Thelonius Monk’s Shapeshifting Chord

monk-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.

screen-shot-2017-01-15-at-2-43-39-pm

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.

dominant_seventh_flat_five_chord_on_c

 

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.

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.

51tPEpSto+L._SX322_BO1,204,203,200_

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.

kalidphoto-color-homeHe 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-Lockhart2Paul 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

Screen Shot 2015-12-18 at 9.49.07 AMInstead 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.

screen-shot-2015-12-17-at-4-03-15-pm

Squaring

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

complex plane gravitation

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/

Screen Shot 2015-12-17 at 3.57.18 PM

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:

mult

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.

Virtual Reality is Biologically Inevitable

Musicophilia_front_coverI am reading Oliver Sacks’ Musicopholia. He discusses several patients he has seen who suddenly become obsessed with music, or suddenly start hearing music in their head as a result of a brain injury. These are called “musical hallucinations”. He describes temporal-lobe epilepsy patients who have musical hallucinations just before a seizure. Fascinating stuff.

It reminds me of how our brains are in the habit of “playing” things – not just music, but scenarios, stories, past experiences, and experiences we wish we could have.

virtual_reality_helmetThe term “Virtual Reality” is usually accompanied by high-tech images of people with clunky things stuck on their heads.

But there is another way to understand virtual reality: it is an inevitable fact of biological evolution on Earth.

What? Virtual reality is more than just a technological innovation?  A gimmick? Yes. Absolutely. Virtual reality has its roots in the early formation of life on Planet Earth.

Years ago I read Daniel Dennett’s book Kinds of Minds. I remember looking at diagrams of how animals form internal representations of the external world.

089a

Dennett shows how the evolution of nervous systems gave way to brains and ultimately consciousness. And along the journey, internal representations became increasingly sophisticated and better at predicting the outcomes of potential actions.

Throughout the history of biological evolution, animal brains became increasingly complex and adaptive to the complexity of the environment (which itself became more complex because of the brains of other animals…and so on). From genetic adaptation … to consciousness: all animals build internal representations of the word in order to function within it. This might be considered the very basis – the original impetus – of intelligence.

images

There was an amazing discovery that I leaned about from reading On Intelligence: The neocortex at the top of the human brain sends information DOWN, as well as information being sent UP from the senses. In other words, while the senses are passing information from the ears, eyes, fingers, etc. up to the higher levels of the brain, the higher levels of the brain are also sending down “expectations” of what might be coming up.

To put it another way. The brain is constantly projecting an internal virtual reality and checking to see if this matches up with the signals coming in.

Picture1

If everything matches up, and if the colliding signals are in agreement, the brain interprets this to be “business as usual”. But if any differences are detected, then various neural networks kick into action, and attempt to process this difference. This may seem strange if you are used to thinking of the brain as a passive recipient of information from the senses.

But consider this: it would be very inefficient to try to soak up the entire gamut of high-resolution reality as it floods-in through the senses. Who has time for that? It is more efficient to run an internal virtual reality based on expectation in parallel with actual reality and only jump into action with something doesn’t match up. Apparently, the six layers of neocortex described in the book are in the business of doing just that. And the higher the cortical layer, the more abstract the processing.

Think about it: the higher-region of the brain is projecting as much virtual reality down toward the senses as the senses are sending signals up to the higher regions of the brain.

Thus: the brain is a virtual reality engine.

And the collective of all animal brains have a major impact on the environment. It’s a feedback loop. The biosphere is a gigantic feedback loop of internal representations, which constantly change reality and subsequently adapt to it.

This massive cross-projection of multiple virtual realities within the biosphere started even before there were animals with brains. One could say that biological evolution has always been in the business of mapping reality into various internal representations – stored in the genes of organisms – as well as in the extended phenotypes that adorn the environment. Human brains are just the most sophisticated version of the self-reflection that emerges from the fabric of the biosphere.

Screen Shot 2015-04-02 at 12.21.09 PMSo, consider the musical hallucinations that Oliver Sacks describes. Consider the unfortunate individuals who fall victim to schizophrenia. Consider the anxiety of playing out the evening’s events before your first date. These are internal virtual realities gone awry.

 

Sony-virtual-reality-headsetWhen I see images of people with big-ass chunks of technology stuck on their faces, I wonder whats going on in the scope of the big picture – in terms of the evolution of brains. Is our internal virtual reality not sufficient enough? Is technological virtual reality just a continuation of the human instinct to tell stories, paint pictures, make movies, and games?

Perhaps the evolution of virtual reality is just that: a continuation of something that we have been doing since we became human: extending our inner-virtual reality with more and more artificial layers on the outside.

Humans are not content with plain old “natural” virtual reality. We have to take it to extremes. And given that we are not content with reality as it is (both internal and external), I guess it’s inevitable.

Pi is Meaningless

Ladies and Gentlemen. Introducing…a completely random series of numbers:

 3.11037 55242 10264 30215 14230 63050 56006 70163 21122 01116 02105 14763 07200 20273 72461 66116 33104 50512 02074 61615

Those are the first 100 digits of Pi in base 8.

“Base 8?” you screech. “Why base 8”.

Why not? We humans use base 10 because (scientists conjecture) we have ten fingers, and our ancestors used them to learn how to count. Having five digits at the end of each appendage is common in most animals we are familiar with.

bat-pentadactyl-limb-7411

But if the octopus had become the dominant species on Earth, and developed complex language, math and the internet (underwater), it is quite likely that it would have come up with a base 8 number system.

1597092

Therefore, octopuses would celebrate Pi Day by reciting its digits in base 8.

Or not.

Maybe they would think Pi is boring.

Like me.

34177-Octopus-Baby

No I’m not an octopus. And no, that’s not me. But it’s cute, don’t you think?

The point is:

I don’t understand why people pride themselves on being able to recite the digits of Pi (in any base). It is a waste of valuable gray matter that could be used for something useful.

According to Michael Hartl, “some people memorize dozens, hundreds, even thousands of digits of this mystical number. What kind of sad sack memorizes even 40 digits of π ?”

It has been found that the digits of Pi are indistinguishable from a random sequence of digits, no matter how high you count. If you select any sequence of digits in Pi (like, say, the first 100 digits starting at the billionth digit), you will find no particular bias or pattern. In fact, the likelihood of any digit (or sequence of digits) occurring is statistically flat: evenly-distributed. It’s as random as it gets (although there is no PROOF yet of the “normality” of Pi).

This is why I suggested in a previous blog post that the music in this video:

Screen Shot 2014-12-17 at 9.33.19 AM

…is meaningless. This guy Blake (who is a fine musician) could have just as easily used the digits from a random number generator.

By the way – I now see that there was a legal battle regarding copyright infringement in a case of using Pi as the basis for a melody.

Two unfortunate first-world preoccupations rolled into one.

Instead of fetishizing the digits of Pi (or any irrational number), why not explore the teachable aspects of Pi such as this:

nyu_pi_activity1_image1

…or this:

tauism-9765a6f7ca5ca1b120533f81f759db37

 

 

 

…or this:

670px-Calculate-Pi-Step-6

According to Wolfram,

Screen Shot 2015-03-13 at 7.23.17 PM

What’s interesting is how chaos is formed – whether in an abstract number system or in a natural system. The digits of Pi should be understood as the result of a dynamical process that emerges when we try to find relationships between circularity and linearity. The verb is more meaningful than the noun.

-Jeffrey

Our Colorful Mathematics Revolution

math

Education bureaucrats are trying to gently and safely tweak a broken system so that fewer students fail math.

bbb

Meanwhile, a colorful revolution is taking shape outside the walls of a crumbling institution. A populist movement in creative math is empowering an unlikely crowd.

Authors of Wikipedia math pages aren’t contributing to this populist movement. They are intent on impressing each other; competing to see who can reduce a mathematical concept to its most accurate, most precise (and least comprehensible) definition.

Screen Shot 2015-01-31 at 11.22.53 AM

A debate rages on a “new way” to do subtraction. Oh does it rage. But step back from that debate and consider that these tricks, algorithms, processes, hacks, become less relevant as new tools take their place. When calculators entered into the classroom, something started to change. That change is still underway.

Do students no longer need to learn to do math by hand? No. But calculators (and computers) have changed the landscape.

Rogue amateur mathematicians, computer artists, DIY makers, and generative music composers are creating beautiful works of mathematical expression at a high rate – and sharing them at an even higher rate. This is a characteristic trait of the “new power“.

math art

Technology

(1) Computers are better at number-crunching than we are. If used appropriately, they can allow us to apply our wonderfully-creative human minds to significant pattern-finding and problems that we are well-suited to solve.

(2) Computer animation, generative music, data visualization, and other digitally-enhanced tools of creativity and analysis are becoming more accessible and powerful – they are helping people create mathematically-oriented experiences that not only delight the senses, but express deep mathematical concepts. And they also help us do work.

(3) The internet is enabling a new generation of talented people (amateurs and professionals) to exchange mathematical ideas, discoveries, and explanations at a rate that could never be achieved via the ponderous machinations of university funding, publishing, and teaching. There will never be another Euler. Mathematical ideas now spread through thousands of minds and percolate within hours. It is becoming increasingly difficult to trace the origins of an idea. Is this good or bad? I don’t know. It’s the new reality.

Five things You Need to Know About the Future of Math

According to Jordan Shapiro:

1. Math education is stuck in the 19th Century.
2. Yesterday’s math class won’t prepare you for tomorrow’s jobs.
3. Numbers and variables are NOT the foundation of math.
4. We can cross the Symbol Barrier.
5. We need to know math’s limitations.

math-fail2

We can (and will – and should) debate how math should be taught. Whether the “symbol barrier” is a actually a barrier, and whether memorizing the multiplication tables is necessary, no one can ignore the seismic changes that are rumbling underfoot.

110511_4lo


-Jeffrey

Pi is Random. Stop Trying to Turn it into Music.

I have seen and heard several attempts at turning the digits of Pi into music.

Screen Shot 2014-12-17 at 9.33.19 AM

The highly-flourished music in this YouTube video is well-crafted. But I agree with the way one comment sums it up…

“So basically we’ve learned that any random sequence of numbers will sound reasonably pleasant if interpreted as notes in a major scale…”.

Yes. It is oh so convenient that the digits 1 through 8 can be mapped to the ever-so agreeable, politically-correct notes of the diatonic scale.

Don’t confuse this talented musician’s performance with anything remotely meaningful about the digits of Pi. Because……

Pi is INDISTINGUISHABLE from a sequence of random numbers. Extensive statistical analyses of the first six billion digits have been done to try to find tendencies, frequencies, repetitions, ANYTHING that constitutes a feature or a pattern.

newslogo

Nothing.

Perhaps there is something in this apparently-random sequence that will someday reveal the existence of an alien intelligence. Yea, right.

Here is Vi Hart’s reaction to some of this musical Pi insanity…

Screen Shot 2014-12-18 at 9.14.19 AM

At least Jim Zamerski has the sense to consider that Pi can be expressed in other bases than 10, for making music. But again, regarding the use of Pi as the raw input into this musical treatment, how much musical content is there?

NONE.

Personally, I would love to hear the results of a search algorithm that finds segments of Pi that come close to mimicking a famous melody. I have no doubt that “Happy Birthday To You” … using the digit 0 to represent a rest, and the digits 1 through 8 to represent the notes G3 through G4 in the C-major scale … can be found somewhere in Pi.

Screen Shot 2014-12-18 at 9.28.56 AM

Just for fun, I figured out what those digits would be. Here they are:

112010403000112010504000118060403020776040504

Sure, it might require wading through billions of digits using a special-purpose pattern-finding algorithm to get statistically close to this exact sequence. But at least the musician would have done some work. And it would be just as original.

What about “Pi Art”?

links-pi-cristianIt’s beautiful.

But meaningless. The digits of Pi are statistically no different than a random sequence of digits. Why not use the golden mean, or e, or the square root of 2? The digits in these irrational numbers are just as meaningless as Pi.

I submit that the best way to get creative over Pi is to think about its real meaning: Pi is the ratio of a circle’s circumference to its diameter. This ratio shows up in some interesting, and sometimes unexpected, situations – like Buffon’s needle:

Screen Shot 2014-12-18 at 10.53.28 AM

I will close with a simple, elegant expression of Pi. It may not be beautiful, but it says it all.

circdiam_42104_lg