How Pythagoras Broke Music (and how we kind of fixed it)

[@OliverLugg]

CORRECTIONS AND CLARIFICATIONS - READ BEFORE COMMENTING: This video has blown up, and was never expected to reach such a large audience. I've become acutely aware that I may therefore be widely spreading misinformation - partly through simplifications and omissions, partly through my own ignorance - and want to correct that as best I can. This list will be updated as I'm inevitably corrected further. 0:09 - Pythagoras was born (if he was born at all) on the Greek island of Samos, but spent much of his life in colonies in Italy. Supposedly. 1:36 - I want to emphasise again that all the stories I tell about Pythagoras in this video are legends, including the fact that he "invented" a system of musical tuning at all. Like many Pythagorean discoveries, Pythagorean tuning likely appeared many times around the world throughout history. 1:57 - Sources disagree on whether it was the hammers or anvils that varied in size, and whether they had twice the dimensions or twice the weight. 2:15 - Some people noted my sine waves sound a bit distorted, maybe even triangular. They absolutely are sine waves, but I did add some reverb for the ambience. Also KZbin compression is a thing. So they perhaps aren't totally pure. 4:15 - I didn't mention that size of hammer is *inversely* proportional to frequency. Size determines wavelength, the reciprocal of frequency. Everything I said here is true, but thought I'd add this just in case you thought I meant the larger hammer produced the higher frequency. 6:03 - I had not explained enough by this point in the video to introduce diatonic scales in a natural way, so saying 'it's the fifth because it's the fifth note of the diatonic scale' would not have been possible. Instead, I made a bad joke. Please stop taking this as evidence I don't know what I'm talking about. This comment is filled with more legitimate claims to that. 7:29 - Due to limitations with the VST I used, I could only tune this instrument's notes to the nearest two or three cents. So none of the plucked notes are perfectly Pythagorean, they're all a tiny bit off. All the sine wave notes I use are however precisely in tune as I made the VST for that myself. 7:30 - My version of Pythagorean tuning starts with a base note at the lowest frequency and works up. In reality, some Pythagorean systems start with a base in the middle and create equal numbers of fifths above and below. The harmonic relationships in these systems are functionally identical. 7:36 - In case you were wondering why all the intervals of the same ratio are the same length when they're being multiplied, this diagram is logarithmic. Multiplication corresponds to addition on this diagram, so multiplication by equal values corresponds to equal length intervals. 12:06 - Obviously x = y = 0 is a solution, but then you'd have a scale with only one note. Hardly useful, so I left that out. 12:15 - There are also systems with 53 notes per octave that do about as well as 12, and many other alternatives. 12:55 - Pythagorean tuning (and meantone temperament) are examples of a larger class of tuning systems called Just Intonation, where note intervals are defined to be rational multiples of one another. Pythagorean is just the version where the multiplicative ratio is a fifth. Many musicians in this period used other types of Just Intonation, but they all suffer from similar issues. Further, in the original Greek period, nobody was using polyphonic harmony, at least in the modern sense. Grating dissonances mattered less as a result. Harmony only became a more concrete thing later, which necessitated the introduction of meantone temperament. However, the equivalent to the wolf fifth in meantone is more of a problem, and that version of the wolf fifth is the more common example than the Pythagorean one in this video. 13:11 - The Catholic Church DID NOT ban tritones. I've since learnt this is a widespread myth that even some of my sources fell for. Made even worse by the fact that I mixed up tritones and wolf intervals in my script despite knowing they were different (but not that different). Massive brain fart there, I apologise. 14:33 - Stevin was only one link in the chain that led to equal temperament. Many people contributed to its development. I simplified it to one person to save time, but that was definitely a mistake. Also, as is often the case, a system like it may have appeared in China even while Pythagoras was alive. 16:23 - The piano was not the first instrument to use equal temperament, merely the one that popularised it. I didn't say otherwise in the video, but I didn't make it clear either. 16:48 - Oh boy. I got a lot of "well actually"s for this one. Some cases I knew already - instruments without fixed pitch, such as violins or even the human voice, can switch between tunings on the fly - but some I genuinely didn't know. Turns out many brass instruments play perfect ratios, and it's only by the skill of composers and performers that we don't notice. Plus, performances of older pieces are often tuned to pure ratios for authenticity's sake. All this does not detract from the fact that equal temperament is the overwhelming standard, and I still stand by my intended point that the average person in the Western world has so rarely been exposed to Just Intonation compared to equal temperament that it might as well not exist for them. 17:00 - Some have claimed this video is biased in favour of equal temperament (with a surprising and depressing amount of vitriol). I think it probably is, mostly because I am. But the system you prefer is entirely an artistic choice. Both are mathematically flawed compromises, I just prefer the one that gives more standardisation and harmonic flexibility. Modern musical hegemony agrees with me, but you don't have to. Electronic music synthesis means you can create music tuned to any frequencies you want. There is Pythagorean music out there if you choose to look for it, but beware: it often gets caught up in New Age, psuedoscientific mumbo-jumbo. If you find anything that claims Pythagorean tuning can heal physical ailments or that it's being denied from you as part of a global conspiracy, steer well clear. The Cult of Pythagoras is alive and well.

[@SuperLol]

very humble and even more knowledges spread. Huge applauds! I knew most of things you described and noticed a few mistakes as you mentioned in this comment, yet the clarity and fluidity of your narration and how well organized all the infos are really impress me. Your correcttion comment further earned my respect. Hope you all the success, which ofc hhelp more people get to know better about music.

[@mahaputera7]

When reddit pro user jumps into YT~ Gee~ what a tl;dr 😂 If this me, I just put : "please put comment down below~ but sorry if keep you waiting for some corrections" WHICH I'll never sees those for eternity 😈haha😈

[@andythedishwasher1117]

Damn dude. I was super psyched to pull a giant "actually" out of my otherwise mostly useless half a music degree, but you had to ruin the fun with your corrections.

[@epichdsheep]

You’re confusing pythagorean tuning and just intonation too. In a pythagorean tuning everything is generated from a fifth. So you get a big third from going four fifths up, and you can close it to a temperament if you want by flattening one fifth by the comma. Just intonation refers to the whole number ratio of pitches. Choirs intonate to just intonation all the time, because we feel the beating in our throats when singing. Of course within the limits of staying in key, because just intonation will cause a comma drift fast. And furthermore on the historie of temperaments, we tuned meantone until the 18th century because we favoured the pure thirds. Meantone too often gets confused with anything other than equal. But there’s a whole world of other temperaments, including well temperaments and pythagorean temperaments. Keep in mind that «just intonation» and «pythagorean tuning» are not explicitly temperaments, although you can build a scale that follows them from a fundamental. We tuned mostly well temperaments up until the 20th century. Equal temperament was not tuned or desired up until the 20th century.

[@AnglosArentHuman]

Local music nerd suffers death by a thousand cuts from other, bigger music nerds.

[@azabiphetamine]

I truly love how Pythagoras is mentioned at the start as "the one who did the *_triangle thing"_*

[@FruityPebbles-420]

Pythagoras was very multifaceted.

[@justaskin8523]

When I was growing up, there was a mom in the neighborhood who would ring a triangle dinner bell every day at 5:30 pm. I never knew who's mom it was, so I think they lived a block or two over. That triangle sound really carries!

[@robertlongtin5003]

Worth noting that all attributions of mathematical work to Pythagoras or his followers (including the Pythagorean Theorem, Pythagorean Triples, and general work with right triangles) is blatantly false. It is generally accepted that Pythagoras' obsession with Pythagorean Triples (*groan*) came from a visit to ancient Babylon where stone tablets with massive lists of them were made and kept. The ancient Babylonians already knew about the square root of 2 and had also made extensive tablets on the subject long before Pythagoras was born. Pythagoras wasn't a mathematician, they have it right in the video about worshipping numbers because he and his followers were numerologists. Pythagoras is known for work on politics and philosophy, but Mathematics wasn't actually in his wheelhouse.

[@AV9000x]

I truly hate that

[@moeball740]

Who knew the "triangle thing" had such far reaching applications as predicting the Won/Lost records of baseball teams based on the number of runs scored vs runs allowed? It can be done with Pythagoras!

[@maximillianhallett3055]

Tritones and Wolfe Tones weren't banned by the church. That interval was avoided, but never banned. Bach used tritones and parallel 5ths and 4ths all the time.

[@filiusgulielmi4636]

^^^

[@OliverLugg]

Hmm, there’s a note in my research from a couple of years ago for a different project that says there were some tritone/wolf bans. I don’t have access to the book it came from anymore so I’ll try and get my hands on it and get back to you. It seems plausible I misremembered or mis-paraphrased. Two things to mention about your point though: Bach was part of the Protestant church not the Catholic, and was alive much later than the presumed time of the bans.

[@filiusgulielmi4636]

@@OliverLugg my understanding is that it was not common practice to use tritones since the goals in music were about glorifying God with beautiful melodies that would reflect divine harmony. Therefore, the church and musicians simply wouldn't want to use the tritone. Also wolfe intervals weren't a problem for choirs since they can fully adjust pitch.

[@OliverLugg]

@Maximillian Hallett Right, finally got my hands on the book I took that from (Big Bangs by Howard Goodall). Apologies for taking so long, I couldn't go back home for a while what with the pandemic and all. It seems I might have paraphrased something in an unhelpful way - the important line in the book reads 'combinations of notes that derived from "impure" ratios were frowned upon by the Church' followed by another line describing the intervals as 'forbidden'. So a bit ambiguous and maybe taking some liberties with that 'forbidden' description. Having looked elsewhere, it appears actual bona fide bans may be a myth. I'll pin a comment with your correction, thanks for pointing it out.

[@AFRoSHEENT3ARCMICHAEL69]

Why do you think God made man on the 6th day and rested on the 7th? Because that day was Loco (Locrian loki) it has no resolution and contains the tritone which is B and F the Devils interval. In western functional harmony we omit the 7th scale degree chord because it's diminished. It doesn't have a perfect 5th.

[@Matthew_Klepadlo]

The reason it’s called an “octave,” and a “fifth,” is because there are seven notes in the major scale. When you reach the 8th note, you just go back to your root, so you achieve an “octave.” When you go to the 5th note in the major scale, is a major 5th. An octave (the “8th” note in the major scale) is 12 notes up… And a 5th (the 5th note in the major scale) is seven notes up. We call notes by their placement in the major/minor scale, not by how many semitones it goes up.

[@multiply67]

what the is major scale? 12 notes up what? you don't make sense. what is a note?

[@Matthew_Klepadlo]

@@multiply67 a note is a singular sound that we make on an instrument. Think of pressing a fret and playing a string on a guitar or playing a single white key on the piano. Music is comprised of 12 notes: A, A#, B, C, C#, D, D#, E, F, F#, G, and G#. The next note is A, so twelve notes. A4 has a frequency of 440hz, A5 has a frequency of 880hz, and A3 has a frequency of 220hz. When you double your frequency, you go up one octave, or up 12 notes. A scale is a sequence of notes in an octave that gets you a specific sound. Going C, D, E, F, G, A, then finally B before you go back to C gets you the C major scale. There are seven notes: CDEFGAB. The “eight” note is the same as the first, just up one octave. When I say major 5th, you go 5 notes up the Major scale. CDEFG GGG!!! C and G make a major 5th interval. This is a very easy concept to understand, especially if you have a piano or piano software on your computer, I mean music theory class 101, day 1, lesson 1 in “Music Theory for dummies!”

[@multiply67]

@@Matthew_Klepadlo thank you

[@frafraplanner9277]

@@Matthew_Klepadlo Finally someone explains the scale and chord names without overcomplicating it! Also, using octave for seven notes reminds me how Latin called a week an "octave" in Catholic contexts

[@tomas_nehyba]

Yeah because a perfect fifth is 5th note in the scale but 7 semitones away from the base note.

[@StarryxNight5]

Wolf interval actually sounds pretty nice

[@OliverLugg]

I was wondering how long it would take for someone to say this.

[@Delfigamer1]

It's like a binaural tone, except for each ear.

[@jblakeplays2541]

@@OliverLugg I mean, I thought it sounded alright too...not the "perfect tone" were used to, but it sounds pretty alright by itself imo

[@OliverLugg]

@@jblakeplays2541 I don't have anything to back it up but I'm pretty sure it's because we've been conditioned with music that uses more harmonically complex sounds, just as with equal temperament. The dissonance of the wolf interval is nothing compared to say electric guitar distortion, which involves many more notes with angry ratios, and we hear that in everything these days. To people of the time though it was pretty offensive-sounding, since they were used to simpler instrument sounds.

[@StarryxNight5]

@@OliverLugg So that's why I like really rough chord changes and minor keys

[@odonovan]

So, let me get this straight. If Pythagoras complains to us about equal temperment, we should just...tune him out?

[@justaskin8523]

Except if he's trying to drown you, in which case it's better to have situational awareness. And maybe don't go on boats with your crazy mathmagician friends!

[@jackoneill8654]

Oh no you di'nt!

[@MarshallTheArtist]

@@justaskin8523 it was a pun.

[@garnet1918]

How dare you

[@xenakisrah6872]

Well played haha

[@DanielDinhCreative]

For the last point, some bands and orchestras actually compensate slightly to account for equal temperament being slightly "out of tune" - players are often told to "flatten" the major 3rd interval in a chord, and slightly "brighten" the 5th. It's a very subtle change for most people, but as soon as you do it, it changes from a "very good" band to an "extraordinary" band. But this usually applies to large and long chords, not necessarily every single note.

[@ImB4k4]

Absolutely. This is giving me flashbacks to high school, where in band but especially in smaller ensembles we were always taught to do such adjustments, especially when you were playing the fifth in a chord. Really good vocalists who sing in harmony with each other tend to do this even if they don't do so consciously or know the mechanics behind it, because it just sounds better.

[@johnransom1146]

Choirs too

[@keida8843]

we had a student teacher led our orchestra one of our high school years and I wondered why he told us to play our F# a little flatter in Dmajor

[@arn3107]

isn't this called Just Intonation?

[@DorianMcLean-u9m]

Very precise changes must be made with a piano guest artist.

[@iamtrash288]

Pythagoras being the gigachad he is: I am either an ancient Greek philosopher that lived about 2 thousand years ago or I don't exist

[@mallusaih]

refuses to elaborate

[@ceasefire2825]

@@mallusaih leaves

[@seasons1745]

Ceases to exist*

[@VashdaCrash]

@@seasons1745 Comes back from inexistence.

[@Johnhasa1]

Is, but is not...

[@deathbyseatoast8854]

im absolutely devastated by the fact that this hasn’t got the views it deserves.

[@nickb8755]

I'd say it comes off a bit biased.

[@ironyelegy]

@@nickb8755 how so?

[@nickb8755]

@@ironyelegy in favor if equal temperment, an equally flawed system.

[@smuwug]

too smart for the recommended page

[@frankafer920]

I’m devastated by the fact they banned BEANS!

[@scottgray4623]

I remember being in high school about 27 years ago and learning that a perfect fifth was 1.5 times the frequency of the root. I was in detention one day, so to pass the time, I started with 440 and multiplied it by 1.5 twelve times, and then divided by 2 until I got 446 (with a bunch of numbers after the decimal.) I assumed I did something wrong, and didn't realize it was a fundamental mathematical flaw until probably 20 years later, when I learned about equal temperament on Wikipedia. How I wish I had resources like this when I was a child!

[@AKoMMusic]

What were you in detention for?! You are clearly very clever and surely must have behaved well in class?

[@scottgray4623]

@@AKoMMusic Thank you, and I did! I was in there for tardiness; I've never been a morning person, and when you get to school late enough times, they give you detentions, because they assume it will somehow cause you to be able to wake up earlier or something.

[@ajadeleye4970]

you ruled as a kid 🤙🏿

[@franceswoodford792]

thank you for your help ,I will now go to wikipedia.

[@franceswoodford792]

@@scottgray4623 your channel is as naked as mine but I'm not a brainiac . we both need to share

[@seanfay4900]

This is actually why a really good orchestra is able to tune to themselves. Skilled musicians can hear the perfect intervals (overtones) and mildly change tuning from equal temperament.

[@jimslancio]

Exactly. Bowed, non-fretted string instruments and singers are particularly well-suited to the fine gradations of perfect tuning.

[@chrisschack9716]

@@jimslancio and in a choir, the third of the chord will NEVER be high enough!

[@forestershelton]

They tune to fifths, starting with an a of 440 hz and tuning the lower d to it then g, then for cellos and violas c then violins and bass tune the a to a higher e

[@isi2973]

Me only playing the piano: ლ(¯ロ¯"ლ)

[@NXTangl]

Yeah, I always tune my strings in fifths when alone. The resonance makes the sound better (or at least louder).

[@fishwithafez]

I'm a music education student, and for the most part this is true, but your notion of "people not hearing any instrument play anything other than equal temperament" is false. Although for some instruments, this absolutely true, namely pitched percussion, and Piano/keyboard. However, most instruments in an orchestra or concert band can move pitches around a bit to deal with all the fudging. Any professional musician worth their salt generally will make the chords more in tune with the actual ratios by ear, slotting in closer than the 12TET allows for. Some instruments, namely brass, physically can't be made in the 12TET tuning, as the pitches played on trumpet deal with the harmonic ratio, I.E. 2/1, 3/2, 5/4 etc. (That pattern does get messy up higher in partials) When I play Trumpet (my principle instrument), I am keenly aware of this, and have to alter my pitches to compensate for this to get the overall chord in tune, with some pitches having to use slides to physically get rid of the really dissonant intervals that come about, such as the Wolf's Fifth. Overall, still love the video! Great job.

[@cameronblanchette6687]

Omg I've never thought about how that would effect brass instruments, I was wondering why trumpets have a tuning slide.

[@OliverLugg]

I genuinely did not know that brass couldn't play in 12TET, so thanks for that information. I'm aware of that point about non-fixed pitch instruments playing other tunings at will though.

[@D34nerd]

Can confirm as a trombonist, dunno how you guys lip bend it to be in tune, since we can always adjust the slide on our end. I believe the principle also applies to fretless strings because of how the harmonics of strings work.

[@eriknatanael]

And not even pianos are generally tuned strictly to 12tet. They sometimes are, but the octaves are often stretched because the high tension on the piano strings stretches their harmonic series, actually making the fifth closer to JI and the major third fursther away still.

[@landsgevaer]

Presumable you mean that brass instruments can't be tuned to play multiple 12TET notes *simultaneously*. Like a single string cannot be tuned to play two such notes simultaneously, since they are not harmonics. But separate strings, or separate tones on a brass instrument, can in principle be in such ratios, surely.

[@ivanaldorino]

"i want to go to musical college to avoid mathematics" The Lecturer :

[@vlastimil-furst]

Hard to avoid math if you go for a university or college. Although, there is this "social studies" thingy. Even though you don't avoid maths completely, you can get to use it a bit more loosely there, after all, it's never exact science when some level of human psychology is involved :)

[@KyrieFortune]

math rock, about to destroy this man:

[@johnransom1146]

I had to take acoustics, an honours physics course, to get my music degree. It was torture

[@yesseru]

Yeah, music is literally just math but you can hear it.

[@bobmcbob8044]

Fun fact, despite the fact that some brass instruments naturally play perfect intervals because of the way they work, we often shift the note slightly to make it fit in equal temperament, because it sounds nicer. So that's the other reason you don't normally notice it.

[@joshuabroyles7565]

Curved brass instruments don't necessarily default to real harmonic intervals anyway. The real world of acoustics is necessarily a lot sloppier than P man seemed to think. Consider real spectra of any 2 hammers hitting any 2 anvils and P man is already off to a bad start.

[@jazzdirt]

Exactly this.. Shifting the pitch ever so slightly with embouchure.. So funny when non musicians are trying to tell someone who's been playing for 40 years how his instrument works...

[@qirqqirqly8398]

I am a painter and my brother is a musician. The similarities between the disconnect in instrument tuning and what we encounter in colour theory is striking, and I feel like I might understand him a little better now.

[@tomaskoptik2021]

The space itself is consciousness. Movement. Geometry of light. Temperature, color, sound, emotions, thoughts...all the same geometry, all the same system. Movement is a scalar structure of light and therefore all phenomena you percieve is the same geometry of the same movement only percieved under different angles (phase shifts and polarities). The geometry of physical mind itself is the geometry of all things created in the physical mind...

[@pastass8857]

@@tomaskoptik2021 sounding like a Pythagorean lol

[@tomaskoptik2021]

@@pastass8857 Pythagorean school is not dead :)))

[@cyberneticbutterfly8506]

@@tomaskoptik2021 That's making something falsely profound of something trivial. It's not that space itself is consciousness, but rather that the brain is the common denominator and all types of perception is filtering of patterns via the senses. This translates to information patterns in the brain that are easier for the brain to process while less easy to process patterns feel discordant.

[@cyberneticbutterfly8506]

Just like how both music and painting can follow the principles of design; Repetition, contrast, variety, and so on. Only repetition is boring, only variety is chaotic, contrast tells of the intent of the piece, etc. The common deniminator is the mind.

[@Ramberta]

As a musician who's never been good at maths, this was so validating to watch. Equal temperament / focusing on interval tuning just comes naturally to me, whereas I'm always mystified by people claiming to have "perfect pitch". Hope you got a good grade on your project because this is good stuff!

[@alhfgsp]

Equal temperament just shows us another reason why perfect pitch, although real nice and impressive, isn't needed to be a great musician. It's all about the relativity of the intervals and being able to hear/identify those contextually.

[@YoungWillem]

I have known about this for years (it is usually formulated as "it is impossible to tune a piano"), but this video has finally made me understand it. Thank you!

[@granteubanks]

But you can tuna fish

[@jeremyhillaryboob4248]

@@granteubanks ba dum tsss

[@ShadowZero27]

the piano tuners i know do not do just temperment on consumer pianos

[@slevinchannel7589]

@@granteubanks People dont recommend each other enough Science-KZbinrs. I try to change that. Anyone has and/or wants a Recommendation?

[@echodelta9]

This beyond temperament. A string has inharmonicity, it is out of tune with it self. The harmonics are not "harmonic" to it's fundamental. This is why we stretch tune.

[@d.c.8828]

Pythagoras: **Hears hammers striking anvils** "Ayooo, this shit slaps!"

[@cooldebt]

😂

[@dafoex]

Why does your avatar look like the Republican Security Forces emblem?

[@d.c.8828]

@@dafoex It is the symbol of the Iron Front, an anti-Nazi paramilitary organization in Germany in the 1930s.

[@dafoex]

Oh, so despite looking like a fascist, paramilitary organisation, it states that it isn't? They really need a better graphics designer.

[@d.c.8828]

@@dafoex Each arrow represents opposition to a different form of authoritarianism: fascism, monarchy, and Bolshevism.

[@stevenatoli2000]

As both a mathematics and a music teacher, I loved your explanations. I find them clear and logical. Thank you for sharing! I also like the way you let us hear the actual pitches of the fractions.

[@OliverLugg]

^ See the new pinned comment for a more thorough list of corrections. CORRECTION: It's been brought to my attention that the Catholic Church didn't really 'ban' complex ratios so much as frown upon them. It looks like the existence of such bans may be a myth.

[@Nukestarmaster]

It's also pretty important to note that all modern Western music theory has it's roots in counterpoint, which was the system the Church came up with to write holy music, so any bans in intervals were bans of using those intervals in music meant to be played in the Church. The Church didn't really care that much about music that was played somewhere else than the Church.

[@nobodywilleverknow8371]

They did ban the tritone though, as the so called "devils-intervall". ;)

[@Nukestarmaster]

@@nobodywilleverknow8371 No they didn't, that is complete revisionist history perpetuated by edgy metal bands. It was rarely used because it's hard to sing, not due to some association with the devil (in fact it was only called the devil's interval several centuries later for that exact reason).

[@nobodywilleverknow8371]

@@Nukestarmaster Welp, the more you know... I learned it in school a while back and have seen it mentioned on a couple websites and in a couple books, so I kinda assumed it was true, but I guess we know less about those times than we'd like to think... ^^'

[@ThreeProphets]

It is. Adam Neely broke it down really well

[@fredhughes4115]

When I was young, I was introduced to the practice of tuning a guitar using harmonics. I thought this was a great idea, since harmonics are relatively easy to get more or less perfect - by listening to the "beats" they produce. So I had this idea that, if I was careful enough, I could get my guitar pretty much perfectly in tune. However, in practice I never could. Some chords sounded downright awful - and I couldn't understand why, given the precision which the harmonics afforded my tuning process. Imagine my surprise many years later when I, by way of a youtube video, was informed that this practice actually makes a guitar precisely OUT of tune - for the reasons that you've explained here. Good to know. Thanks for your work. I appreciate videos like yours that carefully explain interesting and useful information to us - the masses.

[@DonnyBurbage]

Oh hey! That isn't why its out of tune with certain chords. It's the intonation of your guitar that isn't correct. Which is unfortunately caused due to the saddle not being set for the gauge and tension of the strings. There is a lot of systems to sort of rectify this, (fanned fret, Buzz Feitan, true temperament frets, etc) but a harmonic is a close to a pure fundamental note you're going to get on a guitar, and shows you the pitch exactly as it pertains to the tension of the string and the force you struck it with. Listening to the beats is how a piano tuner tunes a piano and it is a good system, but due to most guitars having bad intonation, I can see how that would make someone assume that tuning and listening to the beats makes a guitar out of tune. The guitar is in tune to it's open strings that way when tuning to the beats, but the intonation of the frets is not correct when the saddle is not set for intonation. So it's the bridge, not you or tuning to the harmonics. Just wanted to clarify!

[@Nonononono_Ohno]

@@DonnyBurbage Assume that you tune the A string via harmonics to a fifth above the open E string note. Assume that you do it in such a way that there is zero beat frequency, or in other words, in such a way that you get *exactly* the same frequency. Then you have in fact tuned these two strings in (perfect) Pythagorean tuning, right? Now, when you play an A note on the E string (5th fret on the E string), then the note you get is *not* a perfect fifth above the open E string note, but instead it's a fifth in 12TET. So, when you then play the open A string at the same time, then you have two *different* A notes sounding at the same time, in harsh dissonance with each other. I think this is what Fred Hughes tried to point out, and what had been shown in that video (which I haven't seen). The consequence would be to *not* tune guitar strings via harmonics to perfect fifths, but instead to embrace the fact that a standard guitar is a 12TET instrument. For example, tune the A string by playing the open A string together with pressing down the E string in 5th fret, and by eliminating any beat frequency, until zero beat frequency is reached.

[@storerestore]

The problem with this tuning method is that you're tuning the open intervals to perfect intervals, but your frets are placed for equal temperament.

[@TheDragShot]

I remember simply memorizing the default tunes of each string, and then trying to replicate them as closely as I could by tuning them by ear. That gave me the desired results 99% of the time.

[@DonnyBurbage]

@@storerestore yes, the opens strings are in tune with each other as open strings. And the guitar frets are equal temperament, but that’s why the saddles are not perfectly lined up. They are adjusted to compensate for the frets not being in tune. The only problem you will find is that the 1st fret and 2nd fret are sharp depending on the string gauge. But the Buzz Feitan tuning system adjusts the saddle to be closer to the first fret to address this. Other systems exist also to address this same problem like I mentioned. But tuning by beats is how a piano tuner tunes a piano. I also just want to point out that in-tune is kind of the wrong word to use honestly, due to several other cultures that have different tuning systems that are deemed pleasant to them. Balinese Gamelan would be a good style to check out as they use the warbling beats that are perceived on purpose to mimic the shimmering waters of their country.

[@dusbus2384]

What's weird is I've never understood how music works even though I play guitar until you explained it to me mathematically

[@VioletteLani]

The fact that music can be explained through math is amazing.

[@6695John13]

Only the structural aspect of music can be explained through math tho. Its emotional and spiritual aspect will forever remain a mystery, probably.

[@VioletteLani]

@@6695John13 Very true! Like why certain chords evoke certain feelings in people. It's all so interesting...

[@skeptic_lemon]

@@VioletteLani Well I'm sure glad they can. Even for people who don't listen to music, it's still important, because culture as we know it had a large part of it built around music.

[@erpmo3326]

@@6695John13 touché.

[@cheesist6998]

something literally called math rock

[@colinedmunds2238]

The “devils interval” thing is poetic, not literal. Imagine if somebody instead called a tritone “a pain in the ass to use” and 300 years later we think of the tritone as “the interval of pain”

[@moobles2998]

A future historian thinking: hmm the interval of ass-pain? interesting, maybe it resonated with their sphincters causing this pain? I will write this down so others may know of this!

[@Acoustic-Rabbit-Hole]

I have a color for each note, and i noticed that the tritone of each color is the opposte color. For example, C is red (love songs are always in C). But the opposite color of red is green, which to me is the the tritone, F#.) It also makes sense because F and F# where the Pastorals keys in Classical and Romantic music.

[@kp6880]

@@Acoustic-Rabbit-Hole Love songs are always in C? I pity the vocalist who has to obey that rule.

[@Acoustic-Rabbit-Hole]

@@kp6880 Rule? I didn't write those love songs. But no need for pity. There's always The Cure's "Love Song," (A-minor).

[@jazzdirt]

@@Acoustic-Rabbit-Hole So you overlayed the color wheel with the circle of fifths... interesting form of synesthesia... But the funny thing is as for contrasts.. it works about the same...

[@theshacktube4876]

Many years ago, I watched a professional piano tuner at work and it was fascinating. I was surprised that he wasn't just simply tuning each note separately using some digital tone creator (I had a Pythagorean understanding of music without knowing it). He answered my questions by saying that it was important to tune the piano "to itself" based on octaves. I never understood what he was doing until now. Thank you!

[@zongaaa6673]

Fs for Hippasus

[@reviewman]

F

[@evanbjammin]

F

[@freazeezy]

F

[@osteelgen8225]

F

[@timalechkevitch4566]

F

[@steve_account]

This is a sine wave. *KZbin compression proceeds to completely unravel the video at its seams*

[@roygalaasen]

I think the KZbin algorithm must have regressed slightly. There are loads of artefacts on graphics that should be simple that I can’t remember was a problem in the past. I guess developing compression algorithms, you fix something, break something else.

[@piergiorgio919]

@@roygalaasen probably they have new algorithms that save more space in spite of quality of these kind of scenarios

[@Alex-ABPerson]

@@roygalaasen KZbin is absolutely huge - they've got an *extremely* strong compression to hold all the tens of thousands of hours of video they receive every single day. The quality loss isn't a "bug", it's just because they're compressing it _so_ _much_ - but have to given the sheer amount of data they have to store.

[@roygalaasen]

@@Alex-ABPerson you are right, of course. I never claimed it was a bug. What I tried to put in words was that is is hard to balance an encoding compression algorithm to perfectly manage every single situation to an 100% degree. You fix one situation where the algorithm goes wrong, you may create another where it does not work as well any more. It is in the name “lossy compression”. This is a balancing act on how much quality you are able to squeeze out for as much loss of quality as possible. But: when it ends up hurting most of the videos you are watching, maybe you should find other ways to optimise storage. For instance, there are probably loads of videos that are just stored in the system that nobody even knows of or remembers. Maybe move those over to an offline system and bring it back online if ever anyone were to request it again.

[@Alex-ABPerson]

@@roygalaasen That doesn't really solve the problem that they still have to store _tons_ of data, though. It requires less energy... Maybe... But with even then *how* *many* people are using KZbin they'd be *constantly* "turning on" these "offline systems" all the time just to pull data off them because people literally all the time all over the world would be requesting stuff back again. Plus it really sucks if somebody uploaded something rare 10 years ago and they were never seen again after that, and their video went offline and neither they, nor anyone who really paid attention to the video, requested it back... It would just become near inaccessible for forever. Just in general it would be really frustrating - and all of that just to improve video quality. And it's not like it's the end of the world YT heavily compresses videos - because if you were say, a broadcasting company and _need_ high-quality streaming, you can just use other (or even your own) services that don't compress anywhere near as hard. And for educational videos like this, it generally really doesn't matter.

[@rodwoodhouse578]

I found this fascinating - I was a Barbershop singer, and we spent a lot of time FINE-TUNING chords, to make them ring. The first thing we realised was that the note that a keyboard plays is not EXACTLY the same not as the harmony voices need to sing to make a chord ring - it was obvious that the differences were small, but REAL. I knew the mathematics of harmony, overtones etc, but your video consolidates my basic understanding. Thank you.

[@loucannata6384]

The difference between “perfect tuning” (Pythagorean) and “standard tuning” (equal temperament) was demonstrated in a documentary using the church organs of JS Bach time, each which only sounded “in tune” if it was played in one key only. The concept of tempered tuning was subsequently developed with the introduction of the piano-forte. Despite this, stringed instrument players can and do at times move their hand on the fingerboard to produce a frequency that is closer to the “perfect” sounding pitch. Hard to explain here but best demonstrated with the major 3rd and major 7th. On a violin, if you raise these pitches slightly higher, they actually sound more pleasant - especially in the case of the major 7th. In the case of the major 7th, this note is also referred to as “the leading note” because it wants to resolve your expectation of sound back up to the octave or root note and hence it's pitch sounds more pleasant closer to the root note. The Indian culture is much older than our Western culture and of course, they even have many more notes (we call them micro-tones) within their standard octaves.

[@elfeiin]

There's people who actually pull in notes from other keys and make it work to masterful effect. Why didn't anyone try exploring this?

[@ADUAquascaping]

Their culture isn't older than ours. In regards to what? They don't have just one culture and neither do we. Your statement doesn't really make sense to be honest. Even the Indo Europeans migrated from the Pontic Steppe into India, which is modern day Ukraine. The oldest civilizations in the world are those of Neolithic Farmers who lived in eastern Anatolia and the fertile crescent. Who then migrated into Europe. People have been playing music for a lot longer than our current cultures. Just because their ideas are different, doesn't mean they are older.

[@ADUAquascaping]

In regards to agriculture, farming, and horse riding our culture is thousands of years older. So what you're saying just isn't true.

[@jwylde2]

The original Cajun music used what is called "just intonation" (which may be the same as perfect) They used a one row diatonic accordion that covered 2+ octaves in a single key that was tuned just. The Acadians also used the violin which is almost always tuned in fifths. It turns out that the 5'th note is almost the same in just or tempered so you can play open strings on the violin and it sounds OK for Cajun (other than open strings, the violin can play any tone, so it can play them just). So, if you want to hear just music, listen to authentic Cajun music, especially music without a guitar, although the guitar is usually only playing chords and doesn't really detract much from the just sound.

[@PASHKULI]

Buddy, Math describes the world… but the world is Physics, it is not as perfect as Math. The world is discrete, finite. The Just\Pythagoren intonation (temperament) for Music does not care about physical structure of material and even Geometry… yes, it is ironic to say this about the pythagoreans. What 1500 years after Pythagoras was done with the Equal Temperament was the compensation for the imperfection of the real physical world. Also it was about time with the invention (discovery) of the Logarithm as the mathematical method of approximation (Calculus) between sums and multiplications (powers). Pythagoras did not break anything. He just showed us the perfect, linear structure of Musical notes. But Music is produced by imperfect musical instruments with physical characteristics and various geometry (shapes and sizes) emitted through an imperfect medium (air) environment. Music is not static (always starting from the same tone ote) and linear - it is dynamic and intertwined in harmonies and overlaps. The Equal Temperament (ET), through the approximation of logarithms, gave us the best fit for the real application of Music! On certain instruments there are structural (geometrical) tweaks we can implement, such so certain combination of notes to be as close as possible (more so than ET) to the Just intonation. We call this tweaks True Temperament. Also, a fifth, octave and those obsolete music "theory" names were introduced 1000 after Pythagoras, with the establishment of the Church chants (most of which had 6 and 7 notes, called Church modes… somehow derivatives of the old Mediterranean\Egyptian modes of Music at least a few thousand years old). The only one who fuскed op Music were those church monks such as Guido d'Arezzo, favouring 6 (later 7) out of 12 notes already known, assigning their favourite Latin alphabet structure (sequence) to those 6 (7) notes, and also naming them after the syllables of their favourite religious psalm, on the notes of their favourite mode (which had 6 notes, later the 7th was added) and pretentiously calling it "natural" major mode! The mess we can see today in the form of the standard Music notation (score)!

[@Hansengineering]

A few years ago I started wondering at why ALL pianos sounded out of tune. Every recording across time, if it's just a piano, it sounds slightly janky to me. This started a rabbit hole journey into musical theory. This video greatly simplifies that journey, and I'm happy it now exists.

[@nrcarl00]

You are probably one of pythagoeas’s descendants

[@nextlifeonearth]

@@nrcarl00 or just not tone deaf. There are also tuning systems that put the compromise elsewhere. Each key has a temperament that suits it more. Equal temperament only has the octave that's in tune.

[@alexjenner1108]

Pianos use "stretched tuning", where the lowest notes are tuned slightly flat and the highest notes are tuned slightly sharp. This is done to compensate for some imperfections in the harmonic relationships of the physical strings.

[@kennethgarland4712]

Do you have perfect pitch? I wonder if those who do have perfect pitch get irritated by the slight out-of-tunedness caused by equal temperament.

[@secularZoo]

@@kennethgarland4712 Rick Beato questioned his son Dylan (who has perfect pitch) about this a few years back. He said that the notes didn't bother him or detract from the music in any way.

[@bizrael11]

You explained this astoundingly well. It took me an entire three weeks of a physics class in college to understand what you very neatly explained here in less than 20 minutes. Really good job!

[@jamfactory4119]

As a musician I found this to be educational, fun and informative. Being mathematically challenged it was way over my head. Thank you. Loved it.

[@ryanviertel7059]

This is the most clear description I've ever heard of this subject by far. Plus the historical context really helps put things in place.

[@philnewton3096]

6:20. Not forgetting Rollands octave march contains the release of the the 3rd finger to produce the 5th.

[@muchanadziko6378]

They couldn't eat the "Broad Bean", not just any kind of bean. There are better stories about their broad bean ban though. Once Pthagoras went for a walk on a field, and there was a bull standing on that field. It was eating broad bean. Pythagoras came close to that bull, and whispered something into it's ear. Ever since that day that bull never ate bean again. The pythagoreans were raided by the Athens police. They tried to run, but the only wa they could escape was through a broad bean field. All died...but one. When he stood trial, he was asked by the jury: - Why didn't your brethren run through that field of broad beans? - They'd rather died than stepped on broad bean, and I'd rather die than tell you why

[@iansings7428]

You would have to be a Pythagorean to know (or the bull)

[@stacklysm]

KZbin recommended your 5D chess video to me (which I really enjoyed), so I went in and watched two more videos (Diplomacy and Mortal Engines). And I have to say this is now one of my favorite channels, and I, for once, thank KZbin for recommending this to me. I wish more people have the opportunity to watch your amazing content. Keep up the amazing work!

[@ndc5544p]

I too watched the 5D Chess video, now I'm in the same boat. Dude deserves lots more views for the quality of content he puts up. Thanks for all the hard work you put into these videos! @Oliver Lugg

[@hav5n]

why have i also just done that?

[@bjccook1352]

Music Teacher here... I found this very easy to understand thanks for making and uploading it🎵

[@rosom5516]

Great explanation of tempered music. It also follows why untempered instruments sound better in certain keys when played 'naturally' (ie as per Pythagorean)

[@ffggddss]

And why transposing on a tempered instrument can make trouble! Fred

[@luigi77714]

Fun fact: he did not live in ancient Greece, he was born there in Samos, but he lived most of his life, and also founded his school, in Kroton, a city in Magna Graecia, which was basically a group of Greek colonies that became independent in the south of Italy; Kroton's region specifically is Calabria , and the city is called Crotone today. I know it is basically pointless to correct this, but it is the city I live in we are talking about, so I felt compelled to do this, since we are very proud to be the city where he lived and had his school Also, if it may interest you: he didn't ban beans, but faves specifically, and the reason was that he was allergic to them (it is said that he died by wandering into a fave field while being chased by an angry mob). This rule was not for his "followers" as much as for the students who attended his school, who were forced to accept everything the teacher said as truth, without being able to contest it ("Ipse Dixit": "He said it") for their first 5 years of studies: after this, they were allowed to sit next to him and intervene during lessons with their opinions. This was a precursor of modern high schools. Other weird rules include the fact that you were for some reason not allowed to break bread using your hands (never quite understood that one)

[@mikecrowley2472]

As long as we're splitting hairs... To the ancients, anywhere colonized by Greeks was deemed to be Hellas (Greece).

[@luigi77714]

@@mikecrowley2472 I guess it is fair, but it is also true that Magna Graecia was recognized as some sort of independent region after a while It is of course still "Graecia", but was considered as its own entity

[@sticlavoda5632]

Ancient Greece was really just the entire region where there were Greek collonies, or city states. They didn't have to collaborate with eachother to be part of this sphere. Some of them were under foreign rule. I am Romanian, so an example from my region would be the Greek colony of Tomis, which was under Dacian rule, Dacia being a Kingdom/Empire contemporary to the one of the Romans. They were conquered and recognized the sovereignity of Great King Burebista, yet they were still part of the Greek world.

[@luigi77714]

@@sticlavoda5632 it really also depends on the time period

[@AnthonyMazzarella]

My great-grandmother was from the Cena so I was very excited to learn that Archimedes was actually born raised and lived his entire life in Syracuse

[@petedenton9434]

In many contexts I still hear mean tone and Pythagorean tuning as more pleasing that equal temperament. That said, this is just about the best explanation of the maths that I've seen in 40+ years! Thanks for making this video.

[@sirforcer]

There's more than just 12 tone equal temperament and Pythagorean, lots of cultures have their own tunings, and you can also do equal temperament with any number of notes per octave (if you're interested in hearing that, Sevish produces music in a whole bunch of different tunnings)

[@cameronblanchette6687]

Yes finally another person knowledgable in music's in here

[@xxgimpl0rdxx22]

This is a video about European music, sir

[@sirforcer]

@@xxgimpl0rdxx22 if you want to be like that, Bach used well temperament and quarter comma meantone was used more than equal temperament in the 1500s (which is also a meantone temperament). There's also third comma meantone which is roughly equivalent to 19edo (equal temperament but with 19 notes per octave)

[@dickrichard626]

Microtonal music ins't better though and the arguement behind it is a reoccurring epic fail. If it made sense someone would have used it to produce music *successfully* - (key word) by now. The reason why it doesn't work is, because our brains can only percieve 12 distinctly different pitches while all other possible imperfect inbetween notes voice sympathetically to it's nearest more perfectly harmonically related neighbor from the perspective of a fundamental frequency... In other words, microtones always just sound like slightly higher or lower detuned versions the standard 12 tones and the music created is always ugly, unstable, and just gets weird potentially even annoying, everytime. There is no "better" major scale that can be obtained with microtones and instead it only furthers musical deviation from it. Sure you could use microtones sparringly and get some slightly different character out of them, but completely cutting things up totally differently just doesn't work musically and always lacks cohenrence and rythmic pulse.

[@sirforcer]

@@dickrichard626 how does the tone effect the rhythm? also there are other non-western cultures which use "microtones" in their cultural music (as in they have their own tuning system separate from western system). Its nothing about our brains being hardwired for 12 tones, its just the cultural norm and that has an impact on what people are expecting to hear. If you took a newborn and only played them microtonal music, then I would hazard a guess that they would find 12-tone odd sounding (to my knowledge no-one has done this, although as mentioned before other cultures have their own tunings that don't sound odd to them. If you want to talk about successfully using microtones in western music, Grammy winner Jacob Collier uses them quite often, most notably when he modulates to G half sharp towards the end of his version of "In The Bleak Midwinter".

[@ThreeProphets]

Great video. Probably a little late for the survey, but I also wish you would've included Just Intonation, as that was by and large the accepted system before Pythagorean. It also sounds really good when restricted to its key center

[@highpath4776]

I might need to re-read that defn in the video, what were the Egyptians using (if known). It came to my thought that the pythagoras as a philosopher took the (rational) mathematics as a 'dogmatic religion' rather than investigating and noting the wider interestingness of irrationality (in base 10 land) of root 2 and other irrational numbers / golden ratio in nature as being (perhaps) the difference between 'knowledge - forbidden in the Genesis 2 bible story' and the intution of just sitting back and enjoying that which has been created (and dont get me started on electrons and sub particles as a function of creation !!)

[@slatszbresgal5140]

Don't apologize for your simplifications. They communicate. You have them in good balance. I've hung out with musicians and mathematicians and never understood this crap. You got it through to me in seventeen minutes. thanks. slatsz

[@indylockheart3082]

I've been on a mission to find too quality YT content with very few subs so I can be super smug about it, you my friend have easily taken that top spot...you deserve many more subs and I intend on feeding likes and comments for you to the algorithm to help get you there

[@hegelsmonster5521]

That is also my mission! Which are good unknown subs? -- if you're interested in philosophy I can offer you "The Dissenter" -- a dude who does a lot of interviews with academians; Philosopher Julie -- she is discussing some philosophical notions like moral dumbfunding; and "Chapter by Chapter" a sub that tries to give summaries of (philosophical) books . Soph's Notes I think, you already know.

[@elbschwartz]

Pythagoras did not "invent" anything musically. He just put numbers to it, and did some theorizing about how music "should" be. There is archeological evidence that the Babylonians -- and possibly before them, the Sumerians -- used a 7-tone Pythagorean (ugh) scale. And based on what we know about ancient Greek music, a hypothetical 12-tone Pythagorean system had virtually nothing to do with musical practice. The Greeks thought in terms of tetrachords (the interval of a 4:3 divided into 3 parts), which could be stacked to form larger scales. Some tetrachords qualify as Pythagorean, like 1:1 9:8 81:64 4:3. Others do not. Some sources (particularly Aristoxenus) describe dividing 4:3 into some number of equal parts. Some (Archytas and I believe also Ptolemy) describe non-Pythagorean just tetrachords like 1:1 9:8 5:4 4:3 or 1:1 28:27 16:15 4:3. The point here is that the Pythagorean wolf fifth (more on that in a moment) did not factor into ancient Greek musical practice because musicians were not modulating around a closed 12-tone system, nor were they employing harmony (at least not in the sense of European polyphonic music). So the failure of 12 x 3:2 =/= 2:1 was a purely philosophical problem, not a practical one. When it comes to the late middle ages / early renaissance in Europe, the Pythagorean wolf fifth was also not really an issue. For one, while there was some concept of a 12-tone system, composers were typically not writing the kinds of the modulations that would run into a bad fifth. Second, the dominant musical format (at least in liturgical settings) was a capella vocal polyphony; vocal music is never strictly Pythagorean, and singers would certainly correct a bad fifth via adaptive intonation. There seems to be some confusion here about wolf fifths vs tritones. In a strictly Pythagorean system, the wolf fifth is a bit flat of 3:2 (in decimal format, roughly 1.48), but not quite as flat as a true tritone (9:8 x 9:8 x 9:8 = ~1.42). It was the tritone that was discouraged in liturgical music (as others have pointed out, it was never "banned"), not the wolf fifth. It may seem like a minor distinction, but they really are two different intervals. The wolf fifth was a discrepancy of tuning that could be adjusted for in practice, while the tritone appears naturally in the diatonic scale (whether the intonation is Pythagorean, meantone, equal, or something else) and can only be resolved by adding an accidental. The other issue is that the concept of a wolf fifth only becomes truly relevant in the meantone era, when a sense of triadic harmony developed (1/4-comma meantone produces a perfect 5:4 and a good 6:5) and composers were employing more distant modulations. In meantone, the wolf fifth is *wider* than a perfect 3:2. This interval would typically be placed between G# and Eb, or alternately C# and Ab. And that's usually what people mean when they use the term "wolf fifth" -- a wide 3:2 rather than a narrow one. The transition from meantone to well temperament to equal temperament was a progressive attempt to minimize that wide wolf fifth. So ultimately, the practical concern over wolf fifths stems from the shortcomings of meantone, not Pythagorean intonation. Somewhat ironically, equal temperament is mathematically closer to Pythagorean intonation than meantone. Yes, that has to do with the development of the pianoforte (whose timbre is much more forgiving of intonational errors than the harpsichord), but also the symphony orchestra, which brought together many different instruments, each with their own timbre and intonational tendencies. Equal temperament is the "middle ground" that enables all those instruments to play together more or less harmoniously. So...It's an interesting video, and all the math is correct. But I think it presents a somewhat misleading narrative of why Western musical developed in the way that it did.

[@OliverLugg]

Thanks for the clarifications. I've addressed the main points in the new pinned comment.

[@clementbenny]

WOW How is it that F# = 375 000 000 000 000 cps and = the Colour Red and about 88 bpm ¿ Truly fascinating

[@DaveMiller2]

What he said.

[@DaveMiller2]

@@clementbenny Just remember that color is light and an electromagnetic sine wave. While music is sound and a longitudinal sound wave. But the frequency is more or less the same.

[@noop9k]

Yes, he represents 12-note scale as the “correct” one (despite it just being one of many possible approximations), and natural harmonic intervals as “broken” which is ridiculous.

[@xStephism]

You just taught thousands of people basics of music and the formulas of present day music and made it comprehensible. This is something not even schools do anymore. Thank you for this wonderful content!

[@jeanjaz]

I enjoyed this video and learning these basics. I'm not a music student although I learned to play the violin (almost) and the piano (moderately), nor am I a math student so the math was just barely within my grasp - but enough to understand the concepts. Thank you!

[@lmahesh26]

I didnt realize I was going to get a Music history lesson and loved every bit of it, great work!

[@bahrss]

Ahhh, good video. I'm a bit said you haven't mentioned well temperaments: all these are unequal temperaments that keep some pure intervals, while tempering the wolf interval between several fifths made narrower in order to have at least some pure fifths and thirds and to enable key modulation which was impossible in just intonation

[@tomwilson2112]

I loved this. It tied right in to the music theory I’ve been learning. In fact, I told the person watching with me something like “watch - the problem is going to be that Pythagoras based his scale on fifths…” and you said exactly that, 30 seconds later. Made me sound like a genius. 😇

[@RoamingAdhocrat]

I used to be a student at the University of Bath. Then I became a bus driver based in Bath. …be sure to graduate.

[@nicomelmo]

You are a great presenter. You explained how tuning created/works exactly the way I wanted. Awesome job & thanks for sharing this!

[@shaft9000]

Thanks for the info. An exception you could have mentioned about the @17:00 mark : Vocal music sung unaccompanied is not equal-tempered. They use a pitch pipe for a single reference, and (if expertly sung) the harmonic intervals are shifting from root-to-root in more perfect tuning than an ensemble of equally-tempered instruments do. Barber-shop harmony and some a capella chamber choirs are examples of this phenomena. Some African and Asian traditions exploit this to a similar advantage, albeit in different tonal scale(s).

[@vidaredlund8628]

Wow, this is the most interesting thing I have seen today! Congratulations!

[@wergvra]

well, only people interested in both math and music would click on this video

[@travisroberts4498]

this video deserves the views.... it's perfect length, easy to follow, and keeps your attention throughout..... well done

[@MefilioDeAggelo]

12:43 is called Seikilos Epitaph, the oldest surviving complete musical composition.

[@roguefirework]

The wolf interval sounds like the sound you make when your trying to tune your instrument but your not quite there.

[@shaft9000]

when "not quite" sounds = nearly half an octavre beer-muffs are on 😅

[@rhizoid]

for some reason it sounds normal to me maybe im too used to bad tuning or something

[@cobgod1415]

yeah the wolf interval sounds fine, it sounds less weird than a tritone at least

[@ValkyRiver]

@@rhizoid I've listened to 40:27 so much recently that it no longer feels too dissonant to me, especially in the low register.

[@Ludwighaffen1]

Many thanks for this great video! You made such a great job at visualising and explaining the maths behind the music. Just a comment though: this video focuses on western music. There are other musical cultures not using equal temperament (like arabic, indian, ...)

[@johngreer4436]

Huge bonus points for the quick nod to Countdown while multiplying fractions.

[@TrueShadowMR]

I remember Jacob Collier once saying that all pianos are out of tune or rather that they dont produce perfect thirds/fifths. Now I finally know why, thanks for the official mathematical answer. For me as both a music and mathematic fanatic, this video is insanely interesting.

[@logicalsatire6043]

Musical things like that is why I love trombone. It's literally just a huge tuning slide so one can quite easily mess around with any given tuning system

[@smartpowerelectronics8779]

So interesting, being an electronics engineer and playing piano I wondered why the octaves are split in twelves and the ratio between tones seems odd. Great explanation, and so cool to know we are conditioned to accept this to be in tune😆

[@rodwoodhouse578]

I noticed an error in my comment - I wrote not instead of note - .."the note that a keyboard plays is not EXACTLY the same not as.." I will re-submit with correction. " I found this fascinating - I was a Barbershop singer, and we spent a lot of time FINE-TUNING chords, to make them ring. The first thing we realised was that the note that a keyboard plays is not EXACTLY the same note as the harmony voices need to sing to make a chord ring - it was obvious that the differences were small, but REAL. I knew the mathematics of harmony, overtones etc, but your video consolidates my basic understanding. Thank you.

[@fdagpigj]

Finally someone explained the foundation of western music theory in a way that I could easily understand! Thank you!

[@barbaracorsini6304]

Thank you so much for the clip. I can almost understand this. Now I have so many questions about imperfection and irrationality. I love that a little bit of imperfection and irrationality help to make music make sense to our ears. This must be true about everything...

[@Nirvash2010]

I heard the Seikilos Epitaph in the middle there, and I was pleased. I love the melody of it, and it doesn't sound out of tune at all. Bonus points for being the oldest known song.

[@jhdgdyhi8]

bonus points? Maybe you should watch TV less.

[@francomarrongrigio]

Finally, music theory I can understand! Love this, thank you so much!

[@TheBlueCrunchmaican]

This is easily 5 Million Subscriber content dude. Keep going. The algorithm should find you in no time.

[@tomrybak3074]

I really like the way you depicted the characters, stick man bodies with what look like carved marble heads is a relatively unique representation I haven’t seen used before. I realize you had to simplify things to keep it succinct but thought it was very effective. I’m an engineer, artist, and musician so I enjoyed it on many levels and learned a few things!

[@Nonyanomo]

The filth and wolf actually sounds pleasant and I like it more than the first. Wolf one sounds like ambients than can totally be used for sumthin. I guess I like it because my taste in music is pretty ADHD drivin and wacky.

[@Jason-re4dw]

Really enjoyed that. It puts into perspective why we have to consider intonation when making music to force the notes into the right places.

[@hummel6364]

I honestly don't mind the wolf interval (I believe you called it that). It sounds interesting and even intriguing to me, however that might be the fault of my headphones or KZbin's compression. This is certainly the weirdest video I've ever seen. Seeing music explained in terms of mathematics makes it so much more tangible to me, which may or may not be due to my aspergers. And in the end everything goes back to what I call "the law of close enough". It is not a real law, but it's something the brain does with many things, for example when you see a jacket in a dark room and it looks like another person. Either way thank you for providing this video, you've earned my subscription, and I wish you a good day from Germany 💜

[@connecticutaggie]

Great job!! I am an Engineer and have been fascinated by music theory and how and why notes are where they are since I was a teenager. This was great as it added some history to my technical understanding. I always thought it was Bach that introduced the even tempered scale. Looks like I have some more learning to do. Thanks.

[@Geopholus]

Tuning is indeed an interesting subject and even experts seem to disagree about a WHOLE LOT ! The subject of tempered tunings goes back to perhaps the 1500's. Apparently Galileo's father spent a lot of time developing a temperment based on complex fractions, as part of his research as a mathematics professor in where ever he was in what we call Italy today. (1500's). Bach was a champion of some kind of "Well Tempered" tuning, which according to most musicologists of the 1960's WAS EQUAL temperament (12th root of two). According to several video's I have seen recently, it is suggested that Bach was using a temperment more like Kirnberger's, where the circle of fifths is started at "C" and is worked up and down with slight flattening of the fifths, so that the worst 5ths occur in the key of "F" sharp or "G" flat, but only worse by a a couple of cents (1/100th semitone) or so of the 'best 5ths" which are only out by about 1 cent ("G" & "D"). Since at the time, the best one could do was count beat notes. (we didn't have crystal oscillators High Fi amplifiers, or even easy ways to calculate irrational numbers to many places), judging by the nature of the "Well Tempered Clavier" by Bach , i would still assume we are talking about the best approximation possible of equal tempermant. Bach wrote pieces that modulate to distant keys , that really only sound good when played on equally tempered instruments,.... then again the differences are slight. Apparently when Bach performed on an organ, that was well renowned in the presence of the organs builder, who favored a different tuning system , Bach played a piece that favored that instrument, because it contained no modulations that brought out the weeknesses of that particular organ's tuning, for instance moving to an "F" sharp major chord,.. of "B" or "D" flat, major. Until i did lots of research, and looked at lots of waveforms, as a recording engineer, I didn't even realize that sometimes natural acoustic instruments create waveforms, that change slightly in pitch (frequency) as they are struck, and that the longitudinal resonances and transverse resonances , can also not be perfectly aligned. This is what causes the bass notes on a piano to sound 'grundgy", the upper harmonics are out of tune with the fundamental tone. Also an instrument like Bass Viol has asymmetrical waveforms, in which the top and bottom waveshape is different. Like the top plate of the viol meets restraint when the vibration attempts to compress the convex shape of the instrument. One gets a similar effect with asymmetrical clipping in electronics.

[@kaloarepo288]

@@Geopholus Galileo's father Vincenzo was also a member of the group of people who inadvertantly invented opera -they were trying to revive ancient Greek plays and one of them,noticing that a chorus was involved,suggested that Greek plays were sung -origin of recitative in opera and later composers added the arias or songs.

[@amazingmasteryes9892]

amazing video man, the algorithm wasn't so kind this time around but the effort put into this is great, can't wait for next vid

[@mollywillo]

my mind is blown, i've never understood tuning systems! Like how are different culture's scales different? Didn't even realize that not all scales were equally tempered before this. Showing how Pythagoras built the scale based on repeating the one interval, and then showing that example with other intervals, that was very helpful.

[@Prezzen77]

@6:04 that fifth demonstration perfectly captures the tone and starting notes of The Soft Parade by The Doors (after Jim Morrison's monologue)

[@MrStarchild3001]

Well done! I think you could've mentioned 2^(5/12) = 1.334, 2^(7/12) = 1.498 approximately match 4/3 and 3/2, and that's how this system works to approximate Pythagorean tuning (or what you might call perfect rational multiples).

[@kingoimkenya]

Beautifully explained, with just the right dose of lightheartedness to retain interest in a pretty complex subject.

[@pewpewpandas9203]

"It's rather difficult to drown an entire planet" ExonMobil: Hold my beer

[@hedgehog3180]

He outplayed us in the end.

[@ДаниилРабинович-б9п]

God: well, I have kinda done it already

[@OgamiItto70]

To be fair, it wasn't easy, at least not at first...

[@emiroercan]

I've laughed to this harder than I should

[@shouryansingh1110]

global warming: am I a joke to you

[@LucienLew]

Wonderful explanation of the evolutionary shift from Pythagorean temperament to Equal temperament. Loved it. And thanks for clarifying your errors in the comment section.

[@theclocktower3258]

Awesome video, visuals and numbers really help me learn and I've been struggling to wrap my head around these music concepts x.x I've been wishing for a comprehensive video like this for a while lol

[@chilidogcowboy]

Brilliant video and very effective at communicating something that is extremely difficult to communicate. Thank you.

[@vp21ct]

Man, nobody tell Pythagoras about . . . all of the other music in the world.

[@DennisBLee]

Awesome video, well produced and love the humor. Would love to see a video on the history and usage of the pentatonic scale someday!

[@cavemandanwilder5597]

Really excellent, informative, and engaging video. As a math tutor myself, I think you did a great job explaining the maths, making it understandable, and even putting it all into historical context. One point I’m still confused on: I do think that for non-musician types like myself a brief explanation of the differences and links between a “tuning method” and a “scale” in the modern sense would help. It seems like when you say a tuning method is broken, that means you can’t use it to produce a consistent scale that works for all starting notes, correct? Meaning that with a flawed tuning method would you sometimes have to re-tune the instruments between songs or when you move to a different starting note to minimize bad note combos?

[@OliverLugg]

Now that you mention it, I was a bit vague in my description of what that meant. You’ve pretty much got it though. The problem with Pythagorean tuning is that it creates really nice intervals for some sets of notes and really nasty ones for others. So provided you keep to the nice ones, it works, but that’s not a whole lot. Changing key mid-song (in non-music terms, shifting the set of notes you’re choosing from up by x notes) for instance is pretty much impossible. Equal temperament finds a compromise where every interval is a bit detuned, but at least it’s the same level of detuned everywhere. So there’s symmetry between keys and it’s a lot easier to move to new ones without having to retune your entire instrument. Hope that clears things up. Thanks for the feedback!

[@cavemandanwilder5597]

@@OliverLugg Yes that clears it up, thank you!

[@cetterus]

@@cavemandanwilder5597 Roughly said, physics, more than mathematics determine what we hear as pleasing (consonance) or displeasing (dissonance). Each interval has its own level of consonance/dissonance quality. For technical reasons (vocal range, instrumental range, historical means of measuring sine waves...) whole world of limited range melodies has been created much before Pythagoras. Simple ones (today living in some folk music and children songs) are called "pentatonic" - simple agglomeration of "fifths" as close as you can get them (DO- RE- MI- SO- LA). Again, roughly said, The more complex the ratio of intervals (in todays living music)- the more they are adjusted to fit mathematical instead of physical principles. Octave is not adjusted at all, 5th is adjusted for just 2 cents (1200th of an octave), thirds are adjusted few cents more... there is also complexity of joined notes (intervals and chords) through history- the more music became complex- more adjusting it would require. This is why piano is so important- it was the first instrument widely used where musician could produce multiple waves- it made people discuss, search and apply new solutions for Pythagoras comma. It creeped up slowly and shy from Robertsbridge codex and it still baffles people today. There are "Microfest" music festivals where microtonal music is played using intervals lesser than the smallest ones on 12 tone system. It is big and interesting world...

[@-LTUIiiin]

A tuning system is like all the ingredients you have, a scale is like all the ingredients you will actually use to cook the dish(the song)

[@Cris022]

How does this video only have 1000 views??? The algorithm did you dirty man

[@SweetShire]

Excellent video. This was the most thorough, yet easy to understand video on the math behind intervals that I have seen.

[@cjdubble]

I'm wondering if this has something to do with why "bending" certain notes, sounds incredible

[@levigoodwin3522]

It absolutely does. Check out Indian and Arabic music. A lot of microtonal bending is getting the note to truly line up in the harmonic series to account for the constant "fudging" of 12-tone equal temperament tuning. There are also some other tones that sound good but aren't playable on, say, a piano, due to the static 12-tone equal temperament tuning. These can however be achieved by bending one of those 12 notes up to the target tone.

[@laurencesiegel9362]

Also a reasonable conjecture is that blues intervals come from the natural desire to "fix" the annoyingly out-of-tune well-tempered third.

[@RCAvhstape]

Bending a note adds tension by making things momentarily dissonant, and when you release the bend the tension is released, which is pleasing to the ear.

[@defkon7]

I wasn't expecting to learn a math today, but here I am. I thought the math-y stuff was pretty easy to follow!

[@theidealist1019]

wow, very good explanation of Pythagorean/just/equal tuning you have there

[@ZILtoid1991]

Pythagorean tuning is actually very useful when calculating frequency ratios for FM (or PM as it's often implemented) synthesis.

[@serbiansleeperagent]

As a metalhead, the wolf interval was honestly really nice.

[@andrebergeron330]

Nice! Very well done! I stumbled upon this equal temperament when looking for an exponential function model (I'm a Math teacher) to have students manipulate and explore. Without too much musical history knowledge, I started with frequency doubling between two notes an octave apart and the assumption that half-tones were evenly spaced on that exponential (logarithmic) scale. I later heard of these ratios and quickly realized that there would be discrepancies between the two models. Now I know the contexts. Thanks for bridging this gap with this insight!

[@Just.Kidding]

13:15 This is absolutely untrue

[@nocklepockle]

very much so it's actually pretty upsetting

[@existenceispain_geekthesiren]

yep. appears to be a variation on the "church banned the tritone" myth, which is also untrue and originated from its nickname, "the devil of music", which was just people complaining about how it was hard to sing.

[@JasonMartineauMusic]

Pleased to have been included in your research! Enjoyed the video!

[@IsaacGreear]

What an amazing video. This video answered questions that I had for years! Excellent work, and thank you!

[@Wilipeidia]

A video on music and maths, two things I barely understand, that is good and enjoyable? And I even mostly understand? Damn.

[@amazingworldadventures325]

I'm not a musical person. I can't read music, I've never really understood most musical theory, even the definitions of terms used to describe music often escape me. As such, I viewed this video as being about math and simply using music to explain the different ratios. So, did you trick "music people" into learning a bit of math by cloaking it in musical theory? Or did you trick "math people" into learning a bit about music by explaining the formula theories behind it? Ultimately, does it really matter? In either case, you have broadened many people's view of the world by including examples from two disciplines that only sometimes overlap. Well done!

[@theflyingnon8546]

most "music people" i know are also "math people". most of my band class, if not all, (maybe not the slackers) are in the more advanced maths, and many "math people" enjoy music, to a greater extent than others. it may be because of music education playing a bigger part in someone's life, often leads to heightened intelligence. music is awesome lol

[@XxMadermanxX]

@@theflyingnon8546 not necessarily true in the least lol

[@coffee-lc]

I love watching videos like this, because im watching, im mostly understanding, im happy because im learning. Then i finish the video, and all the learning falls out of my head and vanishes. So then i get to watch the video again :)

[@akamai897]

Just as with contemporary music, I enjoyed this with the flaws. All of it, including the corrections, is fascinating to consider. Thank you.