# Thelonius Monk’s Shapeshifting Chord

One of my part-time hobbies is being a Monk interpreter. A Monk interpreter not only learns how to play Monk’s compositions, but also makes a point of getting into the head of this eccentric man. The reason to do this is that Monk was an improvisor – and he was driven by an inner vision. If you can tap that inner vision, then you can generate Monk-like music – and improvise on it…even while playing Beatles songs.

I wrote a piece in 2013 about Monk as a mathematician.

Math can be about patterns (visual or sonic). Math does not always have to be expressed in numbers. Monk once said,“All musicians are subconsciously mathematicians”.

A Symmetrical Chord

The chord I’m talking about has four notes. It is typically used as a dominant chord – which naturally resolves to the tonic. Unlike the classical dominant-seventh, this chord has a flatted fifth – which makes it slip into a symmetrical regime – as shown in the picture above – inscribed in the circle of fifths.

According to Wikipedia, this chord is called the “Dominant Seventh Flat-Five Chord“. The cool trick about this chord is that it can resolve to either of two different tonics – each being a tri-tone apart.

So for instance, a chord with these notes:     Eb   F   A   B      can resolve to either Bb or E as the home key.

This chord also happens to contain 4 of the 6 tones in a whole tone scale, which Monk famously used (often as a dominant arpeggio).

If you are not familiar with music theory, you may still appreciate the beauty of sonic geometry and how it can generate such variety. If you apply similar concepts to rhythm as to harmony then you have a wonderfully rich canvas for endless musical expression. I like the way Monk wove these geometries together in a way that makes the foot tap and the ear twinge – and the brain tweak.

Monk was of course not the only one to apply these ideas – but he did accomplish something remarkable: the application of embodied math. If you have spent as much time as I have learning his language, listening to him improvise can cause a smile – or the occasional giggle – to pop out. Like an inside joke.

There is plenty of material on the internet about Monk. Here’s one voice among the many who have acquired an appreciation for Monk:  How to Listen to Thelonius Monk – by George H. Jensen, Jr.

# Science writers who say machines have feelings…lack intelligence.

I saw an article by Peter Dockrill with the headline, “Artificial intelligence should be protected by human rights, says Oxford mathematician”.

The subtitle is: “Machines Have Feelings Too”.

Regarding the potential dangers of robots and computers, Peter asks: “But do robots need protection from us too?” Peter is apparently a “science and humor writer”. I think he should stick with just one genre.

Just more click-bait.

There are too many articles on the internet with headlines like this. They are usually covered with obnoxious, eye-jabbing ads, flitting in front of my face like giant colorful moths. It’s a carnival – through and through.

I could easily include any number of articles about the “terrifying” future of AI, “emotional machines”, “robot ethics”, and other cartoon-like dilutions of otherwise thoughtful well-crafted science fiction.

Good science fiction is better than bad science journalism.

Here’s Ben Goldacre:

Now, back to this silly subject of machines having feelings:

Some of my previous articles express my thoughts on the future of AI, such as:

No Rafi. The Brain is not a Computer

The Singularity is Just One in a Series

Why Nick Bostrom is Wrong About the Dangers of Artificial Intelligence

Intelligence is NOT One-Dimensional

I think we should be working to fix our own emotional mess, instead of trying to make vague, naive predictions about machines having feelings. Machines will – eventually – have something analogous to animal motivation and human states of mind, but by then the human world will look so different that the current conversation will be laughable.

Right now, I am in favor of keeping the “feelings” on the human side of the equation.

We’re still too emotionally messed up to be worrying about how to tend to our machines’ feelings. Let’s fix our own feelings first before giving them to our machines. We still have that choice.

And now, more stupidity from Meghan Neal:

“Computers are already faster than us, more efficient, and can do our jobs better.”

Wow Meghan, you sure do like computers, don’t you?

I personally have more hope, respect, and optimism for our species.

In this article, Meghan makes sweeping statements about machines with feelings, including how “feeling” computers are being used to improve education.

The “feeling” robots she is referring to are machines with a gimmick – they are brain-dead automatons with faces attached to them. Many savvy futurists suggest that true AI will not result from humans trying to make machines act like humans.  That’s anthropomorphism. Programming pre-defined body language in an unthinking robot makes for interesting and insightful experimentation in human-machine interaction. But please! Don’t tell me that these machines have “feelings”.

This article says: “When Nao is sad, he hunches his shoulders forward and looks down. When he’s happy, he raises his arms, angling for a hug. When frightened, Nao cowers, and he stays like that until he is soothed with some gentle strokes on his head.”

Pardon me while I projectile vomit.

Any time you are trying to compare human intelligence with computers, consider what Marvin once said:

# No Rafi. The brain is not a computer.

Rafi Letzter wrote an article called “If you think your brain is more than a computer, you must accept this fringe idea in physics“.

The article states the view of computer scientist Scott Aaronson: “…because the brain exists inside the universe, and because computers can simulate the entire universe given enough power, your entire brain can be simulated in a computer.”

Who the fuck said computers can simulate the entire universe?

That is a huge assumption. It’s also wrong.

We need to always look close at the assumptions that people use to build theories. If it can be proven that computers can simulate the entire universe, then this theory will be slightly easier to swallow.

The human brain is capable of computation, and that’s why humans are able to invent computers.

The very question as to whether the brain “is a computer” is wrong-headed. Does the brain use computation? Of course it does (among other things). Is the brain a computer? Of course it isn’t.

# The Singularity is Just One in a Series

I’m reading Kurzweil’s The Singularity is Near.

It occurs to me that the transition that the human race is about to experience is similar to other major transitions that are often described as epochs – paradigm-shifts – in which a new structure emerges over a previous structure. There are six key epochs that Kurzweil describes. (The first four are not unlike epochal stages described by Terrance Deacon and others.)

1. Physics and Chemistry
2. Biology and DNA
3. Brains
4. Technology
5. Human Intelligence Merges with Human Technology
6. Cosmic Intelligence

When a new epoch comes into being, the agents of that new epoch don’t necessarily eradicate, overcome, usurp, reduce, or impede the agents of the previous epoch. Every epoch stands on the shoulders of the last epoch.This is one reason not to fear the Singularity…as if it is going to destroy us or render us un-human. In fact, epoch number 5 may allow us to become more human (a characterization that we could only truly make after the fact – not from our current vantage point).

I like to think of “human” as a verb: as a shift from animal to post-human, because it characterizes our nature of always striving for something more.

There are debates raging on whether the Singularity is good or bad for humanity. One way to avoid endless debate is to do the existential act: to make an attempt at determining the fate of humanity, rather than sit passively and make predictions.  As Alan Kay famously said, “the best way to predict the future is to invent it”. We should try to guide the direction of the next epoch as much as we can while we are still the ones in charge.

In a previous article I wrote that criticizes some predictions by Nick Bostrom, I compare our upcoming epochal shift to a shift that happened in the past, when multi-cellular beings evolved. Consider:

Maybe Our AI Will Evolve to Protect Us And the Planet

Billions of years ago, single cells decided to come together in order to make bodies, so they could do more using teamwork. Some of these cells were probably worried about the bodies “taking over”. And oh did they! But, these bodies also did their little cells a favor: they kept them alive and provided them with nutrition. Win-win baby!

I am not a full-fledged Singularitarian. I prefer to stay agnostic as long as I can. Its not just a human story. Our Singularity is just the one that is happening to us at the moment.

Similarly, the emergence of previous epochs may have been experienced as Singularities to those that came before.

# The Miracle of My Hippocampus – and other Situated Mental Organs

I’m not very good at organizing.

The pile of papers, files, receipts, and other stuff and shit accumulating on my desk at home has grown to huge proportions. So today I decided to put it all into several boxes and bring it to the co-working space – where I could spend the afternoon going through it and pulling the items apart. I’m in the middle of doing that now. Here’s a picture of my progress. I’m feeling fairly productive, actually.

Some items go into the trash bin; some go to recycling; most of them get separated into piles where they will be stashed away into a file cabinet after I get home. At the moment, I have a substantial number of mini-piles. These accumulate as I sift through the boxes and decide where to put the items.

Here’s the amazing thing: when I pull an item out of the box, say, a bill from Verizon, I am supposed to put that bill onto the Verizon pile, along with the other Verizon bills that I have pulled out. When this happens, my eye and mind automatically gravitate towards the area on the table where I have been putting the Verizon bills. I’m not entirely conscious of this gravitation to that area.

Gravity Fields in my Brain

What causes this gravitation? What is happening in my brain that causes me to look over to that area of the table? It seems that my brain is building a spatial map of categories for the various things I’m pulling out of the box. I am not aware of it, and this is amazing to me – I just instinctively look over to the area on the table with the pile of Verizon bills, and…et voilà – there it is.

Other things happen too. As this map takes shape in my mind (and on the table), priorities line up in my subconscious. New connections get made and old connects get revived. Rummaging through this box has a therapeutic effect.

The fact that my eye and mind know where to look on the table is really not such a miracle, actually. It’s just my brain doing its job. The brain has many maps – spatial, temporal, etc. – that help connect and organize domains of information. One part of the brain – the hippocampus – is associated with spatial memory.

User Interface Design, The Brain, Space, and Time

I could easily collect numerous examples of software user interfaces that do a poor job of tapping the innate power of our spatial brains. These problematic user interfaces invoke the classic bouts of confusion, frustration, undiscoverability, and steep learning curves that we bitch about when comparing software interfaces.

This is why I am a strong proponent of Body Language (see my article about body language in web site design) as a paradigm for user interaction design. Similar to the body language that we produce naturally when we are communicating face-to-face, user interfaces should be designed with the understanding that information is communicated in space and in time (situated in the world). There is great benefit for designers to have some understanding of this aspect of natural language.

Okay, back to my pile of papers: I am fascinated with my unconscious ability to locate these piles as I sift through my stuff. It reminds me of why I like to use the fingers of my hand to “store” a handful of information pieces. I can recall these items later once they have been stored in my fingers (the thumb is usually saved for the most important item).

Body Maps, Brain, and Memory

Last night I was walking with my friend Eddie (a fellow graduate of the MIT Media Lab, where the late Marvin Minsky taught). Eddie told me that he once heard Marvin telling people how he liked to remember the topics of an upcoming lecture: he would place the various topics onto his body parts.

…similar to the way the ancient Greeks learned to remember stuff.

During the lecture, Marvin would shift his focus to his left shoulder, his hand, his right index finger, etc., in order to recall various topics or concepts. Marvin was tapping the innate spatial organs in his brain to remember the key topics in his lecture.

My Extended BodyMap

My body. My home town. My bed. My shoes. My wife. My community. The piles in my home office. These things in my life all occupy a place in the world. And these places are mapped in my brain to events that have happened in the past – or that happen on a regular basis. My brain is the product of countless generations of Darwinian iteration over billions of years.

All of this happened in space and time – in ecologies, animal communities, among collaborative workspaces.

Even the things that have no implicit place and time (as the many virtualized aspects of our lives on the internet)… even these things occupy a place and time in my mind.

Intelligence has a body. Information is situated.

Hail to Thee Oh Hippocampus. And all the venerated bodymaps. For you keep our flitting minds tethered to the world.

You offer guidance to bewildered designers – who seek the way – the way that has been forged over billions of years of intertwingled DNA formation…resulting in our spatially and temporally-situated brains.

# bodymapping.com.au

We must not let the no-place, no-time, any-place, any-time quality of the internet deplete us of our natural spacetime mapping abilities. In the future, this might be seen as one of the greatest challenges of our current digital age.

# Why is it a Color “Wheel” and Not a Color “Line”?

`This blog post was published in May of 2012 on EyeMath. It is being migrated to this blog, with a few minor changes.`

I’ve been discussing color algorithms recently with a colleague at Visual Music Systems.

We’ve been talking about the hue-saturation-value model, which represents color in a more intuitive way for artists and designers than the red-green-blue model. The “hue” component is easily explained in terms of a color wheel.

Ever since I learned about the color wheel in art class as a young boy, I had been under the impression that the colors are cyclical; periodic. In other words, as you move through the color series, it repeats itself: red, orange, yellow, green, blue, violet…and then back to red. You may be thinking, yes of course…that’s how colors work. But now I have a question…

Why?

Consider five domains that can be used as the basis for inventing a color theory:

(1) the physics of light, (2) the human retina, (3) the human brain, (4) the nature of pigment and paint, and (5) visual communication and cultural conventions.

(1) In terms of light physics, the electromagnetic spectrum has a band visible to the human eye with violet at one end and red at the other. Beyond violet is ultraviolet, and beyond red is infrared. Once you pass out of the visible spectrum, there aint no comin’ back. There are no wheels in the electromagnetic spectrum.

(2) In terms of the human retina, our eyes can detect various wavelengths of light. It appears that our color vision system incorporates two schemes: (1) trichromatic (red-green-blue), and (2) the opponent process (red vs. green, blue vs. yellow, black vs. white). I don’t see anything that would lead me to believe that the retina “understands” colors in a periodic fashion, as represented in a color wheel. However, it may be that the retina “encourages” this model to be invented in the human brain…

(3) In terms of the brain, our internal representations of color don’t appear to be based on the one-dimensional electromagnetic spectrum. Other factors are more likely to have influence, such as the physiology of the retina, and the way pigments can be physically mixed together (a human activity dating back thousands of years).

(4) Pigment and paint are very physical materials that we manipulate (using subtractive color), thereby constituting a strong influence on how we think about and categorize color.

(5) Finally: visual communication and culture. This is the domain in which the color wheel was invented, with encouragement from the mixing properties of pigment, the physiology of the retina, and the mathematical processes that are formulated in our brains. (I should mention another influence: technology…such as computergraphical displays).

Red-Green-Blue

Consider the red-green-blue model, which defines a 3D color space – often represented as a cube. This is a common form of the additive color model. Within the volume of the cube, one can trace a circle, or a hexagon, or any other cyclical path one wishes to draw. This cyclical path defines a periodic color representation (a color wheel). A volume yields 2D shapes, traced onto planes that slice through the volume. It’s a process of reducing dimensions.

But the electromagnet spectrum is ONE-DIMENSIONAL. The physical basis for colored light cannot yield a higher-dimensional color space. The red-green-blue model (or any multi-dimensional space) therefore could not originate from the physics of light.

DID HUMANS INVENT PURPLE IN ORDER TO GLUE RED AND VIOLET TOGETHER?

An alternate theory as to the origin of the color wheel is this: the color wheel was created by taking the two ends of the visible spectrum and connecting them to form a loop (and adding some purple to form a connective link). I just learned that Purple is NOT a spectral color (although “violet” is :) Purple can only be made by combining red and blue. Here’s an explanation by Deron Meranda, in a piece called…

PURPLE: THE FAKE COLOR – OR, WHAT REALLY LIES AT THE END OF A RAINBOW?

And here’s a page about how purple is constructed in the retina: HOW CAN PURPLE EXIST?

Did the human mind and human society impose circularity onto the color spectrum in order to contain it? Was this encouraged by the physiology of our eyes, in which various wavelengths are perceived, and mixed (mapping from a one-dimensional color space to a higher-dimensional color space)? Or might it be more a matter of the influence of pigments, and the age-old technology of mixing paints?

Might the color wheel be a metaphorical blend between the color spectrum and the mixing behavior of pigment?

Similar questions can be applied to many mathematical concepts that we take for granted. We understand number and dimensionality because of the ways our bodies, and their senses, map reality to internal representations. And this ultimately influences culture and language, and the ways we discuss things…like color…which influences the algorithms we design.

# Enough with this Square Root of -1 Business!

Like so many other people, I was kept from appreciating the beauty and utility of mathematics because of the way it was taught to me.

The majority of introductions to complex numbers start with the elusive and mysterious square root of -1, denoted by i.

A number that has an i stuck on to it is called “imaginary” (a convenient differentiator to “real”). Being asked to learn something that is called “imaginary” is not very motivating to young learners who work best starting with concrete metaphors.

The imaginary number is counterintuitive and confusing. And it’s not the coolest part. Sure, i was an important invention at a critical stage in the history of math when there was no good way to express z2 = -1. And yes, it makes a good ending to a long story (which happens to be true): math has advanced through several expansions of the concept of “number” … from the counting numbers to the wholes – to the negatives – to the fractions – to the irrationals – and finally to complex numbers – where i came along and saved the day.

But…does this mean that invoking i is the best way to explain complex numbers to novices – to everyday people? I join many others in saying that there is a better way to learn about the wonderful world of two-dimensional numbers. One voice among those is Kalid Azad.

He speaks in metaphors and freely engages the visual mind to help us grasp math concepts using our whole brain. In his explanation on complex numbers, Azad says this about i: “It doesn’t make sense yet, but hang in there. By the end we’ll hunt down i and put it in a headlock, instead of the reverse.”

…..

When you get an intuitive, aesthetic feeling for why certain mathematical ideas are being taught, you become more motivated to learn the notation. The corollary: learning math notation without understanding why is like learning musical notation before ever being allowed to listen to or play music.

Paul Lockhart, in A Mathematician’s Lament, compares the way math is taught to a nightmare scenario in which music is taught to students using sheet music notation only (no actual music is played or heard) – until the student is advanced enough to start “using” it.

What is a Two-Dimensional Number?

When I read that complex numbers are really no more “imaginary” than real numbers, I decided that I would start dismantling my old worldview. Why should I assume that numbers have to be one-dimensional? Over time, I became more accustomed to the notion that a number can occupy a plane (the complex plane) and not just a line (the number line). Learning how to make images of the Mandelbrot Set helped a lot.

Think of Multiplication as Rotation

Instead of trying to wrap your mind around i, and how it magically makes equations come out right, let’s start with geometry. Think of multiplication as rotation and expansion. In the blog Girls Angle, Ken Fan introduces complex number multiplication in a nice visual way… here.

Here’s a video explaining complex numbers in terms of physical metaphors, and eventually explaining why the square root of -1 becomes a necessary part of the notation.

Squaring

Consider the following diagram showing what happens when you square certain complex numbers that lie on the unit circle:

The dot on the right represents the complex number (1+0i). When you square it, it stays the same (no surprise: 1×1=1). The number at the left is (-1+0i). When you square that, it becomes (1+0i). But when you square the number at top (0+1i) it “rotates” by 90 degrees to (-1+0i). Finally, at the bottom, the number (0-1i) rotates…but would it be correct to say that it rotates by 90 degrees clockwise to (-1,0i)? Depends on how you look at it. Rotating by 270 degrees counter-clockwise has the same result. This is the nature of rotation and angular reality: it is periodic – it cycles…it repeats.

What an awesome idea. Multiplication is like doing a whirling dervish jig.

Animated Squaring

Here’s an interactive tool I made that allows you to play with 200 dots (complex numbers) randomly scattered on the complex plane. You can experience what happens when complex numbers are squared. It also allows you to multiply the dots (using a complex number dot that you can drag along the screen).

http://ventrella.com/ComplexSquaring/

This interactive tool might make you feel as if the dots on the screen are obeying some sort of gravitational law of physics. Well, in a way, yes, that’s what’s happening. When you add, multiply, or exponentiate numbers, you get a new number. In the complex plane, the space where that change takes place is two-dimensional. That’s cool! We like images.

Here’s another visual tool: when we multiply two complex numbers, such as (a+bi) and (c+di), we can visualize the operation in this way:

In pseudocode:
``` realPart      = (a*c) - (b*d); imaginaryPart = (a*d) + (b*c); ```

This explanation of multiplication does not require i.

To this day, I STILL do not feel very much music when I think about the square root of -1.

On the other hand, the more I play around with visualizing and animating complex numbers, the more intuitive they become, and the deeper my sense that these numbers are as real as any old one-dimensional number.

They are not imaginary at all.