Music And Measure Theory

A connection between a classical puzzle about rational numbers and what makes music harmonious.

Thanks to these viewers for their contributions to translations
German: Josh
Russian: e-p-h

[binarymessiah]

You are secretly tricking music lovers into learning epsilon/delta proofs for convergent series and sequences.

[patrickhodson]

I knew you were a mathemagician, but I didn't realize you were a mathemusician, too. That's awesome!

[MrBeiragua]

I love the fact that you don't expect us to learn it easily, but rather to actively try to understand what you're presenting us. You're one of a kind here in youtube. I have grown sad about the fact that many youtube content providers have given up using maths in their explanation of maths and physics, and when some of them tries to use it, they're hit with a backlash. Maths can be used to help understand itself (that isn't so obvious, even though it's maths for math) and I am certain you're going the right direction. Love your videos!

[tonidoom]

You, 3blue1brown guy, are a genius.
Never have I seen such beautiful math videos.

[soniczdawun]

Absolutely wonderful. First, I get a MinutePhysics video about approximating harmonics, then you come in and tie it in with interval coverage in Measure Theory. I just have one request: Please, for the love of God, don't ever stop making these. These videos are becoming a major thing I look forward to every month.

[thedocta_certified]

My music teacher told the class a story about a music professor he knew that trained himself to always be extremely in tune. As a result of this he began to go crazy because he would always hear the slight out of tuneness in music which made all music sound bad to him.

[macteos]

Here in december 2021, the improvement in your animation and recording quality has been massive over 6 year, but I'm impressed to see how you could present such complex and interesting topics even trough less fine tools. Now I need to see every video in your channel. Keep it up!

[Felipemelazzi]

9:10 "(...) the Proof has us thinking Analytically (...), but our Intuition has us thinking Geometrically (...)."

The wisest words I've heard in a long time.

[Xeroxias]

The deal with simple ratios sounding pleasant has to do with something called overtones. If you get a drum vibrating at f1 = 220 Hz, it might also tend to vibrate at modes of 2*f1 = 440 Hz and 3*f1 = 660 Hz and so on. Those higher frequencies, multiples of the fundamental, are overtones. A drum vibrating at f2 = 330 Hz will also have a relatively strong mode at 2*f2 = 660 Hz, so the 220 Hz and 330 Hz tones (separated by the simple fraction r = 3/2) together reinforce each other and have a pleasing effect.

Overtones get faint really quickly, which is why fractions with large denominators sound dissonant.

[terryendicott]

I liked your presentation on Lebesgue measure. It would have distracted from you narrative but I want to point out to your audience that you actually proved a more general theorem. Every countable subset of the reals has measure zero

[tryrshaughroad]

Could you imagine that my task was to prove that today, when I had watched your video yesterday? I aced my exam of course, thanks to you.

[patrickhodson]

Cacophonous is not the same as dissonant. The word you want is dissonant.

[franzluggin]

You just helped me solve a topological problem with your approach to measure the rationals in the interval [0, 1]. Which made me realize that the rationals, even though they only consist of single point sets that are not adjacent to each other, are not closed under the usual topology of the real numbers, which in turn made me realize that the theorem of Heine/Borel that closed + bounded = compact is something truly special about the real numbers.

[Mammutinc]

Wow, just wow! That is seriously a ridiculously beautiful proof and consequence! Man, I was just sitting staring dumbfounded at my computer screen because of the elegance I was presented. This is why I love and adore math and find it one of the most beautiful things we humans can perceive and understand.

[hannahalsouqi]

This was beautiful and mesmerising. Without being cliche, this opened my eyes and made me see measure theory in a way nobody else has explained it (at least to me). And it harmonised with me on a deep level. Your videos are so well thought out and graphically concise. Thank you for all your work, and keep it up 👍

[LiamHaleMcCarty]

“Suppose there is a musical savant who finds pleasure in all pairs of notes whose frequencies have a rational ratio” ... gotta be Jacob Collier

[silpheedTandy]

constructive criticsm: the musical sounds given as examples need to be LOUDER. they are much too quiet relative to the narration, and i found myself straining and a little annoyed by this. i comment because i otherwise love the videos i've seen so far on this channel!

[Lorkin]

m/n, where {m, n} is rational, m<n and m->infinity, n->infinity

...Right? There's no structure to it though but that algorithm should include every possible outcome

[smrd]

3 blue + 1 brown= uncountably infinite magnificence. Thank-you for your ongoing contributions in teaching us all.

[marioouellette]

La suite de Farey d'ordre n>0 dans lN est l'ensemble des fractions dont le numérateur et le dénominateur sont premier entre eux, appartenant à l'intervalle fermé de 0 à 1 et dont le dénominateur est inférieur ou égale à n, le tout ordonné par l'ordre usuel sur lR les nombres réels. Pour chaque ordre n chaque fraction a/b à donc une fraction qui la suit dans la suite de Farey d'ordre n. nous appellerons la successeur de a/b.
Considérons l'ensemble de toutes les fractions réduites dans l' intervalle fermée de 0 à 1 comme les points d'un graphe G pour laquelle il existe un unique morphisme entre une fraction a/n, laquelle apparait pour la première fois dans la suite de Farey d'ordre n, vers une autre fraction c/d si et seulement si c/d est la meilleur approximation de a/n avec d<n ou si c/d =a/b .Par exemple 2/3 est la meilleur approximation de 3/5, on a donc par exemple mais en particulier une série de flèche 13/8 vers 5/8 de 3/5 vers 2/3 vers 1/2 vers 1/1. Désolé pour les gens qui ne sont pas familier avec le langage des catégorie. Ceci étant considérons, la catégorie des chemins de ce graphe que nous noterons CG . Cette construction donne lieu à une structure d'ensemble ordonné partiellement . Remarquons que les petits éléments sont les fractions avec des dénominateurs petits. Ma conjecture est que la perception de chaque individu et un foncteur de la catégorie CG vers elle même. intuitivement le foncteur correspond à l'acuité auditive de l'individu. Celle -ci est nécessairement limitée pour des raisons purement physiologiques. On peut donc accorder un piano avec le tempérament en racine douzième de 2 notre perception sera toujours construite sur des approximations rationnels . Nous sommes très sensible aux écarts trop grands pour les octaves ou les quintes justes car les rapport de fréquences sont respectivement 1/2 et 2/3 ce qui n'est pas le cas pour des intervalles comme la septième majeur qui a un rapport autour de 15/16 . La meilleur approximation d'une fraction s'obtient par son développement en fraction continue.
Q'advient-il si on considère des accord de trois ou quatre sons? J'ai bien une hypothèse à ce sujet mais je n'ai pas jusqu'à aujourd'hui développé un algorithme simple qui s'apparente au développement des fractions en fractions continues. Les traités d'harmonie classiques nous enseignent que l'accord de septième de dominante demande un résolution sur l'accord parfait majeur. La théorie que j'ai élaboré illustre bien cette loi d'harmonie.