
Ask HN: How many *playable* combinations of piano keys are there? - jacquesm
Given a span of 15 notes for one hand; and two hands I make it out to be less than 49 billion but the exact number eludes me (fingering constraints, duplicate notes between the two hands when they overlap).<p>The 49 billion is 3003 (15 choose 5) * 73 (the number of positions you could do that in) squared.<p>Anybody good at math with a more accurate number?
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detaro
Some overlap gets reduced by always assuming that the "lowest" finger is
choosen to a specific key and seeing how many combinations can be reached from
there upwards.

Also allowing for not all fingers of a hand being used. Since one finger is
set, we only look what the remaining 4 can reach:

14 choose 4 + 14 choose 3 + 14 choose 2 + 14 + 1 = 1471

this works for 74 positions for the lowest finger. For the top 14 positions we
have to allow all fingers of the hand to be inside the 14, so here we have to
add 14 choose 5 = 2002

so 1471 * 74 + (1471 + 2002) = 112327 should be accurate for one hand (or + 1
if we consider the "no fingers of this hand used at all"). That squared is
12.6 billion, but of course there's overlap in there too between the hands.
Don't have a clever idea for that one right now that's not just enumerating
scenarios...

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jacquesm
Ah clever, totally missed that not all have to be used, thank you for this! I
think you meant 12 billion?

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detaro
yes, edited.

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gshdg
Might also note that when the hand is stretched to its full range you can’t
necessarily hit all possible combinations of notes within the range. (That is,
the index finger can’t reach everything between the middle of the range and
the thumb’s full extent.)

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jacquesm
Yes, this is the complex bit that I can't get a handle (pun intended) on, the
degree to which the geometry of the hand constrains the problem.

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gshdg
It’s also dependent on the hand. I can just barely reach a full octave. I know
others who can easily do an octave plus a fifth.

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SamReidHughes
You mean as a chord? It depends on the particular pianist's finger geometry,
but also, the span of notes depends on the bottom note -- sometimes you can't
reach that black or white key, and note that the black keys aren't exactly in
the middle of the white keys, so the distance changes. The span from B to D
flat is easier than that from C to E flat.

Also, one finger can press multiple keys. For example, this is necessary to
play three FGDF/FGDG chords in Chopin Op. 10 No. 12.

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jacquesm
I thought of that one but could not imagine any composer actually used it.
Ugh, there goes my lowest bound.

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ta0987
Chopin sometimes requires the thumb to play two notes, e.g. Prelude in C
minor, bars 2, 6 & 10.

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NewDayRisen
It’s close enough.

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NewDayRisen
49.1 billion.

