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Hey guys welcome to twelve-tone today we're going to talk about twelve tone matrices and if you stick around till the end we'll have some tips on turning a row into an actual composition like a lot of serialism this gets a little mathy but don't worry it doesn't really go beyond basic arithmetic on that note let's talk about notation up to now we've been using standard note names with letters and accidentals but those names are derived from tonal systems we use seven letters because our scales have seven notes and accidentals signified deviations from those scales but twelve-tone music is a tonal every note is equal so when we get deep into it those names stop being useful instead we just use the number zero to eleven signifying how many half steps above see the note is so zero is C 1 is C sharp or D flat 2 is D and so on once you get to 12 you're back at C so we reset to zero and continue up again this system lets us easily visualize intervals which is more important here than whether a note is sharp or flat this is just for composition though when you're transcribing a piece to be played you should switch back to regular notation that's way easier to read with that in mind let's talk about the matrix this is a useful tool for visualizing the structures available to you and it works like this you start with a 12 by 12 grid in the top line you write your prime rows starting on 0 then going down the first column you write your inversion a handy trick to find this is to take each number in your prime row and subtract it from 12 so this 4 inverts to an 8 this 3 becomes a 9 and so on then for each of the remaining squares you add the number at the top of the column to the one at the beginning of the line so this 4 plus this 8 would be 12 or 0 this 3 plus 8 is 11 this 4 plus 9 is 13 or 1 and so on this takes a little time but once you're done it should look something like this so what's the point well if you read across each of the lines you have every possible transposition of your primer row if you read down the columns you have every inversion reading a line backwards gives you your retrogrades and reading columns from bottom to top gives you the retrograde inversions this grid shows you all 48 possible transformations of your starting row making it a useful reference for quickly determining your available options and that's a 12 tone matrix I promise we're done with math for now some of you have already asked what the next step is to quote one viewer many twelve-tone guides just say here are the row forms now get out and.

FAQ

Do conservatives understand liberals more than liberals understand conservatives?
According to Jonathan Haidt research, yes.And according to my personal experience as someone that sees them both from the outside thanks to a somewhat peculiar upbringing he is right.He writes on his book “The Righteous Mind” from page 334:When I speak to liberal audiences about the three “binding” foundations ‡ Loyalty, Authority, Sanctity ‡ I find that many in the audience don’t just fail to resonate, they actively reject these concerns as immoral. Loyalty to a group shrinks the moral circle, it is the basis of racism and exclusion, they say. Authority is oppression. Sanctity is religious mumbo-jumbo whose only function is to suppress female sexuality and justify homophobia.In a study I did with Jesse Graham and Brian Nosek, we tested how well liberals and conservatives could understand each other. We asked more than two thousand American visitors to fill out the Moral Foundations Questionnaire. One-third of the time they were asked to fill it out normally, answering as themselves. One-third of the time they were asked to fill it out as they think a “typical liberal” would respond. One-third of the time they were asked to fill it out as a “typical conservative” would respond. This design allowed us to examine the stereotypes that each side held about the other. More important, it allowed us to assess how accurate they were by comparing people’s expectations about “typical” partisans to the actual responses from partisans on the left and the right)‡ Who was best able to pretend to be the other?The results were clear and consistent. Moderates and conservatives were most accurate in their predictions, whether they were pretending to be liberals or conservatives. Liberals were the least accurate, especially those who described themselves as “very liberal.” The biggest errors in the whole study came when liberals answered the Care and Fairness questions while pretending to be conservatives. When faced with questions such as “One of the worst things a person could do is hurt a defenseless animal” or ”Justice is the most important requirement for a society,” liberals assumed that conservatives would disagree. If you have a moral matrix built primarily on intuitions about care and fairness (as equality), and you listen to the Reagan [i.e., conservative] narrative, what else could you think? Reagan seems completely unconcerned about the welfare of drug addicts, poor people, and gay people. He’s more interested in fighting wars and telling people how to run their sex lives.If you don’t see that Reagan is pursuing positive values of Loyalty, Authority, and Sanctity, you almost have to conclude that Republicans see no positive value in Care and Fairness. You might even go as far as Michael Feingold, a theater critic for the liberal newspaper the Village Voice, when he wrote:Republicans don’t believe in the imagination, partly because so few of them have one, but mostly because it gets in the way of their chosen work, which is to destroy the human race and the planet. Human beings, who have imaginations, can see a recipe for disaster in the making, Republicans, whose goal in life is to profit from disaster and who don’t give a hoot about human beings, either can’t or won’t. Which is why I personally think they should be exterminated before they cause any more harm)3One of the many ironies in this quotation is that it shows the inability of a theater critic-who skillfully enters fantastical imaginary worlds for a living-to imagine that Republicans act within a moral matrix that differs from his own. Morality binds and blinds.
What are the most important qualities for a CEO?
The CEO absolutely must be able to: Establish the vision This may be formulated de novo or built upon the vision of a founder, if the CEO is not the founder. This includes the ability to:Communicate the vision to a broad set of audiences - internal and external stakeholders alikeFrame the vision appropriately to scale, time frame, and even zeitgeistEstablish the culture This critical ability includes the following key activities:Hire the right executives for their positions. This one can be a book in itself, but at the fundamental level, the executives should be qualified for the specific tasks their job titles demand (charisma should be secondary) and have the capacity to translate the CEO's vision appropriate to scale and relevance for their particular rank in the organization.Install a value system upon which the organization's culture adheres. This would then be implemented at the functional levels by the qualified executives for their ranks.GET THE HELL OUT OF THE WAY. If you've hired the right people for the jobs who are competent, then let them do their jobs by letting them act on the authority you've given them.Establish the basis of metrics used (work with your COO/CFO)I realize that this is broad and depends on the function, but that's why I didn't say "establish the metrics" but "establish the basis of" metrics used. In other words, if you say that your organization's metrics is all about sales, well, everyone will take you literally and focus on this at the expense of everything else. If you say that your org's metrics will be about return in investment that has been created for long term sustainability and competitive advantage - not just to address short term quarterly investor expectations, then your executives may actually think "long term" and execute the strategies as such. In summary these qualities do figure into the "leadership" expectations of the CEO but I specifically focused on the fundamentals of "why are we even here in this business at this company" (Vision), "who are we, inside, as a company" (Culture), and "how do we know we're doing what we said we're doing and whether we should do something different" (Metrics).
How is it possible to fill a 2D NxN matrix in spiral form?
The 2D NxN matrix can be though of as 1D array of length N^2 to keep things simple. If not, you can get the 2D coordinates by x = pos / N, y = pos % N.// Constants const int N = 4, const int left = -1, right = 1, down = N, up = -N, int grid[N * N], // Get next clockwise direction int next(int dir) { switch (dir) { case left: return up, case up: return right, case right: return down, case down: return left, } } // Assumes default is all zero void fill(int grid[N*N]) { int pos = N-1, dir = right, for (int i = N * N, i 0, i--) { // Fill the block grid[pos] = i, if (pos == 0 || pos == N - 1 || pos == N*(N - 1) || pos == N*N - 1 || grid[pos + dir] != 0) { // if corner point reached or block already filled, change direction dir = next(dir), } // Move to next position pos += dir, } }
If someone with whom you have discussed an idea writes a scientific paper on that sole idea, what authorship does the person with the idea get?
Depends on your contribution. An idea is not science. Science is all about shaping that one idea.An idea is: ‘Ooooh I wish I could have wings and just fly high. If only I could make a body like a bird, and then use it.‡ This is definitely an idea that can be converted to reality (as we know now). But it is still an idea.Science is knowing that this idea is feasible. That you can make a streamlined body. That you can make it aerodynamically stable. That you can use Bernoulli’s theorem to your advantage.Science is about understanding how an idea can find its base outside one’s imagination, even if it means in ideal conditions.So if someone writes a scientific paper on your idea, you should definitely get credit. It was your idea after all. But you need to know whether your discussion of that idea was as good as the science written in that paper.If your idea was all science, and the person writing the paper has equally contributed to the discussion which helped in shaping this idea in a scientific manner, then it should mean equal authorship.If it was all your idea, your science, your discussion, then you should be the one writing that paper.So if two people are involved in the paper, then you both should also discuss how much each of you have brainstormed for it.It is all a matter of self-realization, on both ends.
Are we living in a simulation?
Google Prof Nick Bostrom’s work on this question. It’s a possibility…Elon Musk recently stated that we could be in a simulation. He said think how realistic games are now and imagine how realistic they will be 10,000 years from now.[In other words, at some point it will probably be possible to create a simulation that is indistinguishable from reality. Of course the simulation can model any era, so although the technology might be developed in the year 12020, one of its simulations might be for our year 2021. It might even be that the technology belongs to an “alien” species, and our experiences of evolution are one of their experiments, studies, or leisure activities.]Peter Diamandis accepted that we’re probably living in a simulation, but so what. (We're in a Simulation - YouTube)A few physicists are even conducting experiments to see if this might be the case.So there is a group of professionals that consider it to be a real possibility.If you know anything about computer games, models and simulations you’ll know that they are an approximation. One of the reasons for this is so that a computer with finite processing ability can run the game, model or simulation. So what would we look for to prove we’re in a simulation? We’d look at the details and see if there were any approximations, or look to see if there was a point at which the simulation stopped simulating.When we look at some of the smallest objects in us and our universe (atoms) we find that strange things are happening. In science this behaviour is described by Quantum Mechanics. It turns out that very weird things happen at this level of detail. For example, particles don’t have specific values until they are “observed”. Before the observation they can be in a super-position of all possible states (or wave functions). This means that they can be doing paradoxical things - like being in all places at the same time! It’s only when they are observed that they take on a specific value, like a specific location and momentum.So in effect, we could argue that they don’t become defined until they are observed. Now that’s interesting because computer simulations in use today only render (define) objects that the user can see on the screen at a given point in time.So here’s something for you to ponder‡ Is the interpretation of quantum mechanics the following?We are living in a simulation.(Perhaps)
What was ITA's competitive advantage?
I was co-founder and COO (until 2021. at ITA Software. Obviously many things contributed to ITA's success, including being in the right place at the right time as ticket sales moved rapidly onto the Internet, but here, in my personal opinion, are a few key differentiators:1) The core algorithms of ITA's QPX search software -- originally conceived and implemented by co-founder Carl de Marcken -- approach the search problem fundamentally differently than predecessor systems. In particular, Carl sought to find all valid solutions to a given airfare query, represent them using a compact data structure, and enumerate solutions meeting specific criteria (e.g, "nonstop", "on United", "before 9am") from this compact representation. All other airfare pricing systems I am aware of, to this day, use a heuristic or "rule of thumb" approach to find a relatively small number of good solutions, an example heuristic might be "when flying from Boston to Los Angeles, consider stopping in ORD and DFW."Carl's comprehensive approach sometimes lets QPX find lower prices than other search engines. But the disruptive advantage is that such a deep search allows user interfaces that let the user assert his/her travel preferences dynamically -- for example, by "drilling down" in a single cell in a matrix of solutions. Intuitively, QPX also creates a "fully filled-out" matrix, as opposed to a sparse matrix, which gives users the positive feeling that QPX really is examining all the possible solutions without bias.2) As a new entrant, ITA Software had no existing business to protect. Therefore, we focused entirely on perfecting search. In contrast, GDS companies like Sabre gave away search as a loss leader to generate lucrative booking fees, so booking -- rather than search -- was (and remains) their primary business.3) ITA had no preconceived notions of how things were supposed to work in the travel industry. We therefore viewed, for example, the problem of getting seat availability for, say, a 500-solution result set as a challenge rather than an impossibility, and used (for a while, at least) simple caching approaches to seat availability that industry insiders refused to consider, on the basis that these would not produce accurate enough results.4) ITA placed a very high cultural value on honesty with customers. For example, if we had an outage that a customer hadn't noticed, we would proactively notify the customer of the outage, even though it would then cost us money under our SLA with the customer. Typically when things went wrong, the long-established vendors would blame each other and spend a lot of effort trying to minimize the damage, rather than trying to fix the problem. Our approach at ITA was much more collaborative (and, when it was our fault, accepting of blame). As a strategy, this was costly short-term but very productive long-term, as we built deeper relationships with big customers that ultimately generated far more revenue as a result.5) The skill sets of the founders and early team members were highly complementary. In particular, Carl was a spectacularly productive programmer, Jeremy was an exceptionally creative negotiator with an excellent mind for legal complexity, and I was able to keep the trains running on time. Early developers (Greg, Greg, Jim, Justin, among many) were superstars, each of whom brought a particular, complementary technical/business focus to the team.
What is word embedding in machine learning?
Broadly speaking, machine learning algorithms are “happiest” when presented training data in a vector space. The reasons are not surprising: in a vector space (including the infinite-dimensional Hilbert space generalization), one can compute angles between vectors, which allows doing “projections”, one of the fundamental operations underlying most machine learning algorithms.But, the major challenge in machine learning is that data come from all over: text documents, images, graphs, sensor streams, and so on. Most of these entities do not “live” in a vector space. For example, a graph is not a vector. You cannot “multiply” a graph by a scalar, or add two graphs.Similarly, words in English do not “live” in a vector space. I can’t multiply 3.5 times “happiness” to create more “happiness” (sometimes I wish I could!). Similarly, I cannot add “more” and “money” to increase my bank account.So, whenever you see the word “embedding” in machine learning, what that means is that the authors are exploring some technique to take non-vectorized data and “embed” it into a vector space.Let’s take a couple of examples to get the hang of the concept. For example, let’s say I want to “embed” a graph in a vector space. How would I go about doing that? A simple way to think about this concept is to imagine drawing the graph on paper. That exercise forces you to take every vertex and plonk it down somewhere on the paper. So, in effect, you have constructed a mapping from the set of vertices V of a graph G to pairs of real numbers (the (x,y) coordinate of the point in space with respect to some arbitrary coordinate system). In other words, you have constructed an embedding of the graphSimilarly, for word embeddings, suppose I want to know what the word “Democrat” means. Well, one way to ascribe meaning to this word is to find some way to map “Democrat” into an N-dimensional vector (say, N = 100), so that every instance of “Democrat” in a text document is replaced by that 100-dimensional vector.In both these examples of graph embeddings and word embeddings, the trick of course is to construct an embedding that is “meaningful”. For example, if you map “Democrat” to some N-dimensional vector, and do so for every word in your vocabulary, then the embedding is useful if when I compute the nearest neighbors of “Democrat”, I get words like “Obama” or “Nancy Pelosi” (two famous Democrats!), and I should certainly not get “Trump” ( a famous “Republican”!).How about for graphs? Well, suppose you are trying to infer some function defined on a graph (e.g., some property of users on a social network, which defines the graph). You are given the graph, but only a small number of vertices are labeled (say the social network labels some vertices “Democrat” and some vertices “Republican”, but most of the vertices are unlabeled, meaning the political affiliation of the individuals corresponding to these vertices is unknown).So, how should we design an algorithm to “fill in” the labels of the unknown vertices, while exploiting the structure of the graph? A random method would be to plonk down the social network on paper, look at the (x,y) coordinate of every vertex with labels, create a training set of the labeled vertices, and then voila’, you have a simple 2D classification problem. This works rather poorly for the simple reason that it completely ignores the spatial connectivity of the graph.So, you’d like to construct an embedding that respects the “geometry” or the “topology” of the graph, meaning that vertices that are “close” to each other in terms of the graph distance should receive “similar” labels. This captures the property that in a social network, people with similar tastes tend to hang around each other. So, “Democrats” tend to hang around with other “Democrats” and similarly, “Republicans” tend to hang out with other “Republicans”. For a real-world example of why this labeling is smooth on the graph, let's look at this image from the polling website FiveThirtyEight, which did a “what-if” scenario prediction of the midterm results from 2021. assuming that only women voted (that’s why it’s a “counterfactual”, meaning a “what-if‡ type of analysis): as you can see, the red “Republican” labels are very smooth across spatial neighborhoods, as are the “Blue” democratic labels.OK, how does one construct an embedding of a graph such that smoothness of the label function across the graph topology is preserved. This leads us to one of the most beautiful results in machine learning of the last 25 years: the graph Laplacian. The Laplacian has been termed “the most beautiful object in all of mathematics and physics” (Nelson, Tensor Analysis). The reasons are not hard to see: every major equation in physics has the Laplacian operator (sometimes called the div-grad operator).In a discrete graph, the graph Laplacian is simply L = D - A, where A is the adjacency matrix of an undirected graph, D is a diamatrix of the valencies of the graph (meaning the degree of each vertex). It is about as simple a matrix as one can define, but my, what beauties does it reveal! One never wants to analyze a graph by looking at the adjacency matrix! That tells you very little. Now, the graph Laplacian, ah, that's an entirely different kind of operator. It’s as revealing as a high precision microscope into everything that you'd like to know about a graph!It would take too long to get into the beauties of this topic, but for those interested, I highly recommend Fan Chung’s brilliant monograph on “Spectral Graph Theory”. This brings the full bore power of differential geometry to the discrete setting, and Fan Chung’s book goes into much depth on this topic. I also recommend Spielman’s lecture notes at Yale, which have some nice examples of graph embeddings.So, how does one construct graph embeddings using the Laplacian L = D-A. Simple. Just compute the eigenvectors of L, where each eigenvector is a vector of dimensionality equal to the number of vertices. The first eigenvector is all 1’s, and corresponds to the constant function. The associated eigenvalue is 0. The second eigenvector is a thing of beauty, the so-called “Fiedler” eigenvector after the mathematician who first studied it. There are many beautiful theorems about the second eigenvalue associated with the second eigenvector (it tells you how fast a random walk on the graph “mixes”, which is useful for so many things, like routing, or how information spreads in a social network, or how diseases spread across populations).With this example out of the way, let's get to constructing word embeddings. How do we embed “Democrat” into a 100-dimensional space? Well, now we are dealing with sequential datasets, namely text documents. Suppose we read through all of Wikipedia (many gigabytes!), but a fast PC can scan all of Wikipedia in several hours now. So, every time the word “Democrat” appears in Wikipedia, we keep track of the surrounding “context” words (say, the 5 words preceding it and the 5 words succeeding it).So, we build a gigantic matrix M, whose rows correspond to words like “Democrat” and whose columns correspond to all possible contexts C. This is a very large matrix, since there are hundreds of thousands of words in Wikipedia (including proper nouns), and millions of contexts. No matter, we have lots of memory and powerful machines. We take this giant matrix M, and find its singular value decomposition M = U S V’, where S is the diamatrix of singular values. The matrix U gives us an embedding of words.Of course, we’d like an incremental way of computing word embeddings, which is far more efficient. Can’t we use a “neural network” to do this, and approximate what our linear algebraic approach does? This was exactly the approach taken by Mikolov in his beautiful work on “word2vec”, and you can download his rather elegant C code that computes embeddings of all the words in Wikipedia in one afternoon on a reasonably fast PC. This is an amazing piece of work, and has brought new life into NLP and ML. Many extensions have been explored, but Mikolov’s original paper is still worth reading.https://arxiv.org/pdf/1301.3781.pdfHere’s a nice picture of the concept of word embeddings and how to reason with them (spatial vector displacements allows one to compute analogical relationships, like King is to Queen as Man is to X, where X is of course “woman”).The Tensorflow web page explains the basic ideas nicely.Vector Representations of Words  |  TensorFlowNow, are word embeddings Euclidean? That would be amazing, if true. A few years ago, while on sabbatical at IBM Research, when I was working with the Watson Deep Learning group, I explored how to reason about word embeddings using a non-Euclidean “manifold” approach. This produces far better results than Mikolov’s original approach. My paper on Arxiv on this topic can be downloaded here (warning: the math is a bit more heavy going than Mikolov’s work!).Reasoning about Linguistic Regularities in Word Embeddings using Matrix ManifoldsHope this has been helpful! Good luck exploring the wonderful world of embeddings. What does the brain do? How are meanings stored in our brains? Philosophers have been pondering this question for centuries. This is what ML research on embeddings is trying to also explain.
Human Brain: Would scientists be able to invent a machine to upload specialized knowledge directly into the brain, and when that happens, would it render the current education system obsolete?
We have had tools for doing this every since we learned to exchange, store and copy information using language and drawings. The smart phone in your pocket, while not being directly connected to your brain yet, serves as real-time memory and knowledge database for your brain everyday. There is currently a lot of research going on with computer/brain sharing and uploading right now and the results are pretty amazing. Seems like it will become a reality with humans within the next few decades:With an Artificial Memory Chip, Rats Can Remember and Forget At the Touch of a Button Scientists link rat brains together over the internet to transfer sensory information http://www.sciencedaily.com/rele... Meet the Two Scientists Who Implanted a False Memory Into a Mouse Remember, technology improves at an exponential rate so if some of our current tech would have seemed far fetched to someone 50 years ago, we can scarcely imagine the technological magic there will be in 2065. At that time, tech will be growing at such an astonishing rate, we will HAVE to have augmented memory and learning in order to just keep up with the basics. This picture illustrates this principle perfectly. As a boy he watched as his country fought each other with cutting edge black powder rifles, horses, and cannons and before he died he saw his country go to war using supersonic jet fighters, attack helicopters, nuclear warheads, and satellites. The amazing changes he will be dwarfed by our own experiences.