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Q: Mathematical Probability ( Answered 4 out of 5 stars,   3 Comments )
Subject: Mathematical Probability
Category: Science > Math
Asked by: lairbear-ga
List Price: $10.00
Posted: 01 Oct 2005 17:34 PDT
Expires: 31 Oct 2005 16:34 PST
Question ID: 575168
What is the mathematical maximum number of songs that could be written?

Request for Question Clarification by pafalafa-ga on 01 Oct 2005 19:27 PDT
There's no real practical limit, but with some assumptions, I suppose
one could come up with a number.

However, are you asking about songs, meaning only a combination of words? 

Or do you mean songs, as a particular combination of words and melody?  

If the latter, I'm not quite sure how a guesstimate number would be
arrived at, but could certainly give it some thought.

Let me know what you think.


Clarification of Question by lairbear-ga on 02 Oct 2005 09:51 PDT
I mean just music/melody:  only western scales (no quarter tones)
Subject: Re: Mathematical Probability
Answered By: pafalafa-ga on 02 Oct 2005 10:26 PDT
Rated:4 out of 5 stars

This was quite the question!

Let's consider one-finger songs...melodies that can be played with
just one finger on a piano, one note at a time.

There are 88 keys to choose from.  Each key can be played in say, 10
ways (whole note, half note, stacatto, etc ... I told you, one needs
to make assumptions for these sorts of things).

So, there are then 88 x 10 = 880 ways to start a one-finger song on the piano.  

Similarly, there are 880 options for the second note, which means that
there are 880 x 880 = 774,400 combinations for just the first two

For the third note, another 880 options, and so on.

Let's say (another assumption) that a typical song is 100 notes long. 
Then there are 880 raised to the 100th power note combinations, which
is roughly equal to 10 ^ 294 (10 raised to the 294th power).

This represents the number of possible one-finger piano melodies with
a length of 100 notes.

This number is ridiculously large.  For comparison, the universe is
estimated to be about 10 ^ 17 seconds old.

Even writing millions of one-finger songs per second since the
beginnning of time, one would not yet even be close to exhausting the
number of possible songs that could be written.

Obviously, once we add notes in combination (chords, rhythym, back
melody, and so on), the numbers grow much, much larger.  So much so,
that there is no real end to the number of tunes than can be written.

Just as obviously, not all the combinations would be recognized as
melodic.  In fact, most of the combinations would not be very pleasing
to listen to, I'm sure.  However, what is or isn't a melody is a
matter of taste, which varies from person to person, and which goes
through considerable cultural change over time.

However, even if only one in a million note combinations is a viable
melody by some sort of definition, then there is still, for all
purposes, an endless number of tunes waiting to be written.

I trust this information fully answers your question.  

However, please don't rate this answer until you have everything you
need.  If you would like any additional information, just post a
Request for Clarification to let me know how I can assist you further,
and I'm at your service.

All the best,


search strategy -- Searched my own small supply (about 10 ^ 10) of brain cells.

Request for Answer Clarification by lairbear-ga on 02 Oct 2005 16:59 PDT
Based on the songs in the western hemisphere written to date, I would
think that 10 to 40 notes would pretty much define a unique melody,
and the melody would be limited to essentially one octave (12 notes).
To qualify my question to only include the number of realistically
written tunes, would you agree the maximum number of melodies possible
= 12*10^40 = about 10^83 (well under a googol)?  Also, I believe the
rhythm variations are included in the assumption of 10 values for each
note and that chord variations really would only change the
interpretation or variation of a unique melody.

Clarification of Answer by pafalafa-ga on 02 Oct 2005 17:44 PDT

Yes, I agree with your math, though I can't say I agree with all of
your assumptions.  But that probably doesn't matter much.  In this
sort of exercise, the important thing is articulating what assumptions
are being made.  Then, others are free to do exactly what you did --
rewrite the assumptions to suit their own sensibilities, and then
re-do the math.

A lot of music borrows riffs from existing tunes.  This has always
been true, but is especially so these days in a lot of hip-hop and
other popular music.

So...does the new song with borrowed riffs count as a new tune, or an
old tune?  To me, it's a new song, since it has a lot of individual
riffs mixed together in a new and unique way.  But if that's the case,
then there's a lot more than just 10 or 20 or 40 notes involved, so
I'd be tempted to stick with my 100-note number.

However, that's just an individual judgement.  As I said, I think the
numbers you came up with are readily defensible.

But in the final analysis, it's still an awfully big number anyway you
look at it.

There are plenty of tunes yet to be written, I'm sure.

All the best,

lairbear-ga rated this answer:4 out of 5 stars

Subject: Re: Mathematical Probability
From: myoarin-ga on 02 Oct 2005 05:29 PDT
Paf has raised the critical question of "assumptions", which is an
intriguing question of its own.

Define song:  just words, or words and music?
Just words:  just in English (or in any one other language), three
verses of four lines, no scat ("dubadubadu"), a certain minimum of
logical meaning, ... ???

Just music/melody:  only western scales (no quarter tones), ... ??

There are umpteen variation of some old folksongs using different
words and variations of the melody  - as well as entirely different

"Onward Christian Soldiers" can be sung to the melody of "Pistol
Packing Mama", and vice versa.  Are these two or four "songs"?  Of
course, there are very many other words to songs that can be sung to
another melody with the same meter.

Personally, I don't think that there is any limit to the number of
melodies or of texts, - even with the limitation to a single language
or a single system of scales, respectively.
If the field is opened to more languages and/or the whole range of
musical scales, I would be even more certain that there is no limit.
Subject: Re: Mathematical Probability
From: lairbear-ga on 02 Oct 2005 09:52 PDT
Yes, Just music/melody:  only western scales (no quarter tones)
Subject: Re: Mathematical Probability
From: myoarin-ga on 02 Oct 2005 22:07 PDT
To keep the song in the range of singable, really on a ab out two
octaves are possible, let's be generous, 30 notes, whereby it would
take some high-powwered musician to explain why not all the half tones
within that range could not be used.

Anyway, back the Bach,

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