Playing the piano taught me math (2016)
xiaoyunyang.medium.com
xiaoyunyang.medium.com
For instance, on piano you can see AND hear the intervals on a scale.
Strangely though, outside of elementary school there’s less emphasis on tactile learning—and no football is not the same thing. Art teaches the mind how to think outside the box and in creative ways. It also taps into the subconscious and allows us to learn and communicate better.
And then I had to unlearn that so that no one else heard it.
- Exponentials and logarithms
- Harmonic series
- Combinatorics
- Chaos theory
- Some calculus
- Resonance
- Linear algebra
It also taught me about physics, cognitive psychology, learning theory, history and even some sociology.
There's some stuff with group actions going on - I think you can think of a dihedral group acting on sets of triads or something along those lines and there are some articles out there about that, but I feel like I don't have a good mapping in my head as to the relationship between music and mathematics.
Perhaps this is not what the GP meant though, I’m curious about however it applies. No idea about where linear algebra could fit in, I have yet to see those parallels myself yet but interesting to think how it could relate
W.r.t. linear algebra: mostly in relation to sound synthesis, but also in relation to music composition, theory of overtones, piano tunings... You ask, I'll explain.
Music composition: this is a bit more far fetched, but I always thought of composition as a combination of vectors which can vary semi-independently. We want to pick some plane through that vector space and move within the plane. For example, we can vary in harmonic complexity, melodic complexity, rhythmic complexity, dynamics, tempo. But it wouldn't make sense to do all at the same time. So we pick one plane, say harmonic complexity and dynamics, and move within that plane. This allows the listener to get used to the 'space' that is opened by the composition. We can then slowly add more vectors, to keep the interest.
A very visual representation of this is the trackpads that you can see on certain modular synthesizers. Often there is a complex setup, but the variables of the setup are reduced to two dimensions, which allows the composers to vary within this projected plane.
Piano tunings: well, basically you have a vector of 230 piano strings of which the overtones are all slightly out of tune. To tune a piano 'correctly', you need to minimise set of equations of strings and overtones (matrix).
> I was called out during class when my teacher saw me fiddling with my fingers during a classroom exercise
From their LinkedIn profile, it appears that the author went to school in the US. I'm not from the US, but my impression of US schools is the complete opposite of one where students are taught to "memorize math facts" or one where students get "called out" for doing something in a different way.
In fact, from the many workbooks that I see on the net, my impression is one where a lot of pain is taken to ensure that students understand the material without resorting to rote memorization.
Also, my impression is that students, if they show independence of thought or potential in an area, are given a lot of support to pursue it from within the schooling system in the form of being put on accelerated paths, freedom to attend classes of higher grades, etc.
In contrast, many other countries have rigid grade structures, where students are forced to stick to the level of discourse in their grade in school, and can only pursue their interest outside the schooling system.
Perhaps some of the European systems have it better, but this flexibility to rise beyond your grade within the school system is something that isn't available in a LOT of countries.
You can't generalize US schools. There are a lot of them, and they generally answer to different authorities, local school districts, state school districts, religious (Catholic) schools do their own thing, etc.
The only exception to this for me was pre-calc, in which they taught us trigonometry rules without explaining it's basic relationships to the unit circle. Our calculus teacher was dumbfounded the next year when we had no intuitive understanding for these things. It took an hour to correct. No big loss. Shrug.
Was forced to memorize times tables, long division, random formulas. Rarely was proof based instruction provided.
Some teachers encouraged creative thought, others would mark a correct answer wrong because you didn't use the specific method they taught in class.
Ironically, having graduated college with a minor in mathematics - I still don't have my times tables memorized.
As an adult, I have mixed feelings about this experience. My mother used to brag about this she had me do it in front of her friends, other parents with kids my age. I distinctly remember during this in a Perkins while trying to enjoy my breakfast. For me, this was simply a trick involving a larger “working memory” as opposed to “rote memory”. Being that it was still a matter of being a “trick” that required practice albeit a different kind of practice it becomes something I thought of as a trained monkey scenario.
This has come up a lot I’m my life. I worked in construction for a while out of high school and my boss would treat me this way as well. He would have me be his human calculator to help him figure out measurements. One time this was going on in a complicated octagonal room where we were trying to lay out brackets for a projection screen. He asked me to run some numbers for him and I refused. I did this because I understood what he thought he was doing but knew that his method was flawed. So I refused to give him the wrong answer, even though it was the right answer to the numbers he gave me. It caused I really big fight where he began talking down to me and tried to put me in my place.
Another Forman was there who witnessed the whole thing and I ended up doing a job with that other guy like 6 months later, but he had not forgotten the incident. He brought it up and gave me a long speech about how I should have just given my boss what he asked for instead of the correct measurements. Something I, to this day, steadfastly disagree with. Fast forward a year or two and my boss was fired due to his incompetence in measuring and ordering the materials that cost the company many tens of thousands of dollars.
Intelligence and thinking skills, in general, are practiced and exercised. However, they are not trained-monkey tricks. It is a method of living well and garnering fullness from life. And to no other end is it acceptable to waste such a thing.
4 + 3 = 7
#define TRUE 1
#define FALSE !TRUE
void print_key();
void show_scale();
double power();
char notes[13][3]={"A ","A#","B ","C ","C#","D ","D#","E ","F ","F#","G ","G#","A "};
char scale_name[8][12]={"Ion(M)","Dor","Phryg","Lyd","Mixolyd","Aeol(m)","Locr"};
int scale[16]={2,2,1,2,2,2,1,2,2,1,2,2,2,1};
int main()
{
int key=3,i,its;
printf("\nWelcome to Scale Printer.\n\n");
do
{
print_key(key); printf("\n");
for(i=0;i<=6;i++)
{
show_scale(key,i,scale_name[i]);
printf("\n");
}
key = (key + 7) % 12;
its++;
printf("\n");
} while(its<12);
return 0;
}
void print_key(int key)
{
int i;
printf("Key of %s\t:",notes[key]);
for (i=0;i<13;i++) printf("%s ",notes[(key+i) % 12]);
}
void show_scale(int key,int mode,char \*modename)
{
int i,j,scale_note;
scale_note=key;
printf("%s of %s\t:",modename,notes[key]);
for(i=0;i<8;i++)
{
printf("%s",notes[scale_note]);
scale_note = (scale_note + scale[i+mode]) % 12;
printf(" ");
for(j=1;j < scale[i+mode];j++) printf(" ");
}
}Not really as most professional musicians understand the problem, but it shows that no instrument can be “perfectly” tuned.
(3-1) + (3-1) = (5-1)
The artsy side of music is very much un-sciency and mostly rationalized post-hoc.
I'd truly like to know how Von Neumann remembered everything.
I've been hearing claims of von Neumann's perfect memory for a long time on HN, but I've been unable to find primary sources to back them except for a single quotation by Herman Goldstine. Could the claim of "perfect memory" have been an exaggeration?
Surely von Neumann had an exceptional memory, but truly remembering everything is extremely rare (some say impossible). If he did, I imagine that the psychologists at Princeton or the IAS would have caught wind of the mathematician with a perfect memory and wouldn't have missed a chance to examine him; all the more so since "true photographic memory has never been demonstrated to exist" [1].