Benedetti's Puzzle (mathematically impossible music)
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- Опубліковано 31 тра 2024
- Italian mathematician Giovanni Battista Benedetti wrote several letter to the composer Cipriano de Rore, the contents of which contained several puzzles that demonstrate the issues of tuning music to pure intervals.
“Comma Pump,” the the phenomenon that Benedetti describes. What is it? And why should we care??
Just Intonation in Renaissance Theory & Practice, Introduction
casfaculty.case.edu/ross-duff...
Pitch Drift in Choral Music
ccrma.stanford.edu/~hiroko/pi...
Os justi - The mouth of the righteous - A Bruckner - 2015
• Os justi | The mouth o...
Schoenberg on Comma Pumps
music.stackexchange.com/quest...
Benedetti: example for comma pump in a letter to Cipriano de Rore
• Benedetti: example for...
Cipriano de Rore, Giovanni Battista Benedetti, and the Just Tuning Conundrum
Ross w. Duffin
www.brepolsonline.net/doi/abs...
Diversarum speculationum mathematicarum et physicarum liber (1585)
echo.mpiwg-berlin.mpg.de/ECHOd...
Dictionary of the History of Ideas MUSIC AND SCIENCE
xtf.lib.virginia.edu/xtf/view?...
JUST INTONATION by Joe Monzo
www.tonalsoft.com/enc/j/just.aspx
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Peace,
Adam
Hey you in the comment section! you're looking pretty acute!
Spinal Tap
gee im blushing :3
Pumped for this episode
**winks**
Thanks man, you're looking pretty sharp yourself!
This is like a calendar without Leap Year.
Without a day subtracted every 128 years
Potentially one of the best analogies!
:)
Leap years are a perfect analogy. It's so easy: There's a leap year every 4 years. Well, except every 100 years, when it isn't. Well, except every 400 years, when it's a leap year again. And we still have to add random seconds here and there to be in sync with the solar year, with weird side effects like 23:59:60 being a valid time on Jun 30 or Dec 31. Sometimes.
No wonder music then!
Lets all start using the iranian calendar system then, where the beginning/end of a year is determined by an actual astronomical event, where by the sun passes through the plane of the earths equator or the plane of the earths equator swallows the sun (the vernal equinox) depending on your astronomical outlook on the earth ;)
some kid in school: ugh i hate math problems im gonna be a musician instead
music:
😂😂😂🤣🤣🤣
exactly it
here I come xddddd
Austin Martín Hernández then you're half gay
If you play guitar for example, you don't need too much math.
“You can’t actually have mathematically pure music without the pitch drifting.”
Great. Now the sopranos have an excuse. THANKS FOR NOTHING ADAM.
Thanks for making a choir geek cry laughing, my dude.
@@TheCubologist lmao aww, hope it happens a lot more often sense
If a soprano could make their pitch drift as described in this video, I would want to meet them and marvel at their incredible ear.
What's more impressive is that despite the pitch drift, the arrangement is designed to start and end in the same key!!!!!!! Literally the math is insane for that
Actually you can... but it would limit creativity. The writer of the melody would have to actually take the math into account, and ensure that there are an equal number of notes which cause a sharpness drift as notes which cause a flatness drift, plus to not have too many of each in a row at any given time.
“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.”
*laughs in extraordinarily intelligent music theorist with enough information about music theory to make a person depressed
I would like to have that problem
Adam is getting crazier and crazier. Just like Vsauce before Michael forgot his password
Soon Adam will give us existential crisis at the end of his videos
Michael Heare is back making vsauce videos finally, after his experiment with paid content that nobody watched, though now he's made all those vids free to watch. The vsauce videos he makes these are ALL about maths, like it's fascinating but he used to talk about a lot more varied stuff. His old videos were essentially video versions of XKCD's "What If" series and books, like the question "what would happen if the sun disappeared". Mr. Heare made a video on that, and it's exactly the kind of question XKCD would cover, albeit he'd answer it with more actual physics, but either way.
Tuning theory is crazy? Hm ... .
freeform144 ok Benedetti
@@duffman18 Ok
Me: *already not comprehending*
Adam: "Of course it can't be this simple"
jamesu Music 😢
Science and Music - book by Sir James H. Jeans - 2012 - Science
Sir James H. Jeans. "On taking the ... clock-face is that shewn in fig. 55; it extends to infinity in both directions, and all simplicity has disappeared." That is the truth of reality as the Perfect Fifth/Perfect Fourth/ Infinity or what Fields Medal math professor Alain Connes calls "2, 3, infinity" as the Unified Field of relativity and quantum physics. The ancient nonwestern cultures realized this truth of reality. Adam Neely is covering it up with his Archtyas 5/4 b.s. 6/5 (harmonic mean) x 5/4 (arithmetic mean) = Perfect Fifth as Geometric Mean Squared. 3/2 is NOT Geometric Mean Squared as Adam Neely is claiming.
Since 81/80 is the amount by which Didymus corrected the Pythagorean major third 81:64, to a just major third 5:4. Archytas will have assigned this interval the ratio 5:4 (the nearest epimoric smaller than the ditone: (9:8)squared = 81:64 and 5:4 = 80:64).
The Monochord in Ancient Greek Harmonic Science
books.google.com/books?isbn=0521843243
David Creese - 2010 - History
Yet we still watch the whole video and try to understand wtf he’s talking about. Even when I don’t, I still feel satisfied by the end of his videos because I’ve learned at least one new thing. 😊
I'm a late learner and I'm 100% self taught from UA-cam videos. I'm going to be completely honest, I feel so proud that I've finally got to the point that I fully understand an entire Adam Neely video! Lol
I remember just a year ago around this same time not understanding anything on this channel. So you can get better quickly if you push yourself. Get a notebook, use the notebook, draw in everything music related. Circle of 5ths/4ths, scales, chords for each key, etc. That helped me alot because I can mentally reference everything by visualizing a page from my notebook. Also as plenty of other people have said stop noodling around on your instrument and *actually practice* or *actually play.* Noodling is a waste of energy and time. Plus it forms very bad habits.
@@the_original_Bilb_Ono So in his book, Kyle Gann is incorrectly claiming that the 5/4 as Major Third is from the 5th overtone harmonic. This is not true at all - it's from Archytas use of geometric mean. "However, he [Archytas] noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5."
A Truman State University review on Scriba, Christoph J. “Mathematics and music.” (Danish) Normat 38 (1990), no. 1, 3-17, 52.
So that's the ORIGIN of Adam incorrectly using 5/4 (and the 16th C. WEstern music theorists) as an extension of 3/2. It's not a "circle of fifths" but in fact it's an infinite spiral of fifths as noncommutative phase (2/3 is C to F while 3/2 is C to G). 6/5 (harmonic mean) x 5/4 (arithmetic mean) = Perfect Fifth as Geometric Mean Squared. 3/2 is NOT Geometric Mean Squared as Adam Neely is claiming. "Archytas will have assigned this interval the ratio 5:4 (the nearest epimoric smaller than the ditone: (9:8)squared = 81:64 and 5:4 = 80:64)."
The Monochord in Ancient Greek Harmonic Science
books.google.com/books?isbn=0521843243
David Creese - 2010 - History
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
Sounds like something that should only be done in private...lol
lmao
The sad thing for me was I started splitting the columns and trying to figure out what the chords were before realizing I shouldn't be analyzing the piece like a Bach chorale. Then it took me a few more minutes to realize this was a tuning puzzle that affects choirs and not bands and orchestras. :) (To be fair, my instrument is trumpet, not voice. So not the tuning issue I normally deal with.)
BTW, I got V 5/3, A unaccented passing tone into vi 5 that resolves into I if you're curious.
i'm very curious if you could cause a choir to do this... like, we often do warmups where you must "fill in" the chord without hearing it on piano
@@Cloiss_ Sounds like an awful lot of practice just to fuck with the choir director during warmups.
I need an endlessly looped version of this so I can hear the drift over a longer period of time.
ua-cam.com/video/FT71tggrrQc/v-deo.html This one goes up an octave for a few of Benedetti's puzzles. Pretty interesting!
I would love that. Just imagine it's like A9 and your ears are screaming
I would listen to that until I couldn't hear it
ua-cam.com/video/FT71tggrrQc/v-deo.html
this is actually until you cannot hear it :)
"It's very pretty. What do you call this?" "Oh, this piece is called Licc My Comma Pump".
Nigel Tufnel
Just intonation was really the saddest key all along, it makes music theorists weep instantly to hear it
No greater comment has ever existed. Cheers.
In D minor: the saddest of all keys.
that one is quite good sir :D
"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.
xD
I’ve never seen a top UA-cam comment use such a clever vocabulary word: “succinct.”
@@henrycavalierkingcharlessp6064 Do you only watch channels for children?
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.
ummm Christian Killgore the math is not *_nearly_* as symmetrical (and "tame") as you described it... For one, that "If you play all twelve, there is only one pattern." doesn't take into account the *_order_* in which you play those 12 notes, and there are 479 million order(ing)s. And you're missing out on all the 12-note sequences that don't include _all_ the different notes, i.e. have some notes repeated while others are missing. And the number of *_these_* sequences is 8.9 trillion. :-B
And this is without ever messing with durations, which would boost the total number way out of the feasibility realm.
Good luck trying to wade through any of those by hand, or even with a computer program. xD
Hey dude. Hope you're doing well. I liked your ideas, it resembles my way of thinking sometimes. Dunno what to say, I'd like to chat some time :)
One amazing story there.
Take care
Nice. I hope you’re doing better. That happened to me too.
Isn't it ironic that the key to wellness is acceptance of Uncertainty? If Perfection was the epitome of 'Knowing What You're Doing' then the invention of Kintsugi by the Zen-influenced Japanese and the Stoic philosophy in the West, is the admission that Chaos is the rule rather than the exception? That we are leaves in a floating world where we don't know whether we we will drift into a raging torrent, or slowly come to rest in a quiet pool until the wind picks us up and blows us elsewhere. All that stuff is beyond our control, and all we can do is focus on the quality of the moment. Slaintè!
I want a girlfriend that compliments me like adam neely complements a wierd musical interval
"You might find a use for that"
“Spicy”
Wow this hit
So you are a weird musical interval? Everything I've strived for! Congrats! I'm just a mr. Bungle power tritone :)
Everyone asks how is music, but no one asks why is music 😞
BEgan
Harmony?
Or just - like in and other art form - an abstraction from reality humans project their thought and feelings onto?
"How is music?"
"I'm fine, but sometimes my temperament gets the better of me."
I'll do you one better! WHERE is music???
if you're not constantly asking yourself "when is music", you'll never find your groove
the answer is "on the 1" btw
@@aloisjanicek293 That's the scientific answer I was looking for. :)
4:10 “Multiply by the inversion.” That, my friend, is called dividing.
Multiplying the inversion, adding the difference... it works, doesn't it?
Or "reciprocal" - if only music theory words weren't already mathematical words with different meanings in so many cases!
Dividing, my friend, is multiplying by the inversion. And so the world keeps spinning.
@@scottfreeland3242 And yet if you repeatedly multiply and divide, funny things happen too ;) The “floating point comma pump”, perhaps?
0.1*0.2/0.1/0.2 = 1.0000000000000002 🤔
@@yinge101 I mean, no this only happens in digital environments with imperfect floating point representations. If you calculate 0.1*0.2/0.1/0.2 with "infinite precision" you just get 1. But still I appreciate your comment it made me chuckle ^^
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 !
Just noticed, it's the video representation of replacing a 4th by minor 7th.
One of the smaller, less insane sins regarding internet video.
At least he didn't add blurry moving shit in 2/3 of the video.
He did it fair and *ahem* /square/
It's still within the 5-limit.
Sexy Piano Notes
/square/ is not comfortable to pronounce.
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-
nice
Why did you have to comment this and force me to find out that it was All Star? I was perfectly content before (in all serious though good job noticing that because I had no idea)
xDDDD
wow good ear!
Whoa, it was so slow I didn't notice...
Therapist: "The Lick Comma Pump isn't real, it can't hurt you"
Lick Comma Pump: 9:59
BASS
He had to to it to 'em (us). The man was cosmically obligated.
It hurts you for sure!
It needs more autotune.
Wasn't that a song by Spinal Tap?
"Fortunately enough 12√2 ^7 is close enough to 3/2." - The big brain guy who created equal temperament.
Or in other words: 12 just-intonated fifths are extremely close to 7 octaves so what if we just stretch the fifth by 2 cents and then it will be sweet as.
Edit: I got it the wrong way around, you would actually need to squash the fifth to get an equal tempered fifth
@@Danicker Hahahah, spot on.
probably a physicists
unfortunately he thus went "so bugger the thirds and sevenths". harmony is always a compromise.
Well (5/4)^3 is close enough to 2
(It's actually 1.953125)
Me after school: "Finally I can rest from maths and watch Adam Neely"
Adam Neely:
Adam Neely: Imma become a math channel.
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”
Don't worry! Not many people would notice. Noticing would require either perfect pitch or really good relative pitch (being able to hear and quantify differences between different notes). So basically, you're not alone.
here, let me help you out:
[music]
you have to listen for a while (atleast 1 minute\60seconfs
@@oscargill423 lots of people wouldn't hear it, for sure, but I don't think you'd have to have brilliant pitch to hear it either
@@tenor1190 For some people, brilliant pitch is being able to tell when a song changes key
Is "Lick My Comma Pump" the spirtual successor to Spinal Tap's "Lick my Love Pump", in Dm, the saddest of all keys?
These go to eleven
In a just intonation tuning in C, Dm is much sadder and slightly dissonant. Equal temperament made every key Vanilla.
There's a fine line between clever and stupid!
Adam was influenced by Mozart and Bach and this licc is somewhere in between there ... a kind of “Mach” piece.
But after a few repetitions it moves to E♭m, then Em, then Fm…
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)
Yes, all organ stops higher than 8' essentially act as harmonic synthesis to enhance the base 8' tone, just as in analog synthesizers. In fact, it has been said that the pipe organ was the first synthesizer. So, when tuning a 2-2/3' stop, it should be exactly 3 times the frequencies of an 8' stop, and a 1-3/5' should be exactly 5 times the frequencies of the 8'. As an organist, I've always found the mixture stops especially magical in how they sound so glorious by adding quints along with the octaves, and only because the quint ranks are tuned justly to the foundations stops.
Some of my best examples started as tests.
Had the same job for a few years! Very interesting experiences... Thanks!!
8:02 " you can't have mathematically pure music without the pitch drifting" unless you make one-note EDM bangers :P
El Sondito babyyyy
*djent
Adam: "Music is too easy, let's add multiplying fractions to it."
I think it is closer to, "We're running out of chalk. Let's do our math with vibrating strings instead."
@@StephenMercer Isn't math with strings (and every other physical phenomena in our universe) just physics?
@@TheWizardMyr Is that why we call it "String Theory"? :hmmm:
You can learn to multiply fractions in an hour. Music can take a lifetime. I think it's obvious which one is easier.
You just multiply the tops and bottoms together, it's really easy
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
that is how it is in english though~
@@Chozal It's French ...
Merci pour le fou rire !
@@MK-zl7hj I know it is. I am French. But "coup de grâce" in English is spelled "coup de gras", for whatever reason. Yes, it's silly and incorrect. Just like us talking about doing one's "footing" for jogging, or going to the "parking" for the parking lot. Adam's not wrong, it's just the way the language solidified.
@@Chozal There's a fundamental difference there, it's that your examples show a shift in meaning, as is natural, while the "coup de grâce" is a mispronounciation.
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.
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
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.
I agree, I was so confused because the second one felt unnatural af
It may be more custom to instruments we usually hear in just intonation. If a group of singers did this, it would seem more natural.
Heck, it might even be due to digitally altering the piano, which can cause errors in the overtones.
You can tell the difference by listening to the volume: in the first, you can have sort of the "wawawa" effect, unlike the second where the sound just goes down uniformly.
yeah, i think it's like the slight dissonances between the notes and the sort of wavering tail end of the sound they produce has become an integral part of the sound of piano to us. listening to the just intonation version again i can see it being more locked-in, more stable, but it also sounds just a bit less like 'piano'.
Having them play in succession doesn’t help, I thought the same. If you listen to the second one a bunch of times then the first will sound out of tune 🤯
"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.
Dude, totally same. I've tried starting multiple instruments over the years but never got anywhere because I always felt so lost and the pieces my teachers had me play felt more like drills than actual music. But I feel like I have some direction now because of all these theory vids, and I'm starting to pick up the guitar again, and it ACTUALLY makes sense to me now.
Ours constantly teaches about old dances like minuet and gavotte, most modern music we got was Prokofyev
Kim Andaya not every method book is created equal, but they are all written so you learn fundamentals of musicianship on that instrument in a way that progresses the student’s knowledge so that the skills become an after thought. Every good musician has to practice their scales, know their intervals,fingerings,bowings,articulations etc before they can begin to make music, it’s just part of the proven process.
@@devonc99 Yeah, I understand that the basics are super important, I'm not knocking that down. I know music is a discipline. But I just had so many unanswered questions, and I have a hard time being dedicated to something I don't understand (in terms of why and how). I think it's just the way that I learn.
Kim Andaya that’s fair, it sounds like you need a better teacher or would benefit more from private instruction if you haven’t tried that already. I’m currently watching Adam instead of doing my music education course work because it’s just more enjoyable, but the fundamentals learned through those drills can be invaluable at a young age with a good teacher. Unfortunately it’s an underpaid, undervalued profession so it’s very common to find bad teaching examples
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.
Hence the phrase "close enough for rock and roll" when trying to get the band in tune ...
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.
WE NEED 10 HOURS VERSION OF THE LICC COMMA PUMP PLEASE ADAM
I would watch for ten minutes at least, think of the ad revenue Adam
I checked Adam Neely 2, it's still not there.
oh s$&%
You could pull this off with the Comma Pump / Shepard Tone combo. The licc never ending rising octaves.
yeah, with licc slowly ascending above the human preception.
"Your musicians were so preoccupied with whether or not they *could* that they didn't stop to think if they *should*."
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.
"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
I never meta-comma I didn't licc.
"You can't have mathematically pure music without the pitch drifting."
Have you tried playing it in A=432Hz? It might help.
How dare you.
I honestly can’t tell if he’s serious or joking
Pro tip: remember to have you synth adlfhaig pe aaheh tt
goddamn it, EazyRun 😂
I‘ve heard of that, but im very dumb when it comes to music. It‘s some sort of conspiracy theory right?
People often talk bad about equal temperament, but it’s genius that people figured it out, and made it so simple
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!
Well, if you instead try just listening to pure 5ths, one based on equal temperament and the other on just intonation, just is more sonorous. The equal-tempered 5th will have beating as the phase shifts over time.
You do realise that we'll need the lick comma pump pumped through 8 octaves and looped for 5 hours now, right?
Ditto!
Yes please
I think this is inevitable and necessary.
Yes please someone make this.
Get Collier to play it live on a range of instruments.
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.
"You can't have both".
Damn you Schrodinger.
What about Heisenberg?
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!
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
This video makes me want to sing one of these as a canon, and throw everyone off by RAISING the pitch instead of the everpresent sinking pitch. How often does the conductor get to complain we're drifting too sharp?
@@drekfletch not much cause like Adam said, music was composed with that in mind
Does anyone know a video in which you can actually hear a choir drifting?
@@ivyssauro123 exactly. It'd be a new and different lecture from being yelled at for going flat. lol
0:57 Lord Vinheteiro, is that you?
Lol
Looking left is watching for malfeasance
Did he ever get to finally impress girl with music? That little sketch was hilarious.
This and the one about rhythm is harmony are the best videos on your channel, Adam!
These are things that move us forward.
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.
this has ruined my saying of "I'm a musician, not a mathematician"
@@tuna5618 You ever just calculate the integral under Mozart? It gets weird fast.
Theory
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 UA-cam'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.
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").
if we play d and a at the same time we change one so that the fifth is harmony
My doctor: Arnold scheonerrberger isn't real, he can't hurt you
Arnold schoenerrberger: 7:22
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.
@Gustavo Campos Google "Daniel Quasar pride flag" - it's a recent redesign riffed off the 2017 Philadelphia Pride flag.
@Gustavo Campos The explanation is that the youtuber 12tone, whose video adam featured in 7:03 always uses Elephant drawings in his videos, he is really great
Gustavo Campos Pretty sure it’s a pride flag with a focus on trans people and POC in the LGBTQ+ community
@@santotiago80 The good thing about Adam Neely, in relation to 12tone, is that you don't need to snort cocaine to keep up with the pace of his talking.
@@FernieCanto AHAHAAHAHHA made my day man! 🤣🤣🤣 some twelve tone vids have subtitles though
Imagine jazz in just intonation where on top of everything, you have to keep track of how much your notes are making you drift.
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.
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.
other musicians in quarantine: making new music
Adam and Ben: mathematically impossible music and fake guitar rap videos
Adam: "God indeed does play dice"
Wasn't expecting Einstein to be called out like that in a music theory video.
It's not quite the quote to pull, though, because this isn't about just intonation being non-deterministic, it's about its internal inconsistency.
Perhaps the quote has been misused often enough that now it simply means "things aren't as simple as we'd like them to be". I don't think Adam was going for the direct analogy, but it's cool to know he's read some physics.
@@martinkrauser4029 I don't think he was going for a direct analogy. He was trying to explain that things are not what they seem in first glance. So "God playing dice" is a great example as there wasn't enough experimental evidence at the time to prove phenomenon like superposition and the uncertainty principle which Einstein thought were false.
It's great to see Adam knows about quantum physics which shouldn't be that surprising considering the types of complex stuff he talks about on the channel. It's highly likely that someone who likes to research, understand, and talk about a complex subject will like to learn about other complex subjects. Adam talked about free will in one of his QnA. His answer was really concise and to the point. It's like he knows the subject so well. Maybe he is into philosophy too.
@@manan-543 i for one wants to hear sonata for electric bass and electron superposition in b flat.
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.
As a mathematics major, this is now among my favorite UA-cam videos of all time.
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
Adam Neely posts this video:
*Jacob Collier Arrives* -Nice
*laughs in G half sharp*
Pretty sure JC use this technique in his "In the Bleak Midwinter", as explained on his interview
@@aloysiuskurnia7643 Didn't know, I knew he used four chords to modulate up to G half sharp, but I didn't know it was this
@@jonocour Actually it isn't this, is not comma pump what Jacob uses, he plays with interval ratios to keep in tune every chord while raising up the pitch, David Bruce composer explains it clear as water
San Tiago yeah I have watched that video. Just a while back so I thought I had forgotten
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.
you're pretty much keeping me sane during this quarantine bro, what a great video
‘Okay! *clap*’ - 10 minutes of incomprehensible wizardry
Me: *Hears theme*
Brain: "Minecraft?"
not at all. just because it's piano you say it's minecraft music. but c418's piano songs are totally different
@@toniokettner4821 It's a joke lol you're taking it too seriously
ERBOCH was i supposed to realize that this crap was a joke?
@@toniokettner4821 Yes?
It has that slightly 'out of tune' sound to it. Absolutely. Wonder if C418 was purposely using some justonic theory in his pieces?
Another great video. I am amazed by Neely's abilty to consistantly deliver quality content.
You are a brilliant communicator and your videos are fascinating.
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
That's an interesting point about rising in pitch. I think this is because a JI fifth is slightly wider than an ET fifth, so if you are ascending in fifths then the pitch will slowly rise. However, if you descend through fifths then surely the pitch would drop... interesting
You can drop in pitch in a very similar way that Benedetti's example rises in pitch. For example: G5-G5sus4-Amin-Dmaj-G drops by a syntonic comma every repetition.
Can I ask what Ji stands for. Dose that mean just intonation?
@@kadendiggs9267 Yes, JI = just intonation.
Concerning playing C-D-C, keep in mind that in just intonation, a "whole step" can be 8:9, or 9:10, or one of several other options as well (although, those two are the most commonly found). Similarly, a minor seventh can be 9:16 or 5:9 or another set of options. Note that 5:9 and 9:10 are a kind of octave-reduced inversions of each other. But, it sounds to me like you're trying to do something like move up by 8:9 and then down by 9:10. Of course you'll end up in a different place.
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
This was an incredible presentation. Can't afford not watching your show!
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.
@@snowwsquire Well, your comment made me fall out of my chair...and land on my head!
I think I have to show this video to all my “I got perfect pitch” friends 😂
Perfect-enough..
love it when the soprano has perfect pitch but you're singing a chorale in a church and they REFUSE TO TUNE TO ANYONE ELSE
Adam: plays a few notes in just intonation
My ears: No no no no NO non nonononononono
Just intonation is interesting, because while on a *piano* we definitely hear it acutely, but it's used everywhere else in music. We even sing in just intonation - your voice pretty much automatically, almost instinctually seeks those 3/2, 5/4 ratios because they sound more...sound... Than 5√2/12.
Just intonation is really important for getting things like resonating overtones, which musicians can use for tuning purposes.
@@TheTauntalus
I sometimes tune my digital piano to the just intonation and am surprised how boring harmonies sound. They lose their "bite" and brilliance. This is probably because the richness of the harmony played on the piano comes from the clashing overtones.
@@ryofurue that's fair. Both systems have their merits - I tend to play music in the world of wind bands and small ensembles, where having that locked in sound helps to clear up the musical sound and avoids some of the unnecessary destructive interference that comes from out of sync tones; but it ain't perfect, and the crunch of clashing overtones definitely can bring out flavors and emotions you just can't get from just.
More power to you, man!
@@ryofurue You must have quite the ear to feel such an emotional difference between just intonation and modern equal temperament. If you gave me the same song tempered differently, I might be able to tell they're not the same, but I wouldn't be able to say one is full of "bite and brilliance" compared to the other.
My musical tastes are deeply ordinary. I'd rather have a sleeve of Oreos than, say, creamy shrimp risotto with mascarpone, or turbot and morels. I'm inclined to believe this is a bad thing -- a "good" person can perceive and appreciates these kinds of musical differences.
eh just intonation is better.
Quasi-pythagorean wolf ET thirds are the definition of ugliness.
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!
great work adam...always taking me down another rabbit hole
The Cosmic Joke pertaining to Western tuning can be summarized very succinctly: 3^12 != 2^19.
Pythagorean comma? lol
Not quite, with 3^12 ≠ 2^19 you are talking 3-limit (only using octaves and fifth) but omit the next prime (5), which is neccessary for well tuned thirds and is used in classical music theory. These lead to different kinds of commas, e.g. 2^7 ≠ 5^3
Some modern theorists go even further and include 7th, 11th and 13th harmonics...
@@foo0815 What are you even talking about?
@@foo0815 Yes, of course there are other commas. I attempted to define the Cosmic Joke very succinctly by simply pointing out the most fundamental inequality.
@@AbhiBass96 Adam talked about 5-limit tuning system at 2:05. Does it suffice to say that the comment about "3-limit" is (gasp) talking about a tuning system, too?
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 . . .
Or do like Jacob Collier and use comma pumping as an expressive tool!
Something like ultra hyper meta mega lydian, right? :)
Yeah, when he said that "you can't have both", I would argue that you simply have to compose with that in mind, just as one composes with the location of the tritone in the diatonic scale in mind.
And if people naturally do it, why bother with tonality anyways?
@@taras-ablamsky the hyper ultra meta something mega lydian is just the first four intervals of lydian repeated, e.g. CDEF# GABC# DEF#G#A ...
Surely somewhere, Bach wrote something like that.
This is way too cool! It has revolutionised the way I look at melodies...
u deliver high quality content. really enjoy ur vids!
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.
I didn't know that gluons could spin upwards in their sockets, but (as a mathematician) I do think roping in Gödel didn't make much sense. Never mind, on the whole the video was very good.
I paused the video at that line, but I think it makes sense broadly. They are both arguably the "you can't have your cake and eat it too" theorem of their respective fields. Cue math people saying "Godel's theorems don't apply to all of math", etc.
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...
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Or maybe I just haven’t been practicing enough and have been listening to way too much Ben Johnston.
Deeply awesome video. Thanks Adam!
I listened to this before bed, and all I can hear now is an infinitely drifting puzzle playing in my ears.
Oh you’ve done it again Adam! This is exactly what I needed to fuel some isolation practice and research!
Becca Deegan thank you so much! Please like and subscribe :) I appreciate all the support x
I love the editing of this video! Keep up the nice work!
The lick at the end is just supreme. I could listen to it for days
I'm writing a thesis paper on different systems of tuning within music and it's always super fun to watch a youtube video mention something about tuning and I know *exactly* what they're talking about
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
yeah like everything is in time
This is brilliant man. You earned a new subscriber.
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"!
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 >:]
Doesn't make it less enjoyable!
I have relative pitch and you're right. this video is painful for me to watch because I keep having a knee-jerk reaction to notes that my brain interprets as "out of tune". the harmonization that Adam calls 'locked-in' and 'resonant' I only hear as off-key, sharp, and warbly... like something playing from a warped record. I want to enjoy it but my brain's just not ready. :(
Nah, they're not really arbitrary. Well, the choice of system may be arbitrary, but the systems themselves make perfect sense. Just intonation uses simple ratios which are more easily recreatable using simple materials and construction methods, which means it makes perfect sense to choose it. 12TET divides the octave equally so every semi-tone is the same and all keys are viable, which means it also makes perfect sense to choose it. Choosing one doesn't validate the other (or any of the other tuning systems), it's just...different.
@@BryM. this is because "perfect pitch" has the commas "tempered out"?
I don't see why it would be less enjoyable. Everyone has "perfect hue", but color gradients aren't somehow worse than solid colors.
Thank you for this wonderful demonstration!
Outstanding video, Adam.
My instructor: play in tune
Me: about that...
My instructor: play in tune
Me: Excuse me, I thought this school accepted other cultures. Why are you infringing on my cultural rights and insisting that your American tuning system is superior?
Ethan Rops lol
"Coup de grâce" is pronounced as in "grass". Otherwise we hear "coup de gras" that would translate, more or less, to a "glimpse of grease" or even a "fat shot"
He said that?? There's no excuse for getting that wrong. There's ce at the end of the word, clearly showing how it should be pronounced.
@@rosiefay7283 yeah no excuse other than, i dunno... like not being a native speaker or whatever, but other than that, absolutely.
Uuuh.. Mardi Gras is "Fat Tuesday", kinda like that?
I think its hypercorrection due to the many French words with a silent final s.
eg. Faux Pas.
@@abyssalboy8811 i think you're onto something
I loved the sound of the just intonation, it's like how I wish the piano to sound like.
in ancient Chinese this imperfection is called the "wolf tune"