a mirror and that means that this is not a Mobius
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strip it's actually something more complicated because it's got singular points in it that act like mirrors I
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just want to remind you that the logic here is in fact the musical logic of two note chords so so the simple wallpaper
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example is showing you the different kinds of singularities we actually confront in news
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okay so these spaces are are what's called quotient spaces or orbitals in
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these spaces points represent objects like chords paths represent Voice leadings or mappings between the chords
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and Boundary points can be associated with specific musical Transformations so
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uh in this case the um the horizontal boundary at the top
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and bottom can be associated with what musicians call Voice exchanges while the left and right boundaries can be
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associated with what musicians call transposition along the chord so I'm going to come back to that later in the
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talk so the cool thing is in two Dimensions we can actually just write this thing
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down we can come up with a an interactive model that contains all possible two note chords you can go to
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my website madmusical science.com mobius.html you can click on it
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and kind of travel around in this space [Music]
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okay foreign
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as we saw the Mobius strip has the structure sorry this space has the
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structure of a Mobius strip which means that if you disappear off the left Edge you reappear on the
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right but lower is glued higher and vice versa um so let's
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in the same way there are paths in this space that seem to bounce off the upper
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and lower boundaries as if they were mirrors sorry that was not a good one
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I should choose this one
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called voice exchanges they they exchange the two notes of the chord
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Okay now what's cool about this space is there's one of them for every scale so
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if you want to work just with the white notes on the piano keyboard
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you can do that the underlying field of pitches matters much less than you might
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think and if you want something really sort of obscure and weird if you happen
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to be a kind of uh perverse musical mind foreign
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you can create microtonal scales that that involve fractional values here that
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lie between different piano notes so so the underlying geometry is kind of
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completely agnostic about what your scale is now
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in the same way we can get one of these things for a uh for a three note chord
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let me pick that up [Music] um
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so here's the geometry of three note Chords it's a three-dimensional space and any three notes I play on my piano
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[Music] Devon they determine a path in the space
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and you can zoom in you can see the labels of the things you could rotate them and and so on and
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so here we have a real tangible model of all possible three note musical chords
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okay so this was an a kind of success this
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gave us a uh it subsumed and unified all these pre-existing models of voice
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leading structure the circle of fifths the tone that's considered as a voice leading space
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these various models of chord types which can be viewed as projections of the spaces we were looking at and many
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others it also revealed some deep principles that underlie musical
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practice I'm trying to get the participant screen out of my way here um
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including the importance of symmetry and near symmetry so I was very proud of
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being part of this work along with so many others and I really felt that that we reached a plateau of new
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understanding of music uh and it also showed us not just the importance of symmetry in general but it
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highlighted a particular set of symmetries that really are important throughout music on the other hand
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there's lots of problems here um the chief among them being that musical geometries are generally High
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dimensional right you need one dimension for every note in your chord so two Dimensions two note cards you can do
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great three note chords you can sort of do with computer Graphics but once we start getting into four note chords five
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note chords seven note scales which live in a seven dimensional space that becomes pretty hard to visualize and
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understand and remember the whole point of this Enterprise is to try to provide simple and intuitive pictures that
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musicians can use I would say because of this complexity the geometry did not obviously suggest novel musical
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directions and in fact some people rightly felt that there's something a little bit tautologist here the the
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geometry said okay chords near each other on the piano are near each other in some higher dimensional space in some
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sense it just translated what we already knew from the piano into this alternative and equivalent High
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dimensional form and and so while I was proud of all this work and super exciting I recognized that for Outsiders
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especially but even for me as an Insider these structures were a little bit too
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unwieldy to be able to manipulate intuitively when you're dealing with a
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even a three-dimensional singular geometry just sort of drawing pictures and looking at lines is not going to
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lead to Musical I understand so that leads to part three of my talk which involves the attempt to simplify these
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models and extract from them actionable musical information this is all material that's coming out in a second book
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that's in production with Oxford University press the general approach here is to avoid representing the higher
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dimensional spaces and all that complexity and to look for more abstract or symbolic alternative representations
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and here the inspiration comes from topology which tries to disregard
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geometrical detail in favor of a much more General conception of spatial
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structure this is a world in which the donut and the coffee cup are the same shape so so we're going to be looking
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for a musical analog of that kind of way of thinking
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more specifically we're going to look at one type of chord at a time not show all
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possible chords but just the major Triads we can if we want you know look at a
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handful of chords there's no problem looking at two at once but but instead of looking at everything we're also
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going to eliminate some of the Transformations linking chords and in particular we're going to link The Voice
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exchanges that just sort of inertly swap two notes in accord
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um okay and the the reason for doing this is that it really simplifies the picture
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you end up with and here we're following in the footsteps of the enormously perceptive and enormously problematic
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music theorist Heinrich Schenker Heinrich Schenker was a flat out racist who had a lot of
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um terrible ideas about music but also had some deep insights into musical structure and one of his Str insights
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was that these voice exchanges can be considered decorative uh it turns out
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mathematically they're they're what's called a normal subgroup and that means we can potion out of that out by them
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without without problems okay so the upshot is where we end up if we do all
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this work we end up with a set of two-dimensional models that show the basic voice leading capabilities of any
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chord in any scale these models can be extended so that they cover multiple
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chords and they focus our attention on particular relationships that can be otherwise very hard to see and this is
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really the dream of musical visualization I believe it leads to actual new musical ideas and techniques
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one of the most interesting features of these representations is their universality you have the same graph for
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chord and scale regardless of their internal structure the only thing that matters is the size of the chord and the
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size of the scale so that's really cool that shows that that is a manifestation
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of this topological perspective where little differences of musical structure don't really matter what really matters
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is you know how many notes you're dealing with at both the chord level and the scale level so let me show you where
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these things come from here's the Mobius strip we looked at earlier that represents all possible two note chords
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chords that sound the same the major thirds
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these are all found on horizontal lines a pair of horizontal lines whose left
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Edge is glued to their right edges but with a Twist so here's what that gluing
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looks like I've taken this dotted line you can follow it around and glue it here and you travel around the solid
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line glues you there so it's pretty clear that this structure
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here can be rationalized and regularized as this
34:48
um spiral diagram here here so where we're going to replace the full Mobius
34:53
strip with this two-dimensional spiral diagram that shows the voice leading
34:59
relations of all possible major thirds found in the chromatic scale once again you can go to the to my website you can
35:07
find these things you can plug in let's say show me c and e in the 12 note
35:12
chromatic scale press draw here's my graph foreign
35:21
and you can play around with the space and and and and just explore it and it's
35:27
much simpler than the full Mobius strip okay with three note chords the miracle
35:34
is basically we just do the same thing here's that three-dimensional space I was showing you earlier the major chords
35:41
live on these three line segments here the top is glued to the bottom with a 120 degree rotation that gives you a
35:49
triple spiral here where um each time you go around a loop you go
35:54
outward by one spiral Okay so we've added a dimension to our chord but we're
36:00
still dealing with a two-dimensional representation and that's true no matter how many notes
36:06
you use here's the circle of fifths we started with this is the Kellner Circle right c g d a e are all next to each
36:14
other here is the spiral diagram representing seven note scales in a 12
36:20
note chromatic collection and you'll see the circle of fifths is right here c g d
36:25
a e and again if you want to you can go to my website
36:33
there we just added the sharp to go from C major to G major foreign
36:41
and you can play around with this structure so this thing is actually a
36:46
line that winds through a seven dimensional orbifold but it's no more complex than than the two-dimensional
36:54
representation that represents two note chords what's really cool is I did not I did not put anything in here I just told
37:00
my computer literature of a major scale that you can play on the piano and this
37:07
circle of fifths Arrangement that the earlier theorists sort of built in by hand it just falls out automatically
37:14
it's it's just a consequence of the geometry um um Okay so
37:21
chord space of every size has one circular Dimension when we are ignoring voice exchanges it turns out the other
37:27
dimensions don't matter very much they're just a big euclidean ball with a trivial homotopic group and homotopia
37:34
classes are what's really important here so we can use these models to understand
37:40
a huge range of musical practices let me just give you two examples this here is
37:46
the uh is the model of two note chords two note thirds diatonic thirds in the
37:53
white note scale so uh notes with one white notes with one white note between them as well as their upside down forms
38:01
if we just toodle around this space foreign [Music]
38:15
we start making something that sounds kind of like classical music [Music]
38:24
and I believe that a lot of classical syntax comes from this two-dimensional
38:30
geometry we tend to think of classical music as built around three note objects
38:35
that we call Triads but I believe that there's a whole hidden layer of musical
38:40
thinking that actually exploits this two-dimensional geometry and I talk about this in this new book I'm I've
38:46
just finished perhaps even more surprisingly is if we take the space that represents major
38:53
chords that you can play on the piano so major chords in chromatic space and just
38:59
try sort of noodling clockwise if we just move clockwise here
39:09
we start to find all the progressions of classic rock so that what I just played
39:14
is Neil Young's helpless along with a zillion other songs
39:23
that's Sympathy for the Devil and a zillion other songs
39:29
that's Pictures of Matchstick man
39:35
sorry that's eight days a week
39:43
that's the air near my fingers by The White Stripes ah and then
39:49
this
39:55
is Hey Joe made famous by Jimi Hendrix so I believe that CLA that rock
40:02
musicians have internalized the structure of possibilities that's shown by this
40:09
diagram so when they are intuitively composing they are navigating a
40:16
possibility space that is really well captured by this simple diagram a lot of
40:22
these progressions are not progressions that are allowable under
40:27
um uh traditional the rules of traditional Western tonal syntax but they're incredibly natural they involve
40:34
ways of combining simple descending Melodies with harmonies that sound
40:39
similar and are closely related and so you can build a kind of grammar of
40:44
intuitive Rock practice out of this geometrical model and you really start
40:51
to understand the structure common to a lot of these
40:56
progressions and even the relations among particular songs so that's a good example those so we have one space that
41:03
we can use to understand classical music another space that we can use it to understand rock music in both cases a
41:09
certain kind of clockwise motion is is important because it represents descending Melodies which are kind of
41:15
musical Norm so that's something we can do with this space that actually increases our understanding we can also
41:22
solve some more complicated theoretical questions so Rick Cohn says three note chords have a toroidal geometry I say
41:29
they have an annular uh geometry which one of us is right
41:36
well it turns out that if you if you look at things very carefully one of the
41:41
loops on the tone Nets is musically inert so so cones tone Nets which seems
41:48
on the surface to be a Taurus actually um has a secretly amular structure
41:53
because if you go along these solid line here you actually end up exactly where
41:59
you started and what that means is that that direction is what's called a
42:04
contractable loop so here we start in C major and if we just so here we have
42:09
these notes c e g in exactly these positions and if we just go in a loop
42:16
like this we end up with those notes exactly in the same position so we haven't done anything
42:22
by contrast if we take this Loop
42:28
we have moved every note up by one so this C used to be down here and it's
42:34
moved up by one that Loop actually has a function so that's why of one way of
42:40
adjudicating between the Taurus model and the spiral model it turns out that
42:45
some of these paths on the Taurus are inert and when you eliminate the inert paths you end up with my spiral
42:51
something very similar can be said for the Taurus that Weber used to represent
42:56
keys in western music remember Heineken Kelner and I all say that the geometry here has a annular structure a circular
43:04
structure it's not a Taurus who's right well once again it turns out that one of the loops on Weber's Taurus
43:11
is actually musically inert if you if you take these solid lines here you
43:16
actually end up mapping C to C double sharp which is otherwise known as D and
43:21
that's a real genuine change if you follow the dotted lines you engage in
43:26
this series of Transformations here you map C right back to where it started this Loop doesn't do anything it's
43:33
homotopically equivalent to the point it's trivial it's not musically meaningful the underlying structure here
43:40
is not the structure of a Taurus it's the structure of the Spiral Kelner and Heineken right here however you don't
43:48
want to stop there because Weber actually made an important point he said that a minor is close to both a major
43:54
and C major and that's why he abandoned the spiral and he's certainly correct
43:59
musically that a minor is perceived as being close to these two other Keys these keys are not near each other on
44:07
the circular diagram so so here it's not enough to say well you know topology shows that you're wrong you kind of have
44:14
to go one step farther and and come up with a way of representing the
44:19
relationship that that labor was concerned with and it turns out that if we if we make a
44:26
spiral diagram um that contains that more accurately
44:31
represents minor Keys we can find out what's going on the the issue here is that every minor key is actually
44:37
represented by three different scales three different seven note collections this is a slightly more abstract version
44:44
of my spiral diagrams that that collapses every spiral to a circle which you can do in some circumstances and it
44:51
turns out that the D Minor T is related by three D Minor scales there's what we
44:57
call natural minor harmonic minor and melodic minor these are all different objects and here I've connected the
45:05
objects that represent a single Key by dotted lines here I've made them into a
45:11
series of concentric circular arcs and here I've abstracted this picture so I
45:18
now have two arcs the upshot is when you think about how this minor key is
45:23
represented by scales you realize it's actually represented by multiple scales
45:29
and and a minor key is a spatially extended object so it's not that the
45:35
keys have a toroidal geometry it's that it's wrong to try to localize a minor
45:41
key in a single point and here you can see the a minor scale stretches so that
45:46
it's kind of close to C and kind of close to a and I think that's the answer
45:52
to Weber's problem um and and this is an example of the kind of deep and subtle modeling that we
45:59
can do once we're thinking rigorously about musical geometry I'm just going to show you this picture because it's cool
46:06
this is a two-dimensional model of the space of chord types these are the kinds
46:11
of things that we started with started section two with the cliff calendar was making the tetrahedron that Rick Cohn
46:18
was making we can represent their musical content with simple polygonal
46:24
graphs so this Square represents Rick cone's tetrahedron this Pentagon
46:30
represents the four-dimensional space that that models all possible types of
46:37
five note chord these are higher dimensional highly singular super
46:43
complicated geometries uh it's kind of hopeless even with the four note tetrahedron to really understand its
46:50
musical content but we can break it down to this more topological representation
46:55
of its underlying musical content and actually start to understand what's going on with these spaces again you can
47:02
go to my website and play with these things so what's cool is we are now in the position where we can evaluate models
47:08
that are more than 300 years old we can make choices between the circle of fifths and the Taurus of keys we have a
47:16
family of very similar two-dimensional diagrams that represent contrapuntal relations among a very wide range of
47:21
chords and scales and these are the sorts of simple surveable maps that musicians can actually use
47:28
there's no mysticism here whatsoever we're just doing math there's a little bit of a joke because this is a model
47:34
you can interact with on my website and it actually looks a little bit like an occult diagram that represents the
47:40
Philosopher's Stone so so I know mysticism diagram is is is actually
47:46
an ancient occult symbol but it actually has sort of strict musical content there
47:52
so now I wanna I wanna end this slightly extended talk by explaining where we are now what does it
48:00
all mean okay so what is the take-home message from all this geometry what can
48:05
it actually teach a musician and the answer is
48:13
wait for it [Music]
48:23
the answer is that music has one more hierarchical level than we think
48:28
a huge amount of implicit musical knowledge involves moving along
48:34
collections so we're familiar with this in the case of scales we have doe a deer
48:41
where we take a pattern and move it along the scale
48:46
but it turns out that we can do this with any collection okay and the idea of
48:53
motion along a hierarchically nested set of collections is really intrinsic to
48:58
the fundamental geometry of music and the basic message of that geometry is that chords act like little scales so
49:05
just as we can do where we kind of take a similar pattern
49:11
and move it along a scale we can also do that along a chord
49:20
or we can really do this kind of motion with any collection okay so here is the
49:29
picture that that uh I think you end up with here we have a nested set of
49:36
musical collections at the top here we have the c major chord
49:42
and if you move systematically left and right you move along this collection here's up
49:48
here's down here's up two down one
49:55
okay so yeah that's motion at this level this guy is then embedded in this guy
50:04
where you have the same set of possibilities here's up two down one
50:09
here's up here's down okay and this guy is embedded inside the
50:16
chromatic scale here's up two down one
50:26
so in each case you have the same set of processes available at these multiple
50:32
hierarchical levels patterns up here are most naturally expressed in terms of numbers that refer to these guys
50:39
sequentially so up two down one plus two minus one is basically most naturally
50:45
represented by operations at what I call input scale degrees the this the chord can be thought of as
50:52
a little mapping that turns input degrees into output degrees these output degrees then become the input degrees of
50:59
this next collection where you have a similar set of models here so we have things like translation and reflection
51:06
available at input degrees at multiple levels okay and this is actually a
51:12
computer architecture what I'm saying here is that a musical collection is a recursive object that can be nested
51:19
inside itself so almost every computer representation of musical objects if it
51:25
works like this at all it Maps input degrees directly into notes we can hear
51:31
I'm saying that's actually not how Music Works the way Music Works is every collection Maps input degrees into
51:37
output degrees that then get mapped into the input degrees of the next level and
51:42
so you can Nest collections inside each other and I've actually built a little python nestable scale object that allows
51:49
you to do this and what that means is you can have all these different processes which might be structurally
51:54
similar happening at the same level uh simultaneously and that's where you
52:00
start to get things that sound like music so a little terminology I call this thing the intrinsic scale it is the
52:07
scale formed out of a chord's own notes I call this thing the extrinsic scale
52:13
which is the scale form like it's the scale that contains our scale that's what we normally think of as a scale the
52:20
big idea here is that every chord is its own little intrinsic scale my dream is to be able to think
52:27
musically using arbitrary hierarchies of arbitrary collections to sit down at the piano choose some chord I've never
52:33
played before embedded in some five note scale I've never played before and start to move fluently within that new musical
52:40
space it's really hard to do but it's not impossible to do Okay so
52:46
every scale defines its own notion of translation which musicians call
52:52
transposition so if we take just the c major Triad and look at these two Notions of
52:58
transposition what do they look like
53:04
so here what we've done is we've taken the c major uh chord and we move it up along the white note scale here that's
53:11
transposition along the extrinsic scale here we move it up along itself
53:21
that's intrinsic transposition and now the thing is these things these two operations get combined sometimes they
53:28
combine to produce a near identity transformation so there are objects on
53:33
this space that are quite close together even though they're not close together in this representation now the spiral
53:40
diagrams I showed you earlier represent the combination of hierarchically nested
53:46
transposition so if you slide along the diagram that's transposition along the the big scale the extrinsic scale and if
53:54
you Loop along the diagram that's transposition along the intrinsic scale so let me just show you that here we go
54:02
so if I um if I slide downward
54:09
okay everybody um
54:16
everybody Moves In Parallel along the scale and meanwhile if I Loop I move along the little chord
54:26
now and what this graph is showing us is proximate combinations of those two so
54:31
it turns out if I if I do slide down here and then counteract that with a loop
54:39
voices those two Transformations counteract each other and the voices don't move by very much and that's the
54:45
key to a lot of musical activity now here I want to say my friend and colleague Jay Hook back in 2003 he
54:53
emphasized the importance of hierarchically nested transposition and he really jump started my interest in
54:59
music theory in this way I now think he really put his finger on the Core lesson of musical
55:04
geometry so I want to make sure I include him in my Pantheon here I'm going to end now with a little lightning
55:10
tour of what we can do with this kind of new way of thinking the first idea is
55:16
that musical objects which we call motives often Move Along Accords intrinsic scale here's an example from
55:22
Louis Armstrong
55:27
okay so Armstrong takes a little motive here
55:33
that's his motive [Music]
55:39
into the E flat major Triad and it moves down along the E flat major Triad what's
55:45
cool though is that the first note doesn't belong to the E flat major Triad at all it actually is a what's called a
55:51
neighbor note that belongs at the level of the scale so here I've represented this with a Network that has two kinds
55:59
of transposition in it and um we can actually represent it on the
56:05
hierarchy in this way I have to change Keys here but these are the harmonic notes at this
56:13
upper level and then here's the neighbor note at a different hierarchical level [Music]
56:20
and that figure moves down
56:26
so the idea that musical objects are constituted by
56:31
hierarchical networks of intervals is a really new and cool idea I've built a
56:37
little calculator that actually allows you to play with these relationships so I've built a Syntax for musical motives
56:47
you can play around with these ideas because it's actually hard to do it with your brain
56:52
a very similar process happens in harmonization so this is kind of normal
56:58
Episcopalian him tune Harmony [Music]
57:03
What's Happening Here is this melody is moving systematically along the scale
57:08
these inner voices are shadowing the melody in parallel along a chord that is
57:14
changing so once again we have a hierarchical network of intervals we can represent the melodic note as moving at
57:20
this hierarchical level it's giving rise to a chord that is then being shadowed
57:26
One Step lower at this chord level and what makes this tricky is that this guy is actually changing
57:34
here's that Beatles song eight days a week I talked about earlier I'm gonna just play it here
57:42
[Music]
57:54
okay so what's going on here is we have a set of chords that can be read as
58:00
always descending
58:07
so we actually end up in a different place from where we started we don't have a Melody but systematically hopping
58:13
back and forth between two notes of the scale
58:19
[Music] foreign the melody then leaps up one
58:25
extra scale so three two three two two
58:30
three four but what happens is the melody ends up
58:36
exactly on the same note that it started even though it's a different scale
58:42
degree at the level of at the top hierarchical level so what we have here is perpetually descending motion at the
58:49
level of the chord that is counteracted by ascending motion at the level of the melody to create
58:56
a kind of psychologically complex repetition if you think about this naively it's
59:02
just repeating repeating repeating but the reason that repetition is interesting is that it's it's actually
59:09
more subtle you have ascent and descent counteracting each other this happens all over music
59:15
and it leads to what I call the quadruple hierarchy where we have voices those are notes you know what
59:22
um what the Beatles are singing moving inside possibly scale like chords that move inside scales that themselves move
59:28
inside a chromatic collection every adjacent layer of this hierarchy is represented by one of those spiral
59:34
diagrams I showed you in part three here's a wonderful example from Beethoven it's the opening of the
59:39
volstein Sonata like eight days a week I see the basic harmonies as perpetually
59:45
descended [Music]
59:53
it's actually kind of a rock progression meanwhile against that the right hand Melody is ascending
1:00:03
foreign [Music]
1:00:08
psychologically complex repetition this creates a kind of gigantic exciting
1:00:14
wedge that is the perfect curtain opener for one of the great pieces of the classical tradition here the primary
1:00:20
notes of the melody are shown in open note heads you'll see them ascending to
1:00:25
the right upward along the chord but then as the music progresses the chord changes shifting downward from one staff
1:00:33
to the next so you'll see the result is a kind of diagonal uh down and right
1:00:38
motion let's listen I think it's pretty clear [Music]
1:00:52
thank you [Music]
1:01:03
all right we can also represent this with my spiral diagrams so here is the motion of the voices inside the chord
1:01:10
and here's the motion of the chords inside the chromatic scale over here
1:01:15
you see large counterclockwise motions which correspond to the ascending pattern and over here you see small
1:01:21
clockwise motions which correspond to the descending path here the spiral motions you see here
1:01:28
recreate exactly the notes that are in the music okay it's not an analogy it's
1:01:34
not a sort of it's it's not just a symbolic representation it's more like a
1:01:40
musical instrument in the in the sense that it really does if you started in the right place you can find the Motions
1:01:46
that that make any desired output so we're going to listen twice first we'll look at the counterclockwise motion over
1:01:52
here uh which is the ascent relative to this uh the ascent relative to the chord
1:02:01
foreign [Music]
1:02:18
now we can look at the more Placid descending motion on the right [Music]
1:02:33
foreign
1:02:41
and then I have one last example which comes from one of my heroes McCoy Tyner this is a line he plays in his solo uh
1:02:49
on passion dance from The Real McCoy and what happens here is he plays a series of Triads that are changing their
1:02:55
inversion that that means they're moving at the top hierarchical level but they're also moving relative to the
1:03:02
scale so they're moving on the middle hierarchical level and then the scale itself is changing from mixolydian to E
1:03:08
flat minor pentatonic as you as you go so that's another level of motion and
1:03:13
this is all improvised and it goes by super fast so we'll listen to it one time here
1:03:26
and now we'll watch it on a spiral diagram I didn't try to do multiple diagrams here I just do uh the motion of
1:03:32
the chord inside the scale you can look over here on the musical notation you'll see exactly the notes that Tyner plays
1:03:42
[Music] [Applause] [Music]
1:03:47
so here I think the geometry is really starting to help us understand what it
1:03:52
is that musical competence and musical expertise consistent so McCoy Tyner is an
1:03:58
incredibly uh thoughtful and and intellectual musician
1:04:04
who is able to internalize these relationships and if you look at his playing carefully he does this not just
1:04:09
with the major chords but with all sorts of chords he he's a really um he's got a great combination of being
1:04:16
a very principled and cerebral player while also having like a real visceral
1:04:21
expressiveness okay so the final picture we end up with is one of meals within Wheels we have voices moving inside
1:04:28
possibly scale like chords that move inside scales that move inside a chromatic aggregate each of these pairs
1:04:36
of levels is represented by a structurally analogous topological diagram here this is a very simple
1:04:42
organizational strategy that generates music of extraordinary psychological complexity we've seen rock examples
1:04:48
classical examples Jazz examples when you start to think these way in this way maybe those stylistic differences aren't
1:04:55
quite so important it's actually really hard to manage three separate levels of motion along a
1:05:01
collection particularly with unfamiliar chords and scales musicians do this by internalizing uh a set of routines and
1:05:09
habits that are tied to particular collections so if we want to expand Beyond those internalized routines this
1:05:16
is an opportunity for music theory we can make music musical calculators computational assistance we can lay out
1:05:23
this whole structure in ways that that that explain the underlying principles and make it clear how you might
1:05:29
generalize these procedures to arbitrary cord and scale environments so thanks a
1:05:34
lot if you want to look at this PowerPoint madmusicalscience.com spatiality.pptx there's also videos
1:05:41
calculators and a lot of other stuff there so that's it
1:05:47
foreign