HOW MANY SONGS?
                           
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Page Modified 30 Jan 2016
 
From time to time, we all hear two popular songs which sound similar to each other.   And we wonder, have almost all possible songs been written?

So let's try to calculate the number of possible songs.   Start by assuming they are a maximum of forty notes long.   If you look at songbooks (or just try singing or humming tunes), you will see that most songs are about this long or longer.   So the number we calculate will be a lower limit.

Further assume that the first note is always C.   This is equivalent to making all songs so that they are in a particular key.   (It doesn't have to be the key of C, it's any key that forces the first note to be a C.)   And assume that the notes can be quarter notes, half notes, or full notes.   So we're ignoring eighth notes, three-quarter notes, three-eighth notes, sixteenth notes, etc.   Also let's restrict ourselves to one octave.   So the notes we are going to examine are C, D, E, F, G, A, and B, in one octave.   And let's not consider flats or sharps.

Assume that we play only one note at a time – no chords, no double notes.   Assume further that the second tone has to be different from the first tone, that the third has to be different from the first and second, that the fourth has to be different from the first three, and the fifth has to be different from the first four.   This restricts the number still further.   Let's assume that the sixth tone in the song can be any tone, but that tones six through ten have to all be different.   And let's also assume that tones 11 through 15 have to all be different, tones 16 through 20 have to all be different, tones 21 through 25 have to all be different, and so on.

Here are our assumptions so far:
 
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ASSUMPTIONS
 
  Tones per song:   40 Tones used:   C, D, E, F, G, A, B  
  Notes used:   Quarter note, half note, full note  
  First tone:   C Second tone:   D, E, F, G, A, or B (six possible)  
  Third tone:   Five possible Fourth tone:   Four possible  
  Fifth tone:   Three possible First five notes: 3 x 18 x 15 x 12 x 9 possible  
  Where we have used the fact that the first note must be a C, but can be a quarter note, half note or full note.   The second note has six possible tones but can also be a quarter note, half note or full note which gives 6 x 3 = 18 possibilities.   The third note can be one of five possible tones but can also be a quarter note, half note or full note which gives 5 x 3 = 15 possibilities.   The fourth note thus has 12 possibilities and the fifth note 9.  
  Tone 6: C, D, E, F, G, A, or B Tone 7: 6 possible Tone 8: 5 possible
  Tone 9: 4 possible Tone 10: 3 possible
  Notes 6 through 10: 21 x 18 x 15 x 12 x 9 possible

Notes 11-15, notes 16-20, notes 21-25, notes 26-30, notes 31-35, and notes 36-40 have the same number of possibilities as notes 6-10.
 
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NUMBER OF POSSIBLE SONGS
 
Now we are ready to calculate the number of possible songs.   There are 3 x 18 x 15 x 12 x 9 = 87,480 possibilities for the first five notes.   For the next five there are 21 x 18 x 15 x 12 x 9 = 612,360.   And there are 612,360 possibilities for each of Notes 11-15, notes 16-20, notes 21-25, notes 26-30, notes 31-35, and notes 36-40.

 
  TABLE 1.   POSSIBLE COMBINATIONS FOR EACH SET OF FIVE NOTES  
  SET
1-5
6-10
11-15
16-20
21-25
26-30
31-35
36-40
   
  NO.
87,480
612,360
612,360
612,360
612,360
612,360
612,360
612,360
   
     
 
So the total number is (I have used Scientific Notation):

87,480   x   [612,360]7   =   8.8 x 104   x   [6 x 105]7   =   8.8 x 1039 x 67

Now 67   =   279,936   =   2.8 x 105       so that the total number of songs is

8.8 x 2.8 x 1044   =   2.5 x 1045

How many songs is this?   Let us assume there are ten billion people on earth (there are actually about 7 billion), and that each person writes one song per second.   How they could write one down this fast, even if they created it this quickly, I don't know.   But bear with me.   There are about 30 million seconds in a year, so ten billion people writing a song per second would write, in one year, this many songs:

1010   x   3 x 107   =   3 x 1017

To write   3 x 1045   songs would require almost   1028   years.   The age of the universe is about 15 billion years, or   1.5 x 1010   years.
 
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A SLIGHTLY DIFFERENT APPROACH
 
Suppose I say this is too many.   Suppose I have two songs with the tones C - A - G - D, but one has two quarter notes followed by two half notes, and the other has four half notes.   According to my assumptions, these songs are different.   But suppose I don't allow this – I only consider songs to be different if the tones are different.

Then for notes one through five I have 1 x 6 x 5 x 4 x 3 possibilities, or 360 possibilities; for notes 6 through 10 I have 7 x 6 x 5 x 4 x 3 = 2520 possibilities.   Notes 11-15, notes 16-20, notes 21-25, notes 26-30, notes 31-35, and notes 36-40 have the same number of possibilities as notes 6-10.   So the total number of songs is:

360 x (2520)7   =   3.6 x 102 x 2.57 x (103)7   =   3.6 x 610 x 1023   =   2 x 1025

Writing this many songs, at 3 x 1017 songs each year, would still take almost 108 years, or 100 million years.
 
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ALLOW LONGER SONGS
 
Let's repeat this analysis, but assuming songs can have 50 notes.   Then in Table 1, there will be two more columns, for notes 41-45 and notes 46-50.   Each column will increase the number of songs by 612,360 (roughly 6 x 105).   So the number of songs will be, using the first method, approximately 3.6 x 1010 x 2.5 x 1045   =   9 x 1055.

If we calculate the number using the second method, the number will be 2520 x 2520 x 2 x 1025  , which is approximately 1.2 x 1032.

What if songs can have as many as 100 notes?   Then using the second method, the number will be 360 x (2520)19   =   3.6 x 102 x 2.519 x (103)19   =   3.6 x 102 x 3.6 x 107 x 1057, which exceeds 1067.

I leave it as an exercise for the reader to calculate the number of songs with 100 notes using the first method.   Also, you can see what effect the assumption the first five tones have to be all different has.   You can try changing it to four, or six, or three.

 
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