String Instruments 1 Updated Nov 24, 2009

Outline A.Piano B.Guitar C.Violin D.References 2

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 ( ) 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) 6

1b. UPRIGHT PIANO ACTIONS FULL SIZE CONSOLESPINET 7

1b GRAND PIANO ACTION 8

1c Pedals Left: soft pedal (or una corda pedal) Middle: function varies with piano Right: sustain (damper) pedal 9

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. 10

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 7 th harmonic This is only approximately true 11

Railsback Curve: Octave Stretching Metal Strings are slightly inharmonic (e.g. 2 nd harmonic of string is slightly greater than 2x fundamental). Hence to eliminate beats, the note an octave above must be slightly tuned sharp! 12

B. Guitar 1.Strings 2.Frets & Tuning 3.Plucking 13

History of Guitar In ancient times there was the kithara These evolved into Zithers. Some have “frets” 14

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 15

History of Guitar “modern” guitar designed by Antonio de Torres circa Antonio de Torres

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] 17

1b String Details Approximate Mass Densities of Steel Strings (gm/cm) E B G D A E

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 19

2b. FRET: Rule of 18 (GREATER ACCURACY IS OBTAINED BY USING RATHER THAN 18) 20

3a. Tuning Your Guitar Using frets will give equal temperament tuning (e.g. 5 th 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] 21

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. 22

3c. Vibration Modes x 23

C. Violin 1.Construction 2.Family 3.Playing 24

History of Violin Evolved from ancient stringed instruments. In middle ages similar instruments were the “Rebec” and “Vielle” 25

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 26

27 Mills Violano Virtuoso (1907) Mechanically played violin!

1. Construction NO FRETS The violinist must do it by feel! 4 strings tuned in perfect 5ths G3, D4, A4, E5 28

1b. Construction Scale length (playing length of string) is exactly 327 mm Each string tension is approximately pounds Like guitar, lower strings are bigger in diameter 29

1c. Resonances of Cavity & Plates MODES OF TOP AND BACK PLATES MODES OF A VIOLIN CAVITY 30

31 1c. Resonances of Plates

2a. VIOLIN OCTET 32

2b. VIOLIN OCTET 33

2c. Overlapping Tuning 34

3a. Plucking String 35 Plucking string in the middle makes a triangular waveform (odd harmonics only)

3b. Plucking String 36 Plucking string at 1/5 its length suppresses the 5 th harmonic

3c Helmholtz Theory of Bowing 37 string sticks to bow, creates kink, then slips

3c. Helmholtz Theory of Bowing 38

3d. Bowing and Harmonics 39 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: Rossing: The Science of Sound String Tension Calculator: 40

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. 41