How to understand the keyboard of a piano (or any such instrument)?

Question

I would like to ask about something like theoretical foundations or fundamentals of an 88 piano keyboard. My thinking is

a) there is a range of pitch a human can hear (lets assume that there is no interpersonal differences)

b) there is a number of tones (not sure if this is a right name) a human can differentiate (out of this range in #a); lets just assume this. My question is - what is or could be the number of these "tones" and what is the best of objectively measuring them (is it in Hertz)?

c) as far as my knowledge goes (but it is not complete yet, as I have asked the above questions) this is laid out on a piano keyboard (and lets assume that we are talking only about the main 7 octaves (like there is these two white keys and the beginning and one white at the end, lets exclude these; unless you could explain why they are there). Now every octave is 12 keys - wbwbwwbwbwbw (as colors) - Do we have exactly the same distance between these as far as the number of units that we would use to measure these sounds or tones (as in the question #b)? I am talking about laying it out one after another - lowest tone to the highers, so the black ones would also be in the mix (like the first black one after the first one). Basically kind of going in the direction how this range of pitch "a human can hear and differentitate" translates onto this keyboard and why.

d) Is the above, how this translates onto a keyboard arbitrary or not? What are the reasons for doing it this particular way (I know that this could be a lenghty disuccion, but maybe... :) )?

e) Looking at these 88 keys they are different but have something in common (in this "music theory system" that I dont yet understand, I will add) as we have (as an example) C1, C, c, c1, c2, c3, c4 - and these applies to all the other ones white, we can make sets of this. So my question is - what is this "occurence", how this can be explained? Like a range of pitch a human can hear and differentiate and then there is tones or pitches in the mix that are different (in these units that we would measure) but also have something in common too? What is the nature of this relation, how this can be explained?

f) What is the difference between white and black keys? Are the black keys less important (it has always seemed to me this way; if we could pick just one type it would definitely be the white keys)?

g) So we have octaves. Could we have a system where this would be 14 sounds or 16 sounds as a mix (and also with these repetitive tone names, like the C1, C and so on, example above)? (Is this something like, yes, there could be but the sounds would be too similar to each other so more than 12 is not a needed thing - like getting into this aspect of differentiation - like 12 is a good pick, more would be too close to each other in how they sound to a human and it would sound [closer to] the same [than not, lets say, to a human]).

Thanks a lot for the reading.

Answer 1

b.

My question is - what is or could be the number of these "tones" and what is the best of objectively measuring them (is it in Hertz)?

This is what psychoacoustics studies. Per Wikipeida:

Frequency resolution of the ear is 3.6 Hz within the octave of 1000 – 2000 Hz. That is, changes in pitch larger than 3.6 Hz can be perceived in a clinical setting.[6] However, even smaller pitch differences can be perceived through other means.

c.

Do we have exactly the same distance between these as far as the number of units that we would use to measure these sounds or tones (as in the question #b)?

There are many different ways to tune a piano, and I do not claim to understand it. In general, no it is not that simple. This answer discusses some of the complexities involved.

d.

Is the above, how this translates onto a keyboard arbitrary or not?

Read up on temperament

e.

So my question is - what is this "occurence", how this can be explained?

If pitch X is an octave higher than pitch Y, then the frequency of pitch X is double that of pitch Y. Consonant intervals tend to have similarly "nice" ratios.

f.

What is the difference between white and black keys?

I don't know how to answer this simply. Perhaps someone else can. The simple answer is that there is no significant difference. Studying music theory can help clear this up.

g.

So we have octaves. Could we have a system where this would be 14 sounds or 16 sounds as a mix (and also with these repetitive tone names, like the C1, C and so on, example above)

Yes. Keep in mind that when we are discussing the 12-tone system, or the concept of keys, we are specifically talking about Western music. Indian and middle-eastern music uses quarter tones (i.e. notes that fall "in between" the keys on a piano.) Much Chinese music is based on a pentatonic scale (though they also coincidentally had a series of 12 notes similar to ours)

Closer to home, I have heard recordings of either Baroque or Rennaisance music for a quarter tone piano. I forgot where I heard it and am therefore unable to provide any references.

Answer 2

There are too many sub-questions here, and a lot of them already have answers. I suggest you look across to screen right and plough through them - it will take some time!

To attempt to clarify some of your queries.

The pattern of black and white keys is for convenience as much as anything. With this patter, which repeats every 12 notes, it keeps things simple: the white keys between the two blacks are always D - just octave copies of each other. The 88 note is convenient too. Most pianos are not well enough made to cope with making higher or lower notes than that range sound that good. Human hearing goes way above and below that, but practicality has to come into play, so to speak.

In 12tet, which is what the majority of instruments tune to now, each octave (C>C, G>G etc is split up into 12 equal parts. It's a compromise, but works quite well, particularly allowing pieces to sound 'in tune' in any key - previous tunings made one key sound perfect, the other 11 not so.

Each octave note has a pitch 2x the Hz of its lower same name note. Thus A4 is (generally) 440Hz. The octave A lower is 220Hxz, while the octave above is 880Hz. Think about this: the higher we go, the bigger the band of frequency, and vice versa. So those 12 notes in one octave need to be spread differently for each octave. There's also the oddity that pianos need their tuning 'spread' a little - it's a vagary of the instrument. That's enough to be going on with, it'll soon be Christmas...