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

[@binarymessiah7670]

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

[@cdsmetalhead99]

This doesn't really have anything to do with epsilon-delta arguments. Epsilon is just the standard symbol for a small positive real number.

[@mattfoster669]

Abstract Messiah huh huh stupid random symbol of meaning

[@High_Priest_Jonko]

It's Measure Theory

[@alielhajj7769]

cdsmetalhead99 actually it do, it is a proof that the measure of the set of rationals is zero

[@SalekeenNayeem]

It's amazing! I'm doing my Real Analysis course this semester And as a music theory enthusiast, this is a treat!

[@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!

[@psilocyberspaceman]

+Mosco Monster Except, Physics is not about Maths; Physics is about objects.

[@MrBeiragua]

+psilocyberspaceman sorry, but physics is a long tradition of using maths to explain "objects" (whatever you mean by it). If one doesn't use maths, it isn't physics.

[@psilocyberspaceman]

Mosco Monster " If one doesn't use maths, it isn't physics." - What a ridiculous and religious proposition. An object is that which has shape. On the other hand, what is a point (you know, the axiom that maths use extensively but can’t really explain)?

[@3blue1brown]

+Mosco Monster I agree that math is best appreciated and most beautiful when staring into the heart of its substance, rather than glancing only at the results. That said, I understand why most math youtubers tend not to walk through the reasoning itself. The unfortunate chicken-and-egg problem is that people will not appreciate the heart if they aren't relatively fluent in the language of math, so the only way to evoke the curiosity necessary to develop that fluency is to give an unexplained peak at the rewards that will come. Anyway, thank you for the kind words.

[@MrBeiragua]

+psilocyberspaceman You might say it is a "religious position", but i am just stating what physics is, or rather what physics was constructed to be in the last centuries. Galileo said mathematics is the language with which the book of nature is written; Newton called his book "Mathematical principle of the Natural philosophy". Mathematics is a part of theoretical physics that simply cannot be taken away. If you take maths away from the science of physics, you're left with something other than physics. We can talk about physics without maths, but we can only really talk physics in mathematical language. That's how the science was made.

[@patrickhodson8715]

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

[@8948380]

We need to tell ViHart

[@henryg.8762]

@@8948380 But of course!

[@seeseefok7659]

magic

[@ValkyRiver]

I am a microtonalist, and I recommend to you 53 equal temperament

[@segmentsAndCurves]

@@ValkyRiver Haha.

[@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.

[@Frisbieinstein]

I have read that all people with perfect pitch are bothered by this. Glad I don't have it.

[@Rudxain]

I have a similar ability, I can hear some of the harmonics generated by a guitar string and some electrical devices (mains hum). However I can only hear 3 (fundamental plus 2) sines or a max of 5 if I concentrate enough lol. I'm losing this ability because I'm not training currently. I also can hear the periodic change in amplitude when 2 waves are very near in frequency (periodic interference patterns), even if these waveforms have different timbres (like a sawtooth and a guitar string), but I can only do so if both waves are actually interacting in the air, my brain cannot simulate interference in real time, the waves must "touch" for me to notice how close or far their frequencies are (how tuned or out of tune)

[@Thebiggestgordon]

@@Rudxain hearing the changes in period is how most string players tune their instruments, if the strings are close to in tune but not quite you get a slight pulsing/throbbing sound because the strings are wiggling in and out of phase. It’s very useful.

[@Rudxain]

@@Thebiggestgordon interesting, I thought I was one of the only people that do that lol. However, I later realized that trying to tune a string perfectly and relative to another string is a bad idea, since guitars are supposed to use "equal-temperament", not "justonic", so I was using justonic for tuning the strings, but the "metal plates" were still tuned using equal-temperament, yielding some sort of hybrid tuning system

[@Thebiggestgordon]

@@Rudxain you mean frets? It’s true they’re tuned to equal temperament, but it’s not worth your time tuning the strings to equal temperament. Unless you have insane aural skills, there’s almost no way you’re going to be able to reliably hear to the correct hundredth of a tone to be out of tune by while tuning, and your audience certainly isn’t going to be able to tell. It’s easiest to just tune the 2 E strings so that they are perfect octaves, then tune each of the strings a perfect 4th to the string next to them using the pulsy out of phase technique.

[@soniczdawun1]

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.

[@3blue1brown]

+Guy Edwards Well, that certainly just brightened my day. I'll try to keep them up as I can.

[@martind2520]

Guy is absolutely right there! These videos are awesome!

[@Lashoun]

I can't agree more with what he says: your videos are so clear I can see my reflection on them. Thank you.

[@baladevadasasanchezarias4082]

Great video, thank you very much. And nice pic of

[@henryg.8762]

YESSSSSSSSSSSSSSSSSSSss

[@toni3doom]

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

[@cocoa1996]

His name is Grant Sanderson :) He wrote the entire animation library himself. It's on github :D

[@valasfar1557]

toni3doom He went to Stanford as well.

[@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.

[@irrelevant_noob]

@The Scourge Protector yet another instance in which your intuition fails you. :p

[@macteos8539]

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!

[@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

[@kenhimurabr]

Electroacoustic music has branches of works over EXACTLY this.

[@joeyhardin5903]

when he said about the 23:21 polyrhythm i just instantly thought of him

[@pixiecute839]

He came mind exactly to mind.

[@obiwankenobi3058]

Harry Partch baby

[@joeyhardin5903]

@@obiwankenobi3058 h

[@terryendicott2939]

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

[@henrikljungstrand2036]

Is that true though? Is it not rather the case that every countable subset of the reals must be absent of the property of having any well defined measure? You cannot get say just the rational numbers only, as an infinite union of OPEN subsets, yet a measurable set must be defined as a union of open subsets iirc, since the open intervals/balls are the basic measurable sets. Then we have the Banach-Tarski nonsense (doing non-constructive things using dubious axioms like Choice or Law of omniscience or other ill defined infinitary operations) which purportedly allows us to duplicate volume through cutting a ball into a finite number of non-measurable pieces, and then move and rotate them before composing them again "seamlessly", no scaling used.

[@EebstertheGreat]

@@henrikljungstrand2036 It's definitely true. Most measurable sets are not open. The Lebesgue measure is defined in terms of open coverings, and this video shows an example of an open covering for the rationals. You can cover all the rationals with intervals whose combined length is any positive real number. Therefore the infimum is zero, and so is the Lebesgue measure.† There is no need to appeal to "infinitary operations," and nothing is ill-defined. The axiom of choice is not necessary for this conclusion at all. In fact, it should be intuitive that a countable set should have measure zero, since it can always be contained within an arbitrarily small set. Like, a line segment can be contained in an arbitrarily thin rectangle, so it should have a smaller area than any rectangle, i.e. zero area. † More precisely, this defines the rational numbers as a _null set_ in the reals. To prove that these have measure zero, we still have to prove that they are measurable, e.g. by showing that they satisfy the Carathéodory criterion. This is a pretty straightforward calculation. Let X,Y ⊆ *R* with λ*(X) = 0, where λ* is the Lebesgue outer measure, and let X' = *R* \ X. Then λ*(Y) ≤ λ*(Y∩X) + λ*(Y∩X') by subadditivity. On the other hand, 0 ≤ λ*(Y∩X) ≤ λ*(X) = 0 (so λ*(Y∩X) = 0), and λ*(Y∩X') ≤ λ*(X), again by subadditivity. Therefore λ*(Y∩X) + λ*(Y∩X') ≤ λ*(X). Putting these two inequalities together, we get λ*(X) = λ*(Y∩X) + λ*(Y∩X'), the Carathéodory criterion.

[@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.

[@andreadanelli7831]

Mathematics is a universal language!

[@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.

[@SIrLoneRabbit]

This is correct.

[@TwelfthRoot2]

But overtones don’t exist when you’re talking about instruments that produce a single tone per note (e.g. pipe organ, tuning forks, etc.). Here comparing the tone with the beat “tone” is a better explanation, and what should only be considered. The overtone series of overtone-producing instruments are just reinforced by the OT series.

[@kenahoo]

@@TwelfthRoot2 There are plenty of rich overtones in pipe organs & tuning forks.

[@zombiedude347]

When any object vibrates, it doesn't actually create a set of discrete "overtones", but a continuous spectrum of vibrations that peak at it's "overtones". Peaks clear to our ears are tonal, and peaks muddy to our ears are atonal.

[@coqdorysme]

Your comment would have been great if you chose a string instead of a drum. Drums have modes of vibration where the "harmonies" are not integer multiples of the fundamental, and that's why some of them are called "unpitched".

[@gustavomello2207]

The limit of quality when it approaches to perfection is your channel. I totally love your explanations and your animation style. Keep up with the good work! I've watched all your videos and I've already become a fan

[@franzluggin398]

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.

[@02vLxcZF]

Very nice video. Would you consider doing videos on Measure Theory in the same vein as those you have done for linear algebra?

[@3blue1brown]

I won't rule it out, I'd love to do lots of "Essence of" series.

[@jgrecbio]

I'd love to watch it one day. I love to learn the way you teach, thanks for this great work

[@EvanZamir]

3rded! This was a great, fascinating presentation. Would love to see a full series on measure theory.

[@HaouasLeDocteur]

I would definitely check it out!

[@antoniovelazquez9869]

Yes, please! More videos about measure theory!!

[@LucenProject]

13:12 This has helped me get a better grasp of countable vs uncountable infinities far more the many Cantor based videos. THANK YOU!

[@tryrshaughroad551]

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.

[@ha-tsuneko]

“For others, like square root of 2, it sounds cacophonous.” Me: beautiful, amazing lydian modal interchange

[@johnnycochicken]

amazing isn't it? yet it sounds restless either way, even though it needn't feel sour

[@pauljonathan3799]

its also a tritone

[@johnd4345]

@Orion D. Hunter Patrick Hodson beat ya to it

[@WilliamFord972]

Even a perfect fourth sounds dissonant against the bass.

[@klaasbil8459]

This channel is the VERY best to explain complex concepts. The presentation of the logic is very well thought-out and effective, and the animations that go with it are well-chosen and beautifully worked out.

[@matthewsutton377]

Ever since being introduced to music theory I have always wanted to see if there was a pattern for fundamentally why some frequency sounds harmonious with another, but at a fundamental level instead of 'they just do'. I tried various sources but yours explained it so clearly and so fundamentally, in just the way my math oriented side wanted to see it. It's fascinating to see it broken down to rational numbers and find that instead of bogging down the spirit of music and art, it has made me more interested and awestruck at music than I was before. I think the breakthrough for me was when you pointed out that the harmonies between frequencies might break down into an order similar to 5:7 or 3:4, or whatever it may break down to if you zoom in on every oscillation. Thank you so much for making this video!

[@hannahalsouqi7609]

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 👍

[@LunizIsGlacey]

You could say it _opened your ears_ too!

[@Nirmanyu]

I absolutely love your videos, and everything about them. You perfectly bring out the beauty of mathematics! Keep the good work up :)

[@pooyaja]

Just beautiful. I loved the fact that when you were naming a few rational numbers, the corresponding chord was seamlessly playing on the background. Nice tying up with the Lebesgue measure at the end.

[@srineeey]

I was struggling with an example in Functional Analysis, and you sir, just handed me the solution on a silver platter, hats off!

[@patrickhodson8715]

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

[@MatildaHinanawi]

This is what I was thinking too, and it actually threw me off.

[@abdvs325]

Patrick Hodson what is the difference in meaning?

[@ragnkja]

abdvs325 A single dissonant interval is too structured to be cacophonous, while a chaotic sequence of harmonious intervals can be cacophonous without being dissonant.

[@cloyun-hee9564]

Dissonance is only relative anyways. A typically dissonant chord will sound consonant if put in context of more dissonant chords, and vice versa. Harmony ain't simple, that's why we need *psycho*-acoustics to truly understand how we perceive consonance, dissonance, tension, release, etc.

[@franciscoaragao5398]

Patrick Hodson no. You must refine you definition of dissonance, Tyndal, Schoenberg and Helmoltz can help you

[@smrd0110]

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

[@CRGreathouse]

This is jaw-droppingly gorgeous. I knew the measure theory result and that rational and near-rational tone ratios were pleasing but had never connected them this way.

[@DaRealMidnight]

I absolutely love your videos! They show a different side of mathematics from the one that is being taught in college and it helps understanding it on a deeper level. Also they are really well made, you seem to have a talent for this.

[@awsmunicorn7488]

"musical fifths" music theory teacher, sweating violenting and shaking audibly: "what type of fifth"

when omitted, you assume perfect fifth. who in their right mind would refer to a diminieshed or an augmented fifth as just a fifth?

[@awsmunicorn7488]

@ yes, please tell that to all the pedantic and homicidal music theory teachers

@@awsmunicorn7488 im not a music student so idk what they do to you in there, but wow the teachers are sadictic if they are that pedantic :D

[@ALPalmos]

Just call it a perfect fifth and be done with it ;-)

[@pmcgee003]

Love the videos ... especially Sum 2^n. Just a personal perspective, this one had a strange emphasis to my mind. The fact that the rationals take up none of the number line is amazing, bizarre and cool. This is such a different approach to that that I find it a strange and fascinating. Not in a bad way. Looking forward to more, and to promoting this channel.

[@postylem]

Another beautiful exposition! Thank you for all the work that goes into these videos!

[@Songvbm]

Being a longtime student in music mathematics and statistics I just loved your explanation regearding the fact. Upload more like this one.

[@StarmuteVII]

The graphics in your videoa are exceptional. Excellent job!

[@RahulKumar-if9wt]

using python.

[@juneglave5014]

you blew my mind. I want to know math and be like you. thank you for the inspiration

[@spencertaylor6910]

Wow. And Lebesgue integrals suddenly make so much more sense. Thanks Grant, you're the best!

[@tmagrit]

Your vídeos and ideas are absolutely amazing! Thx so much!

[@ruchirrawat8804]

as a pianist and a math lover this is the most enlightening video i've ever watched

[@rarelycomments]

Glad your audio quality is catching up with the visual quality in recent videos! This sounds like it was recorded in a cupboard using a cup on a string

[@pedrocoza]

The work behind these videos is amazing, I love the dedication and the ability to explain hard things on an accessible way for all.

[@DASmallWorlds]

It really is. Only complaint is that his older videos have a poor microphone-voice quality which comes up occasionally with unpleasant 'pops'. Other than that, it's absolutely enthralling.

[@oarevalo21]

Thank you. I've studied math in my teens and adult years but hadn't seen modern math used in such a beautiful clear way to reason about an experience so intimate like music.

[@guitarpaintings9319]

This is a beautiful video, thanks so much.

[@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!

[@Bubdiddly]

Chill out tandy

[@DASmallWorlds]

@@Bubdiddly BIg necro my dude

[@zoroarklover363]

@@DASmallWorlds smol necro

[@TheMartian11]

There was no need to provide a constructive criticism really, it's not that big of an issue, you could have just said that it could be louder.

[@joeybee3718]

@@TheMartian11 lol "no need to provide constructive criticism, you could have just provided constructive criticism"

[@stephanovskyy]

Truly spectacular video ! Went through the Conservatory of Music in Paris and barely ever touched the relation between music and mathematics, even thought they are so intricately linked ! Absolute pleasure to watch !

[@prdoyle]

Mind blown. This video was at the perfect level for me -- I saw each result coming just at the moment you begen to explain it. Also that block of text at the end was super interesting too.

[@Gameboygenius]

The concept of harmoniousness has to do with the distribution of overtones of a sound and how they are "projected" onto the cochlea. The cochlea works in the frequency domain and does something similar to a Fourier transform, where every individual tone (i a very hand-wavy sense) applies pressure to one point along the cochlear spiral. A harmonious interval will have many common points between the two notes played, keeping in mind that the two notes will have a number of overtones apart from the fundamentals. As far as overtone distribution is concerned, most natural instruments will have overtones corresponding to multiples of the harmonic series over the fundamental. You can however synthesize a sound where the whole spectrum is compressed or expanded, such that each frequency component is shifted by a factor, say 0.9 or 1.1. When doing this, you can create a rather convincing major chord where the fundamentals of the chord notes lie at 0.9*1, 0.9*5/4, 0.9*3/2. Some natural instruments do in fact have non-linear overtone distributions, and that's in fact a pet peeve of mine regarding MinutePhysic's video. One such instrument is the piano, which needs a so-called stretched tuning the lower octaves are tuned slightly lower compared to the expected tuning, and the higher octaves are tuned slightly higher. When I saw the title of the MinutePhysics video, Why It's Impossible to Tune a Piano, I was expecting it to mention that phenomenon. The video was about equal temperament, when in fact justly intonated tuning was the norm before ET came around.

[@crimfan]

There are end effects on strings, too. This is why a guitar needs to have its intonation set with fine tuners down at the saddle. People with particularly sharp ears sometimes set it at the nut too (e.g., see Buzz Feiten) but if you make that compensation it won't sound good with other instruments that aren't compensated. A guitarist who really mashes strings into the fretboard will often sound sharp and there are ways for them to end up sounding flat, too.

[@WilliamFord972]

It’s baffling that we “hear” on a logarithmic scale. That is, any two notes separated by an octave sound the same “distance” apart, even though the frequencies double with each octave jump.

[@dudleybrooks515]

Thank you for adding this. I added my comment about William Sethares's experiments with ideal scales for various timbres before seeing your comment -- so apologies for a bit of duplication. I think the moral is that the mathematics of "real-world" music is much more complicated than simple ratios.

[@Gameboygenius]

@@WilliamFord972 it's not actually baffling at all. Most of our senses are in fact logarithmic in nature, like how we perceive light and sound intensity. If we somehow experienced frequency linearly, it would be the odd one out. It also makes sense if you look at either the waveform, or the spectrum, of the signal and notice how they line up. The more they line up, the more harmonious the music sounds, roughly speaking.

[@hizzogg]

I may misunderstand this guy but as I find his math fascinating and well presented, I'm still looking for the connection to his musical claims. I'll watch a few more times and see if I get it. I did just want to supplement this video with some additional music history and acoustics information (not that I'm an expert but maybe someone can help me out if I miss something). The video states that using powers of the twelfth root of two works well because the ratios generated are very close to low-denominator rational numbers. This is very true, however, it should be noted that using 19th, 22nd, 31st, 41st and 53rd roots is fairly common among microtonalists and theorists to achieve some very pure (meaning close to those low-denominator ratios) intervals. Makes sense that as the root number increases you have far more numbers to play with and as a result, some of them are more likely to be closer to those ratios. Also note that not all numbers get good results here. Using powers of 10th, 28th, 45th roots, or just any random root number, won't work to well, as they don't give values as close to the pure intervals. The ones I listed above are some of the values found to work well. Maybe there's a mathematical proof to be worked on there! Btw, powers of 22nd roots are used in India for tuning. Just thought this should be added to his explanation so people don't think there is something especially magical about powers of 12th roots. Sorry for this short rant: I have stated the following in many a scathing response to a would-be musical-mathematical claim (because it really gets frustrating): Pythagoras was only the VERY beginning of music theory as we know it - literally. There have been countless brilliant philosophers, composers, performers, instrument designers and mathematicians working on music theory and tuning in the past 2500 years. There are countless books, treatises, essays and dissertations on the subject. I've come accross a lot of mathematicians and computer scientists who, obviously not having learned much about music theory, decided to attempt revolutionizing it with their radical new idea. Usually, it doesn't apply at all and sometimes reinvents a wheel that we threw away or improved upon hundreds of years ago. The author of this video isn't so far off, however, his lack of musical jargon seems to demonstrate a lack of research. It only takes a couple of short essays or wikipedia articles to pick up some good music theory jargon and certainly to learn the meaning of 'dissonant' and how it is used in context, instead of cacophonous. end rant Lastly, dissonance is subjective and changes through time and culture. There are charts, made by psychoacousticians, showing some near-universal perceptions of dissonance for "all" intervals. If we only look at Western music, we will find a time in history when only the 1:2 ratio was acceptable as a "harmony" (it's really just an octave). Later, the 3:2 ratio became acceptable, then 4:3, then 5:4 and so on, later using all of the reciprocal intervals as well. These denominators are very low. Our culture has been steadily increasing those lower denominators. If it seems that we stopped developing this way, that is, stopped at 12th roots in our equally tempered instruments, it is due to cultural-economic factors. Certainly, there are plenty of musicians using larger denominators (or larger root-based larger octave divisions for equal tempered tunings) and many of them can appreciate and readily identify those intervals. If we go back to the 13th century, before ears were accustomed to 3rds (6:5 and 5:4) played simultaneously as a sustained chordal sonority, does the theory presented in this video break down? If the value of epsilon in this proof is arbitrary, what is the author really saying about our tolerance for dissonance based on distance from the low-denominator ratios? Also, he is making claims about dissonance, especially in regards to his hypothetical savant, that he is not backing up and may not be provable at all. What value of epsilon is acceptable for making this connection at all? Unless I'm missing the point, which is very possible, what I see here is a near-connection or a hypothesis that requires some real experimentation. What I think is presented here is some very rudimentary and incomplete information about tuning and a very nice proof that may or may not be connected. If I'm way off here, help me out! Just to get a sense of psychoacoustics and a few of the hundreds of historical tuning systems: mentalmodels.princeton.edu/papers/2012musical-dissonance.pdf en.wikipedia.org/wiki/Musical_temperament

[@sebreens]

I believe that the part about epsilon was more related to Measure Theory than to music. What he wanted to present with the hypothetical savant was that if there was one with incredibly accurate ears, but also one that enjoys even the most complex rational numbers, his 'acceptable notes' will still cover only an arbitrarily small portion of the whole interval. If we look back at things in history, at first everything was simple and more complex arts were perceived as the so called cacophony, but as we, humans, evolve our culture, art and all other aspects follow. The same as people in the 13th century most probably wouldn't find any joy in surrealism, the same may follow for music. As you may say, there may be many more roots that would produce ratios very close to rational, but they also produce higher denominators, which means that the music gets more complex. This could be the reason why for me as a westerner it is harder to enjoy Indian music as I am not used to the level of tonal complexity. My hypothesis is that if higher root division of the scale gets adapted in a society's culture, the society would perceive the music 'reasonably' consonant. That said, it would be easier for them to enjoy lower root division music, than for the unaccustomed folks - higher division music.

[@u.v.s.5583]

The answer to your torment is very simple. This is mathematics in the spirit of Ian Stewart. The application is purely hypothetical. Very important for a Pythagorean, with probably no application for the musician. (Except when he wants to define a Lebesgue integral for some obscure purpose.)

[@crimfan]

It's not elaborated in the video but musicians have argued a lot about tuning over the centuries. Even temperament (based on 12th roots of 2) enables two musical tasks----modulation between keys and transposition----but it has its downsides and in other applications is often replaced with just intonation. Other tuning systems, such as used in many non-Common Practice musical traditions make use of microtones. However, by doing that, they largely give up the ability to modulate. Even understanding composer's intent from the days before ET is tricky. Keys with many sharps or flats back in the day (e.g., Ab, C#, or F#) were much more dissonant than keys with few sharps or flats (e.g., C, F, or G). A composer writing in one of those keys was, whatever else they were doing, exploring that dissonance. Now the key doesn't matter as much. This is a great help for anyone who needs transposition, such as a vocalist. However, groups like barbershop quartets or brass bands often use JI instead because of how much richer the harmonics sound. Even in instruments tuned in a Common Practice style a lot of playing techniques have been developed to mimic the sounds you'd get in a different intonation. The blues, for instance, explores clashes between minor and major thirds and minor and major sevenths, in big part because the just third and just seventh are in between those notes. (I hope I explained that right.) There are a ton of videos on this online, some of which go into loony territory. I think a followup video might explore how different kinds of approximations can matter. ET makes all notes sound a little out of tune to make things work out on average while JI tries to optimize a different criterion.

[@galek75]

If he has ears, let him hear. You are clearly trying to push a historicist narrative. It really helps to ignore the math, doesn't it?

[@smittymcjob2582]

Great comment! I'm neither a mathematician nor a musician so excuse my errors on terminology and the basics. Following on your comment: -Has anyone ever proved that a people raised only on dissonant harmony (combination of notes whose frequency ratios are irrational numbers) will find our music pleasing? Is liking combinations of thirds and fifths by brain construction or is it learned? I know this sounds like asking if anyone ever proved people don't like pain (some things are just obvious), but I still wonder since as you said there was a time when even combinations of thirds and fifths sounded dissonant and people learned to like those, and maybe this could have unfolded totally differently if musicians back in the old days, faced with extending music to dissonant-sounding thirds and fifths or dissonant sounding irrational ratios, had picked the latter. -As you said, It is important to note that 12th. root is not the only way to achieve even temperament. There are others but perhaps because of practicalities in instrument design and other factors the simpler 12th. root dominated. -I followed the mathematics part of the video intently and I think I understood most of it. It occurred to me at the end that the gist of the math is that there is infinitely more irrational numbers than rational numbers between 1 and 2 (maybe aleph-0 for rationals, aleph-1 for irrationals in the range?) and so even if you like ALL the rational frequency ratios (no matter how large the denominator) there is infinitely more frequency rations you won't like, with the important conclusion that you can't just randomly go assigning frequencies to the notes between two C notes since random selection will certainly lead to irrational rations and dissonant notes. In that sense, even though the conclusion was important to note the mathematical proof didn't seem that necessary, and the time could have been spent delving more into the main topic of the video. -I laughed at your point about using "cacophonous" instead of "dissonant" since a friend having watched this video was telling me that in music theory two notes whose frequencies aren't related by a simple ratio are called "cacophonous"... This is how new terminology comes about I guess... don't be surprised if you see "cacophonous" appearing in music theory books in the next few years... The video was great. I don't mean to criticize, but maybe point out what I felt was missing.

[@niemandniets9010]

i already knew everything in this video, you explained some of the basics really well. I gave it a like and i subscribed, keep up the good work!

[@SiyiZhou]

Great. This solves the question that I am wondering for a long time when learning music theory.

[@peterlim5730]

Many thanks---you have stirred me all up and this I think will be my 1st lesson in this fascinating technology. I am a self-taught melodies-writer----I made a strange discovery in the past year that I could write music from numbers--as you know, the Chinese use the numerical system instead of Western musical notations----still very popular in China and several parts of the world--say 2x 3 =6 which will sound ray, mi lah....or 3 -1-6-2 which will sound mi-doh-lah ( below middle c) ray....then I would write these phrases into my computer software which of course will appear as sheet music (with Western notations) .....last example- 1-6-3-2 --doh lah (below middle C) mi ray.....in every such case, I can compose a piece of music which seems 'right'--- of course, there are almost limitless possibilities . Being self-trained, I depend on my intuition and imagination----I think I've achieved something using my own methodology --some of my music is played in Melbourne and others will be published. Jean-Jacques Rousseau the famous French philosopher wrote about this subject as well--music written in numericals . The advantage of the numerical system is that it is ubiquitously understood by all musicians and at the same music can be played IN ANY KEY without another set of score. I don't know whether Western composers/musicians could make sense of what I am saying. Anyway, I am thrilled to have seen this very rare video. My deepest thanks again-what's the name of the author?

[@333_studios]

I get you bro. I would LOVE to have a numerical system as the standard instead. My mind is better geared toward jumping between intervals rather than memorizing dead-on placements of notes. Transposing keys is annoying too. A numerical system would make that much easier. I'm horrible at reading notes, but I love creating my own melodies.

[@cthomp31]

So excited to find this video, thanks for exploring the real math of music theory. One thing to double check: at 1:36 -- an 8/5 interval is actually a type of minor 6th, not major (which would typically be 5/3)

[@user-yn1mu2eb8t]

yes i was about to say same thing

[@brianevans4]

I found this so interesting (as someone trying to learn more about harmony). Please make more videos on the mathematics behind musical frequencies and the patterns that make for harmony. There is such great depth in this subject, I'd love to know more, and I find your way of explaining do satisfying to my curiosity

[@akilansundaram2181]

Extremely good content once again. Connecting music and mathematics, just amazing. Couldn't have gotten any better.

[@FelipeV3444]

I watched this video some time ago and hardly understood shit. I now learned real analysis and yes, indeed this is beautiful. Thanks for sharing!

[@SOBIESKI_freedom]

You vids are fabulous. Kudos to you, brother. Keep 'em comin'! :))

[@LK90512]

Every time I watch this video it amazes me like the fist one. You are truly an artist.

[@jerryhu9005]

Looking at a video from 3 years ago, I can see just how much better your microphone has gotten

[@sunglow9835]

7:43 42nd rational number. He knows the meaning of life guys

[@p.as.in.pterodactyl1024]

Yeah, but does he know the _question_ to the meaning of life?

[@tom7]

Neat!

[@thomashall4683]

As someone who loves both Maths and Music, I found this very fascinating!!!

[@miguellopezrivera2560]

Hi! I'm an engineering student at GMU. I just wanted to affirm you on your work. It's amazingly portrayed and I like how you ask us to take time on our own to rethink and rediscover. I've not commented on anything on KZbin in years so I'll leave you to interpret that at will and there is so much more I'd like to say but I'll leave it at that. Thanks so much and I hope to see more work of yours

[@msgordito99]

I love your videos. They makes me learn a lot of all kinds of things related with mathematics. Keep doing it. And I have a question, which type of font did you use in your videos?

[@3blue1brown]

+Sebastián Escobar +Arnau Mas is correct, it is the default LaTeX font.

[@geurgeury]

+3Blue1Brown \varepsilon >>>>> \epsilon

[@anfedres]

+3Blue1Brown Could you tell me what kind of video editor or programn are you using. I´d like to use for my presentations. Thank you.

[@3blue1brown]

+Criangulien Man, you are so right. What have I been doing with my life.

[@3blue1brown]

+Andrés Felipe Echeverri Guevara I made my own little animation tool. It's kind of custom to my own use right now, and isn't super friendly as a library to download and learn, but if you feel comfortable with python and want to poke around with it, I'd be happy to point you to where you can find it.

[@dhausmig]

I like this. It reminds me of a study I did with a computer in the 1970s determining which roots of 2 produced values with the smallest error approximating fractions with low denominators. I found that the 19th root of 2 had less total error than the 12th root of 2 without having an excessive number of notes. I wanted to make an instrument based on the 19 note scale but found that if I moved the bridge of a guitar back so that the 19th fret was equidistant from the nut and the bridge I couldn't find guitar strings long enough. Refretting the neck and maintaining the bridge position seemed like it would be more work than I cared to do. I also considered making a piano keyboard with one black key between B-C and E-F and two black keys where conventional pianos have a single black key. Now that I have a 3D printer on order I intend to do this. Knowing that this instrument might not sound right with other instruments, I intend to add a switch which will mute the single black key and make both double black keys play the same note using the 12th root of 2. In other words it would be possible to use it as a normal keyboard. Do you have any thoughts on the 19 note scale?

[@3blue1brown]

+Bill Bohan That sounds awesome! I think I may have read about someone experimenting with different numbers of notes in the scale. I'd be curious to hear how it sounds. Have you tried playing with these sounds just using a computer?

[@dhausmig]

I have just started to explore this with sonic-pi. I looked at csound before but it was too complex to get started.

[@Q3020Q]

Hi Bill, were you able to produce any results?

[@emmanuelamiot]

There was a guitarist invited at a colloquium in Puerto Vallarta in Nov 2014 (icmm.cucei.udg.mx/) who played using a 7-note scale out of the 19-note temperament. Noah Jordan his name. 19/12 is related to 12/7 in the Stern-Brocot tree, which has connections with musical scales and temperament stuff (and word theory), but it is pretty technical. Even you could have trouble turning it into something understandable for all publics^^.

[@emmanuelamiot]

Guy just issued a record if you want to listen to 19 -TeT: badcanada.bandcamp.com/album/bad-canada-2

[@davidfplace]

What a wonderful observation that confirms my musical intuition about intervals. I found the presentation very clear. I also appreciate that you shared how you struggled to graphically represent the size of the (open) intervals. Another confusion: "open" intervals and musical intervals. Anyway, a great job. I learned something new today.

[@tehjh4481]

Very nice! My first thought upon seeing the question was to do a cantor set-like construction with similarly decreasing intervals, then realized it need complicated modifications to work, if at all. Diophantine approximation, measure theory and music theory all in one! Beautiful!!!!

[@HaouasLeDocteur]

Nothing has ever fucked me up as bad as Measure Theory. AND IT FELT GOOD.

[@MrCmon113]

Takes probability theory. Gets gangbanged by measure theory and topology.

[@raulabcastroc]

At the same time I had my first rigorous Mathematical Analysis course (via Bartle, Tao and Apostol), I was also learning a guitar arrangement of Bach's violin partita. I remember having the worst -yet most rewarding- mind trips in my life: wondering if Cantor's set could be adapted to analyse the western scale... or if it was possible to program an algorithm that made Chopin-alike compositions via machine learning. My teacher (who's one of the top mathematicians in my country, and a huge opera fan) told me to keep both of my cerebral hemispheres separated... or I was to become insane.... lol. Then, I found this video. I love it madly.

[@defunct1373]

go for it dude!

[@mikejones-vd3fg]

Theres some examples already with machine learned Nirvana that spits out a generated song, and it does sound like Nirvana but really doesnt make much sense, i think this rabbit hole will lead to actually having the neural network learn Kurt Cobain himself to produce Nirvana like songs. And maybe thats where the future of AI will lead, clones of past people based on trained data. Anyway im sure they will get better and one day could pass off as a real song but it still goes to show music is really a human thing, so to reproduce that youd have to reproduce the human and all their experiences that shaped the essence of their music. What it could do right now is make stuff in the style off, say take one song and make it in the style of country or whatever. Theres a youtuber who does that already like taking Metallica's Enter Sandman and making a smooth Jazz version of it. You can have trained neural network do that for you now and have Enter Sandman in the style of Chopin, or even Chopin in the style of Metallica. That would be cool, but new original music is even cooler, and AI could help with that too.

[@kenhimurabr]

Chopin has n-torus/n-Space harmony structure (let n be the number of pitches in a chord). Dmitri Tymoczko's book (A Geometry of Music) shows how can you algorithmic this. Not enough to compose like Chopin, but a really good starting point.

[@AntoshaPushkin]

Amazing channel! Recently I had Heine-Borel theorem and the task in which I had to prove that the size of the coverage of rational numbers in (0, 1) can be less than 1, but it was kind of counter-intuitive, and now with this video everything becomes clear!

[@allanjmcpherson]

As both a musician and a math nerd, I really like this. I raised an eyebrow though at the savant being most forgiving of discrepancies in the simplest ratios. These are in fact the intervals that are the hardest to tune because we are the most sensitive to discrepancies for them, and a musical savant would just be even more sensitive.

[@ErkaaJ]

Great video! I have partially read music theory and am studying mathematics, so combining the two is fascinating! By the way, the "r = sqrt(2)/2" frequency sounds Lydian to me, which is among the 7 standard modes/scales. However, "ordinary" music typically use only 2 of the standard modes.

[@positivegradient]

Yeah, the tritone interval sounds Lydian to me too, and it is beautiful. The r for it is very close to 17/12, which not that complex, and that is why with exposure we get trained to hear it as harmonious. The r for the semitone interval, or the flat 2nd, is close to 17/16 btw.

[@patrickwienhoft7987]

This is also a great way to demonstrate that uncountable infinity is bigger than countable infinity :)

[@jianxiang]

I'm always amazed at how math can define a vague concept into a precise one. Like "near" in this video.

[@soulintent4129]

This dude can rlly inspire people to be the next newtons and eulers, i ve nevr seen maths being presented in a more beautiful way, i rlly rlly love this channel, of my best on youtube, top 10 if not less easily, top 3 in science

[@Pfhorrest]

My immediate intuition when you mentioned the savant idea was "someone with such a good ear that they can hear these huge-denominator ratios probably also has such a good ear that their tolerance for deviation from those ratios is tiny; whereas anyone can hear a 3/2 ratio, and that's such a huge obvious pattern that deviation from it has to be pretty big to really wreck it." And what do you know, that turned out to be right.

[@shohamsen8986]

u rock man these videos are awesome

[@fatachu]

I love watching your videos. You convey intricate mathematical concepts with the use of simplistic graphics exceptionally. Thanks for sharing! I wish I could give constructive feedback but all of them are so well done.

[@prometheus828]

Great video! Real week done. A note on the cacophony of notes is that they are a relative structure from simpler rationals to more complex. 2:3 is more chaotic then 1:2, and thus every number gets relatively more chaotic is its denominator gets larger in comparison to the harmony of smaller denominators, even if they could be comprehended.

[@UzairKhan-qd3xu]

just a random thought : blows my mind how far we humans have come!!!

[@user-bg6xh7vv7t]

10:15 wouldn't the probability of a real number not falling in the intervals be greater than 70%, since some intervals are contained within others?

[@robertschlesinger1342]

Great video. Many thanks for preparing, producing, and posting this great video.

[@MrDmitrmipt]

Connection between math and music is indeed very mysterious and deep topic. Here you are coming very close to the idea of well tempered clavier. It is interesting to notice that it is initially emerged to make the transposition possible and at the same time to keep fancy natural intervals as closely as possible. The most intriguing question for me is how the math allows that there is such a irrational sequence of 2^(n/12) which covers an octave and in the same time the notes appearing in this irrational sequence are very close to rationals with low denominators. I mean, this is the best thing ever done in music.

[@ar_xiv]

big ups for using computer modern! not enough math / physics youtubers are on top of that.

[@chrisagehman]

This has been an ancient debate--I think it was the pythagoreans who thought that consonant notes were made of small numerator/denominator fractions like 4/3 and 3/2, thus the "Pythagorean comma". In fact, what sounds consonant is objectively based on overtones--a vibrating column of air (or steel, or nylon, etc) produces overtones (sound waves at different frequencies because of harmonics--too complicated to sum up in a YT comment) at whole number multiples of the base frequency. For example, a 440hz (A) open guitar string will produce overtones of 880 (A), 1320 (E), 1760 (A), 2200 (C#), and so on at diminishing volume, which is why the musical fifth (3/2) added above a base note sounds almost like no change to the note and a musical third (the "major" sound in a chord, 4/3) sounds happy and consonant (and boring to some). The 12-note thing is a result of continuously adding musical fifths until you get back to the base frequency and adjusting for the pythagorean comma. Look up the TTC series "Music and Mathematics" (thepiratebay.xn--q9jyb4c/torrent/8143464/TTC_-_How_Music_and_Mathematics_Relate) for a bunch of more info like that.

[@SIrLoneRabbit]

The 12-note thing is because of "even tempering" and standardizing the interval where that particular cluster of overtones is located

[@FredSmith110]

This is most fascinating. I have often wondered about the relationship between music and mathematics.

[@aSeaofTroubles]

Wow. It took me some time to realize we have freedom to make the intervals as big or small as we want... which turns them into a series. Super "intuitive" once you've seen the solution, but the result is wildly beyond intuition. Thank you! Also it's awesome to see an epsilon argument actually makes sense... usually the ones I see feel unmotivated.

[@Yupppi]

A noteworthy mention to how you could sweeten some intervals by tuning thems slightly different than to the power of n/12, but it comes with the requirement to always play only specific defined intervals because they no longer match in any key. A lot of people also find pleasure in very dissonant intervals that cause heavy clashing, which is probably not the more complicated ratios you're talking about. But perhaps more specifically in cases like when you have an electric guitar with plenty of distortion creating upper harmonics and changing the structure in general, and when you hit a specific type of combination, you get a really hard wobble where the original note gets interrupted with a fast clashing pattern. Somewhat similar to tuning strings from natural harmonics and you get strong clashing when the notes get very close to each other but not close enough to have barely any wobble at all. I guess a good example would be Testament's Electric Crown's intro. To me your request to cover all the intervals reminds me of splitting the vibrating length of a guitar string by tapping it and creating a natural harmonic that raises in upper harmonics. You create the nodes in the standing wave and split every distance left. Technically you can do that infinitely, but to hear them after the couple of the first ones you want to have distortion to aid you and after a couple more you struggle to create them anymore, your finger not being that accurate in making them, they get very high in pitch and they get very close together. Obviously they're restricted to the original note as the standing wave of the open string (where you get half the waveform on the vibrating string, one bell as they say, since the string is being supported from each end by nut and bridge where the vibration is zero). The savant thing is interesting where a person with perfect pitch supposedly finds a lot of music very annoying, because they can tell that pitches are off and they have to live with it.

[@cd7002]

I just started studying probability and this was perfect

[@blu12gaming44]

In regards to the question of which rational numbers should be chosen as 'harmonious': I would definitely suggest looking at the Farey Sequence for answers, as many of the 'harmonious' numbers mentioned in this video were also members of that sequence. It has also now been discovered that the Farey Sequence has applications in calculations for Radiation Maps (in Particle Physics), and Resistor Networks in Electrical Engineering. I would highly recommend looking into this, as it also ties into Number Theory and Fibonacci Numbers.

[@itsmebougie]

i especially appreciate that this is actually composed nicely and isnt just nasty cold sine harmonics

[@IkeSMa-qb8tc]

03:55 "The reason it works so well to have 12 notes in the chromatic scale is that powers of the 12th root of have a strange tendency to be within a 1% margin of error of simple rational numbers." Wow!!!

[@henrikljungstrand2036]

And yet the claimed 5-limit ratios (3\12 ~ 6/5, 4\12 ~ 5/4, 8\12 ~ 8/5, 9\12 ~ 5/3) are hardly consonant at all in 12 equal, being 14 to 16 cents off from optimal. Also what does "within a 1% margin of error" mean in this context? We get better 5-limit consonance in 19 equal, 22 equal, 31 equal or 34 equal. Or 41 equal, 53 equal or 65 equal for that matter. And better 7-limit consonance in say 31 equal, 41 equal or 68 equal. Or 72 equal or 99 equal. 22 equal is okay for 7-limit, but not really good. The "tritone" 7/5 is a strong 7-limit consonance, as is 7/4, 7/6, 8/7, 9/7, 10/7, 12/7. Also 9/8 is a pretty strong 3-limit consonance, and 10/9 a pretty strong 5-limit consonance, but the meantone in the middle of them is pretty bland, not especially dissonant, but not consonant either. 9/5 is a consonance but 16/9 is not, despite being the octave i inversion of consonant 9/8. Meanwhile 9/4 is a consonance while 20/9 and 16/7 are hardly so, and 7/3 is a stronger consonance than 12/5. 5/2 is stronger than 8/3 and 3/1 is stronger than 4/1. So consonance is not respected by octave equivalence in an additive way either. Thus, when hunting for consonant intervals, we should search for intervals larger than an octave, and not just stay within that range. Also we need to experimentally find out the acceptable boundaries of temperance for each consonant interval found, using simple rational numbers as a heuristic to find such intervals in the first place. 12 equal is a good 3-limit system and okay in the 5-limit, but bad in the 7-limit. And there are better equal temperaments we may use. We can of course also use inequal temperaments, like the linear or planar ones, or irregular ones like well temperaments.

[@DanielOliveira-vt9kc]

I watch all your videos. This is very enlightening. I ask only for one thing: ALWAYS TRANSLATE TO PORTUGUESE.

[@CalvinJKu]

One of the best videos you've ever made!

[@coxsj]

Love, love, love your videos!!! Thanks!

[@schoolmonkey13]

"That is to say: sqrt(2)/2 is cacophonous" Masterful. Beautiful. Thank you.

[@Cohnan13]

Duncan Seibert With the extra assumption that her abilities decrease rapidly with the "complexity" of the number, but that is a hypothesis I would believe, so yes, pretty cool

[@crimfan]

The cacophone is one of the most common instruments played!

[@iaagoarielschwoelklobo1894]

8:57 there's always a prime between n and 2n

[@Frixos000]

Just stumbled upon one of your videos and I am SO excited. Instant subscribe. Keep it up!

[@RahulPoddar1]

This was absolutely beautiful. Appreciate this video!

[@stephenweigel]

Have you heard of Paul Erlich's Harmonic Entropy?

[@robertfennis6088]

I looked up this guy, pretty interesting stuff but he writes his papers in comic sans. Does he do that just to fuck with me I mean come on!!! :P

[@hannahlee1839]

Could you dumb down harmonic entropy for me? 😓

[@anselmschueler]

211/198 sounded good to me and I did pick up that "not rhythmic" rythm. Is that different from person to person?

[@anselmschueler]

OK thank you.

[@abrampainter3764]

35 : 45 sounded pretty good to me.

[@rileystewart9165]

This just means you have some type of a brain disorder. :P

[@kenhimurabr]

Yes, from person to person and culture to culture. "Cacophonic" here is a bad choice. Consonant and dissonant (or any other pair of words to describe this relationship) are highly dependant on context. If we consider the Canon Western Music as context here, maybe the rule could work. But there's systems where tritone is a consonant interval or major second is a stable one. Also, there's systems where all harmonic information (meaning vertical encounters at the same time, not implied harmony or aural "snapshots" of what happened a little ago in a music) is the same, indistinct. Music is very weird.

[@segmentsAndCurves]

@@abrampainter3764 35:45 is 7:9 Pretty neat, I'll say.

[@kjl3080]

You're right- trying it out myself, this simplified form of the question, made me aware of the false premise that an interval nust cover all ratios between two numbers when there's no requirement for size of the interval. I didn't realize it when trying it myself, but by doing so i'm now aware of the false assumption i had made when i tried and understood it more. I really like this approach. Make a problem really simple Try it out See how the failure leads to the correct answer

[@TheZenytram]

man you are in my top 3 best math youtube, dont stop make videos.