String Instruments1

UpdatedDec 21, 2012

Outline

A. Piano

B. Guitar

C. Violin

D. References

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A. Piano 3

• Upright Piano (similar to one in classroom)

Piano History 4

• Evolved from stringed instruments that are struck with hammers, such as the dulcimer

• The harpsichord (late middle ages) had a keyboard where strings are “plucked”

Piano History 5

• The Clavichord (circa 1505) hit metal strings with metal blades

• The invention of the modern piano is credited to Bartolomeo Cristofori (1655-1731) of Padua, Italy, who developed the mechanical hammer mechanism

1. Keyboard: Modern Piano

• 88 Keys (36 black, 52 white)• Start at A0 (27.5 Hz), end at C8 (4186 Hz)• Range: 7 octaves (plus 3 notes)

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1b. UPRIGHT PIANO ACTIONS

FULL SIZE CONSOLE SPINET

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1b GRAND PIANO ACTION 8

1c Pedals

• Left: soft pedal (or una corda pedal) • Middle: function varies with piano• Right: sustain (damper) pedal

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2. Strings

• Upper 68(?) have 3 strings per note• Tenor section (7) have 2 strings per note• Lowest 8 have only one string per note• Lower Strings are wire wound to increase mass

density. Lowest ones are double wound.

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3 Harmonics

• Striking a string at 1/n th its length will suppress the n-th harmonic

• Piano hammers approximately hit string at 1/7 th to suppress 7th harmonic

• This is only approximately true

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Railsback Curve: Octave Stretching

Metal Strings are slightly inharmonic (e.g. 2nd harmonic of string is slightly greater than 2x fundamental). Hence to eliminate beats, the note an octave above must be slightly tuned sharp!

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B. Guitar

1. Strings

2. Frets & Tuning

3. Plucking

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History of Guitar

• In ancient times there was the kithara

• These evolved into Zithers. Some have “frets”

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History of Guitar

• In the Renaissance there was the 3 string cittern or cither. Its has a flat back with frets.

• More strings were added. This is related to the mandolin.

• Lutes had more strings with a rounded back

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History of Guitar

• “modern” guitar designed by Antonio de Torres circa 1852

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Antonio de Torres1817-1892

1a Strings • 6 strings tuned to E2, A2, D3, G3, B3, E4

• Mass: want tension T of each string to be nearly the same (100

Newtons or 22 pounds), since they all have the same length (L=65 cm), need lower strings to be progressively more massive. Hence E2 string is approximately 16x bigger in mass density , or 4x bigger in diameter “d” than E4.[Note is density per volume, is density per unit length]

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T

Ld

T

Lf

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2

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1b String Details • Approximate Mass Densities of Steel Strings (gm/cm)• E4 0.0057• B3 0.0101• G3 0.0209• D3 0.0375• A2 0.0656• E2 0.1017

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1b. Guitar String Masses For guitar, all strings same length, and want tensions the same, so to get different frequencies, the masses of strings must be different

String gm/cm FreqE4 0.0057 330 HzB3 0.0101 247G3 0.0209 196D3 0.0375 147A2 0.0656 111E2 0.1017 82.5

•E2 is 2 octaves lower than E4•Or (1/4) the frequency

•Hence mass bigger by nearly a factor of 16

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F

Lf

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c. Guitar String Diameters For guitar, all strings same length, and want tensions the same, so to get different frequencies, must vary diameters of strings

String Diameter FreqE4 0.010 mm 330 HzB3 0.013 247G3 0.017 196D3 0.026 147A2 0.036 111E2 0.046 82.5

•E2 is 2 octaves lower than E4•Or (1/4) the frequency•Hence diameter is nearly 4x bigger!

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Ld

Ff

2a. FRET PLACEMENT

•Each “fret” raises note by one semitone (equal temperament)•Standard classical guitars have 19 frets•electric guitars between 21 to 24 frets

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2b. FRET: Rule of 18

(GREATER ACCURACY IS OBTAINED BY USING 17.817 RATHER THAN 18)

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3a. Tuning Your Guitar

•Using frets will give equal temperament tuning (e.g. 5th fret of E2 string sounds same as open of A2 string)

•Using harmonics tunes to perfect 5ths, resulting in Pythagorean error (n.b. B3 string). [method explained in class]

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3b. Harmonics• Playing a string at 1/n th its length will suppress the nth harmonic. Hence

playing close to sound hole will result in less harmonics, playing closer to bridge will give more harmonics

• Electric Guitar pickups: often 2 or 3 at different positions to capture differing amounts of harmonics (if positioned at 1/n the length of the string, will suppress the nth harmonic). Pickups nearer the bridge will be rich in higher harmonics.

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3c. Vibration Modes

• x

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C. Violin

1. Construction

2. Family

3. Playing

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History of Violin

Evolved from ancient stringed instruments. In middle ages similar instruments were the

“Rebec”

and

“Vielle”

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History of Violin

• In renaissance there was the 7 string lira da braccio (circa 1525)

• The “modern” 4 string form invented 1555 by Andrea Amati

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Mills Violano Virtuoso(1907)

Mechanically played violin!

1. Construction

•NO FRETSThe violinist must do it by feel!

•4 strings tuned in perfect 5ths

•G3, D4, A4, E5

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1b. Construction• Scale length

(playing length of string) is exactly 327 mm

• Each string tension is approximately 12.75 pounds

• Like guitar, lower strings are bigger in diameter

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1c. Resonances of Cavity & Plates

MODES MODES OF TOP OF TOP AND AND BACK BACK PLATESPLATES

MODES MODES OF A OF A VIOLIN VIOLIN CAVITYCAVITY

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331c. Resonances of Plates

2a. VIOLIN OCTET

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2b. VIOLIN OCTET

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2c. Overlapping Tuning36

3a. Plucking String

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Plucking string in the middle makes a triangular waveform (odd harmonics only)

3b. Plucking String

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Plucking string at 1/5 its length suppresses the 5th harmonic

3c Helmholtz Theory of Bowing39

string sticks to bow, creates kink, then slips

3c. Helmholtz Theory of Bowing40

3d. Bowing and Harmonics41

•Bowing position is at approximately 1/9 the length (suppresses 9th harmonic).

•Bowing at other positions would have different harmonics

References• Some slides from Rossing: http://ccrma.stanford.edu/courses/150/schedule08.html• Rossing: The Science of Sound• String Tension Calculator: http://www.mcdonaldstrings.com/stringxxiii.html

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Notes/Things to Do• This is a very rough draft, created Nov 24, 2008• History added Nov 24, 2009• Show densities of strings match the estimated tension of a string, need string

diameters• Pictures of types of pianos (upright, grand, toy)• Can we show the piano C8 is slightly too high from ideal? Octave stretched• Show more clearly how rule of 18 comes from• How do widths of frets change?• Clarify which instruments out of the 8 violins are the most common.• Comments May 21, 2012:• ADD a recording of Harpsichord• ADD recording showing different amounts of harmonics present in different pickups or

playing guitar at different points• Note our piano has 10 single strings, 20 double, and rest tiple.

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