F rom the as sumpt i ons and the tab l e
above, we obtain 25 notes of two-note tunes
(eliminating C’ to C’ because it is the same as
C to C).
Listing out all the combinations is a tedious
and t ime- consumi ng task . Therefore, we
employ the use of permutations or an ‘ordered
arrangement’. If we only had 2 notes, C & D
to choose from and are limited to a two-note
melody, then we would obtain 4 combinations:
CC, CD, DD and DC. If we had 2 notes to
choose from and are limited to a three-note
melody, we would obtain 8 combinations:
CCC, CCD, CDC, CDD, DCC, DCD, DDC and
DDD. The formulae for these two scenarios
would be 2
2
possibilities and 2
3
possibilities
(number of starting notes
number of notes in the melody
). In
our more complicated scenario, the first note
can be any of the 13 notes; the second note
is similar and so on. If n is the number of notes
in our melody, then the possible combinations
of notes is given by 13 x 13 x 13 x .. x 13 = 13
n
. To
find the duplicates, we apply the same thought
process. For a duplicate sequence that does
not contain a C, there are 12 choices for the
starting note, 12 choices for the second note
etc. We obtain 12 x 12 x .. x 12 = 12
n
. Thus, the
combinations we can obtain from a melody
with n notes without duplicates is 13
n
-12
n
.
We can eas i l y gene ra l i se the above
discussion to find the number of melodies in
any scale. Let “s” be the number of possible
chromatic notes we can use. We obtain s
n
–
(s-1)
n
= s
n
[ 1-(1-1/s)
n
].
A s o n g co n s i s t i n g o f s i x n o t e s h a s
approximately 1.84 x 10
6
combinations and
a song cons i s t i ng of ten notes prov i des
a whooping 7.5 x 10
10
possibi l ities! We’ve
eliminated rhythms, but to account for that,
a rough approximation br ings us to 8.25
19
possibi l ities, or 2.6 t r i l l ion year s’ wor th of
material. Not to mention that this is without
harmon i sat ion, tempo and al l the other
variations possible in music.
Although popular songs tend to gravitate
toward certain patterns of melodies, it is mostly
due to what we find to be pleasing to listen to.
In conclusion, it is safe to say that we will not be
running out of music any time soon.
Going up in pitch 音高向上
Going down in pitch 音高向下
First note
旋律中第一個
音符
Second note
旋律中第二個
音符
Pitch difference
(semitones)
半音程
First note
旋律中第一個
音符
Second note
旋律中第二個
音符
Pitch difference
(semitones)
半音程
C
C
0
C
’
C
’
0
C
C#
1
C
’
B
-1
C
D
2
C
’
A#
-2
C
D#
3
C
’
A
-3
C
E
4
C
’
G#
-4
C
F
5
C
’
G
-5
C
F#
6
C
’
F#
-6
C
G
7
C
’
F
-7
C
G#
8
C
’
E
-8
C
A
9
C
’
D#
-9
C
A#
10
C
’
D
-10
C
B
11
C
’
C#
-11
C
C’
12
C
’
C
-12
