Physical measurements
PRINCIPLE OF A SWINGING STRING
A pitch can be physically described by the frequency of the sounding tone, i.e. by the number of oscillations per given unit of time. In acoustics, Marin Mersenne investigated the connection between frequency and pitch. He found out that the frequency of a vibrating string is directly proportional to the square root of the tension force F and inversely proportional to the string length l and the square root of the cross section q.”
In the following example, all other parameters remain unchanged: If the side length is halved, or the tension force is quadrupled, or the mass per meter is reduced 4 times, the string sounds an octave higher, so the vibrations per given unit of time double.
From this knowledge, Marin Mersenne came up with a formula:
Transformed according to the clamping force F, this formula results:
The mass per unit is calculated using the density ρ and the diameter d:
According to Joseph Fourier, each periodic oscillation can be broken down into individual sine and cosine oscillations and written in mathematical series. It is assumed that the oscillations of the overtones are harmonic or a multiple of the fundamental frequency.
But with metal strings, such as a piano or a Queen Mary harp, the overtones of the vibrations are not harmonic to the fundamental frequency, but slightly out of tune. This is shown in the figure below.
The higher the current nth overtone, the greater the deviation of the frequency in relation to the multiple frequency of f0:
TUNING
The Queen Mary harp is tuned diatonic to F major, C major, B flat major, or G major. Depends on the piece being played. William Macdonald, a harp maker from Isle of Sky, uses 475Hz for the tuning and 80Hz for the lowest string. However, he often builds harps tuned to 440 Hz for his customers. There are various claims as to which tunings were used in the 15th century, but we limit ourselves to equal tunings with the tuning pitch of 440 Hz and 475 Hz.
[8] Diatonic scales are seven-step scales that divide the octave space into five whole steps and two semitones. For example, C major corresponds to the white keys of a piano. In order to be able to play other keys, the corresponding strings must be retuned. In the key of G major, the F strings would have to be retuned to F sharp. Compared to a harp, a piano has 12 tones per octave and all keys can be played without retuning. IV
EQUAL Tuning
[9] "Equal temperament, also called equal temperament, was calculated by Marin Mersenne in 1636 and became more and more popular in the 19th century."
With this tuning, the impurities are evenly distributed over all 12 semitones and are therefore no longer perceived as disturbing. However, this also means that every key sounds the same and there are no longer any different timbres. [10] The following formula is used to calculate the frequency. With this formula it is possible to assign a number to the chromatically ascending tones.