How Pythagoras Broke Music (and how we kind of fixed it) [see comments for corrections]

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 YouTube 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.

^ 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.

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

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.

Wolf interval actually sounds pretty nice

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

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.

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!

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

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

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.

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.

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.

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

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!

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.

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

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.

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!

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