Runescape

Prerequisite: none

Yup, I played Runescape back in 2006. My account has a combat level 90, total level 1220, and some 10 mil. gp in the bank. That is a lot of time spent (no life, hurhur). And moving away from childhood, there is simply not enough time. My commitment lies elsewhere. It has been almost ten years since I started, and some six years since I stopped playing regularly.

Occasionally I would skim the updates, and if something looks interesting I might log in. Sadly, I am not a child anymore and it takes much more to amuse me.

It sounds like the game is receiving lots of bashing recently. I think it is only a matter of growing up. Players have grown and grown out of the game, as a child would grow out children's clothes. Some things were great in childhood, such as Harry Potter. Heck, Harry Potter used to be the best thing evaaar. As a kid there was so much to marvel at. Then the novelty dies out. Only the nostalgia remains.

Will I still play it? No.
Is Runescape a lame game? No.
Did the enchantment wear off? Pretty much. Ya.

Having sacrificed so much time exploring a virtual world, what have I gained? Were my efforts wasted? Was the game worth it? Although I am not known for gaming, I know for a fact that many things had stayed with me.

1) Vocabulary

I boosted my vocabulary terrifically through playing Runescape, being a foreign English speaker. I know the ores from copper to gold (as well as the fictional ores). I know the fish from crayfish to swordfish. I know the trees from oak to yew. I know the folk from barbarian to knight. I know the creatures from goblin to dragon. What is mithril? What is a pike? Where are willows found? What does the apothecary do? What is the difference between a ghost and a revenant (besides that one is mightily stronger than the other)?

How else should I know everything from dagger to longsword, rapier to maul, mace to battleaxe, boots to vambraces? What is the difference between a platebody and a chainbody? A coif and a cowl? What is so distinct about the scimitar? Oh, scimitars..

Most players probably do not bother with the "examine" function, but I examined everything. The thistles on the grass. The ducklings in the river. The mushroom by a rock. The rock itself. I know the difference between a stalagmite and a stalactite. A fern and a shrub. Wheat and grain. I could go on forever..

Ya, the kind of thing I do..

2) Technology

Never will I forget that Cu (copper) and Sn (tin) make bronze. Ever. Or that Fe (iron) and carbon (coal) make steel. Or that alloys with higher carbon (coal) content make sturdier metals. Or that impurities can exist in ores, and cause the iron bar you are smelting to fail. Imagine my delight when I came across these details in AP chemistry.

If you played Runescape, you know a thing or two about ore mining, bar smelting, spear fishing, gem cutting, leather tanning, flour milling, dairy churning, and possibly to your expense, scamming (not a technology, but ya).

3) Just.. things

What does it mean to "blow a raspberry"? What is a kebab? What are runes anyway? And of course there is plenty of nonsense logic as well.

And the things I may have encountered too soon for my age..

And the music. Although the MIDI is awful, the composition is very something. The modality really brings out the European medieval setting. It is nice to hear something else for a change that is not Ionian or Aeolian. I recognize some Lydian, Dorian, and Phrygian..

Just things.

I had gleaned all I could from the game. There is nothing more for me in that bubble. Headed back to reality ever since and called it an experience. 10,051,749 XP to be precise.

Really feeling this way (in real life) throughout high school life and towards university:

Maybe that is the most important thing I learned from Runescape. Got to embrace every waking moment. That XP is going to make a difference. So get out of that virtual world and get some real life XP.. and become some real life skill master. Unless you want to stay a noob.

Neumes

Recommended prerequisite: music theory

For most people Bach is the oldest composer on their repertoire, who lived somewhere between the 1600s into the 1700s. In the mainstream the oldest things out there are probably Vivaldi or Pachelbel. There is nothing wrong with that. It is understandable that fairly modern techniques are not directly transferrable to the Baroque music style.

But this post is not about Baroque music (which is another story to itself). What if I told you that before Bach, even before the Renaissance, their music was written with square notes?

These square notes are neumes. Notice that there are four staff lines instead of five.. and there are no definite measures or tempo. Is this notation outdated, antiquated, primitive, or what? No, this comes to the same issue as Baroque music: the style is simply different.

This notation was often used for plainchant sung in churches. More important to understand is the purpose of plainchant. It was not so much for public performance than for personal prayer. Those European cathedrals had very deep echoes, so sounds delayed, overlapped and bounced back. In that environment there was no need for precise rhythmic notation.

Another quirk about neumes: there is no specified starting pitch. You get to sing with whatever "key" you like. Notice that four staff lines just about spans one octave, which is the average vocal range.

Brilliant. How to sing it?

An Idiot's Guide to Square Notes tells you everything you need to know! In fact it explains better than I can. I like how it explains that you would not notate a symphony with neumes any more than you would a Gregorian chant with modern notation, because the styles require different approaches. The two notations were even invented by the same person. Hur hur.

For a feel of how it sounds: https://www.youtube.com/watch?v=pqDIEjQfdNk

Why are neumes not getting much attention? Perhaps the religious background makes people wary? What I am more interested in is how this style of notation can change our mentalities towards music. Is there some potential we can rediscover from the freedom neumes provide?

G Flute

Recommended prerequisite: music theory, physics

Four months ago I made a G flute from a flagstick.
This calculator takes care of drilling details: http://11wall-west.com/~ph_kosel/flutomat.html. These are the measurements I used:

The holes I made are ugly.. punctured with a sewing needle then “dug” out with pencil and scissors. There is no mouthpiece. The end is just flat like a straw. It is almost the same way ancient people put six holes into a segment of reed from a riverbank.

The way woodwind works is the stream of air must split in half. The recorder is easiest to play since the design splits the air for you. The modern flute is a little harder since you must aim at the edge (or maybe not, I never played one). As for my straw-like flute, it is quite something else..

The sounding concept of my flat brim flute is pretty much like a ney, which I learned from here: http://www.neyzen.com/ney_metodu.html. The technique is veeeeery hard. You are somehow supposed to split your airstream on the flat brim. A subtle twitch in lip shape, ney angle, or tongue placement is enough to extinguish the sound. As you can see I only managed three notes (D, E, F#) after trying for three months:

By the way, there is another sounding technique which requires wedging the edge between your two front teeth and hissing like a viper.. This Persian method creates a unique tone, but no thanks. The Arabic/Turkish method I am using is difficult enough.

Really, during the first three months I only produced air and wispy harmonics. As you may or may not have known from A Cringeworthy Process, I a not one to give up so quickly. This is recently, my fourth month trying:

I achieved two extra notes by overblowing the D and E to get their perfect fifths A and B. The fingerings are the same, but the stronger breath makes the second harmonic (perfect fifth) ring. For some insight to how it works, refer to Harmonic Intervals and Resonance and Timbre and Overtones. I can manage overblown octaves as well, but my breath transition needs some attention first. It is generally not recommended that I learn so many notes before mastering the sound quality.. still working on it!

How was the first flute inspired? My guess is an observation of wind flowing through an empty log. But from this flutemaking experience I find that flat brimmed flutes are really hard to play. Even after making the flute, it takes three months' effort to produce a sound (perhaps even longer for the first flutist ever, since there are no preceding flutists to teach the first). Either ancient people had nothing better to do, or they were increeeeedibly smart. I think they actually were smart.

Five notes (D, E, F#, A, B) is enough to play a couple songs but let me work my intonation first. Then I might play you something decent..

Kauffman Spin and Vassiliev Singularity

Follow up of Knot Theory and Polynomials

Prerequisite: algebra 2

Some more about the nature of knots.

Kauffman Spin

Kauffman assigned spins to each intersection on a knot, either "up"or "down". In Sossinsky's book Knots: Mathematics with a Twist, he uses sticks instead pointing to checkered A or B regions. Notice that if you travel along a positive crossing of a knot, A will always be to the right of B relative to the direction you are traveling, then change places with B after the intersection.

In this example of the figure eight knot, A and B regions are assigned like blacks and whites on a chessboard. Note that the outside is an A region. In this particular example there are three A spins and one B spin, as indicated by the sticks. If you smooth out the sticks it will either join or seal up shapes in the knot. In this particular example there are two closed shapes, so there are two gamma regions.

Plug in these numbers to the equation below and you will have only obtained a partial sum. There are four crossings in the figure eight knot, so there are sixteen possible assignments of A and B spins (given by 2^n, where n is the total number of crossings). For the fifteen other possible spin assignments, count A spins, B spins, gamma regions, obtain all partial sums, then calculate the grand sum. The result should be a polynomial for the knot K.

Reminiscent of electron spins. I like.

Vassiliev Singularity

Vassiliev had another idea of knot crossings. He treated crossings as singularities. A point, as you would with two intersecting lines on a graph. As a knot changes crossing through singularity, it becomes another knot and so passes to another domain on the map. All knots in the same domain are the same knot, and knots on boundaries have singularity crossings.

I. The value of the unknot is 0.

II. All routes produce the same results.

III. Crossing a boundary along its arrow is +1, and crossing a boundary against its arrow is -1.

Starting with O going to C, to F, to H, the corresponding values are 0 +1 -1 -1, the sum of which is -1. Taking the alternate route from O straight to H is 0 -1, which is also -1.

Neat! Except I am not sure how the map is determined. *Scratches head*.

This new rule is similar to Conway's Skein Relation again. This takes even more brain bending to get around. The second row shows unknots, which rule I determines has a value of 0. The two pairs of knots in the third row are zeroes, which total zero.

In the fourth row, the trefoil is broken down, then further broken down in the fifth row. There are three unknots and one trefoil, giving a value of 1. Knot crossings definitely do not form singularities in real life, but this idea does produce fascinating results.

I feel like I can never look at objects the same way again. Some part of me will tell me to mould it, and give it a little twist. There is more in the book that I am still trying to get my head around..

Knot Theory and Polynomials

Followup of Braid Theory

Prerequisite: algebra 2

Mathematics is broad and wide. Nonmathematicians have probably never heard of topology, which is about the properties of deformed objects and spaces. Under topology is knot theory.

Receive an introduction from Numberphile:
https://www.youtube.com/watch?v=aqyyhhnGraw

And take up some basics from this site. I love the commutativity and associativity of knots. Simple yet mindblowing:
http://www.popmath.org.uk/exhib/knotexhib.html

I was at this awkward stage where the understandable information are too easy and the advanced information are too difficult. Then I found Knots: Mathematics with a Twist by Alexei Sossinsky. It tells of ways people applied rules to discover invariants, consistent properties that suggests an underlying truth. I bent my brains a little, but at least it is still topologically the same brain..

Conway's Skein Relation

A knot is topologically the same no matter how you stretch and tangle it. But once you cut and reattach sections, it is no longer the same knot. There are three possible ways to attach two sections, eerily reminiscent of topoisomerase and their work on DNA strands.

In order to attach numerical value to knots, three rules are established (pardon me for using N instead of L). The upsidedown ∆ denotes "the polynomial of" whatever is in the bracket.

I. Two knots are the same if their polynomials are the same.
II. The polynomial of the unknot is 1.
III. Conway's Skein Relation: [the polynomial of N+] minus [the polynomial of N-] is [x times the polynomial of N˚].

Rule III is better understood with this diagram, where the intersection/break is the only point of difference between these three knots.

If we plug in the unknot into Conway's Skein Relation, we get 1 - 1 = 0x, which is just 0. From rules II and III we found that the polynomial of a double ring is 0.

If you still have not figured out how Conway's Skein Relation works, notice how the parts outside the dashed circles are the same, and the insides are N+, N-, and N˚.

Conway's Skein Relation is as easy as algebra. If we plug in the double ring and unknot along with their polynomials, we get ∆(H+) - (0) = x(1). The polynomial of the hopf link is x.

We can bring it even further. You need to stretch your imagination a little more in this example. (1) - ∆(T) = x(x), do the algebra, and figure that the trefoil knot is -(x^2)+1.

Note that rule I applies all the way.. so far. The problem with Conway's polynomials is that a trefoil knot and its mirror image have the same polynomial. They are topologically different because one cannot be deformed into the other without breaking, and Conway's polynomials do not express that. More advances will be made to solve this problem.

Kauffman Bracket

The revised rule I is another form of Conway's Skein Relation. In rule II <KUO> denotes a knot added to an unknot. Rule III states again that the polynomial of an unknot is 1.

You can figure that <00> is (-a^2 - a^-2) by rule II. Let K be an unknot, which is linked to another unknot. (-a^2 - a^-2)<1> is just (-a^2 - a^-2). <00> = (-a^2 - a^-2).

Here are some more polynomial assigning that you can do according to Kauffman's revision. More algebra, but with a knotty twist. You may have noticed that the two knots on the same row as the hopf link are in fact unknots, but their polynomials are not the same! One has a positive degree while the other is negative.

The little formula at the base solves the problem. The w(K) (called a writhe) is equal to the number of positive crossing minus the number of negative crossings for a knot K.

Then Jones did another revision after Kauffman. I am not sure how recent updates work, or if there are even any.

There is something mysterious about knots. It may appear to be just a piece of string, yet it can be so spatially complex. It is a mesmerizing tangle of intraplay. True, simple, aesthetic. I like.

Damn. What will humans decide to number next?

A Cringeworthy Process

Recommended prerequisite: art 1 (I never took it either, so no worries)

When I ask someone, "Why don't you try to draw something?", the replies are always something along the lines of "I am not good at it".

I guess it is nice for others to know about my successes, but perhaps it is more important for them to know about my failures. And I can assure you that failure happens more often than success.

If you ever hear three minutes of perfection at the piano, understand that it took three years of failure to get there. No exaggerating. Sometimes I had to run a measure fifty times over (not that I counted) in order to get it right. Sometimes I had to put a piece off for a couple months so that I might build the required skills before returning to it.

To prove my point here is a not-so-flattering watercolour from grade 8 (I am cringing so hard right now):

The background is supposed to be one smooth gradient. The object spacing is awful. Stark crooked linings, little hue diversity.. and for some reason I left a lot of white edges (still combating the habit now). If there is anything presentable about this, that is because the teacher fixed it for me. I am facepalming so hard right now. Hur hur.

You have no right to say that you are no good at something, until you have tried the following:

1) Give yourself a second chance. It surprises me how many people become dismayed at their first attempt. Really, if your first attempt at something is a masterpiece then you must be a genius. But you are not, and I am not, so we need second chances. Forgive yourself for not being born a genius. Let go of pride.

This was what I was trying to portray:

And here is my first attempt at the bottlemouth:

This is me in the summer of 2015, my first time glazing. Wow so lifelike *cough cough*. There is no hint of transparency or reflection whatsoever! Yuck.

I may have gained some art experience over the last three years, but none from a proper classroom setting, and none with a brush. This is my second watercolour since grade 8 (my first is just a few months before this) but that was not enough to make me "fluent". Of course my first glazing sucked. I would not be surprised to know that your first glazing sucked too.

2) Focus on one section at a time. How do you expect to do the whole thing if you cannot even do a half, a quarter, or one tiny portion? Do not let this process discourage you. If you need to minimize the portion, do what you need to do. Since I did not do the bottlemouth too well, I did repetitions. All ellipses are so off, and the fuzz is so grotesque I just want to crawl into a hole.

Note that you are not just doing blind repetitions, otherwise you will be "practicing your mistakes"! What you are doing is not exercising a procedural memory, as psychology suggests. You are trying to overcome a mental block. Practice is as much thinking as doing. You cannot practice without one or the other. I did one bottlemouth a day and took four hours with each (estimate), thinking a whole lot before my next layer of glaze (I could afford to spend so much time only because it was the summer holidays).

3) Listen to counsel. Experienced people know what you need. If you ever wondered how you can achieve something, they have the answers. To ditch their suggestions is to do exactly what keeps you from succeeding. If someone tells you that in 2x = 1, x = 1/2, and you insist that x is anything but 1/2.. good luck in finding a solution.

This bottlemouth is significantly less fuzzy than my previous ones but according to my mentor, this is not enough (if I had shown him the ones before this it would have made him foam at the mouth, or perhaps gave him cancer). There may have been more that he would like to point out but at least he was honest enough to catalyze the progress you see throughout this post. Otherwise I would be stuck at stage one.

Perhaps I still have issues with my lines, but I agree that fixing this problem will bring me to a whole new level. Maybe I should practice some lines, just lines, and try to overcome my mental block..

What I cannot show you is how I agonized to capture a decent-enough photo for this painting. The photos were long trashed but I can tell you that I submitted five or so photos to my mentor, and captured ten times more photos than I submitted (estimate). The process was a whole week to itself, just to get the one photo for this painting. If I had listened more closely it would not have taken so long.

But then again, if my first shot was perfect I must be a genius.

4) Do not worry about comparing yourself to others. "When I see how amazing they are, I just want to give up". This is one of the most common problems people face. Don't get me wrong, comparison is okay. Comparing the work is good for finding influence, but comparing the people does nothing to help you improve.

So what if someone is younger? Guess what, you will only get older, and today is as young as you will ever be. Get working today while you still have the strength in your limbs and the plasticity in your brain. Do you mean to stay idle for the rest of your life just because someone else is better? So what if someone is better? There will aaaaalways be someone who is better. I do not intend to live my next sixty years without trying anything new, well do you? Are you really going to give up now, and live without it for the rest of your life? So what if someone took up the skill later than you did but still does better than you? That person certainly did not waste time comparing!

(Sorry for the rant)

More importantly, reconsider why you are doing this. Am I painting because I want to earn other people's praises? If the only reward you seek is superficial, you will not get very far. Or am I painting because I like to paint? Sure the process is gruelling, but I do like to create all things pleasant! Then that is what matters. Do something for the sake of doing it, and not for other people's approval. Have some intrinsic motivation.

Rather than wasting time on how people perceive you, why not refine your work so that you do not even need to worry about their perception? Treat the disease, not the symptom.

I have never seen what an art 1 student's work is supposed to look like. Nor do I try to find out. All I relied on was my mentor's feedback, some exemplars, and an honest self-critique. This is a bottlecap on a jar lid.

There is not enough contrast for highlights to stand out. Lines are still fuzzy. The lid is not even properly elliptical. It looks like a dead oyster. I had no idea how an art 1 student should do, but I was determined to fix it.

5) Be patient. At this point in time I had already spent away one month of my summer holidays. And yet I had not started my official canvas. This trial took me a week or two.

The letters do not follow the curve. There are not enough reflections on the jar. The banana has a funny pallor. The marble is not splotchy enough. The wall is too yellow. I am glad to have spent the extra week.

The official took some three weeks. If I had only shown you this, you would be misled to think that this is what you should be able to do on your first attempt. But that is not the case. I had to endure a month of failure.

Of course this is by no means perfect. The table line does not seem to make sense. There is something eerie about those cinnamon sticks. There is something not so convincing about the thermos flasks. The blue bottle at the back is rather stripy. The puddles on the pink jar are awkward. The bottlecap does not look like it fits on the bottlemouth. The axes of the bottle do not match. The splotches on the marble are not quite the right shape..

And my damned lines. Always. Squiggly, fuzzy, drunk lines.
But I am not cringing. At least, not yet..

When I paint now I still cross my fingers and hope that my lines will turn out presentable. It should not feel this way. I know for a fact that I do not "hope" that the intonation of my violin will sound right. If I practice enough I will have more control and confidence. This is not the end of the climb. Actually, there is no end to the climb. So there is no reason to be disappointed that you have not reached the top. This is why you should enjoy the climb, and not feed on praises regarding your climb. Intrinsic motivation.

My goal for the next three years is to be able to look back at this painting, cringe really hard, and crawl into a hole. Truly, that would be a sign of progress.

I still laugh at this.
It is cringeful. It is worthy. It is cringeworthy.

 In summary:
1) Give yourself a second chance.
2) Focus on one section at a time.
3) Listen to counsel.
4) Do not worry about comparing yourself to others.
5) Be patient.

Unless you have done all of the above, I forbiiiiid you to say that you are "no good" at something!

Japanese Pagoda

Prerequisite: none

This is a Japanese pagoda from 700AD:

Japan has intense seismic zones. A couple of its earthquakes made headline news in the past. How come we never hear about fallen pagodas? Ever?

Because they don't. Pagodas don't fall. These buildings remain while modern innovations continue to crumble.

Construction secrets revealed here:
http://web-japan.org/nipponia/nipponia33/en/topic/

To summarize the article:
1) Wood is flexible.
2) Tenon-mortise joints increase flexibility.
3) All levels are independent of other levels,
4) so that each level sways in counter balance
5) on a central pillar.

So there you have it. The pagodas were designed to absorb and shake out the shock. A Chinese idiom describes this very well: 以柔克剛. The literal translation is "to conquer hardness with softness". "To conquer force with yielding" sounds much better with our context.

 Profound eh.

Riemann Sphere

Prerequisite: algebra 2

Remember this little prat?

I hated this guy (especially when it touched my roots). It seemed like a figment of imagination created by man. It was an ugly freak that did not belong in nature. Yuck yuck yuck the imposter..

If I had allowed myself to let go of the real plane for a little while, I might have come to appreciate the imaginary.

My favourite part in Zero by Charles Seife, ironically, was not the historical origin of zero, the development of the number zero, or the philosophical aspects of zero. The Riemann Sphere came into my spotlight.

The complex counterpart of our real unit circle looks like this:

Very neat eh. If the number is outside the circle, raising it to some exponent will make it grow further away from the circle. If the number is inside the circle, it will collapse toward the origin. Sorry for the nonstandard notation.

And here you have the actual sphere. Assign the complex plane at the equator, add infinity at the north and zero at the south. Place the south of the sphere on the origin of the plane.

The way Seife puts it, imagine that there is a light at the north of the sphere. Whatever is on the sphere will be projected onto the plane like a shadow.

Interestingly, circles that intersect the north of the sphere appear as lines on the plane. Seife's book Zero introduces infinity as zero's "twin", in which understanding one leads to a better understanding of the other. Really nice story, but that is for another post.

The sphere is cool because you can transform it. Say you transform a number x by i. Spin the sphere 90˚ then points on the sphere will correspond to their projected points on the plane, accordingly with the transformation.

For example:
1 on the plane = i on the transformed sphere
-i on the plane = 1 on the transformed sphere
-1 on the plane = -i on the transformed sphere
i on the plane = -1 on the transformed sphere

In other words, 1 = i after the transformation, and so on.

Seife was particularly interested in this transformation, regarding the division of zero and infinity.

The Riemann Sphere brought some kind of order to the chaos. I like how the imaginary terrain is mapped out, since it has always seemed an alternate ghostly realm to me. The neat visual tames my fear of the imaginary number. The best kind of math is one that is true, simple, yet aesthetic. The Riemann Sphere has achieved all those elements.

But still, I would rather not have i mess with my roots.

Polynesian Navigation

Prerequisite: none

Many people are impressed that Columbus sailed across the Atlantic in 1500.

But consider this.

It is 1000 BC and you need a way to sail throughout the Pacific. You know nothing about compasses, massive ships, and telescopes. You do not even know of writing. How are you supposed to navigate the largest ocean on a canoe?

The Polynesians certainly found a way.
http://www.exploratorium.edu/neverlost/

This site gives details on how they construct canoes, read weathers, utilize stars, pinpoint latitudes, harness winds, measure traveling speeds, interpret wave patterns.. and much more! It is all very impressive.

There is even evidence of having sailed all the way to South America. Surely these people must have been somewhat sophisticated to have done such feats. How many people before the birth of Christ had managed to travel such distances in their lifetime? Not a lot!

I refuse to believe that interactions only happen across land. Despite being split on our standard world map, the Pacific is just as full of activity and exchange. There is a widespread something from Taiwan, to New Zealand, to Hawaii, and all the lesser known islands in between. When I think of the relation between Polynesian and Aboriginal Taiwanese sea culture, it opens to me a further curiosity for the forgotten islands on the Pacific.

I ought to find out more about these people..

Omniglot

Prerequisite: none

Wondrous site: http://omniglot.com/

"The online encyclopaedia of writing systems and languages".

So what?

I knew that many minor languages exist, but this collection blew my mind. There are sooooo many languages and scripts within the neighbourhood of one region. It is also interesting to compare between proximally close languages.

Above all, foreign scripts are just so pretty. How many of us are aware that there are writing such as these in northern China? Or even that Manchurians have their own script?

Some pages to get started:

http://omniglot.com/writing/types.htm
Difference between abjad, abugida, alphabet, syllabary, and semanto-phoenetic writing systems.

http://omniglot.com/language/articles/index.htm
Articles, articles, and articles. About anything really.

http://omniglot.com/writing/direction.htm
Writing directions can be so odd you just have to see it to believe it.

The Ancient Berber script (in Morocco) runs from bottom to top.. hurhur!

Language is one of the many aspects of life that we take for granted. It is strange to think of the voices we never get to hear, of the records of past civilizations that are no longer intelligible, or simply legacies that no longer exist.. as well as current cultures that are heading down this path.

Timbre and Overtones

Followup of Harmonic Intervals and Resonance

Prerequisites: physics, music theory

Common misconception: play an A on the violin and the frequency is 440 Hz.
Almost true.

Most sounds are actually a blend of many frequencies above the fundamental frequency. Timbre, or colour, is the difference in harmonic distribution that gives each musical instrument its own distinct sound. Timbre can also vary between each person's voice, or subtly between each violin.

Recall the formula f = n*(v/2L) where
f = frequency
n = harmonic number
v = speed of sound
l = length of string or open pipe
and the fundamental frequency is when n = 1

Here is what I mean by "a blend of many frequencies". The higher frequencies are harmonics. The fundamental is most amplified and so the most noticeable.

This reminds me of overtone singing (if you have no idea what that is: https://www.youtube.com/watch?v=7zZainT9v6Q). The funny whistling sound are overtones , or harmonics, and the lowest note is the fundamental. Frequency ratios early in the harmonic series are easier to produce than ratios further down the series. Meaning, consonant intervals have stronger resonance from constructive interference, while dissonant intervals sound fainter from destructive interference.

Not all of us regard the human body as a musical instrument. There are many hollows in the human body that allow strong vibration, which is needed for harmonics to be noticeable at all. Resonance is not only established with others, but with the self as well when you let your other harmonics ring. Maybe not necessarily to produce overtones, but to build a richer timbre.

 Consider the fact that organisms are designed according to their functions. What function does singing serve humans? Why is music even a thing? For me, music is more than an art or a recreation. There is something transcendental about the way music works. I am still figuring things out here..

Harmonic Intervals and Resonance

Prerequisites: physics, algrebra 2, music theory

Polyphony is a sweet phenomenon, even if in physics it just means that two sound waves overlap each other. The first few frequency ratios in the harmonic series sound consonant (stable), and the ratios further along sound more dissonant (unstable).

harmonic series: (n)/(n+1)

Here I have the ratios graphed out next to its counterpart notation in music theory (I flipped the fractions just because it is easier for me to visualize). The dashed line is the resultant wave of superpositioning the two waves. Not be confused with the musical term, the physics term for that gross fluctuating amplitude is beat (and that, my fellow string ensemble mates, is how you can tell whether your strings are slightly out of tune).

Recall that amplitude corresponds to volume, and period corresponds to frequency.

And what about superpositioning the same pitch? If you think about it, the resultant wave is just double the amplitude while maintaining the same shape. When sound waves overlap in a way that amplifies itself nicely, it makes resonance. Consonances are more resonant than dissonances.

These things are indeed taught at school, but no teacher strings them together this way. You may have heard that music and maths are related (now you know it means more than counting beats). This is only one of the examples.

And no, my point does not end here.

Playing in an orchestra or singing in a choir is a humbling experience. According to how resonance works, playing off pitch can cause interferences, decrease the amplitude where waves cancel out each other, and make beats (the physics term). In order to achieve resonance, everyone has to listen to each other and blend in.

Harmony, literally.