Unit 10.2 the Physics of Music Objectives Notes on a Piano

Unit 10.2 The Physics of Music

Teacher: Dr. Van Der Sluys

Objectives

• The Physics of Music – Strings – Brass and Woodwinds • Tuning - Beats

Notes on a Piano

Key Note Frquency (Hz) Wavelength (m) 52 C 524 0.637 51 B 494 0.676 50 A# or Bb 466 0.717 49 A 440 0.759 48 G# or Ab 415 0.805 47 G 392 0.852 46 F# or Gb 370 0.903 45 F 349 0.957 44 E 330 1.01 43 D# or Eb 311 1.07 42 D 294 1.14 41 C# or Db 277 1.21 40 C (middle) 262 1.27

http://en.wikipedia.org/wiki/Piano_key_frequencies

1 Vibrating Strings - Fundamental and Overtones

A vibration in a string can L = 1/2 λ1 produce a standing wave. L = λ Usually a vibrating string 2 produces a sound whose L = 3/2 λ3 frequency in most cases is constant. Therefore, since L = 2 λ4 frequency characterizes the pitch, the sound produced L = 5/2 λ5 is a constant note. Vibrating L = 3 λ strings are the basis of any 6 string instrument like guitar, L = 7/2 λ7 cello, or piano. For the fundamental, λ = 2 L where Vibration, standing waves in a string, L is the length of the string. The fundamental and the first 6 overtones which form a harmonic series

http://en.wikipedia.org/wiki/Vibrating_string

Length of Piano Strings

The highest key on a piano corresponds to a frequency about 150 times that of the lowest key. If the string for the highest note is 5.0 cm long, how long would the string for the lowest note have to be if it had the same mass per unit length and the same tension?

If v = fλ, how are the frequencies and length of strings related?

Other String Instruments

• All string instruments produce sound from one or more vibrating strings, transferred to the air by the body of the instrument (or by a pickup in the case of electronically- amplified instruments). They are usually categorized by the technique used to make the strings vibrate. The three most common techniques are plucking, bowing and striking. • A vibrating string on its own makes only a very quiet sound, so string instruments are usually constructed in such a way that this sound is coupled to a hollow resonating chamber, a sounding board, or both. On the violin, for example, the taut strings pass over a bridge resting on a hollow box. The strings' vibrations are distributed via the bridge and soundpost to all surfaces of the instrument, and are thus made louder.

2 Production of Multiple Notes

A string at a certain tension will only produce one note, so to obtain multiple notes string instruments employ one of two methods. One is to add enough strings to cover the range of notes desired; the other is to allow the strings to be stopped. The piano is an example of the former method, where each note on the instrument has its own set of strings. On instruments with stoppable strings, such as the violin or guitar, the player can shorten the vibrating length of the string, using their fingers directly (or more rarely through some mechanical device, as in the hurdy gurdy). Such instruments usually have a fingerboard attached to the neck of the instrument, providing a hard flat surface against which the player can stop the strings. On some string instruments, the fingerboard has frets, raised ridges perpendicular to the strings that stop the string at precise intervals, in which case the fingerboard is called a fretboard.

An Air Displacement Wave is also an Air Pressure Wave

The nodes of the displacement wave, where the air is not rushing back-and-forth but is doing the most piling-up-and-spreading-out, are the antinodes of the pressure wave. The antinodes of the displacement wave, where the air is rushing back-and-forth the most, but is not piling up or spreading out at all, are the nodes of the pressure wave. Both waves must have exactly the same frequency, of course; they are actually just two aspects of the same sound wave.

http://cnx.org/content/m12589/latest/

Vibrations and Open Tubes

These are the first four harmonics allowed in an open tube. Any standing wave with a displacement antinode at both ends is allowed, but the lower harmonics are usually the easiest to play and the strongest harmonics in the timbre. The longitudinal waves are represented as pressure waves

3 Vibrations in Closed Tubes

Again, these are the lowest (lowest pitch and lowest frequency) four harmonics allowed. Any wave with a displacement node at the closed end and antinode at the open end is allowed. Note that this means only the odd-numbered harmonics "fit".

Woodwind and Brass Instruments The two shapes that are useful for real wind instruments are the cylinder and the cone. Most real wind instruments are a combination of cylindrical and conical sections, but most act as (and can be classified as) either cylindrical bore or conical bore instruments.

Higher Notes in Woodwind and Brass Instruments Just as on a string, the actual wave inside the instrument is a complex wave that includes all of those possible harmonics. A cylinder makes a good musical instrument because all the waves in the tube happen to have simple, harmonic-series-type relationships. This becomes very useful when the player overblows in order to get more notes. As mentioned above, woodwind players get different notes out of their instruments by opening and closing finger holes, making the standing wave tube longer or shorter. Once the player has used all the holes, higher notes are played by overblowing, which causes the next higher harmonic of the tube to sound. In other words, the fundamental of the tube is not heard when the player "overblows"; the note heard is the pitch of the next available harmonic (either harmonic two or three). Brass players can get many different harmonics from their instruments, and so do not need as many fingerings.

4 Tuning Instruments - Beats A 110 Hz sine wave (cyan), a 104 Hz G# sine wave (magenta), their sum (blue) and the corresponding beat frequency, BF (red)

Bf = f1 - f2

A Beat Problem

A tuning fork produces a steady 400 Hz tone. When this tuning fork is struck and held near a vibrating guitar string, twenty beats are counted in five seconds. What are the possible frequencies produced by the guitar string?

Musical Scales http://en.wikipedia.org/wiki/Chord_(music) http://en.wikipedia.org/wiki/Musical_keyboard

• Each octave represent a doubling of frequency. • In western music there are twelve notes per octave. Each note increases in frequency by a factor of 21/12 or a half step. • On the piano, the white keys represent a C scale, or the notes C, D, E, F, G, A, B, C. The difference in frequencies between C, D and E as well as F, G, A and B are full steps or a difference of 22/12, where as the difference between E, F and B, C is only a half step or 21/12 • If two note are played simultaneously and only separated by a half step, the sound will be discordant.