Can you explain to me the relationship between note frequency and wire (string) tuning? I see, for tuning violin, 440 Hz is used and called an A4, while a 432 Hz can also be used and that too is called an A4. But, don't each note in an octave have a particular fixed frequency? What I am implying is, for example, if frequency of C in an octave (taking fourth octave) is X Hz, be it if I tune the wire (string) to 432 Hz or 440 Hz, shouldn't the C's frequency still be fixed at X Hz?
The frequency of C will always be relative to your chosen A. Depending on your tuning system, it will be somewhere around 1.2 times the frequency, a ratio of 6:5. A justly-tuned minor third has exactly that ratio, while the ratio of an equal-tempered minor third is the fourth root of two, which is about 1.1892. A Pythagorean minor third is 32:27, or about 1.1852.
We can easily calculate the frequency of these three definitions of C for any frequency of A:
╔════════════════╤═══════════════╤══════════════════╤════════╗ ║ Frequency of A │ Pythagorean C │ Equal-tempered C │ Just C ║ ║ (1:1) │ (32:27) │ (2^(4/12)) │ (6:5) ║ ╟────────────────┼───────────────┼──────────────────┼────────╢ ║ 415.00 │ 491.85 │ 493.52 │ 498.00 ║ ╟────────────────┼───────────────┼──────────────────┼────────╢ ║ 432.00 │ 512.00 │ 513.74 │ 518.40 ║ ╟────────────────┼───────────────┼──────────────────┼────────╢ ║ 440.00 │ 521.48 │ 523.25 │ 528.00 ║ ╟────────────────┼───────────────┼──────────────────┼────────╢ ║ 443.00 │ 525.04 │ 526.82 │ 531.60 ║ ╚════════════════╧═══════════════╧══════════════════╧════════╝
The violin has infinitely variable pitch. The point at which you stop the string for C will vary depending on which of these Cs you want. This in turn will depend on the harmonic and melodic context. But because the ratios are the same, the point at which you stop the string will be the same regardless of the frequency of the open string.
That is, wherever you put your finger to play the 528 Hz note when the string is tuned to 440 Hz, that will be the same place where you'll put it to play the 518 Hz note when the string is tuned to 432 Hz.
A = 440Hz has become close to a standard tuning pitch throughout several parts of the world. However, A, historically, has been several other pitches in different parts of the world, and is still not exactly 440Hz when one considers various orchestras' tuning in various locations.
However - as long as everyone who is playing together in a particular ensemble all uses the same datum point of A = xxxHz, then it will sound fine. Once that has been established, all other notes, C3, F4, G5 etc., will be automatically in tune with a properly tuned instrument.
I could, for example, tune my guitar down (or up) a bit or a lot, and as long as each string is in tune with the others, playing it in isolation , it'll sound in tune with itself. Should someone else want to play along as well, they would have to use the tuning that my guitar was in, otherwise it's going to sound awful ! This was one of the reasons A = 440Hz was chosen. It could have been 437, 448, and as long as everyone adheres, there's no problem
A4 = 440 is simply a convention. It was selected 1921 as a standard. Some follow it, some not. The orchestras around here, Stockholm Sweden europe seem to preferr 442 as of now.
The frequencies of other tones are relative to each other. The octave above, A5 is double frequency. The octave below is half frequency.
Intervals between the other tones depends on which system you use. Most common today is the Equal temperament. Here the frequency ratio between two tones is the 12-th root of 2.
This answer is only scratching the surface of a very complex subjekt.