How many songs are there?

"Assuming the world doesn't end, will there come a day when all the music it's possible to write has been written? It's finite isn't it?" - Claire W, via Twitter.

Excellent question. Yes, there are a finite number of songs in the universe - with the one condition that no song can last forever. Let's make it more restrictive and say no song can last for more than 5 minutes (though we could choose 20 or 30 mins and come to the same conclusions.)

Given that, though, it turns out it might not make very much difference that its a finite number...

All of these 5 minute songs could be played on a digital player. So we know all songs can be represented digitally: the waveform can be encoded in numbers. This is jolly helpful for answering this question. Digital encoding is ultimately bits: ones and zeros. So each bit can only be in one of two states, like a lightswitch. 1 - 0. If you have two bits, there are more options: 1/1 - 1/0 - 0/1 - 0/0. You have four options for how to arrange two bits. Three bits? 1/1/1 - 1/1/0 - 0/1/1 - 1/0/1 - 0/0/1 - 0/1/0 - 1/0/0 - 0/0/0. Eight options.

The pattern appearing there carries on: 2,4,8. That's 2, 2x2, 2x2x2... and the pattern carries on working for any number of bits. So the number of combinations is just 2 to the power of the number of bits you have. Which is how, incidentally, you get a byte. A byte can be a number anywhere from 0 to 255 (making 256 total options.) A byte is just 8 bits, and given what we've just said, 2 to the power of 8 is 256. You can check this by writing out eight numbers in a row: 1,2,4,8,16,32,64,128. Choose any combination of these numbers - in effect, marking it one or zero - and you can get any number from 0 to 255.

Err, hope that didn't confuse matters. Back to bits again. So: we know a) any song can be described with bits and b) we have a simple way of working out how many combinations you can get for any number of bits - if we have 100 bits, it's just 2 raised to the power of 100.

The next vital question: how many bits in a 5 minute song? Another side-note: if we're able to describe any 5 minute song with bits, we can also describe any song less than 5 minutes, since there can just be a chunk of silence. So - presuming no mp3 compression, and we just have a waveform, how many bits? It depends on how you sample the music, but supposing we're talking about CDs. This works, of course, because you could have *any* music on a CD and be able to tell what it was: having a better quality version wouldn't change the fact of what song it was. The same counts in reverse: you could probably drop the sound quality very low and still be able to identify what the song was. But let's stick to CD quality for the sake of argument.

So - CDs have samples of music taken at 44.1 kilohertz. Sampling at one hertz would be one sample per second. 44.1kHz is 44,100 samples per second. Each sample on CDs takes 16 bits to encode. Which gives 16 x 44100 bits in total per second: 705,600 bits. A 5 minute song has 5*60 seconds = 300 seconds. So - the total number of bits in a 5 minute song: 300 x 705,600 = 211,680,000. Just over two hundred and eleven million bits.

We now know that its possible to work out the finite number of combinations for 5 minutes worth of music: it's 2 to the power of 211,680,000. But how big a number is that? Well, there's the famous legend of Krishna and the King, in which Krishna, posing as a sage, asks for only a meagre prize if he wins a chess game: "One grain of rice shall be placed in the first square, two grains in the second square, four in the third square, eight in the fourth square and so on." Which is, of course, just the problem we were working out with bits. It's 2^64 since there are 64 chessboard squares. That turns out to be 18 billion billion grains - enough to cover the surface of India one metre deep.

So if that's 2^64, what might 2^211,680,000 look like? It turns out to be a number over 63 million digits long. As a comparison, the total number of atoms in the Earth is "only" about fifty digits long. One calculation of the total number of hydrogen atoms in the universe comes in at a number 80 digits long.

Another attempt at perspective: if you played 5 minute songs back to back for 1 year, you'd get through 105,120. The universe is between 13.5 and 14 billion years old - so that's about 1.4 million billion songs, back to back, if we got started at the Big Bang. That's a number 13 digits long - some way off our 63 million digit total.

But this isn't entirely accurate: the number I've described is all possible combinations of those bits. Clearly, a lot of them would just be noise. Is there any way of knowing what percentage would count as music? Well, partly that comes down to one's taste, of course! But its worth remembering that, within all the combinations that don't count as songs, you'd be able to find every other 5 minute chunk of sound that's ever existed: somewhere in that number-space, there's you as a 3 year old child talking to your parents. There's also a whole lot of stuff that never happened: you as a three year old child talking to Jeremy Paxman about cheese on toast. Paxman as a three year old child talking to *you* about cheese on toast. Etc. Anything, everything.

Which is odd, isn't it, because we know it's a finite number. What this means, I think, is that finite numbers will do us fine on a day to day basis, thank you very much. I mean - you can also digitally encode video. How many possible DVDs are there in the world? Again, somewhere in that number space there's a DVD where 3 year old Claire plays Aragorn in Lord of the Rings. "By nightfall, these hills will be swarming with orcs!"

So, to sum up: yup, there are a finite number of songs - they're a subset of the finite number of total bits on a CD. But there are soooo many of them, we could never, ever write them or listen to them in the lifetime of this or many more universes.

Comments

A smaller finite number

[philosophical throat-clearing]

We can get the finite number of possible songs down quite a bit more by considering the identity conditions covering songs. The number of songs is going to be much smaller than the number of five-minute noise-chunks that count as music. Why? Well, that set of noise-chunks contains, for example, every actual and possible performance of "Summertime" -- Coltrane's, Nina Simone's, mine in the shower this morning, etc. Unless you want to adopt an implausibly strict notion of identity for songs, where any difference in sound --> different song, it's clear that these are all versions of the same song.

So now consider that all the possible performances of "Summertime" include performances by every distinctive voice that's ever existed or will, and on every instrument; and for each voice and instrument and combination, the possible performances include every performance by that voice etc that differs slightly from another in pitch, timbre, tempo, etc etc. So in the set of musical five-minute noise-chunks, there's going to be billions (zillions? more?) of performances of "Summertime". But they're all versions of the same song.

My maths isn't good enough to see exactly what we can do with that, but if the set of musical noise-chunks is guaranteed to contain zillions of versions of every song, perhaps we can safely divide the number of noise-chunks in that set by zillions and achieve a more credible (though still incredibly large) result.

Things get tricky when we try to specify exactly what the identity conditions are for songs. How far do we have to deviate from the template before we don't count as playing the same song any more? Until you resolve that, you're never going to be able to give a precise answer to the question of how many possible songs there are (similar to the point you make that which five-minute chunks count as music is partly a matter of taste).

What about variable lengths ?

This matter has always fascinated me.

Your post is interesting, but it omits the notion of variable length.

I'd like to know what your take is on calculating the number of different possible songs
if the songs can be of variable length (i.e., of any size between 1 and 211,680,000).

I'd calculate that by summing all powers of two between 1 and 211,680,000 inclusively.

Variable lengths

I kind of got around that by assuming that all songs could be described by a set length string, but that many of those would include sections of silence. That seems more straightforward to me than considering how many different lengths there could be and then summing each. Indeed, if you tried to do that, you'd then have to subtract any song-portions or songs-plus-silence where they were duplicated.

Er, that's if you were trying to get an accurate number, which I suppose we're not... just an order of magnitude. I think.

The reasoning behind my

The reasoning behind my "summing" idea was considering the portion subsequent to the end of a song as not being made of bits : for example, on a CD of a given size, the remaining unrecorded space is left empty, not made of zeroes or ones.

Music can be of any length,

Music can be of any length, for example, John Cage wrote For As Long as Possible, which has few notes, sustained for an indefinite length, but it makes sense to narrow it down to recorded music. Classical pieces can go for hours and some have lyrics, such as church music, but also secular music and then their are Operas (is that one long song or several different songs, as they segue into each other, but I guess all CDs do that to an extent), but I suppose we're talking about popular music. Jazz compositions and even some live recorded rock songs can go on for 30 minutes and even in pop, Cat Stevens had Foreigner Suite, which goes on for 18 mins, but I don't think that's really one song. As the post says about one song, it may be best to think in terms of singles and I think the longest of which was Jesus of Suburbia by Green Day at 9mins, so it does raise the question of why 5 was chosen, but we get the idea.

To answer that in the main post about different versions of the same song, are rip-offs and knock-offs included as the same song or not? I think the only way is to listen to every version of Summertime and analyse the digital code. The furthest apart may define the boundaries, but that means at one point a different song is only going to the same distance away as another version of the same song, if it exists yet and we don't know yet what noticeable difference one bit makes in terms of pitch, tempo, rhythm, etc.

Actually, I found out that

Actually, I found out that Bauhaus' 1979 debut single; Bela Lugosi's Dead is about 30 seconds longer than Jesus of Suburbia.

Considering Bela Lugosi's

Considering Bela Lugosi's Dead by Bauhaus, 10 minutes might be a better figure, which would obviously create a number with 127 million digits or 254 million in stereo.

how many songs

Wouldn't the number be smaller if you consider that there are only 12 notes? As a musician I often notice several songs will have the same cord progression and melody. It comes up a lot. For instance people will here chicago's song "25 or six 2 four" and green days "Brain Stew/Jaded". It happens so often that Ive taking to saying that everything has been done since there are only 12 notes.

It often sparks debate about if the song was stolen. Whole websites like www.thatsongsoundslike.com are dedicated to this phenomenon.

Interesting, and relates to

Interesting, and relates to the question of how small you can compress the information that would describe a specific song, or all songs in general. The answer's very different if the goal is recognition by a human. How would you compress an acid techno track that had no melody at all? A midi description perhaps. Or you could cheat and say it wasn't really a song...!

At any rate, yes - the number is much smaller, given that some form of compression or map should be able to describe it (like sheet music.)

Some cultures consider there

Some cultures consider there to be more than 12 notes. So there would be more than 12, but it would still be a much smaller number than considering any combination of bits as songs.

Error

This article made an appearance on the VSauce YouTube channel. Michael based half of his video on this article.
Now, I hate to be pedantic, but there's a mistake with your calculations. You overlooked one small thing; CD quality audio contains two channels; left and right. Your math should be something along the lines of 44100*2*16*300. You'd know that if you've ever produced music. Thanks and Sorry for being so pedantic.

Cripes.

Here's the Vsauce video. Huh! If I thought anyone was going to be paying attention I might have spent more time thinking this through...!

Infinite/Finite paradox

This is a little frustrating when trying to fit this into the idea of infinity. When you mentioned the same principle also applies to video, this really hit home for me.

Since, as far as we know, time can continue on indefinitely - doesn't this mean that there is also an indefinite amount of viewable events that can transpire? How is it possible that all of these infinite amount of viewable events can be depicted in a finite number of frames per second?

This seems, at the very least, like a paradox to me.

What's worse, as you already mentioned, is that the frame rate or bit rate could also be reduced, and we humans would still be able to tell what was happening, albeit at a much lower quality. This means the number of possible viewable/hearable events would be significantly smaller!

How does that jive with an infinite universe?

MY MIND IS BEING TORN APART FROM THE INSIDES

Number of bits in a 5-minute song

Actually, the math you did calculating the number of bits in a 5-minute song is correct only for mono. It turns out that CDs are actually 16 bits *per channel* per sample; if you wanted to calculate the number of bits for stereo audio, it'd actually be twice as large (423 million instead of 212 million). That said, very few songs actually utilize stereo audio to its full extent, and nearly all songs would be recognizable from their mono versions alone. But very interesting article, and thanks for sharing!

Paradox?

Here's where I'm confused...

In the total number of 5-minute audio recordings is the fight I had with my girlfriend over ketchup. Also included is the first time I said a word, and the subsequent reaction of those who heard it.

Suppose both incidents were recorded and then played together on my stereo. Suppose I recorded both of those recordings and made a new recording of them. That new 5-minute recording is also in that finite number, no? Then, of course, so is a new recording of that 5-minute recording with my simultaneous commentary on how interesting both incidents sound together.

In fact, a recording of all the possible recordings is, or should be, one of the possible recordings. But how is THAT possible?

Godel keeps popping up in my head for some reason.

Re: Paradox?

Try thinking of it as just 0's and 1's. Really it just comes down to what audio you can make with a certain allowed number of bits. Anything that would be possible to be recorded into a 5 minute CD audio file is in that finite number. One of those possible combinations of 0's and 1's is the audio recording of me listening to myself listening to myself listening to myself reciting the Gettysburg Address in Icelandic while playing golf on Mars.. Seems impossible, and that surely it would have to have infinite possibilities, but when you really think about it, there MUST be a limit to how many unique 5-minute audio files there can be.

Phenomenally interesting topic! Vsauce brought me here. ;)

Well if u have managed to

Well if u have managed to split that 5 mins into infinite. Than, the only way to differ it from one another is the vocal quality of we humans that differs from one another. Besides, get out!, get out?, get out.. Could be said in different expressive tone whereas in general its gona still b get out,, all of them the same.
Human mind's not that brilliant to split that 5 mins into trillions or so on and give it a lil twist changings so called '0's and '1's to form a new version. For human brain ther ain't bits, its all wordings and similarity we match, dnt we? Than one day comes we hardly get different words or a different rhythm, but than the durability of the song matters, no song could b listened after a certain limit, yh but wen in a new video format or new musical format, could be.
Errmmm Yh, as u mentioned technology can differrentiate a lil ups nd downs in that huge figure we get splitting those 5 mins,
but songs ain't meant for technologies.. Are they?

Interesting signals are not too random. 211,000,000 bit crypto..

It's just like trying to crack a password, only that it's one-time padded. How would you know that a certain 5 minute chunk of sound was minutes 2 through 7 of the 18-1/2 minutes of the Nixon tape that got erased, or some other conversation that Nixon had when he was talking to Abraham Lincoln, and could that have had the sound of a steam train or a Toyota Prius's generator overlaid in the background. The sound would exist in the set of all possible 5 minute songs, but you'd never know when you unlocked the right one. The United States does not forbid the listening to VLF radio transmissions intended for submarines for this same reason. It's probably really classified military information, even if you found a meaningful sequence of characters, you would have only discovered one of many sets of messages, some of which aren't even secret. Likewise it's theoretically possible to generate all possible 5 minute songs, though it's quite computationally infeasable.

And there can be different songs on 2 identical vinyl records pressed from the same master mold, one was just played one more time beforehand. Or it could be a microphone recording it from a cell phone speaker and another one through a $10,000 audiophool system.

stereo

The calculation described assumes a CD contains a single mono audio channel. But CDs, of course, are stereo, so it actually takes TWICE as many bits to store a five-minute song. So if two songs that have distinct stereo mixes but downmix to the exact same mono song are to be considered different, then there are actually 2^423,360,000 possible 5-minute songs: a number with over 127 million digits!