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

pythagoras you know the one who did the triangle thing it's quite the man of legend we're not even sure whether he existed at all but if he did he was a philosopher who lived in ancient greece about two and a half thousand years ago [Music] he had a cult-like following known as the pythagoreans who were supposedly really quite mad [Music] as just one example it said that pythagoras banned his followers from eating beans only to die at the hands of some soldiers he refused to run away from because the only escape was through a bean field in spite of the madness or perhaps because of it the pythagoreans made incredible strides in the newly developing field of mathematics they worshipped the subject believing numbers and geometry were the keys to understanding the beauty of the universe today we credit them with several foundational discoveries even though pretty much all of them were discovered thousands of years earlier by someone else the one you're all familiar with is the pythagorean theorem in any right angled triangle the square of the length of the side opposite the right angle is equal to the sum of the squares of the other two sides a squared plus b squared equals c squared one of pythagoras's followers hypasis is said to have realized that plugging in one for each of the shorter sides tells you that the third has length square root of two and that this number can't be written as a fraction today we call numbers that can't be written as fractions irrational numbers legend says pythagoras believing in a pure and consistent world of perfect ratios promptly drowned hyposis for daring to suggest such heresy the legend we're focusing on today though tells the story of how pythagoras invented a system of musical tuning and in the process kind of maybe sort of broke western music for 2000 years thanks pythagoras walking past a blacksmith one day he couldn't help but notice the sound of hammers striking anvils and that different sizes of hammer produce different notes in particular striking two hammers at the same time one twice the size of the other produced a very pleasing combination of tones to see why we need to ditch pythagoras for a moment and begin with the basics [Music] [Laughter] this is a sine wave the purest form of a musical note every note with a pitch you play or sing has some associated sine wave called the fundamental that determines that pitch let's slow things down sound in the simplest terms is a vibration could be a vibration in anything your headphones or speakers are currently vibrating causing the air to vibrate causing your eardrum to vibrate causing you to hear a charmingly knowledgeable voice inside your head we usually represent that vibration by drawing a graph of the wave like this one peaks correspond to areas of high pressure and troughs to areas of low pressure if the wave repeats we can measure its frequency by counting how many peaks past a given point in a given time span the repeating segment of the wave between two peaks is then called a cycle in this case we have a frequency of two cycles per second or two hertz as we call it compare to a note with double the frequency four hertz for every peak that passes our marker in the lower sound two peaks pass in the higher one when you draw both ways on the same graph you get a pretty pattern but not merely in the visual sense let's speed things back up [Music] it turns out that frequency is the same thing as pitch a higher frequency gives a higher note and a lower frequency a lower note and when you combine a note with another that has exactly twice the frequency well you can hear for yourself the nice simple pattern in the graphs becomes a nice simple pattern in the sounds this is what pythagoras noticed the fundamental of a musical note is determined by the size of whatever vibrates to produce that note so if one hammer is twice the size of the other you'll get two notes with a frequency ratio of two to one we can also write this as a fraction two over one which is clearly just two but writing it like this will come in handy shortly the interval between two pitches a ratio of two to one apart is known as an octave it's the most pleasant sound you can create with two different notes which comes from the fact that two to one is the simplest ratio [Music] in fact it's such a perfect match that our brains often treat two notes an octave apart as being the same but let's face it you can't write much of a song with only two notes we need a few more and this new mathematical connection tells us where to look what's ever so slightly more complicated than two well three of course so let's consider a three to one ratio that is one note with a frequency three times the other as you can hear it still sounds rather pleasant but maybe a tiny bit less than an octave something we can do to simplify things from here on out is to move this new note down an octave because we perceive octaves as being so similar we can basically treat this new note as the same note how do we calculate its frequency though well just as two over one is an octave up one over two or a half is an octave down so we can multiply three by a half to get three over two our new note has three over two times the frequency of the original think of it like this for every two cycles the lower pitch note completes the higher one completes three we called the interval between our original note and this new three over two note a fifth cause why not if two's company then three's a fifth fifths are nearly as nice to hear as octaves but with extra interest and utility suddenly we have music that moves and varies in emotion instead of simply keeping to almost indistinguishable octaves everywhere and we don't have to stop there we can use exactly the same approach to build a note with frequency five over four times the original or six over five or whatever the amazing thing is that our ears are hearing the maths the more complicated the fraction the more we move away from pleasure and the closer we get to pain i am not joking here's this monstrosity called the wolf interval pythagoras and his followers inanimate as they were with mathematics saw this connection between simple maths and beautiful music as proof of a perfect mathematical universe the story goes that pythagoras inspired by his blacksmith encounter developed a new method of building a musical scale that is coming up with a set of notes that will work nicely together to make music he decided to fit a set of notes within one octave using fifths here's how he did it starting with any frequency which we'll call the base you go up a fifth then go up a fifth again giving a new note three halves squared or nine over four times the pitch of the original but nine over four is greater than two so in fact this new note lies outside the octave range we want to fill as we did earlier we can simply move it down an octave by multiplying by a half now we're rolling go up another fifth to make another new note then upper fifth again down an octave to fit it inside the original octave and voila yet another and you keep going and going and going until you've gone up 11 fifths to end up with 12 notes including the original [Music] now you've filled your octave with 12 notes you can go anywhere make the octave above by multiplying every frequency by two or the octave below by dividing everything by two [Music] and go on like this until you have as many mathematically perfect musically beautiful octaves as you desire this method of constructing a scale is called pythagorean tuning using only pure fractions we've constructed a bank of notes for all our musical needs we can place them all on a piano and give them names and start playing whatever we want saving the knowledge that we have the power of maths to guide us [Music] that can't be right that interval's supposed to be a fifth but that that sounds horrendous uh pythagoras what have you done what is this ratio all right let me let me just check some stuff and multiply those fractions together in those fractions and carry the two and yeah yeah it's a wolf interval [Music] why pythagoras why okay so i and pythagoras did a bit of hand waving somewhere in there let's go back and see what really happened remember we went up in fifths 11 times to create our final pitch i'm going to ignore the shifting up or down by octaves so things are a bit clearer but as we established octaves don't change the character of a note much so everything i say here will apply whatever octave you're in what happens if we go one step further and add another fifth onto this final note if we have created a truly useful tuning system then this must equal a note we already have somewhere otherwise we wouldn't be able to play a fifth starting from any note we wanted going up another three halves gives three halves to the power of 12. in decimal form about 129.74 times the frequency of our original bass note did we create any other notes with this frequency well nearly for example every octave we go up multiplies by 2 and 2 to the power of 7 is 128. so exactly seven octaves above our bass note we have a note 128 times its frequency that's quite close in fact it's the closest we can get the next note up is 136 times the base frequency which overshoots our new note we have discovered what's called the pythagorean comma this is the ratio between 12 fifths and seven octaves or 129.74 over 128 about 1.014 pythagorean tuning works or doesn't by assuming this ratio is 1. that is 12 fifths is equal to seven octaves it's basically a fudge to ensure we can stick octaves end-to-ends while keeping octave intervals themselves nice and pleasing this is what caused the wolf fifth because even though it looks like we can go up a 12th perfect fifth after we went up 11 to get back to an octave it's not perfect at all we actually go up a smaller frequency ratio than that one that is definitely not pleasing okay you might say well what if we went up even further why did we only make 12 notes inside each octave where did 12 even come from if we want our scale generated by fifths to circle background to an octave of the bass note somewhere we need to find whole numbers x and y such that 3 halves to the power of x equals 2 to the power of y that way going up x fifths would get you to the same place as going up y octaves but this is impossible you can't get to any whole number by multiplying three halves by itself several times let alone a power of two you can get close and in fact there are systems with 665 notes per octave that do much better than 12 but at that point it's so hard to hear the difference between each note you might as well not bother 12 notes is on balance the best compromise but a compromise it still is our quest to build a tuning system with both perfect fifths and perfect octaves everywhere has failed the upshot pythagorean tuning is broken now that's not to say it's broken everywhere if you don't mind not using a few of the notes or intervals we created in your music then you can make it work and that's exactly what musicians did for thousands of years western composers had to restrict themselves to using only a certain subset of these notes for each song and even then they had to be careful not just because they ran the risk of offending their ears but because they ran the risk of condemnation to hell since the catholic church banned the wolf interval during the middle ages at this point you might well give up on building a tuning system from fifths and wonder whether generating everything from a different interval might fix everything composers at the time had the same idea creating a system called mean tone temperament here everything revolves around making the ratio 5 over 4 perfect in as many places as possible this is the ratio of a major third but for the same reasons this can't generate a consistent tuning system either turns out any tuning system built to preserve mathematically beautiful ratios everywhere cannot exist alas pythagoras was wrong music is ruined and we're all doomed to live a cold musicless life forever what's that uh we're not millions of people listen to music that uses complex intervals every day and don't even think about it but how well remember hepasis poor drowned hypasis what if i told you he was on to something eventually mathematicians decided irrational numbers weren't so irrational after all if the square root of two appears in something as essential as a right-angled triangle maybe we should stop denying its entire existence they decided in the 17th century a dutch guy named simon steven wondered if they might hold the key to fixing this musical system the replacement he created is called equal temperament in equal temperament you stop worrying about fifths or thirds or anything more complex than an octave instead your goal is to make the ratio between any two successive notes the same everywhere because a 12 tone system almost worked already stefan began with that he wanted to keep octaves with their 2 over 1 ratio pure but nudge everything within an octave into a better position to get 12 equally spaced notes inside each the ratio you need for equal spacing is the solution x to this equation if there are 12 notes in an octave you're going up by 12 of whatever your equal interval is to get to the next octave so you're multiplying the equal ratio by itself 12 times you then need x to the power of 12 to equal 2. unsurprisingly the solution is an irrational number the 12th root of 2 or about 1.059 if each note has frequency 12 root of two times the previous one then after 12 steps you've gone up one full octave and there you go you've just built yourself an equal tempered scale where a given interval never changes no matter where you play it and you'll never encounter any grating wolf-like dissonances so what's the catch well apart from the octave none of those intervals are mathematically perfect they are all irrational their sound waves don't line up neatly at all the thing is our ears don't actually care so long as it's close enough to a nice round ratio it sounds somewhat possible [Music] and in fact in our modern world it sounds more than passable following stefan's invention somebody thought it would be a great idea to integrate equal temperament into this newfangled thing called a piano the piano then took the world by storm and everywhere it went equal temperament followed suddenly everyone in the western world was hearing equal tempered music so much that it stopped sounding out of tune you may have thought that even the purest of the pure pythagorean ratios i played earlier sounded weird and this is why you've likely never heard any instrument play anything other than equal temperament and yet they sound fine even though they are all according to pythagoras's definition at least out of tune we have been conditioned to hear equal temperament as in tune and pythagorean as out of tune which is rather mind-blowing when you think about it [Music] pythagoras mind would also be blown had he known this but in a rather different way fortunately it's quite difficult to drown an entire planet thank you very much for watching i made this video as part of a university project and i need your feedback on how well it communicated maths you can either leave a comment below or fill in the survey linked in the description

Channel: Oliver Lugg

Views: 957,424

Keywords: Pythagoras, Pythagorean comma, Musical tuning, Equal temperament, Ratio, Fraction, Mathematics, Maths, Math, Maths communication, Music, Ancient Greece, Simon Stevin, Hippasus, Irrational numbers

Id: EdYzqLgMmgk

Length: 17min 40sec (1060 seconds)

Published: Fri Apr 23 2021