Tell me about the mathematical ratios of different notes on the piano.?

Question

For instance, C-F, C-A- C-B, etc.

Answer 1

http://www.phy.mtu.edu/~suits/notefreqs.html

C0 16.35
F0 21.83
ratio = 16.35 / 21.83 = 0.748969308 or about 3/4
You can do the rest of the math.

Answer 2

Here is the easy version. I don't know how much math you know but if you know roots and and geometric series then what I am about to tell you will make sense (hopefully).

The frequencies of the notes on the piano are in a geometric sequence. The reference key is the middle A which is 440Hz. Every key after that (going up that is) is multiplied by the 12th root of 2 which is about 1.0594630943592952645618252949463...

When I talk about the next key, I mean the next half-step which may be a black or a white key. So mathematically, going from A to A# is the same as going from A# to B or going from B to C. The ratio of any key on the piano to the next is key is always the 12th root of two.

So the middle A=440Hz
A# = 466.16 Hz
B = 493.88 Hz
C = 523.25 Hz
C# = 554.37 Hz
D = 587.33 Hz
D# = 622.25 Hz
E = 659.25 Hz
...
G# = 830.61 Hz
A = 880 Hz
and so...

So when you are going to the left, you simply divide by the 12th root of two. Now notice that since an octave actually has 12 keys in it (7 white and 5 black) after every 12 keys, you have used up all 12 of the roots so the frequence simply doubles. Middle A is 440Hz and the next A is 880 Hz and the previous A was 220Hz.

Another thing to notice that this geometric sequence is never zero (it gets closer and closer to zero but never reaches it). But the reason that we stop with a certain number of keys on the piano because the human hearing range is only between 20Hz and 20,000 Hz so when the frequencies of the keys gets above 20,000Hz or below 20Hz, those notes are obviously not included in any music instrument, not to mention how BAD it would sound.

FYI, this holds only for pianos in America. I think that different instruments are tuned differently AND Europeans tune differently than Americans. I think that the base frequency for pianos in Europe is slightly different.

Answer 3

IN the 'true tempered' scale, each of the 12 notes (half steps) in an octave are broken into equally weighted ('tempered') steps and are rational numbers (e.g. The root triad, consisting of the 1'st, 3'rd, and 5'th tone of the scale would be the root frequency, 16/12 times the root frequency, and 19/12 times the root frequency) The *big* problem with this sort of scale is that it's impossible to change keys in the middle of a composition (which was why most most 'ancient' music was written in what are, today, called 'modal' scales). But in the early 18'th century a pretty fair amateur mathematician named J.S. Bach developed what is known as the 'well tempered' scale in which each tone is the 12'th root of two greater than the tone before it. This makes key signature changes possible since the interval between, say, the root and the dominant will be 5 times the 12'th root of 2 no matter what the frequency is of the dominant tone.


Doug

Answer 4

An octave is a halving or doubling of frequency, and there are 12 notes (including half steps) per octave. Therefore the distance between successive notes is the 12th root of two, or 1.05946.

Middle A is 440 Hz. The next note is 440 (1.05946) = 466.16 Hz, the next is (466.16)(1.05946) = 493.9 Hz.

Divide by 1.059 for each step below A.

This is for even tempered, American scale.