Physics Midterm 2
Waveform - wave shape controls quality/timbre, sum of sine waves/harmonics
waveforms played at same time are added
beat frequency - difference of two frequencies, useful for tuning instruments
harmonics - have same period therefore they're multiples of the fundamental frequency
1st harmonic = fundamental, 2nd harmonic = 1st overtone, 3rd harmonic = 2nd overtone, etc.
fourier analysis - tells you how much of each harmonic is in a waveform
square and triangle waves - odd harmonics only
sawtooth wave - all harmonics
many high frequencies (aka high harmonics) = bright sound
many low frequencies (aka low harmonics) = "warm sound"
sound identical if harmonic content is the same (Ohm's law of acoustics)
waves with same frequency and opposite phase cancel out
Hearing
human frequency reception range: 20-20,000 Hz, can hear differences as small as 5 musical cents
high end of hearing range decreases w age (young ppl 18 kHz, older people 12 kHz women 5 kHz men)
structure of human ear
ossicles amplify sound, safety mechanism that protects against hearing damage (>90 dB)
tensor tympani (attached to hammer) and stapedius (attached to stirrup) activated by autonomic reflexes and prevent ossicles from moving
psychoacoustics - study of how we hear sound
auditory processing: sound waves picked up by nerve endings connected to cochlea hairs, ear machinery particularly basilar membrane processes, auditory cortex and then cerebral cortex process
basilar membrane - continuum that is excited by different frequencies (excited regions can overlap)
two different theories (read them first) --> place theory: basilar breaks up initial sounds into frequency bands that get sent along different nerves to brain // temporal theory: timing allows brain to make sense of things and perceive pitch more accurately
temporal theory - takes into consideration timing of nerve signals, explains accurate pitch discrimination, pitch identification in complex waveforms, and pitch identification with missing fundamental
timing also plays role in sound localization, sound travels x further to reach one ear (compared to other)
place theory - pitch perception dependent on areas of basilar membrane
notes closer than the critical bandwidth (~minor third or 3 half-steps) excite the same nerve endings, results in dissonance
Musical Instruments - contains some part/element that vibrates which causes air to vibrate and make a sound
3 features of vibrations: equilibrium position, restoring force, and inertia (think of a pendulum)
Newton's first law: inertia // Newton's second law: F=ma
change frequency of pendulum by changing restoring force (F) or by changing inertia (m)
doubling the frequency means doubling the velocity in half the time
doubled frequency = quadrupled acceleration
vibrating strings, columns of air, or solids
vibrating strings = "standing waves"
equilibrium = undisturbed string, restoring force = tension, mass, bending force (add to restoring force) = stiffness of string
vibrating strings follow Mersenne's law: f = 1/2 sqrt(T/Lm)
frequency doubles when you...half the length, half the diameter, quadruple the tension, decrease density by a quarter
notes decay mainly due to friction
air friction - esp. important for thin/hard strings,string loses kinetic E when moving air molecules, decay time depends on frequency as 1/sqrt(f)
internal friction (aka heat) - esp. important for softer strings, decay time depends on frequency as 1/f
modes of vibration, 1st mode being fundamental, freely plucked strings exhibit all of the modes
moving points = anti-nodes, non-moving points = nodes
"nth" mode has "n" times the frequency, i.e. moves of vibration are essentially just harmonics
amt. of specific modes depend on where string is plucked
middle pluck = odd harmonics (hits nodes of even harmonics, therefore no sound)
1/3 of the way pluck = no 3rd, 6th, 9th, etc. harmonics
1/4 of the way pluck = no 4th, 8th, etc. harmonics
near end of string pluck = almost all harmonics --> bright tone
Soundboards - perfect ones are not too hard and not too soft, wood and plastic work best
(acoustic) impedance - object's ability to vibrate depending on its stiffness, i.e. how much it moves when a force is applied to it
impedance matching - nothing is stopping wave from spreading, eg. string attached to another string
mismatched impedance - object has higher impedance than string
important function of soundboard --> prevents strings from losing energy too fast, we want a balance (don't want too much mismatch)
Guitar: E2, A2, D3, G3, B3, E4
pitch bend (why does note get sharper?): string gets longer which lowers pitch BUT tension goes up which increases pitch --> % change in tension is much larger than change in length, therefore pitch goes up
Piano
each node operates hammer thru series of three levers: key, jack, and hammer
hammer strikes strings, vibration transmitted to soundboard thru bridge, sound decays, damper stops sound when key is released
timbre
strings struck ~10% of way along string --> increases amt. of high harmonics, brighter sound
harder strikes = brighter tones, softer strikes = warmer tones
hammer can't be in contact w string for more than one vibration (otherwise vibration will be dampened), ~5 ms for large hammers and 0.5ms for smallest hammers
hammers smaller/faster at higher end of piano where vibrations are faster
notes don't have as many high harmonics because hammer can't move fast enough
playing notes faster or slower affects harmonics
Transient - distinguishing tone at beginning of note, central to sound of instrument, most of note is in the transient, particularly important for piano
piano development focused on increasing volume: larger soundboard, more strings per notes, longer/thicker strings, faster/heavier hammer action
more pliable soundboard decreases acoustic impedance BUT string needs to provide more energy, therefore we need to increase the velocity with which it vibrates (make hammer faster/heavier) and increase mass (make string heavier)
octaves stretched slightly larger than factor of two so that harmonics coincide (avoiding inharmonicity)
bending force greater for higher frequencies
more bending force for shorter strings/higher notes => inharmonicity gets worse
Modern Violin Family (first three tuned in fifths, double bass tuned in fourths): violin (G3, D4, A4, E5), viola (C3, G3, D4, A4), cello (C2, G2, D3, A3), and double bass (E1, A1, D2, G2)
fourth-tuning on double bass allows player to switch to a higher string instead of moving their hand, convenient for switching to bass guitar (which is also tuned to fourths)
harmonics form sawtooth wave, fall off at 6 dB per octave
2 types of resonance: air (comes from hollow interior) and wood (comes from violin itself)
main wood resonance = 400 Hz/~G4
main air resonance = 260 Hz
good violins have separate resonances spread across frequency range of instrument so that every note excites as resonance
in a resonating pipe, wavelength is 2x length of pipe
Tubes of Air/Cones
length L of tube is half a wavelength (λ = 2L)
1/2λ = fundamental mode, 1λ = 2nd mode, 3/2λ = 3rd mode, 2λ = 4th mode
pipe closed at one end = quarter of a wavelength (starts with pressure anti-node and ends w pressure node) --> harmonics determined by whether shape is cylindrical or conical
1/4λ = fundamental, 3/4λ = 2nd mode, 5/4λ = 3rd mode, 7/4λ = 4th mode
nth mode has (2n-1)/4 λ in pipe w length L
conical pipes - flared, gets wider toward bottom, amplitude decreases with width, all harmonics present
ways to drive vibrations in wind and brass instruments: reed, lips, fluid-flow instability (aka edge-tones)
fluid-flow instability - unstable/jet-like air flow interacts w sharp edge that disrupts flow (some air goes into instrument and some goes out)
flute, organ pipe, recorder
driven end = velocity antinode => open pipe
frequency of edge tone: air flows back at ~40% of outgoing velocity
time = half of period P (refer to sheet for formula)
overblowing increases frequency by 3x
edge-tone frequency that is close to the natural frequency of the pipe facilitates a mutual relationship in which both will feed each other
organ: pipes divided into ranks, each rank spans 5 octaves from C2-C7, longest pipe = C2 w frequency 65.4 Hz
recorder: frequency = 365 Hz w all holes covered
flute: instead of mouthpiece that generates edge tone, it is actually generating by blowing thru mouthpiece
overblowing + register hole allows you to get higher notes - register hole bleeds E from 1st harmonic (and if placed right, won't bleed any E from 2nd harmonic bc hole is at node of 2nd harmonic), emphasizes 2nd harmonic and makes it easier to hit by overblowing
Reed Instruments
based on Bernoulli effect - faster airflow means lower pressure
reed produces bursts of pressure in mouthpiece => mouthpiece is pressure antinode (open end = pressure node), reed opens bc of pressure and bernoulli effect pulls them closed again when air passes thru
clarinet = closed pipe (closed at one end), pressure vibration at closed end (mouthpiece) drives reed at natural frequency
straight cylindrical cone ==> only odd harmonics, overblowing gives you third harmonic
normal playing uses 1st, 3rd, and 5th harmonics
hard vs soft reeds: hard reeds have sharp resonance and only plays one note. soft reeds have broad resonance and don't have a clearly defined note
hard reed (eg. harmonica)
harmonica - metal reed for each note, sounds depend on whether you blow or draw air
blues harmonica/blues harp plays only major scale (no sharps/flats)
10 holes, 3 octaves - not all notes present, low G occurs twice
chromatic harmonic has slide that allows you to play all 12 notes
soft reed (eg. clarinet)
divided into single and double reeds - not much difference in sound, more for convenience of playing
single reed: saxophone (conical)
double reed: oboe (concical)
flare of pipe ~3 degrees --> wider flare allows for larger tone holes => louder sound
click to edit
flare of pipe ~1 degree --> narrower flare produces more high harmonics and brighter sound
Brass Instruments = reed instruments where "reed" is player's lips (pressure antinode, closed pipe), usually metal
bugle, trombone, trumpet
change frequency of trombone by sliding pipe --> longer pipe = lower frequency, how far to slide pipe depends on how much more length you need to lower by half-steps
advantages: simple to make, perfect tuning possible
disadvantages: hard to play in tune, hard to play fast/legato, requires cylindrical pipe
problem w cylindrical pipes is that it only produces odd harmonics => less bright sound, fewer basic notes to work off of BUTBUTBUT trombones (and trumpets) give you ALL the harmonics excluding the fundamental
explanation: initially odd harmonics are shifted by the bell (which slowly widens and flares suddenly at end), makes pipe effectively shorter for low notes and longer for high notes (i.e. pushes low notes up and high notes down)
net result: first five harmonics approximate well 2nd-6th odd and even harmonics, fundamental way off (can be played if you try hard, called "pedal tone")
cylindrical bc of slide - 7 slide positions that lower sound by half-step increments (perfect 5th w 7 half-steps), choose harmonics by changing lip tension and air pressure
standard tenor trombone lowest harmonic = B flat in first position
trumpet gets rid of slide in favor of valves --> valves divert air thru extra sections of pipe to make total length longer, plays 2nd-8th harmonics
2nd valve has shortest crook, 1st valve has 2nd-shortest, 3rd valve has longest crook
first and second valves can be pressed together to get third valve BUT note will be slightly sharp --> corrected for by "lipping down" (this is actually preferable to playing 3rd valve alone)
combine valves to get different lower notes
resolved by compromised tuning, relaxation of lips, third valve slide (almost always used except for shorter notes where you won't notice the tuning as much)
lowest note out of tune by 20 cents
most common # of valves = 3
types of valves: piston and rotary
tuba can play up to 11th harmonic, can play fundamental via extra valve/crook
The Voice - most versatile instrument, narrow pitch range but capable of wide range of timbre
wind instrument: lungs supply air, air passes thru trachea to larynx/voicebox, passes thru vocal chords (pressure antinode) and causes them to vibrate, vibrations filter thru pharynx, mouth, and nose to produce voice sounds
muscles around vocal chords allow you to loosen/tighten to make different notes
don't completely close when singing/speaking softly, close and cut off air flow with louder noise --> close for considerable length of time when very loud