The Fibonacci Music Box ( 

the thing that always annoyed me about those “digit songs” is that they always seem to use base 10. If you’re gonna use digits then surely base 7 or 12 is a more elegant solution than a number as arbitrary as 10.

I have been writing this sequence of 60 numbers down for over 20 years never really understanding what it was. 1793 for short hand, I've been obsessed with it. Every 5th number is a 5 or a zero, when typing them on a number pad they form a fantastic 4 sided symmetrical flower, I've been plotting them in 15 pointed stars using string art and holy fuck you have a video that just breaks it all down. I've never gotten deep into mathematics and I didn't have any education past high school, but this has life long been my favorite set of 60 numbers that I found one day in highschool art class trying to draw spirals. I am beside myself right now.

I can really see this being inspiration for video game music. The segment between 10:21 and 10:43 specifically gave me the feeling of discovering a mystic cave, with every different cycle being more mysterious and magical than the last, especially with that echo effect.

=== Table of Contents === 0:00 Intro 0:20 Part I: Disappointing Fibonacci Music 1:36 Part II: In Defence of Digits 3:28 Part III: The Fibonaccis Modula 10 6:00 Part IV: Pigeonholes and Loops 8:06 Part V: A Sonic Interlude 9:16 Part VI: A Multitude of Moduli 11:50 Part VII: The Extended Fibonacci Cinematic Universe *Nick Burns:* _You're welcome!_

That final musical composition which utilized multiple pisano periods in various combinations to create a unique tune was my favorite part. Because it truly demonstrated a way to combine these various periodic elements together with the software you've created in a way where it's not just a loop of notes at an even rhythm. This added more variety and interest to the sound. Extending this app to account for such combinations of patterns by duplicating the boards, and allowing you to set entry and exit points at certain parts of the period (which can then loop based on some bpm and time signature) combined with the ability to select which scale each grid operates on... having these features for each grid would be pretty incredible. Because you could synchronize them, or cause them to diverge and see how the various periods intersect and then recombine. And also some pitch adjustment. So you could set a starting note for your scale as the root note of a grid. Combine that all together, and you have a generative music system that can inspire so many new kinds of sounds that nobody would ever likely think to compose all on their own.

Thank you. For the simple "digit separated sequence" at the beguinng i thought of playing them simultaneously, it would do chords, and bigger and bigger chords as numbers would get more digits

It is very interesting how even bird song sounds like the Fibonacci sequence. The last "song" just sounded like nature.

Tool's song, "Lateralus" uses the Fibonacci Sequence it its rhythm. Beautiful song. Really cool video!! Great job!

I enjoyed this a lot.. I'm a musician, composer with a love of maths.. and of combining the two.. the difficulty for me with sequences like the Fibonacci is that it is fundamentally not about the numbers.. It's about relationships.. and anything exponential is difficult to convey in both time and pitch past a certain point.. my investigation recently has been about nesting which works well with the fibonacci.. if we let the layers of composition be determined by the expanding series but we nest the previous series in as detail we gan being to get a sense of how the relationships work

I can't believe we finally discover how Nintendo made the music for BOTW/TOTK 🤯

A few others made similar comments, but this vid was stressing me out for the first 55% of its runtime. The whole time i was watching, i was trying to mentally compose a tactful complaint comment about it being a base 10 centric vid. I felt much relief once that was addressed. 😊

Here's an interesting thought: Each of your loops starts with 0,1. But once you're throwing away the direct use of the fibonacci numbers, there's no reason you have to do that. Try starting with another pair that isn't in your existing cycle. It too should cycle. But since you have 100 digit pairs, and the original cycle used 60 of those, there's only 40 left. So the new cycle must be smaller. There are likely more than 2 cycles for any given modulus. In mod 10, (0,5,5) is a cycle. Another is (0,2,2,4,6,0,6,6,2,8,0,8,8,6,4,0,4,4,8,2). Note that this one can be broken into 4 phrases of (0,x,x,y,z). The rest are (1,3,4,7,1,8,9,7,6,3,9,2,), (2,6,8,4), and of course, the trivial (0,0). I did all these "by hand" (in a spreadsheet). To explore arbitrary moduli, you'd definitely want a program that can find all cycles. And then I unpaused, and you got into this topic as well…

Another way of looking at it is, the Pisano period is the order of [[1, 1]; [1, 0]] in the group of invertible 2 x 2 matrices mod n, since if you don't reduce mod n, the powers of that matrix give matrices with consecutive Fibonacci numbers.

11:17 WOW that looks so similar to a graph of the totient function. I wonder what relationship the two gtaphs have, if any, especially if the appearance of the "lines" on each have anything in common. I suppose the lines are really a natural expectation of ANY function with ties to modular arithmetic and expected values under particular and common inputs. The main visible line in the totient function is the direct result of totients of primes outputting 1 less than said prime, as well as some other expected values, and this likely occurs for much the same reason, but by different means.

I dont know anything about any of this but my brain has rotted to the point that this thumbnail has become by far the funniest thing ive seen all day.

I really enjoyed your Fibonacci music at the end of the video. The video itself is also very nice - clear animations, clear explanation of why the modulo sequences have to loop (which was not immediately obvious to me before you explained it :)). Great stuff!

i CANNOT get over the song you play us out with at the end (15:58).... listened to it on loop for maybe 30 mins now. is there a fuller version available anywhere?

Another answer to the question of "Why there's no orphaned sequence of nodes?" is that the process of generating a next Fibonacci number is deterministic, which means a point in a grid can only be generated by a *single* point. By the way, huge thanks for making this awesome video

10:46 sounds like something you'd hear in a purposefully glitchy boss song

You'll find a couple of explorations of Pisano Periods on Notes From The Analytical Engine by Beat Frequency (on Bandcamp) - specifically tracks 3 and 23 - Pisano Bebop and A Traveller's Reverie. (One of Fibonacci's nicknames was The Traveller.)

I'm definitely gonna use something like this for my minimalistic game.

Very interesting. Something to keep in mind, the sequence gets closer to phi, the farther you go the more precise this proportion gets, so it wouldn’t make so much sense to take into account the first numbers of the Fibonacci sequence since the golden ratio is just not as precise to phi from the start.

-What music do you like? - It's... complicated...

For our exercise of showing why the loop must always return to the initial point for the fibonacci terms, it's relatively easy to show actually. Assume we have 3 fibonacci terms (or their last digits, specifically), A+B=C. In order to find a looping point later in in the sequence that does NOT require you to first go through A and B (thus creating a boken loop in this sense), two things must be true; We must have another value of C further down the line that is NOT created by the sum of A and B (one example could be that A and B are 1 and 4 with C is 5, but C could ALSO be made by 2 and 3), AND the NEXT term in the sequence must ALSO be the same as the term that comes after C, otherwise it obviously isn't a loop. This however, leads to a contradiction, because if we assume the next term after C- we'll call it D for consistency- is equal to C+B, and our assumption is that C AND D must be unchanged, then B must ALSO be unchanged. In other words, B=D-C, and we are saying D and C are fixed constants, and so must be B. But, we said that C must be created by a DIFFERENT pair of values A and B than our old set, but since we've determined that B and C are both constants, we can use the same basic difference identity to show that A=C-B, and therefore A ALSO can't change if both C and B remain unchanged. This means it is impossible to repeat any sequence of two terms C and D without FIRST going through the A and B that led to C to negin with, enforcing that no fibonacci terms can return to a previous point in the sequence without entirely cycling through it again.

7:35 It’s obvious that any one point will always lead to the same point, but what about in reverse? Every point shows two numbers. The current digit and the previous digit. This means you can go backwards by subtracting the previous digit from the current one, which in turn means that the mapping can be done backwards as well. No two points can lead to the same point, because otherwise the mapping couldn’t go backwards (you’d reach a point that diverges into two which is impossible)

"The Extended Fibonacci Cinematic Universe" floored me

Modulo 81 sounds like how they made futuristic computers sound in the movies from the 60s :D

11:00 this looks like the path a captured animal would pace

Super underrated

I would love to see a continuous version of this, with all points in the grid coloured by the number of steps required to return to the start.

This is why I love SoME

marc.. you are a very naughty man making this machine free.. I didn't sleep until 4am and now I am on it again.. I have a life to live.. ;8)

The song you made at the end reminds me of the game simtunes. If you have never seen the game before I'm sure you'd love it even though it's old AF. The song I'm thinking of is the one made out of the picture of a guy in a laboratory.

Finally someone do it in the right way

9:57 This could easily be the default ringtone of a phone

Why am i getting the imagination of someone who drank the all the energy drinks and is *_JUST_* shy of dying from caffine OD and is playing whatever comes to impulse on the xylaphone and this alone just sums up the music box?

I found a reletivly coherent infinite song Sum up the number of 1s in each sucssessive binary number. Then, travel that far away from middle C on the C scale. If you REALLY want it to stay a bop after note 16, you take the change in the sum of ones in the successive binary number number and play the note which is the Nth fibbinachi number higher if the sum increases by N, and the Nth fibinachi number lower if the dum decreases, where N is the absolute difference between them plus 2. Absolute fire.

Why do I get the bad feeling of listening to these will open some portal to another dimension...

Brilliant, this is enough to turn anyone into a musical numerologist! When playing on the FibonacciMusicBox, once you move into the higher end of the modulo range, some of the some of the repeating patterns look like chaotic attractors in phase space. Lyapunov exponents anyone?

14:44 "How many distinct cycles are there for a given modulus?" See A015134 on the OEIS. There are a couple of interesting patterns in this sequence: a(2^n) = 2^n. Also it looks like a(k^n) = (k^n + 1) / 2 for k in {3, 5, 7}. That patterns breaks down for 11 though.

This is absolutely wild, thank you so much for this, and for just putting that tool out there to play with, it's astounding. I have been tweaking the intervals and decays to turn it into a more conventional "instrument" and seeing what kinds of cascades I can create. Cannot wait for my iPad to finish charging so I can try it as a physical instrument. Thank you!

Woop a musicpy some submission!!!!

The reason why the loop has to go back to its beginning is best explained by reversing the sequence. If it is true that the n+1th point is entirely determined by the nth point, the converse is also true, because F_(n-1) = F_(n+1)-F_(n). To make it even clearer, the sequence is defined as X(n) = A^n X(0) where A is an invertible matrix. X(n) = X(m) means that A^n X(0) = A^m X(0), which, assuming n > m, means A^(n-m) X(0) = X(0), i.e. X(n-m) = X(0) : the pattern loops back to the start.

A fascinating video thankyou! I did something in the same vein using the digits of repeating decimal fractions see the chalkdust article fractograms and a follow up where sound is included. Thanks again for a great article!

Oh Great! Guess what I'm going to be messing around with for the next couple of months! Well, I'm going to go get my tablet and whiteboard now. Seriously, thank you. [from an old disabled U.S. Army Infantryman who just happens to like to do recreational math]

Oh yeah, there is a soundtrack called gilded Runner from the video game "Genshin impact". Instead of the melody, the Composer uses Fibonacci sequence into the rhythm. And it sounds really unique

The damn observation skills this dude was able to make about his box!!

Great video connecting number patterns, music, and coding.

you've gotten me very interested in this type of math

2:10 - 3:00 I believe the first digit of the Fibonacci numbers follow Benford's law with a logarithmic distribution, i.e. about 30% of the time it'll be a 1, about 17% of the time it'll be a 2, etc.

I think the reason you cannot have a forking path is because of the way fibonacci numbers are calculated. In order to get the the pair (3,5), for example, you must go through a point (x,3), as that is the only way to get f(n-1) to equal (3,5) in the next step. Of course only one number adds to 3 to make 5 and so we get the pair (2,3) and the logic repeats. However at this breaks down at (0,1) as that number has no predecessor so the one possible combination that gets to it, (1,0), allows it to loop. If I'm correct, i believe that means that every position on the pegboard has exactly one position that leads to it, no more and no less.

Having listened to a whole bunch of them now, I have to say my favorite is still the original Fibonacci mod 10 on the major scale. That one is just charming!

Some of these straight up sound like a Marimba solo excerpt

You can get a mandlebrot with it too it seems because if you look at the growth you see a thing similar to when you increase the expontet

We can go the Fibonacci numbers backward e.g. 13 - 8 = 5 ; 8 - 5 = 3 and so on. If we don't loop back to the beginning we would have a fork where two different outcomes would be possible if we go backwards. But that is impossible because the Fibonacci sequence is determined by the previous numbers. We can not get a fork going forward and we also not get a fork by going backwards. There will always be only one possible next/previous solution.

Nice Video, I wonder what you think about the Fibonacci sequence inside the Tool song Lateralus?

7:57 exersise proof (SPOILER) . . . . . . . . . . . . . . . . . . . each pair of numbers in the mod fib sequnce has only 1 pair that leads to it: say you want to get to the pair [a, b]. you will need to get before to x + a = b (mod 10), and x has only one possability (for 0

ooh this could make some interesting waveforms, taking the value on each axis and interpolating them in some way

8:04 -- since the fibbonaci numbers are deterministic, only one pair of points can lead to the other (eg A,B can only be preceded by C,A and there is only one such pair) a "fork" in the path would require 2 different points leading to the same one, which is not possible.

So *this* is how Nintendo designed the GameCube startup sounds.

Finaly. So intresting

I think you will always reach a loop because this seiries is reverseable For each numbers b and c you can always find the number a = c-b. If there are 2 ways to reach the same point, then this point would no be reverseable. Because there is only one way forward and one way back, and we know there must be repeating numbers then it must be a loop

Im having major tom and jerry flashbacks at 8:52 to 9:05

This is the first video on mathematics and music that I see that does not seem mathematically naive to me. You address the point that random sequences are a boring source for notes and you also consider different mappings between numbers and notes. I have no idea of music AT ALL! So this was refreshing. Thank you.

I think an interesting idea would be to have the x axis of the mod represented different types of chords, and the y represent different base pitches for the chords. this would restrict you to mods that are multiples of the amounts of pitches in scales (e.g., multiples of 5, 7, or 12)

Try to click 3 times really fast at position (5, 11) modulo : 12 tempo : 179 cycles : 5 scale : double harmonic median pitch : 62 for reference i click 14+ times/sec (touchpad + mouse)

"fast", before the "very fast", sounded like some windows vista music sample lol

The problem isn't fibs in music it's the fact that certain aspect of the music don't follow the same pattern, Inner Rythm and dynamics need to match and really theres an unlimited amount of characteristics that can be bent.

would be even more interesting if each grid also had a transformer function (rule) for the duration and octave of each note in the sequence

this is amazing! only suggestion id have is an option to show the colours on the dots before you click them. thank you so much it is oh so inspiring

Wow, you've got some talent to keep the interest! :)

Great video, so much great information! I want to see if adjusting the algorithm to be base9 will yield more 'harmonic' results. Or really any base divisible by 3. Super curious about 9-simplex propagation through space/time rotation and how to generate music from that.

Awesome, a new TOOL song!

Try to get your hands on a DS and a copy of *“Electroplankton.”* I’m confidant you’ll enjoy it a great deal. It might be expensive if it’s mint with the original packaging, as it has a pretty nice instruction booklet, like games used to have, and, IIRC, it’s on really nice paper with metallic ink. Not essential to enjoy the game, but weirdo collectors would probably pay way too much for it, thereby inflating its price. TLDR: *Electroplankton.* Oh, be sure to use a nice pair of headphones. Regular jack, nothing proprietary (looking in your direction, Apple…).

This could be used as a random melody genarator!

my favorite was the overtone scale, i wish you did one where it wasnt so fast

1:09 yea base10 stuff too

I just played with it and plugged these intervals into the custom scale setting and got beautifully musical results with Mod 8 and Mod 10: 2, 4, 5, 7, 8, 9, 11, 12

The "inversion" you talk about presumably refers to the 31st digit on being (10-1st digit) ie 11235.., maps to 99875... and 0 becomes 0. This is curious and always struck me as "musical" at least when I noticed reciprocal prime repetends of even length show the same thing however in the reciprocal case its "9's complement". To clarify: 1/7=0.142857 142857 etc the period is (p-1)/k where k=1 in this case and is therefore maximal. Splitting the repetend in half 142 is the 9s complement of 857. And this structure is universal (always when repetend has even length which is most cases though just because all primes>2 are odd it can be that (p-1)/k is odd). Despite reckoning there's a musicality in that inversion it was never obvious to me what the appropriate mapping ought to be and I sense you share that. The theory about about all this stuff is there:-- cyclotomic polynomials and things and I just have a peripheral understanding of number theory. 1/89 is a prime with maximal length period (in base 10) and seemed to me back in the day a decent sized candidate (because 1/7 won't keep people interested for long without a lot of operator involvement) and it has a curious decimal expansion reading it forward: 0.01(check it out for homework). Its proven by the way. Good luck. 1/61 has period 60(b10) and moreover every digit is represented the same number of times. I ought to point out this does NOT mean its normal in the formal math sense of normal.

Tigran Hamasyan has entered the chat

cursed math music video, perfect

Interesting stuff. Music not numbers, notation arbitrary and culturally created, excellent first ground clearing.

Great video! However, there's one little thing I may point out. You are using the standard* 12 notes per octave tuning system (12 TET). Thus, the result is restricted to the 12 notes of this set. But it is important to realize that it is not the only possibility. As I see, there is nothing universal about our 12 TET system, but it is a cultural choice (as opposed to the Fibonacci sequence). I think it would be an interesting idea to "hear" this sequence of numbers in other tunings (This can be seen as using different glasses to see the same object). (*In Western culture). - The Xen Wiki is a wonderful resource. I would like to offer you some examples and other related links, but RU-vid thinks I'm spamming you... XD. - Anyway, love this concept of using the (mod). Such an interesting approach! :)

Brilliant video!!!!

esta wea es el mejor video que he visto 👏🏿👏🏿👏🏿👏🏿👏🏿👏🏿👏🏿👏🏿👏🏿👏🏿👏🏿👏🏿

It kind of reminds me of the N*3+1 equation

I noticed in passing that mod 5’s cycle covers a full 80% of all possible pairs, and it got me thinking. What is the highest fraction of these nodes that any Pisano sequence passes through? In other words, what is the highest possible value of pi(n)/n^2? n=3 seems hard to beat, since at another glance it hits all points but the degenerate sequence at 0, 0, but can it be proven to be the highest? That’s still only 8/9; if a higher value of n were to also hit all such points, it would still score higher.

Love it!

If you're gonna do it in base 10 surely you gotta use the pentatonic scale 🤔

At the beginning you mentioned the ever increasing feel of the first digit, and I feel like using that as an X coordinate with the last digit as the Y coordinate, you'd somehow maintain the feel of the sequence even more.

you should consider making this a vst/clap plugin to make it maximally accessible to musicians

8:51

Now let's assume that every state of infinite universe after itself has one next following state Is universe a infinite loop? Could you travel backwards in time by traveling fovard enough? Could be there near identical states of universe on the timeline created by possibility of time travel (if you timetraveled to your past and you killed in the past your grandparents that wouldn't be your past)? Are there any other unexplained time travel paradoxes???

Lateralus - TOOL

Commenting for the algorithm...!

Teeeechnically the song at the end is less Fibonacci and more Lucas, because (0,1) isn't a part of many of the loops. Wait a second, this video taught me something. Heck!

Could there be an algorithm where, given a valid music sheet (single notes, non-cyclical, maybe?) it could find a valid (family of?) modulus and start point that defines the melody? 🤔

How would it sound like if, for each modulus (up to some limit), you would divide the octave into that many equal pitches? How about unequal temperaments?

This is so fun to play with! I'm experimenting with 31TET (interval=12/31~=0.3871)

truly magical