The math behind music | Linear algebra episode 3

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#vectors #linearalgebra #innerproduct #orthogonality #correlation #fourier
I'm very excited about this video, because it contains some experiments with sound and music. We explain Fourier analysis from the ground up, step by step. The final formula is not as difficult as you may think: it's just a projection of a sound signal onto a set of basis waves. We build a simple low-pass filter and some simple synthesizer wave forms. And then we explain Fourier analysis all over again, from two completely different perspectives!
[SOUND 1] / geert-van-damme-289809712
You can find the music in this video on SoundCloud. The artist is Geert Van Damme. Go and listen to his compositions, they're amazing!
If you want to support our channel financially, you can do so on Patreon, where you also get early access to new videos: www.patreon.com/user?u=86649007
Here are some extra links where you can find more information:
[3B1B 1] • But what is the Fourie...
A beautiful explanation of Fourier analysis as a search for maximal correlation between 2 signals.
[MTB 1] • Why {1,x,x²} Is a Terr...
Pavel Grinfeld explains why the powers of x are a very bad set of basis functions. The video includes diagrams of these basis functions, and provides a better alternative.
[WIKI 1] en.wikipedia.org/wiki/Fourier...
Cool animation that shows how a time signal breaks apart into frequencies.
[WIKI 2] en.wikipedia.org/wiki/Triangl...
Information about the triangle wave and its spectrum. You can even listen to a triangle wave being built up from sine waves, one by one. Also shows the convergence of the Fourier series as the value of n goes up.
[WIKI 3] en.wikipedia.org/wiki/Square_...
Information about the block wave and its spectrum.
[STEX 1] math.stackexchange.com/questi...
The proof of the orthonormality of sine & cosine waves.
0:00 Introduction
1:14 Treating polynomials as vectors
5:59 Treating all functions as vectors
8:15 An infinite-dimensional vector space
10:21 The inner product of 2 functions
14:43 Waves: an orthonormal basis
20:15 How to find the coefficients
23:01 Sine & cosine waves are orthonormal
26:26 Building a low-pass filter
28:34 The spectrum of a sound
31:34 Correlation between two sound signals
32:56 The Least Squares method
36:12 Outro
This video is published under a CC Attribution license
( creativecommons.org/licenses/... )

Опубликовано:

1 июл 2024

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Комментарии : 19

@eNicMate Месяц назад

The relations shown in this video are completely mindblowing.... As a hobby mathemathician and professionalelectronic engineer this has been a bit lifechanging .... thanks a lot.

@mike2884 6 месяцев назад

Music AND math ? Im sold. 😊 thank you!

@oukid2633 3 месяца назад

This is so good I was always wondering how orthogonality relates to Fourier series thank you!

@AllAnglesMath 3 месяца назад

You're welcome. Always happy to clarify something.

@linuxp00 6 месяцев назад

Nice, QM fundamentals from scratch haha. Better than my university's mathematical techniques course

@cangrejoxidao 6 месяцев назад

This is an awesome series! I love the pace, and the perfect ratio between the density of information and the clarity of the steps and examples..

@AllAnglesMath 6 месяцев назад

Thank you so much! It's often difficult to make a good trade-off between going too fast and going too slow. I'm glad you like the balance we found.

@parsahamidi 6 месяцев назад

Beautiful video as always!

@AllAnglesMath 6 месяцев назад

Thank you!

@nanamacapagal8342 4 месяца назад

Beautiful video as always Also I did try and experiment to see what happens when the spectrum of frequencies is continuous, and tried a continuous interval from 0 to 1 of cosine waves (Yes this breaks the nice inner product space but I got curious) Had to use riemann sum to account for partial multiple frequencies, the result looked like sin(x)/x. Now I can see why all the engineers are excited about this function, to the point they gave it a name sinc(x)

@AllAnglesMath 4 месяца назад

Wow, that's a really useful experiment. What you've (re)discovered here is called the (continuous) Fourier transform. Basically, what you did is calculate a Fourier series for a function that isn't periodic at all. The official way to do this, is to let the period T go to infinity. As you do so, the different frequencies in the spectrum get closer and closer together, until they become a continuum. The result is a transform that takes a non-periodic continuous function to a non-periodic continuous spectrum.

@aaronmartens2903 6 месяцев назад

Thanks for the video

@rylieweaver1516 6 месяцев назад

These videos are so good! I refer them to all my math friends

@AllAnglesMath 6 месяцев назад

That's great to hear, thank you so much.

@aflons8893 4 месяца назад

Is there any other inner product for functions than the integral in the video, which also has all the nice properties? Or is that one the "best" one? Great series 😀

@AllAnglesMath 4 месяца назад

Great question! There are many different inner products that you could define. For starters, you can introduce a "weight function" under the integral. Different weights give different results, which basically makes the number of possibilities infinite. In engineering, these weights can be used e.g. to put more emphasis on a certain part of the interval. If you want your approximations to be better near the edge points, you just use a weight function which is larger there. I'm pretty sure that you can also come up with inner products that don't use an integral. e.g. If you sum up the function values at a discrete set of input points, you probably already get a good inner product, although I'm not sure.

@coredullgraph 14 дней назад

Thanks for the video. Am I understanding correctly that any function can be written as a linear combination of 1,x,x^2.... (or the terms in the Fourier series)? If so, is there a proof for this?

@AllAnglesMath 10 дней назад

Thanks for the question. The answer is no: there are many exceptions. Plenty of functions don't have a Taylor series, and plenty more have a Taylor series that doesn't converge for all real inputs.