How many sounds does the piano make?

Question

Mathematics

I am learning to play the piano.
If there are approximately 10 ^ 118 unique combinations in chess, with a layout of 8 * 9 = 64, then here, on the piano, there are 88 keys (52 white and 36 black, do not count: sound length, rhythm, pauses, but we take 10 fingers when calculating hands, and the range they can reach).
How many possible combinations with 88 keys?
How many of them are humane? (according to the calculation, 88 keys simultaneously pressed sound worse than a correctly set chord, it is possible to calculate this indicator using all possible combinations that are used on stage).

Answer 1

as far as I know, musicians need to train to be able to press a note in one octave with one hand and the same note in the next octave. that is, 1 and 13. we assume that he can press any keys between them (in life, not any will turn out, at least not for everyone).
thus, how many combinations can be typed with one hand, starting from any conditional first note (this note will not necessarily be pressed):
2 ^ 13 = 8192
, that is, these are combinations of everything. 13 notes where each can be pressed.
but with one hand we can only type 5 keys from them.
let's write a simple script that counts the number of combinations with a maximum of 5 clicks by enumerating.

<?php
$cnt = 0;
for ($i = 1; $i <= 8192; $i++) {
    $bitCount = substr_count(decbin($i), '1');
    if ($bitCount <= 5) {
        $cnt++;
    }
}
echo $cnt; // 2380

but in fairness it must be said that some neighboring notes (clean notes, skipping sharps) can be pressed 2 keys with one finger. which slightly expands this range. we will not take this effect into account.
Let's call this the maximum possible number of chords played with one hand.
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playing in this way (grabbing an octave with one hand) you can start from the first note up to 76. That is, starting from each note, you can play all the chords.
with this calculation there will be intersections, that is, one combination of two starting positions with the same keys.
in order to exclude them, we will take into account only achords that begin with 1 conditional selected key.
such combinations will be 2 times less. 8192 / 2 = 4096 maximum number of unique chords with one hand.
total chords with one hand on the entire canvas of keys:
76 * 2380 = 180880
with two hands you can type any chord with one hand anywhere and any chord with the other hand.
180880 * 180880 = 3.2e10
but there are also intersections when two hands are in the same or close positions.
let's try to count the number of intersections:
intersections will be when the initial keys of both hands match or are within the octave.
------------------
hereinafter "pleasant consonances". this thing is very dependent on the culture, in any case it will be established chords. Let's take these chords as an example
https://pianocovers.ru/chords
there are a lot of them, 33 chords are given for each key. (I calculated that there are only 27 combinations with 5 keys pressed, but all these achords can be typed with one hand, as I wrote by pressing adjacent keys with one finger)
33 * 12 = 396 total chords in one octave.
7 octaves
396 * 7 \u003d 2772
are combinations of "human-friendly" ones with one hand on the entire canvas.
with two hands it turns out:
2772 * 2772 = 7 683 984
but here again, intersections are obtained when two hands are nearby.
secondly, not every chord with another will be "pleasant". probably more pleasant will be chords of one note and chords containing the same notes in different octaves, but here everything is really very subjective, and depends on culture, customs, etc.
Of course, the calculation turns out to be approximate, but music is a very subtle and informal thing. I don’t know why you need this, but I hope my calculations will be useful.
Happy New Year ;)

Answer 2

How many of them are humane?
The answer also depends on the BACKGROUND.
Those. the same chord in a jazz composition will sound good and out of place, but in the classics it will sound terrible. Some kind of C7b5 is typical for bossa nova, and you won't find it in academic music.