1. String Instruments General comments
Vibration of strings - standing waves
Fundamental:
Harmonics:
The fundamental frequency and harmonics are determent by length, L, tension, F, and linear mass density, μ.
Usually, length and mass of the string remain constant, and turning is accomplished by changing tension.
Timbre is mainly determined by harmonics
The combination of harmonics depends on the method of string excitation (plucked, bowed or struck by a hammer; exited in the center or at the edge), and on the resonances (ear and wood resonances)

2. 2. Vibration of a plucked string
Some properties:
String plucked at its center contains only odd harmonics. Intensity of these harmonics decays as 1/n2 , where n is the harmonic’s number
If distance from the end is 1/n (n=1,2,3…) then n-th harmonic is missing
Plucked at its center
Plucked at 1/5 of length
Harmonic
Ampliude
Phase 1 + 2
none 3
1/9 - 4 5
1/25 6 7
1/49
Harmonic
Ampliude
Phase 1 + 2
0.40 3
0.18 4
0.06 5
none 6
0.03 -

3. 3. Vibration of a bowed string
Displacement and velocity
During the grater part of each period, the string is carried along by bow.
Then it suddenly detaches itself and moves rapidly beck until it is caught by the moving bow again.
Frequency is determent by length, L, tension, F, and linear mass density, μ.
The beginning and the end of slippage depend on position of the bow.
For each position there is maximum and minimum bowing force. Closer to the bridge these forces are stronger.

4. 4. Violin
Violin Construction: four strings (of steel or gut) tuned to G3, D4, A4, E5 (196.0, 293.7, 440.0, Hz)
What is the interval?

5. Vibration of the violin body.
Important factor in sound quality
To determine normal modes forces can be applied to the bridge
Most of the sound that a listener hears from a violin is radiated into the air from the top and back plates
The sound quality can be changed considerably by the place where the string is bowed:
If bowed close to the bridge (sul ponticello) then the sound is brighter with more harmonic content
If bowed further from the bridge (sul tasto) then the sound is darker, more mellow with less harmonic content
The bridge
Transforms motion of vibrating strings to the top plate
Has strong resonances (typically around 3000 and 6000 HZ)
Mute (additional mass of about 1.5 g) can be attached to the bridge to lower frequencies of resonances
Forces
Combined tension of strings 220 N (50 lb)
Downward force on the top plate
Sound post and bass bar support the top plate

6. Benade's resonance curve for a violin
The main air resonance enhance the D string.
The main wood resonance would enhance the A string.
It should be compared with Hutchins' measurements for a Stradivarius.

7. Resonance curve of a 1713 Stradivarius violin from the measurements of Carleen Hutchins
The prominence and frequencies of the air resonance and wood resonance are taken as important indicators of the quality of the instrument.

8. 5. Other bowed string instruments
Violin: G3, D4, A4, E5
Viola: C3, G3, D4, A4
Cello (violoncello): C2, G2, D3, A3
Bass (contrabass): E1, A1, D2, G2
5a. The Viola
The viola is the alto of the violin family, and it has the responsibility of playing the tenor part in the string quartet. Larger and heavier than the violin, it is tuned a fifth lower and has a darker, somewhat nasal tone. The strings are tuned a fifth apart at C3(130.8 Hz), G3, D4, A4(440 Hz).

9. 5b. The Cello
The cello, or violoncello, is the second largest member of the VIOLIN family of musical instruments. It is tuned an octave below the VIOLA and serves both as a melodic and bass instrument in chamber and orchestral music. The body of the cello is approximately 76 cm (30 in) long and is much deeper than those of the violin and viola.
The strings are tuned a fifth apart at C2(65.4 Hz), G2(98 Hz), D3(146.8 Hz), A3(220 Hz) if tuned in equal temperament to the A4(440 Hz) standard.
The top plate of the cello is made of spruce or pine because of their good sound radiating qualities. The sides and back plate are made of maple. The neck, pegbox and scroll are made of maple. A thin piece of ebony is glued to the top side of the neck to form the fingerboard. At the end of the neck where the pegboard is attached, a small grooved piece of ebony is attached for holding the strings in position. It is called the "nut".
The strings pass over a bridge made of maple which transmits the vibrational energy from the strings to the body of the instrument. The top of the bridge is curved to match the contour of the top plate and to allow one string at a time to be bowed.

10. 5c. The Double Bass (bass Viol or contrabass)
The largest and lowest-pitched member of the orchestral string section. The standard double bass has four strings and a range from E just over an octave below the bass staff, upwards for nearly three octaves, although some instruments may have five strings in order to extend the range downward.
The strings are tuned a fourth apart at E1(41.2 Hz), A1, D2, G2(98 Hz).

11. 5. Acoustic Guitar
Usually one of the families: classical, flamenco, flattop (or folk), archtop
Guitar construction:
Six strings (of steel or gut)
Tuned to E2, A2, D3, G3, B3, E3 (82, 110,147,196, Hz)
What are intervals? (A fourth except for the major third between B and G)
A matched set of strings have carefully adjusted masses so that the strings are tuned to the correct intervals when the tensions are the same (to keep strings from putting any distorting torque on the instrument)
Top plate made of spruce or cedar (thickness about 2.5mm)
Brazilian rosewood is favored for making the backs and sides of a guitar because it is very hard and tends to contribute to a brighter sound (also mahogany, or maple)
A round hole 3 1/4" to 3 1/2" in diameter in the top plate creates a cavity resonance which strengthens the sound produced
Top and back are braced to strengthen plates and to transmit vibrations from the bridge
Bridge transfers vibrational energy of strings to the top plate
Frets (metal inserts) placed by fret rule: "one-eighteenth the remaining length of the string“

12. Fret Rule for Guitars
To provide for definite pitch relations between notes, metal inserts called frets are inset in a fretboard on the neck of guitars. The raised edges of the frets provide fixed lengths of string when the string is held down against them with a finger. The interval between successive frets is normally one equal tempered semitone.
Frets on guitars are placed by the fret rule "one-eighteenth the remaining length of the string". This makes them approximately a semitone apart. If the musical interval produced by this rule is expressed in cents, then a string of length 17/18 of its original length will be higher by an interval of 98.9 cents compared to a precise semitone of 100 cents. Since the just noticeable difference in pitch is about 5 cents, then the fret rule could be applied for one change of fret with no problems in intonation. One would have to be careful about cumulative errors if the rule were applied repeatedly, so it should be checked at intervals such as the musical fourth which should be precisely 3/4 of the length of the open string

13. History
The Spaniard Antonio de Torres Jurado is credited with considerable enhancements of the modern classical guitar in the mid nineteen hundreds. The body and the sound hole were enlarged and the fretboard widened. Perhaps his most important contribution was the development of "fan strutting", a series of struts which diverge from the sound hole on the top plate of the instrument. This design gave it a considerably stronger, more sustained tone.

14. 6a. There are several varieties of electric basses
6. Electric Guitar
Electric guitar construction:
Most often made with solid bodies since they depend upon several electromagnetic pickups and amplifiers to produce the sound and are not dependent upon the resonance of the hollow body like the acoustic guitar
Several pickups allow the player to adjust the quality of the sound by choosing the amounts of signal used from each pickup
There are volume controls for pickups
Position control switch allows a quick selection of pickup combinations. It is possible to choose front pickup only, front and middle, middle only, back and middle and back only
6a. There are several varieties of electric basses