I question whether you mean songs in general or how many arrangements of one song are possible? In music, arrangement tends to stand for a composer taking another composer’s work and rewriting it - usually for a simpler ensemble or a different style or genre. The answer to both is infinite.
There are 12 notes in the chromatic scale (C-C#-D-D#-E-F-F#-G-G#-A-A#-B) and 10 possible numerical digits (0–1–2–3–4–5–6–7–8–9.) I’m going to assign the first 10 notes to the first 10 digits (i.e. C=0, C#=1, D-2, etc.) Now give me any 3 numbers and I’ll write a short melody with it. 016 is C-C#-F. 784 is G-G#-E. You get the idea. With these constraints, there are 10*10*10=10^3=1000 different melodies. What if the melody had 7 notes. 10^7=10 million different melodies. There’s no real limit on how long a melody can be but they tend to be 4 bars in length, usually in 4/4. That’s 16 different notes if there’s one on every beat. 10^16 is a large number. But no melody is that boring, and multiple notes can fit into one beat. Let’s say there are 49 different notes now, because there’s an interesting melody. 10^49, which is huge. Theoretically you can divide one beat infinitely but realistically no one could ever sing that. Let’s say that 16th notes (4 per beat) are the fast we can sing, so let’s try all 16th notes. 4 measures*4 beats*4 notes per beat = 64 notes. 10^64 different melodies is realistically the cap.
But wait!
We’re still missing 2 notes (11 and 12!) so it’s really 12^64. But wait! If you’ve ever looked at a conventional keyboard, there are definitely more than 12 notes. There are 88! However since we’re going for realism, realistically one person could only sing about 22 different notes (an octave and a 6th.) But there are 4 different voice types so now we have (22^64)*4. Ok but all of the melodies before were still different therefore need to be accounted for. Therefore the number of melodies that can be reasonably sung in a 4 bar phrase is:
[(22^64)+(22^63)+(22^62)+…+(22^2)+22]*4
= 344,599,743,649,857,348,667,895,143,440,526,146,264,357,563,546,771,509,781,408,176,172,900,914,212,106,966,975,240
Which is rather large but not infinite…
But consider for a second all that goes into a song. We have dynamics, we have changes in dynamics, we have different tempos, changes in tempo, accents, ornaments, articulations, slurs, etc. that’s all in the written music. But then we have different timbres of singing: do you want the high stuff in chest or head voice? But all of those things could potentially be exactly quantifiable and added to the equation. But there are still some more things that can’t be such as:
Pitch. I said there were 22 but only if you can sing those pitches exactly. Odds are you won’t be in tune and will sing the A# a little flat. But how much? Pitch is measured in Hz so you could be flat by 10 Hz, or 1 Hz, or .1 Hz, or .01 Hz, etc. Just the discrepancy of an A# sends us to infinity already, but I guess that’s not entirely realistic to have that good of an ear… (We also use the term cents to describe tuning as it lets us take logarithmic tuning such as Hz and make it linear, with 100 cents in between every half-step)
The number of notes. Yeah I said 64 notes total but some melodies are longer.
WORDS! If we’re talking songs, then there has to be words. There’s 15,831 different syllables in the English language so theoretically every note could be one of those, but generally they need to string together to form coherent words and phrases so the actual number might be lower.
But if we expand our definition of melody to not be so restricted, we get even more possibilities! So just the first note of a song has:
88 notes*198 cents pitch differentiation*6 dynamic levels*240 different tempos*2 accented/not accented*8 ornaments give or take*5 articulations*2 slurred/unslurred*15,831 different starting syllables*4 different starting lengths of the note = 254,213,539,430,400 different possibilities for just the first note of a melody alone. If a melody is just 4 notes long, then that’s (254,213,539,430,400)^4. Now every time we add a note to our melody, the number of possibilities skyrockets.
This was a lot of fun but in the end, regardless of anything else, you can always have a longer or more complicated melody that can add another note, therefore making an infinite amount of possibilities.
It would be fun to calculate the number of tonal melodies possible in a 4 bar space but perhaps for a different time.
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