




There
are a lot of factors that affect the frequency, and thus the pitch, of
a string's vibration. As shown in the graph below, frequency has a
direct relationship with pitch, but it is not linear; as the pitch gets
higher, higher frequency increases are needed. Thus we can talk about how to vary frequency instead of varying pitch. In terms of frequency, the general combined equation for frequency of vibration on a string is: Using this equation, and substituting the letter "p" for "rho," we can analyze the equation. N refers to the mode we are looking for; we are focusing on the fundamental for now, so N=1. T is the tension in the string, determined by how tight the tuning pegs are wound. P is the mass density of the string, which for the same material is effectively proportional to the thickness of the string. L is the length of the vibrating section of the string. Thus, we know that as tension increases, frequency increases because tension is on top; when frequency increases, pitch increases, so winding up the tuning peg results in a higher pitched sound, which we know from experience. The thickness of the string, which is proportional to the mass density, has an inverse relationship with frequency. As P increases, which indicates that the string is getting thicker, the whole T/P unit decreases because P is on the bottom. Thus, F decreases, causing pitch to also decrease. Thus a thicker string causes a lower pitch; this is also intuitively correct, because the D string is always thicker than the A string. Lastly, the effective length of the string has an inverse relation to frequency. If L decreases, then F increases, because L is on the bottom of the equation, so a shorter length results in a higher pitch. This is the property used when playing a stringed instruments, as fingering the string causes different lengths of the string to vibrate, causing varying pitches. Bowing causes standing waves to propagate along the string. Waves propagate along different materials at different speeds; for example, the speed of sound waves in air is around 343m/s at STP. Sound travels faster through denser materials, like steel. Standing waves, however, appear to remain stationary; this is a result of the waves bouncing off the end nodes of the string and propagating back along the string with a halfwavelength phase change, causing patterns of nodes (no vibration) and antinodes (greatest amplitude). The distinctive sound of each instrument is based on the wave pattern created, which is a mix of the fundamental frequency and all the higher harmonics, with different harmonics slightly emphasized. The fundamental is always most highly emphasized in normal playing. However, in some cases, you can play a harmonic, a note in which a harmonic higher than the fundamental has the greatest energy. This occurs when more than half a wavelength fits in the string length, as in the diagram. The fundamental, or 1st harmonic, has half a wavelength according to the properties of standing waves, while the 2nd harmonic has one full wavelength. Thus, the number of wavelengths in each successive harmonic is N/2, where N is the number of the harmonic. With some pitches, it is much easier to hit a harmonic than others; for example, lightly placing a finger halfway down the string will cause a node there but keep the entire string as the effective vibration length, causing the second harmonic, and placing a finger a third of the way down will cause the third harmonic. Suppose that it is the 2nd harmonic that gets played instead of the fundamental. The pitch of that harmonic is always an octave higher than the fundamental. This is easily calculable by using the equation v=fλ. When the wavelength is halved, and the speed of a sound wave through the same medium stays the same, then the frequency must double; by the WellTempered Clavier scale standards, doubling the frequency causes an increase of one octave. Here is a chart of each pitch and the corresponding frequency required to make that pitch:
The big bold numbers at the top indicate the octave, and the notes in that octave correspond to their frequencies, in Hertz. As you can see, each octave up doubles the frequency; look at any row, and each successive number to the right is double the previous one. In the WellTempered Clavier pitch standards, there is also another relationship: for every half step up, the pitch increases by 2^1/12, which approximates to 1.06. Thus, if you start on any note, the equation to find a note's frequency a certain interval above it is F2 = F1(1.06)^H, where F2 is the frequency of the higher note, F1 is the frequency of the lower note, and H is the number of half steps taken to get to the higher note. For example, let's try it out with D4 and A4. D4 has a frequency of 294 Hz. It takes seven half steps to reach A4, so let's plug all this into the equation: F2=F1(1.06)^H F2=294(1.06)^7 F2= 442 Hz This is close to the A4 frequency of 440 Hz and is only off because the 1.06 is an approximation. 

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