Fundamentals of Music Technology

Juan P. Bello

•  Office: 409, 4th floor, 383 LaFayette Street (ext. 85736)

•  Office Hours: Wednesdays 2-5pm

•  Email: jpbello@nyu.edu

•  URL: http://homepages.nyu.edu/~jb2843/

•  Course-info: Tuesdays 4.55-6.35pm (Studio F) E85.1801: Fundamentals of Music Technology Course materials at: http://www.nyu.edu/classes/bello/FMT.html

Lectures tentative schedule

•  Weeks 1-2 What is sound? •  Weeks 2-3 Hearing •  Week 4 Microphones •  Week 5 Loudspeakers •  Weeks 6-8 Mixers •  Week 7 Mid-term exam (30%) – October 20 •  Weeks 8-9 Cabling and Interconnection •  Weeks 10-11 Basics of digital systems •  Week 12 Communication\MIDI •  Week 13 MIDI code •  Week 14 MIDI sound control/synthesis •  Week 15 Final exam (30%) – December 15

Demonstrations schedule

•  Teaching Assistant: Langdon Crawford (demonstrations, assignments + teaching in week 8)

•  Thursdays 4.55-6.10pm, Studio F

•  Week 3: Wave propagation •  Week 5: Transducers •  Week 6: Mixers and signal flow •  Week 10: Building your own studio •  Week 12: Sampling and Quantization •  Week 14: MIDI and Synthesis

•  8 Assignments (weeks 1, 3, 5, 6, 8, 10, 11 and 12). Due a week later.

Evaluation and Resources •  Mid-term exam = 30% •  Final exam = 30% •  Assignments = 40% •  Attendance and class participation

•  All relevant information is (or will be published) on the class website - Please read it carefully and keep checking for updates.

•  Assignments will be announced with sufficient time and published online •  Penalties will apply to delays

•  Book: Francis Rumsey and Tim McCormick (2002). “Sound and Recording: An Introduction”, Focal Press.

•  Further reading will be recommended as the course progresses. •  USE THE OFFICE HOURS (Wednesdays 2-5pm)

What is sound?

Juan P Bello

Sound •  Sound is produced by a vibrating source that causes the matter

around it to move. •  No sound is produced in a vacuum - Matter (air, water, earth)

must be present!

•  The vibration of the source causes it to push/pull its neighboring particles, which in turn push/pull its neighbors and so on.

•  Pushes increase the air pressure (compression) while pulls decrease the air pressure (rarefaction)

•  The vibration sends a wave of pressure fluctuation through the air

Air particles

Waves

•  Waves can be longitudinal (the particles move in the same direction of the wave) or transversal (the particles’ movement is perpendicular to the wave’s direction)

Longitudinal Transversal

Sound waves (1)

•  In sound wave motion air particles do not travel, they oscillate around a point in space.

•  The rate of this oscillation is known as the frequency of the sound wave and is denoted in cycles per second (cps) or hertz (Hz).

•  The amount of compression/rarefaction of the air is the amplitude of the sound wave.

•  The distance between consecutive peaks of compression or rarefaction is the wavelength of the sound wave (denoted by λ)

•  A fast traveling wave results on a greater λ

Sound waves (2) •  If the frequency of oscillation is fixed, then the sound wave is

periodic (with period t, and frequency 1/t) •  The simplest periodic wave is a sinusoid

•  Because of the inverse relationship, the higher the frequency, the shorter the time between oscillations.

•  Humans frequency range: 20-20kHz (20,000 Hz)

t

Compressions

Rarefactions

Sound waves (3) •  The speed of a wave (c), depends on the density and elasticity of

the medium (and thus in its temperature). •  In air, at 70 °F (21 °C), c = 769 mph (344 meters/s). This is slow

when compared to most solids. •  If the speed c and the oscillation frequency f are known, the

wavelength can be calculated as: λ = c/f

wavelength

Frequency and wavelength •  There is then an inverse relationship between wavelength and

frequency •  E.g. for f = 20 Hz, λ = 56.4 ft, and for f = 20kHz, λ = 0.67 in •  Frequency range is behind size differences in, e.g. musical

instruments and loudspeakers

Different types of sounds •  Sinusoids are only one possible type of sound •  They correspond to the simplest mode of vibration, producing

energy at only one frequency •  They are often called pure tones and are extremely rare in real life

(e.g. a recorder produces an almost pure tone)

•  However most sounds are not so simple resulting in complex waveforms

•  The more complex, the noisier the sound is - when the pattern of vibration is random, the sound is said to be noise

•  Demo: ftm_demo1

Periodicity (1)

•  If a waveform pattern is repeated at regular intervals, then the sound wave is periodic and has definitive pitch

•  We can use Fourier Analysis to break down the waveform into a series of frequency components known as harmonics

•  These components can be seen in an amplitude vs frequency graph of the sound known as frequency spectrum

•  Consider a sinusoid: it has a simple pattern that repeats at its oscillating - or fundamental - frequency (f0)

Jean Baptiste Joseph Fourier (1768-1830): French mathematician and physicist

Fourier Analysis f0

Periodicity (2) •  For more complex patterns, more complex configurations of

spectral lines will appear (see ftm_demo1) •  The underlying assumption in Fourier Analysis is that any sound

can be made out of the combination of (many) simple sinusoids with different amplitudes

•  Note that a sound wave is periodic (and pitched) no matter how complex the repeated pattern is

•  Pitch perception occurs as long as the repetition rate is within the human audio frequency range (see ftm_demo2)

+

Spectral Analysis (1) •  What are these complex spectral configurations and what

waveforms do they produce? •  Harmonics (or Overtones or Partials) are frequency components

that occur at integer multiples of the fundamental frequency •  Their amplitude variations determine the timbre of the sound

f

T = 1/f

Frequency Am

plitu

de

f

T = 1/f

Frequency Am

plitu

de

2f 3f 4f 5f 6f

Fundamental frequency

(first harmonic)

Harmonics Overtones Partials

time

time

Am

plitu

de

Am

plitu

de

Spectral Analysis (2) •  Example: Square wave - only odd harmonics (even are missing).

Amplitude of the nth harmonic = 1/n

Harmonic modes •  Most sources are capable of vibrating in several harmonic modes

at the same time •  Examples: a guitar string, this room

nodes antinodes

1st harmonic

2nd harmonic

3rd harmonic

4th harmonic

Complex sounds •  Most pitched instrumental sounds also present overtones which

are not integer multiples of the fundamental. •  These are known as inharmonic partials

Harmonic Inharmonic

Non-periodic sounds

•  Non-periodic sounds have no pitch and tend to have continuous spectra, e.g. a short pulse (narrow in time, wide in frequency)

•  The most complex sound is white noise (completely random)

Frequency Am

plitu

de

time Am

plitu

de

Frequency Am

plitu

de

time

Am

plitu

de

Phase (1)

time

Am

plitu

de

+

-1

-1 1

1

-2

0

2

time

Am

plitu

de

+

-1

-1 1

1

-2

0

2

•  In phase: cycles coincide exactly (sum duplicates amplitude)

•  Out of phase: half cycles are exactly opposed (sum cancels them)

Phase (2)

•  There is a range of partial additions and cancellations in between those extremes (ftm_demo3)

•  What causes phase difference?

•  The phase difference depends on the delay time and the wave’s frequency

time

Am

plitu

de

0 t1 t2 t1

t2 delay

Phase (3)

•  Phase is commonly measured in terms of degrees of the oscillating cycle of a periodic wave

•  The frequency defines the number of cycles per second, thus the delay x frequency x 360° returns the (unwrapped) angular phase difference

0° 90° 270° 180° 0° 90° 270° 180° 0°

90°

180°

270°

Sound power and intensity •  A source (e.g. bell) vibrates when a force (e.g. striking hammer)

is applied to it. •  The force applied and the resulting movement characterize the

work performed by the source (W = F x Δs) •  Power (P = W/t) is the rate at which work is performed and is

measured in watts. •  An omnidirectional sound source produces a 3-D longitudinal

wave. The resulting wavefront is defined by the surface of a sphere (S = 4πr2), where r is the distance from the source.

Sound source

r

Wavefront The original power is distributed on the surface of the wavefront. As r increases, the power per unit area (intensity) decreases: I = P/S

Intensity and SPL

•  The effect of sound power on its surroundings can be measured in sound pressure levels (SPL) - much as temperature in a room relates to the energy produced by a heater.

•  Both intensity (Watts/area) and sound pressure (Newtons/area) are usually represented using decibels (dB)

•  dB are based on the logarithm of the ratio between two powers, thus describing how they compare (dB = 10log10(P1/P2)).

•  This can be applied to other measures (intensity, SPL, voltage), as long as their relationship to power is taken into account.

•  In the case of intensity and SPL, the denominator of the ratio is a reference value, defined according to the quietest sound perceivable by the average person.

•  Thus by convention, 0 dB corresponds to SPL = 2x10-5 N/m2 or I = 10-12 watt/m2

Acoustic fields

•  The previous model of sound energy distribution only applies to omnidirectional sources and free fields (acoustic areas with no reflections)

•  Free fields are extremely rare as there are always reflections (from the ground and nearby surfaces)

•  In rooms there is both direct and reflected sound •  Where reflected sound energy is predominant the field is said to

be reverberant (or diffuse) •  The near field is close to the source, where direct energy is much

higher than reflected •  The distance from the source at which reverberant energy

becomes predominant depends on the room’s volume and absorption.

Useful References

•  Francis Rumsey and Tim McCormick (2002). “Sound and Recording: An Introduction”, Focal Press.

–  Chapter 1: What is sound? •  Dave Benson (2002). “Mathematics and Music”. http://www.math.uga.edu/~djb/index.html

–  Chapter 1: Waves and harmonics