Benedetti's Puzzle (mathematically impossible music)

okay so this is a musical puzzle that was first posed by the Italian mathematician giovanni battista benedetti in 1585 and the music is technically impossible because if the tuning math is applied precisely then the pitch inevitably starts to rise as the music repeats g-major becomes G sharp major becomes a major on into infinity here let me show you [Music] you the phenomenon you're hearing of the pitch rising is the result of something called a comma pump you might think of it kind of like a Renaissance version of a Shepard tone that auditory illusion but the reasons behind it are actually quite different now obviously if you were to play the sheet music on a normal piano you would not get the kama pump effect this videos clickbait it's not possible instead in order to get it i used a series of specially tuned pure piano sounds to demonstrate them I'm going to play you a simple G major triad in two different tuning systems and I want you to tell me which of them feels more quote-unquote in tune to you [Music] the first of these examples is what you adhere on your normal garden-variety equal-tempered tuned piano but the second example uses a special system called just intonation where the frequencies of the notes relate to one another by simple ratios harmony that uses the system feels very stable and resonant to me singers will often talk about the locked in feeling that occurs when they're singing and using these simple frequency relationships it's good stuff I love listening to and playing music with this kind of texture but benedetti spa's 'l aims to show some of the dangers of using this system and it all hinges on the fact that in the West we use something called five limit tuning five limit tuning essentially means that every one of the notes from the chromatic scale can be represented by a simple combination of octave perfect fifth and major third so in order to purely tune a minor sixth you would go up an octave and then down a major third the big problem here is that you can actually get to the same note in more than one way giving you different frequencies for the same pitch so F might not actually equal F kind of like a musicians version of griddles incompleteness theorem Kyle Gann calls this the cosmic joke music will never be completely and consistently mathematically perfectly in tune there will always have to be a compromise somewhere at least if we're using western harmony in five limit just intonation this is what happens invented that these puzzles so let's break it down let's figure out exactly why the music is impossible and what that tells us buckle in my friends because it is about to get theoretical theory so on the first beat we have three notes a low G a middle D and a high G that high G has the ratio of two to one the ratio of an octave and the middle D has the ratio of a perfect fifth or three to two on the second beat the soprano voice moves to the note a a is a perfect fifth higher from D now to add intervals what we can do is multiply their ratios now to add interval is what we can do is multiply their ratios so fifth three to two times a perfect fifth three to two gives us the ratio for a nine to four so yeah we are literally multiplying notes together now if you wanted to you could multiply C major seven sharp eleven times an Augmented fifth and that actually would mean something and tuning Theory not sure why you'd want to maybe just to flex on your math teacher but uh yeah you could so fairy and the next beat we have the notes C and E C is a perfect fifth down from the octave G now to subtract intervals from one another we can multiply them by their inversion to subtract intervals from one another we can multiply them by their inversion so to go a perfect fifth up we'd multiply by three to two but to go a perfect fit down we multiply by two two three so measuring is perfect fifth down from the upper octave gives us the ratio of four two three four C which incidentally is the ratio of a perfect fourth up from the root which we can confirm by checking Wikipedia hey the system works all right e is a major third of from C so we can take the ratio of four two three four C multiply that by the ratio of a major third five to four to arrive at five two three four e which is the ratio of a major sixth up from the root which again wikipedia confirms it sounds good right the singers will be singing this music using intervals that were based off an octave a perfect fifth and a major third and all is right in the world and no of course it wouldn't be that simple God indeed does play dice the world is chaos and Wikipedia is believe it or not sometimes wrong remember when I said that you could arrive at the same pitch in more than one way we can actually get to the same note in more than one way well in this case the soprano voice a is being held over from the previous beat so the other singers are going to need to tune their notes to that a and to get a locked-in feeling the way that we can derive the tenor e from the a is by going down an octave and then up a perfect fifth when we do this we get a twenty seven to sixteen ratio for e which is very different from what we got before in this case e does not in fact equal e what's more is if we take this twenty seven to sixteen E and measure down a major third to derive our see we get the ratio of twenty seventeen to twenty which is not the four to three ratio from C that we got before it's a little bit sharper this interval by the way has a technical name and that is the acute fourth so good for you see you're looking pretty acute so even though the numbers for the second set of pitches look more complicated they will sound more in tune because they were tuned to that upper a that was being held over the coup de Gras of Benedetti sposa arrives on the last beat this is where the sword is twisted the soprano a resolves down to a G G is a perfect fifth up from C and if we measure the interval of a perfect fifth up from C's twenty-seven to twenty ratio we get a value of 81 to 44 that final G which is a little bit higher than that octave G that we started with the distance between these two G's is about 21 cents roughly a fifth of an equal tempered semitone it's called a sin tonic comma there is a whole wonderful wide world of these minor tuning discrepancies called commas if you want to learn a little bit more go check out twelve tones excellent video on them but the pitch rising effect of this comma pump or comma drift is not merely academic it's actually bedeviled musicians for many many centuries now in his book the fundamentals of musical composition composer Arnold Schoenberg would write of how natural semitones defer in size from the tempered a fact which causes choirs to get off the pitch it's very common for choirs singing acapella without any instruments to end in a different pitch space from where they began through no fault of their own it's actually built into the mathematics of many pieces of music they sing perfectly in tune the pitch starts to drift but if they fudge the tuning a little bit the pitch might stay stable but the tuning will suffer that's the cosmic joke we can never have it both ways you can't actually have mathematically pure music without the pitch drifting there have been many attempts at solving some of the problems that benedetti puzzle aimed to illuminate he actually wrote several different puzzles each showing different ways that you could have calm adrift using just intonation the solution that the modern world has kind of settled on is something called twelve-tone equal temperament the commas have been tempered out of the system so we can play in every key the pitch doesn't drift that the tuning is not quite perfect and so this is why benedetti T's puzzle is impossible you can't play it with mathematically pure intervals and have the pitched stay stable at the same time sometimes you just can't have it both ways you know I didn't actually study any of this tuning Theory stuff in music school very few people do in fact because music school is more about how music works today and less about why music works the way that it does today and tuning theory goes a long way in being able to illuminate that question why why is music you know at some point Europeans abandoned it their system of just intonation in favor of tempered tunings that got rid of that nasty comma but that didn't happen in other cultures and it's not because that comma was unknown to them in fact it was Chinese musicians who first described a system of twelve-tone equal temperament before Europeans did it's just they didn't adopt it into their music and that's what's so interesting to me because tuning theory is so deeply ingrained in the musical DNA of many different cultures it influences the kinds of melodies you can write the scales that you can use the instruments that you might build tuning theory cuts right to the heart of all of that and I think that's pretty cool so thank you so much for watching I hope you are as excited about tuning theory as I am if you enjoy this channel please consider joining my patreon I'm gonna leave you guys with one of my finest creations I give you the lick kama pump you please

Channel: Adam Neely

Views: 631,268

Rating: 4.9700747 out of 5

Keywords: adam, neely, jazz, fusion, bass, guitar, lesson, theory, music

Id: TYhPAbsIqA8

Length: 10min 24sec (624 seconds)

Published: Mon Apr 13 2020