Thanks for the maths lesson, however your math is inconclusive in that the possible combinations have nothing to do with creating melodies.
The weak link: you propose the combinations for sequences of only 5 or 7 notes without repetition and only give the top half of the formulae. Moreover, it also looks like you have copied some material from elsewhere and not carefully analysed the import of what it is you have copied.
The true measure of combinations/permutations in creating a melody comes back to the number of notes the melody occupies: it is a case of the
Unknown Formula:
N is the number of choices
S is the number of choices to be selected
Pentatonic possibilities
So, a 12 note pentatonic melody has the following number of combinations:
N = 5 (N is the number of available choices or notes)
S = 12 (S is the number of choices to be made)
substituting into the formula we get
| (12 + 5-1)!
12! x (5-1)! |
| = | 16!
12! x 4! |
| = | 16 x 15 x 14 x 13
4 x 3 x 2 x 1 |
| = | 4 x 5 x 7 x 13 |
| = | 1 820 possible 12 note pentatonic melodies |
Multiply by 24 to include all possible variants (12 starting notes and major/minor pentatonics), for a total of 43680 melodies in all keys
Major possibilities
A 12 note major melody has the following number of combinations:
N = 7 (N is the number of available choices or notes)
S = 12 (S is the number of choices to be made)
substituting into the formula we get
| (12 + 7-1)!
12! x (7-1)! |
| = | 18!
12! x 6! |
| = | 18 x 17 x 16 x 15 x 14 x 13
6 x 5 x 4 x 3 x 2 x 1 |
| = | 17 x 13 x 7 x 4 x 3 |
| = | 18 564 possible 12 note major melodies including all of the 1 820 possible 12 note pentatonic melodies |
Multiply this by 6 to get all modal melodies: 111 384. Multiply this by 3 (original result + harmonic and melodic minor and modes): 334 152.
Chromatic possibilities
A 12 note chromatic (
ALL 12 notes available to be used) melody has the following number of combinations:
N = 12 (N is the number of available choices or notes)
S = 12 (S is the number of choices to be made)
substituting into the formula we get
| (12 + 12-1)!
12! x (12-1)! |
| = | 23!
12! x 11! |
| = | 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13
11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 |
| = | 23 x 19 x 17 x 14 x 13 |
| = | 1 352 078 possible 12 note chromatic melodies including all of the above possible melodies including all 12 keys, major/minor and modal variants |
In other words there are 10.2 as many 7 note melodies as there are 5 note melodies and 72.83 as many 12 note melodies as 7 note melodies (742.9 as many 12 note melodies as 5 note melodies) that are each 12 melodic tones long.
A calculator for checking different length melodies and scale choices
How many different scales can we build depending on how many notes are in the scale?
Not so silly a question as:
C-C#-D-D#-E is a pentatonic scale - it fits the literal definition of penta-tonic - five-tones.
Given the ease with which we can be literal these days (computers make it so easy to generate each permutation), it is probably worth stopping for a moment and considering how many artificial/synthetic scales we can construct for each scale size from 2 to 11 (these scales would also include what we might consider the naturally occurring scales such as major/minor pentatonic, blues, major, all the minors and their modes). The chromatic scale (all 12 notes) whilst it can have a different starting note is the same no matter which note we start on each note is a semitone above or below its predecessor.
| Notes in scale | Permutations/potential synthetic scales |
| 2 | 132 |
| 3 | 1 320 |
| 4 | 11 880 |
| 5 | 95 040 |
| 6 | 665 280 |
| 7 | 3 991 680 |
| 8 | 19 958 400 |
| 9 | 79 833 600 |
| 10 | 239 500 800 |
| 11 | 479 001 600 |
The point of all of this
So to the point at hand; yes the math does trump the logic of the street. How much of the above is actually applicable in a creative manner is a matter of individual opinion and not one of fact - there are 7 melodies that will consist of only the same note repeated (each of the seven available notes repeated 12 times), there are others that will appear to be unusable in a pop context simply because of the way the individual notes are approached and quit.
That you can probably create one of the many melodies that already exist is not in dispute, that it can be exactly the same melody is - the above math does not take into account the need to have a minimum duration for each note - the set of available durations, including silences (rests) is not significantly large, but the likelihood of one type of duration appearing over another means that math becomes more complicated - you are not going to have a melody jump from a 16th to a whole note to an 8th triplet to a half note rest - it will not work to keep the people tapping out the 4 on the floor that they be used to.....
add in dynamics (louds, softs and movements from one to the other), articulations (how the note is played held, short, accented, etc) and the math gets hellishly crazy.
Unless you are into math rock; steer clear of this puppy......
