Benedetti's Puzzle (mathematically impossible music) 

Hey you in the comment section! you're looking pretty acute!

This is like a calendar without Leap Year.

some kid in school: ugh i hate math problems im gonna be a musician instead music:

“You can’t actually have mathematically pure music without the pitch drifting.” Great. Now the sopranos have an excuse. THANKS FOR NOTHING ADAM.

“You either die a musician or live long enough to see yourself become an extraordinarily intelligent music theorist with enough information about music theory to make a person depressed.”

Adam is getting crazier and crazier. Just like Vsauce before Michael forgot his password

Me: *already not comprehending* Adam: "Of course it can't be this simple"

the weirdest part is that i starting singing the soprano part on the piano while imagining the lower notes in my head in order to stay in tune, and after a bit i found my singing was actually becoming sharper and sharper, without me even trying. congratulations adam, you made me comma pump myself

I need an endlessly looped version of this so I can hear the drift over a longer period of time.

"It's very pretty. What do you call this?" "Oh, this piece is called Licc My Comma Pump".

"I'm not sure why you would want to - maybe just to flex on your teacher" I've never heard such a succinct summation of jazz before.

This literally drove me crazy when I was trying to learn music theory. I never knew about this specific puzzle, but when I was studying intonations, and doing a bunch of math that other people have already done for no reason, it occurred to me that you could never mathematically tune a piano. I guess you can, but it at the time I didn't know this was a stylistic choice a musician could make, it just seemed to contradict what I already knew about tuning. This cosmic joke drove me insane, as from my perspective it seemed like every instrument was 'faking it', even if it was just by a tiny fraction of a semitone. The reason I stumbled upon this, was I was trying to discover every unique intonation pattern that could be made using some combination of twelve notes. So if you choose to play just a single note, there is only one intonation pattern. If you play all twelve, there is only one pattern. If you choose to play two notes, that number goes up, but it also corresponds with the number of patterns if you had chose to play 11 notes. I literally wrote out each intonation pattern by hand, and then went about trying to eliminate duplicates. So one pattern might look different from another, but if you shift pitches, they are essentially the same. When you get to 7 note intonation patterns you will find the major and minor scales for instance. That is not 14 intonation patterns, but rather two. I did all of this to try to figure out why we use the major and minor intonation patterns, when mathematically many other patterns exist. I wanted to know what made those patterns different. For instance, using seven notes you could form a pattern where instead of a major scale of C-D-E-F-G-A-B, your scale went C-C#-D-D#-E-F-F#. You could use that scale to form a key, chords, chord progressions, and all that good stuff. It sounds horrible, don't do it, but you could. But I knew that somewhere in all my math were other scales that did sound good. There were blues scales, and jazz scales. I wanted to try to play each of these unique intonation patterns as if they were a musical key, and see if there were any that sounded good, but had never been used before. Then I got admitted to a mental hospital. That is not a joke. The musical/mathematical stuff was more a side effect of my mental health problems than the other way around, but that did happen. I had forgotten about all of this stuff until watching this video. I had gotten into music practice finally, knowing full well my instrument would never and could never be perfectly mathematically in tune, but being content with 'close enough' and 'sounds okay to me'. Thinking back, I'm sure someone could write a program on a computer that calculated every unique intonation pattern, build a 'chord wheel' of sorts around them, and generated a few sample composition to test whether they had any practical merits to them. Unfortunately I threw all my math in the garbage, and decided to learn to play in major and minor keys before exploring the depths of what is mathematically possible. The reason I did that was realizing what was mathematically impossible.

I want a girlfriend that compliments me like adam neely complements a wierd musical interval

Everyone asks how is music, but no one asks why is music 😞

Therapist: "The Lick Comma Pump isn't real, it can't hurt you" Lick Comma Pump: 9:59

1:45 How did NOT I realize that was All Star until now after watching this for the 5th time or so... ADAM YOU CHEEKY LITTLE-

7:31 - Adam, you messed up a ratio! No , not in the music. The aspect ratio of the choir video is 4:3; you have it stretched to 16:9 !

4:10 “Multiply by the inversion.” That, my friend, is called dividing.

"Fortunately enough 12√2 ^7 is close enough to 3/2." - The big brain guy who created equal temperament.

Me after school: "Finally I can rest from maths and watch Adam Neely" Adam Neely:

him: “yeah you can hear that the key has gotten higher we’re in Ab major” me, tone deaf, hears literally no difference: “I mean I’m a little lost but fair enough”

Is "Lick My Comma Pump" the spirtual successor to Spinal Tap's "Lick my Love Pump", in Dm, the saddest of all keys?

I worked my way through school in part by holding keys for an pipe organ tuner and in the process learned a tone about applied tuning theory. The core stop on the organ is always tuned in equal temperament from a reference pitch (except for that one house organ that we tuned in Just C -- The wolf howls in d-flat), However every other rank in the organ is tuned by ear against that core stop. This means that while the intervals between notes on the keyboards are equal temperament, the tuning between stops is Just. It is part of the system that makes a well maintained pipe organ sound like it is playing a single note even though there could be 6 or more pipes sounding together. It creates what this video describes as that "locked in" feeling. There is also a process of adjusting the volume and rate of attack of the pipes to blend together called voicing. (Fun fact for the day the toccata J.S. Bach's Toccata and Fugue in d-minor is a stylized embellishment on what voicing frequently sounds like, leading some to believe that it may have originated as the Maestro's test for an organ)

I can’t be the only person who watches his videos that is not a jazz musician or studied music theory. I mean I’ve played multiple instruments but I only have a basic understanding. I just love watching his videos and I’m honestly so engrossed by learning new things even if I won’t ever use this information

Adam: "Music is too easy, let's add multiplying fractions to it."

Man I love these theories, but your french cracked me up at 6:25 . A "coup de grâce" is indeed a final thrust, but what you said is a "coup de gras", meaning "a fat hit" :D

8:02 " you can't have mathematically pure music without the pitch drifting" unless you make one-note EDM bangers :P

This was one of the things I came to experience after playing guitar for a few years; you can't have every string and every note in tune with every other note. On more than one occasion, I remember spending easily, _easily_ over a half hour at a time doing nothing but trying to tune my guitar. I'd get it perfectly tuned using a piano in my college practice room tuning to, say, the low E string. Then I'd tune all the other strings to the low E, and then compare A-string, D-string etc. to the A, D etc. notes on the piano. I went back and forth like that until I decided I was either a complete idiot (Probably true.) or it's impossible to have perfect tuning that also sounds perfect. (Definitely true.) I'd even learned this in depth at that point; cappella pitch relation being different than other musical instruments as a simple example. Took me a bit to put two-and-two together. Lastly, this is what perplexes me about people with so-called 'perfect pitch.' They can memorize the names of notes to the pitch of notes, but at the same time, what is and is not in tune varies depending on your tonic.

"school is more about how music works today and less why" - reason why I sucked at music back in school right there. Everything constantly felt so arbitrary, and the teacher was in so way over her head I couldn't even dare inquire about anything. Just took the barely passing grade and left. Only over the years and with channels like yours I began to fathom and come to terms with the state of it all. Thank you based Adam.

Out of two versions of G major the first one definitely feels more in tune, probably because I've been conditioned by decades of listening to music in equal temperament.

Very interesting video. I believe this is all why, even with "perfect" intonation on a guitar and "in tune", some chord shapes sound crap unless you slightly detune or overtune a string. For me personally, it's usually the G-string that needs to drop a bit (insert joke here) for many open chord shapes, especially open D.

I'm not sure why, but whenever I hear pianos tuned to just intonation like this, I think the equal temperament version sounds better. Not sure if it's just what I'm used to or of its just that I find the electronic artifacts of retuning the piano sounds bad.

"Your musicians were so preoccupied with whether or not they *could* that they didn't stop to think if they *should*."

this has ruined my saying of "I'm a musician, not a mathematician"

This reminds me of a time I was in choir where we sang a song acapella, but ended up one half step down from where we started. B major was the original key but we ended in B-flat some way somehow. The same thing happened on this other song except we ended up a half step higher than the original key. Started in A-flat major and ended up in A major. There were also times were we ended up in a quarter tone key like E half flat major.

"You can't have both". Damn you Schrodinger.

WE NEED 10 HOURS VERSION OF THE LICC COMMA PUMP PLEASE ADAM

I never meta-comma I didn't licc.

My doctor: Arnold scheonerrberger isn't real, he can't hurt you Arnold schoenerrberger: 7:22

People often talk bad about equal temperament, but it’s genius that people figured it out, and made it so simple

This is fascinating. In my choir, we often sing acapella pieces and I would always get annoyed because we would end out of key. Now I realize that it's us singing in tune that forces us to end out of key. Thank you for this

When you mentioned this as music's version of Gödel's incompleteness theorem, I did a double take. That is such a fundamental theorem in theoretical maths. When you got to "you can't have just intonation and a stable pitch", the connection clicked in such a perfect way.

"I didn't actually study any of this tuning theory stuff in music school..." I was *shocked* when this didn't lead into an ad for Brilliant XD

Interestingly, of the 2 versions of G major you played at 1:20, I found the first one warmer and more natural. I guess this means I'm so acclimatized to equal temperament that anything else sounds out of tune to me!

0:57 Lord Vinheteiro, is that you?

You do realise that we'll need the lick comma pump pumped through 8 octaves and looped for 5 hours now, right?

There is a problem with this explanation and I will try to clarify it. In the just intonation major scale, its degrees have the following frequency ratios: I: 1 II: 9/8 III: 5/4 IV: 4/3 V: 3/2 VI: 5/3 VII: 15/8 I: (VIII, octave higher) 2 One can easily calculate that the same intervals at different positions have different sizes - eg the major second between I-II is larger than the same interval between II-III; the minor third between II-IV is smaller than the one between III-V; also, the fifth between II and VI degrees (D and A) equals (5/3)/(9/8) which is 40/27, not 3/2. Therefore, the ninth between G in the bass and upper A of this example, in just intonation would be 20/9 (2.222…), not 9/4 (2.25). These fine differences enable us to hear DEGREES of the scale. Moreover, this music example is far too simple and I doubt that any choir would rise intonation singing this bar repeatedly. The problem is more theoretical than practical - in practice, all singers and instruments without fixed intonation listen to and are led by the tonal centre and by the scale degrees, not by some manipulative theoretical multiplications of the frequencies (here wrongly supposed that the two fifths, G-D and D-A, are equal). Hence, the assumption that the bass will rise the pitch leaping up from G to C isn't correct. Every string player knows that if he, for instance, plays a major sixth, with open G string and with first finger on E, that E won't sound in tune as a perfect fourth with the open string A. (We meet again those ratios, major sixth + perfect fourth is 20/9, and two perfect fifths, as tuned on a string instrument, make 9/4). In practice, the player would move his finger just a little bit, depending on the tonal context. So the singers change their pitch to stay within the tonal centre. This is done automatically most of the time. In more complicated situations, as mentioned in this video, a cappella choirs and strings do shift from the initial tonality, but this also could be a symptom that, in those particular cases, music may be badly written. Those who want to find out more about this subject can read Paul Hindemith's "Unterweisung im Tonsatz" (English title: "The Craft of Musical Composition").

As soon as he mentioned Gödel’s incompleteness theorem, I thought, “Of course! It’s exactly like Perry Fernalia’s gramixious obstacle!” and it all made sense.

Adam: "God indeed does play dice" Wasn't expecting Einstein to be called out like that in a music theory video.

"You can't have mathematically pure music without the pitch drifting." Have you tried playing it in A=432Hz? It might help.

This and the one about rhythm is harmony are the best videos on your channel, Adam! These are things that move us forward.

As a mathematics major, this is now among my favorite RU-vid videos of all time.

I decided to have a whack at the problem, and I learned some interesting things. My first attempt went like this: Instead of having 3 fundamental ratios, there will only be two. 2:1 (octave) and 3:2 (fifth). The reason I chose this is because it allows notes to be arranged in a hex grid where each step in a direction is always the same ratio interval. Right would be the octave (2:1), up/right would be fifth (3:2), and it turns out that up/left becomes down a perfect fourth (3:4) (down an octave and up a fifth). However, while this hex grid is stable and without conflict, there are still multiple paths to the supposed same note. The third for example has two obvious potential paths: Either down two octaves and up four fifths, which results in 81:64 (funnily enough a ratio of squares), or up two octaves and down three fifths, which results in 32:27 (please ignore RU-vid's tendency to link anything that looks like a timestamp). These are already decently complex ratios, but I'm ignoring complexity and searching specifically for stability. In this case, however, it is obviously not stable because there are multiple paths to the same note. This is when it hit me that the only way to have a just intonation system that is stable is to have an intonation system where there is only one valid path to each note, which in turn requires that there only be one fundamental ratio from which to derive the rest of the notes. But if we pick the octave, we'll never be able to reach the notes in between. So the only way to derive all notes from a single ratio without skipping any is to pick the smallest interval, that being the half-step or semitone. And here we reach the crux: Which ratio should we use for the semitone? Since we are already guaranteeing the stability of the system, we can now focus on how it sounds. Which ratio would sound the best? We could pick 16:15 because it's the simplest ratio that matches what we feel the interval should be, however, that kills the octave, which would be about 2.17 times the frequency of the starting note. Heck, the 7th is closer to a proper octave in this case, being 2.03 times the frequency of the starting note, resulting in a potentially interesting 11-tone system. But no, we want a nice sounding 12-tone system because we like octaves. In fact, the octave is the simplest of ratios, and the most pleasing to hear, so what if we picked a ratio that will result in a perfect octave after 12 iterations? Well, it turns out that that ratio is the *irrational* 2^(1/12):1, otherwise known as equal temperament. Turns out the one stable system that sounds the best is equal temperament. If we want to sound better than the best stable system, we can only choose unstable systems. And thus the paradox returns, and we're back where we started.

Imagine jazz in just intonation where on top of everything, you have to keep track of how much your notes are making you drift.

I adore this channel. Your channel has brought me a lot of these great deeper questions to mull over. Keep on keepin on dude it lights a fire to my imagination.

You are a brilliant communicator and your videos are fascinating.

other musicians in quarantine: making new music Adam and Ben: mathematically impossible music and fake guitar rap videos

‘Okay! *clap*’ - 10 minutes of incomprehensible wizardry

Deeply awesome video. Thanks Adam!

I’m curious how this relates to musical traditions that use a drone. In that context, we don’t get drawn offsides by harmonic progression and are free to play with just intervals with an anchor. Fascinating!

Adam Neely posts this video: *Jacob Collier Arrives* -Nice

Me: Oh boy, he's getting theoretical; am I going to need elephant doodles to understand this? Adam: If you want to understand better, watch this video with elephant doodles.

Thanks for all your amazing videos! As a Theory teacher, it's so great to have student-friendly resources like this to help tackle the "why" of music.

Another great video. I am amazed by Neely's abilty to consistantly deliver quality content.

Me: *Hears theme* Brain: "Minecraft?"

It blows my mind that equal temperament is so ingrained into my brain that when Adam shows both tunings back to back the first one actually sounded more "in tune" to me

This was an incredible presentation. Can't afford not watching your show!

Three times in this video I came down to ask a question only to have you exactly answer what I was typing up. Such an awesome video! I was lucky enough to have a jazz teacher who taught me this stuff through the book "Lies My Music Teacher Told Me" in college. It completely changed how I sing in choir. I'm a low bass, so any time I sing a capella I constantly have to trade off tuning an interval and keeping the root notes in the right key center. It wasn't until I learned about this that I really clicked for me.

I think I have to show this video to all my “I got perfect pitch” friends 😂

Adam: plays a few notes in just intonation My ears: No no no no NO non nonononononono

I don't play an instrument and I don't study music. But since one of your vids came on randomly, I've been hooked. The science, psychology and pattern involved has opened my mind considerably. Thank you for an unexpected introduction to music theory.

great work adam...always taking me down another rabbit hole

I have a friend who sings barbershop quartet. He once told me that one of the songs they would sing would always end up about a semitone sharper than they started. Really interesting to hear the explanation. Thanks Adam

As a non-mathematician I too love using mathematical concepts like Gödel’s Incompleteness Theory in dubious analogies. It can be fun to make mathematician’s eyes spin upwards in their sockets much like a gluon.

Great video, I LOVE hearing the different tuning of the g chords and intervals next to each other. Also the horrific sliding up of the repeated bar! I feel like we lose something with equal temperament, but in our world of chromatic music I think it’s inescapable on keyboards!

you're pretty much keeping me sane during this quarantine bro, what a great video

The Cosmic Joke pertaining to Western tuning can be summarized very succinctly: 3^12 != 2^19.

The solution to this is pretty obvious: Everyone just needs to start composing every single piece in such as way that the upward moving and downward moving comma pumps cancel out, so that by the end you are back at the pitch level you started at. In fact you could take it a step further and consider this to be a sort of second-level tonicization. Say your piece starts in tonic, moves to the dominant key in the central section, and then returns to tonic. At the same time you manipulate your comma pumps so that the underlying pitch levels rises by (say) a quarter-step in the central section and then gradually steps back down to the original pitch level by the end. Though if I had my druthers we would write every single phrase so as to be "comma pump neutral" as landing on a tonic that is say 20 cents higher than the original after just a few measures is really jarring. Composers: Make every phrase comma-pump neutral or I *will* be hunting you down . . .

I was just trying to explain this to someone yesterday! And, clumsiliy stumbling over the concepts, I was just confusing. So I sent him you're video. You really have a gift for taking complex topics and making them understandable.

Outstanding video, Adam.

Hey Adam, I recently designed something I called "Monkey Tuning" (because of jacob collier's tuning system swinging analogy in one of his June Lee interviews) and I've been thinking for a while now about making my own VST. The idea is to allow musicians using midi controllers to change their tuning systems in real time (while performing). I've noticed that the tuning always climbs slowly upwards (as opposed to downwards), but I've never understood why. Mathematically, is it because an interval's ratio to the root is on average greater than the inverse of the ratio which corresponds to its inversion in the octave? For example, if I'm playing in C JI, retune to D JI keeping the note D the same between the two tuning systems, then tune back down to C JI now keeping C the same, the new C is higher than the original C. Is this because the interval ratio of a major second greater than the inverse of the interval ratio of the minor seventh? Thanks for the cool vid. Very helpful to know what to look up now to read more about this. Cheers

Oh you’ve done it again Adam! This is exactly what I needed to fuel some isolation practice and research!

u deliver high quality content. really enjoy ur vids!

This is way too cool! It has revolutionised the way I look at melodies...

9:56, and all of a sudden, tears of joy begin to swarm... So perfect an ending to the imperfection manifest!

This video from 3:14 almost feels like a complete piece of music! That was unusual experience

Thank you for this wonderful demonstration!

Awesome presentation as always🙏

I’ve been looking for the reason behind why I’ve found myself trending sharp over time in unaccompanied singing since being introduced to Just Intonation and Tuning Theory (outside of music school). This may just be part of it... . . . . . Or maybe I just haven’t been practicing enough and have been listening to way too much Ben Johnston.

I love the editing of this video! Keep up the nice work!

9:58 "The Lick Comma Pump" Exquisite work Adam, bravo!

This is brilliant man. You earned a new subscriber.

Wow, imagine having "perfect pitch" only to discover that it's just a conditioned psychosomatic reaction to essentially arbitrary pitches that makes most possible music less enjoyable >:]

This the content we need more of, even if it's more niche, and even if someone has a different opinion... they're wrong

I am as excited!!! maaaan you make me think of so many things!!

So true that you rarely have an opportunity to learn this stuff in music school. I always had a conceptual idea of the difference in just and equal temperament, but never actually learned the underlying mathematical difference, so I’m all about these tuning theory and temperament focused videos.

I love your tuning theory videos, exploring the very foundation of our occidental music tradition is extremely interesting! The fact that changing the tuning opens up new dimensions of soundscapes and then delving into a whole new world of harmonies is amazing. I would love to hear your thoughts on "the well tuned piano"!

'Tuning Theory' is such an intuitive term compared to the phrases other peeps have come up with in places like the Xen community.

Excellent and informative video about my favourite cross-over topic: music math harmony

Loved this video, now I'm going to be obsessed with this