Frequency of the string and length of the string: opposite ratios

   When we use the metronome or tune the piano we use describe the notes in terms of frequencies.  The concert A that most countries refer to has been set at 440Hz. However, in many countries such as Japan and Germany, the concert A is now 442Hz-445Hz. This is to raise the intensity and excite the emotions. These days the A is so sharp that on recordings it is sometimes difficult to tell whether it is tuned to A or Bb. In my opinion this is commercial method rather than anything else and in the long run does not make a big difference to the music itself. I personally tune my violin to 440Hz.    Therefore even if the piano in the concert hall is tuned to 442Hz, it makes little difference to the way I perceive intonation. On the other hand, if students tune the violin to 445Hz to practice, he/she may run into many problems trying to adjust to a lower 442Hz tuning. Naturally the ear can easily react to tuning up to a set intonation but has physical reaction to tuning down.
   A note is created by a shock or wave from a material travelling in the air. 440Hz denotes the number (440) of waves in one second. Therefore if this number was multiplied a similar shaped wave as A is formed, such as 440 x 2 = 880 which would result in an A an octave above, and 440 x 1/2 = 220 would result in an A an octave below. In such manner the Pythagorean theory became today's seven note octave. His experimental instrument was of a very simple kind which was used to study the properties of the string. The stringed instruments of today have the same principles. If one plays a natural harmonic above the open string, the note produced would be an octave above and this is placed half way on the string. If a note two octaves above the open string was desired, one has to place the finger so that the string is a quarter of its length. Therefore all the harmonics on the string is a multiple of the Hz of the open string.
   The wave lengths measured in Hz identifies the name of the note but it also identifies the exact intonation of the note. The violin is a difficult instrument to tune, perhaps even more than other stringed instruments. This may be because the normal limitation of the human ear is only 200Hz-4000Hz. But a well trained ear is very sensitive and can identify the slightest change in the note's wave length. The smallest movements on the fingerboard are made by the tip of the fingers.
   Although the strings themselves speak out the specific notes, the resonance is enlarged by the body of the violin.