People have inner-ear cochleas {hearing, sense} {audition, sense}, with sense receptors for mechanical compression-and-rarefaction longitudinal vibrations {sound, hearing}. Sounds have loudness intensity and tone frequency. Hearing also analyzes sound-wave phases to locate sound space directions and distances. Hearing qualities include whisper, speech, music, noise, and scream. Hearing can perceive who is speaking, what their emotional state is, and whether they are lying.
physical properties
Hearable events are mechanical compression-and-rarefaction longitudinal vibrations in air and body tissues, with frequencies 20 Hz to 20,000 Hz. Sound-wave frequencies have intensities, amplitude, and phase.
Two frequencies can have harmonic ratios, with small integers in numerator and denominator.
Sound waves ultimately vibrate cochlea hair cells.
neurons
At low frequencies, sound and neuron activity have same frequency. At high frequencies, nerve-fiber activity distribution represents pitch. Neuron firing rate and number represent sound intensity.
properties: aging
Aging can shift tone sequence.
properties: analytic sense
Tones are independent and do not mix. People can simultaneously hear different frequencies at different intensities.
properties: beats
Sound waves can superpose to create lower-frequency beats.
properties: habituation
Hearing does not habituate quickly.
properties: hearing yourself speak
Bone attenuates higher frequencies, so people hear their own speech as more mellow than others do.
properties: individual differences
Sound has same physical properties for everyone, and hearing processes are similar, so hearing perceptions are similar. All people hear similar tone spectrum, with same tones and tone sequence.
properties: memory
Melodies ending in harmonic cadence are easier to remember than those that end otherwise.
properties: opposites
Tones have no opposites.
properties: precision
People easily distinguish tones and half tones and can distinguish quarter tones after learning. Adjacent-quartertone frequencies differ by several percent.
properties: tempo
People can perceive sound presentation speed: slow, medium, or fast.
properties: time
Hearing is in real time, with a half-second delay.
properties: tone relations
Tones have unique tone relations. A, B, C, D, E, F, and G tone-frequency ratios must be the same for all octaves. Tones, such as middle A, must be two times the frequency of same tone, such as lower A, in next-lower octave. Without constant in-octave and across-octave frequency ratios, tone A becomes tone B or G in other octaves. For normal hearing, tones relate in only one consistent and complete way. Tones cannot substitute and can never be other tones.
properties: tone similarities
Similar tones have similar frequencies or are octaves apart.
properties: waves
Tones directly relate to physical sound-wave frequencies and intensities. Sound waves have emissions, absorptions, vibrations, reflections, and transmissions.
properties: warm and cool
Warm tones have longer and lower attack and decay, longer tones, and more harmonics. Cool tones have shorter and higher attack and decay, shorter tones, and fewer harmonics.
evolution
Hearing evolved from fish lateral line, which has hair cells. Hearing uses one basic receptor type. Reptile hair cells have oscillating potentials from interacting voltage-gated-calcium and calcium-gated-potassium channels, so hair vibrations match sound frequencies. Mammal hair cells vibrate at sound frequencies and have sound-frequency oscillating potentials, but they add force to increase vibration amplitude. Perhaps, the first hearing was for major water vibrations.
development
By 126 days (four months), fetus has first high-level hearing.
Newborns react to loud sounds. If newborns are alert, high sound frequencies cause freezing, but low ones soothe crying and increase motor activity. Rhythmic sounds quiet newborns.
animals
Animals can detect three pitch-change patterns: up, down, and up then down. Bats can emit and hear ultrasound. Some moths can hear ultrasound, to sense bats [Wilson, 1971] [Wilson, 1975] [Wilson, 1998]. Insects can use hearing to locate mates [Wilson, 1971] [Wilson, 1975] [Wilson, 1998].
relations to other senses
Hearing, temperature, and touch involve mechanical energy. Touch can feel vibrations below 20 Hz. Hearing can feel vibrations above 20 Hz. Sound vibrates eardrum and other body surfaces but is not felt as touch.
Vision seems unrelated to hearing, but both detect wave frequency and intensity. Hearing detects longitudinal mechanical waves, and vision detects transverse electric waves. Hearing has ten-octave frequency range, and vision has one-octave frequency range. Hearing has higher energy level than vision. Hearing is analytic, but vision is synthetic. Hearing can have interference from more than one source, and vision can have interference from only one source. Hearing uses phase differences, but vision does not. Hearing is silent from most spatial locations, but vision displays information from all scene locations. Hearing has sound attack and decay, but vision is so fast that it has no temporal properties.
Smell and taste seem unrelated to hearing.
Some people can name heard tones {absolute pitch}|, and this correlates with learning note names when young.
People can listen to one speaker when several speakers are talking {cocktail party effect}. Hearing attends to one message stream by localizing sounds using binaural hearing and sound quality and by inhibiting other message streams.
Seeing lip movement aids auditory perception {McGurk effect}. In humans, sight dominates sound.
Things can be about ears {otic}.
Equal-temperament tones can form mathematical groups {sound dodeconion}. The twelve octave tones and half-tones have equally spaced frequencies. A regular 12-vertex dodecagram has points separated by 30 degrees and can represent the twelve tones, and rotations by 30-degree multiples result in same geometric figure.
frequency ratios
Tone pairs have frequency ratios. Octave from middle-C to high-C has tone frequency ratio 2/1. Middle tone, such as middle-G, makes reciprocal tone ratios, such as middle-C/middle-G, 3/2, and middle-G/high-C, 4/3.
A tube {Eustachian tube}| goes from middle ear to pharynx, to equalize pressure inside and outside eardrums. Pharynx valves close tube when talking but open tube when swallowing or yawning or when outside air pressure changes.
Area {belt area} adjacent to area-A1 primary auditory cortex can receive from area A1 and respond to complex sound features.
Area {parabelt area} laterally adjacent to belt area can receive from belt area and respond to complex sounds and multisensory features.
Cortical frequency-sensitive auditory neurons align from low to high frequency {tonotopic organization}.
Hearing neurons {auditory neuron} receive input from 10 to 30 hair-cell receptors.
frequency
Auditory neurons respond to one frequency, within several percent. Frequencies are between 20 Hz and 20,000 Hz.
intensity
Auditory neurons respond to low, medium, or high intensity. Low-spontaneous-firing-rate neurons {low-spontaneous fiber} are for high-intensity sound and have narrow-band frequency tuning. With no stimulation, their firing rate is less than 10/s. Firing rate rises with intensity {rate-intensity function, neuron}.
High-spontaneous-firing-rate neurons {high-spontaneous fiber} are for low-intensity sound and have broad-band frequency tuning. With no stimulation, their firing rate is greater than 30/s. Firing rate rises with intensity to maximum at low intensity.
Mid-spontaneous fibers are for intermediate-intensity sound. With no stimulation, firing rate is greater than 10/s and less than 30/s.
Free intracellular calcium ions modulate cricket hearing interneurons {omega interneuron} [Huber and Thorson, 1985] [Sobel and Tank, 1994].
Human hearing organs {ear} have outer ear to catch sounds, middle ear to concentrate sounds, and inner ear to analyze sound frequency and intensity.
Pinna and ear canal {outer ear}| gather and focus sound on eardrum.
Only mammal ears have a cartilage flap {pinna}| {pinnae}, to catch sounds.
A 2.5-centimeter tube {auditory canal}| {ear canal}, from outside pinna to inside tympanic membrane, protects tympanic membrane from objects and loud sounds.
Auditory canal has wax {earwax}|. Perhaps, earwax keeps ear canal moist and/or sticks to insects.
Thin connective-tissue membrane {tympanic membrane} {eardrum}| is across ear-canal inner end. Tympanic membrane is 18 times larger than oval window.
Eardrum connects to air cavity {middle ear}|.
Middle ear has three small bones {ossicles}|: hammer, anvil, and stirrup. Two middle ear bones evolved from reptile lower jawbones [Ramachandran, 2004].
Eardrum connects to middle-ear bone {hammer bone}| {malleus}, which connects to anvil.
Hammer bone connects to middle-ear bone {anvil bone}| {incus}, which connects to stirrup. Anvil bone is smaller than hammer bone to concentrate sound pressure.
Anvil bone connects to middle-ear bone {stirrup bone}| {stapes}, which connects to oval window. Stirrup bone is smaller than anvil bone to concentrate sound pressure.
Muscles {tensor tympani muscle} attached to malleus can tense to dampen loud vibration.
Muscles {stapedius muscle} attached to stapes can tense to dampen loud vibration.
A coiled trumpet-shaped fluid-filled organ {inner ear} {cochlea}|, 4 mm diameter and 35 mm long, is in temporal bone.
Inner ear, nearer auditory nerve, has one straight row of 3500 inner hair cells {hair cell, cochlea} and has three S-curved rows with 3500 outer hair cells each (10,500 total). Outer-hair-cell cilia poke through tectorial membrane. Hairs have long part, medium part, and short part, linked by hairs from small tip to medium middle and from medium tip to large middle. Cochlea hair-cell receptors microscopic fibers and microscopic cross-fibers cause resonance between frequencies.
Oval-window movement makes pressure waves, down vestibular canal, which cause middle-canal vertical movement, which slides tectorial-membrane gel horizontally over upright cilia. If pushed one way, hair-cell-membrane potential increases from resting potential. If pushed other way, potential decreases. Inner hair cells send to 10 to 30 auditory neurons.
Outer hair cells can receive brain signals to extend cilia, to stiffen cochlear partition and dampen sound. This reduces signal-to-noise ratio, lowers required input intensity to sharpen tuning, or sends secondary signals to inner hair cells.
Stapes connects to membrane across opening {oval window, hearing}| at cochlea beginning. Oval window is 18 times smaller than tympanic membrane, to concentrate sound pressure.
At base, tympanic canal has soft tissue {round window} that absorbs high pressure.
Cochlea outside has a canal {tympanic canal} {scala tympani}. Tympanic membrane is over tympanic-canal end. Round window is over tympanic-canal base.
Cochlea outside has a canal {vestibular canal} {scala vestibuli}.
Tympanic and vestibular canals join at cochlea point {helicotrema}.
Cochlea middle has a canal {middle canal} {scala media}.
Cochlea inside has a canal {cochlear canal}.
Membrane {Reissner's membrane} separates middle canal and vestibular canal.
In cochlear canal, a coil {basilar membrane} also separates middle canal and tympanic canal. Close to oval window {base, basilar membrane}, basilar membrane is stiff and narrow. At other end {apex, basilar membrane}, basilar membrane is wider and less stiff.
Basilar-membrane structures {organ of Corti} have 30,000 hair-cell receptors, with stereocilia and fibers. Organ-of-Corti base detects high frequencies, and organ-of-Corti apex detects low frequencies (place code).
Gel membrane {tectorial membrane} attaches to end of, and floats in, middle canal and touches outer hair cells.
Basilar membrane, tectorial membrane, and organ of Corti together {cochlear partition} detect sounds. Cochlear partition is in middle canal.
Sounds affect many hair-cell receptors {hearing, physiology}. Hearing finds intensities at frequencies and frequency bands (sound spectrum).
properties: fundamental missing
If people hear harmonics without the fundamental frequency, they hear the fundamental frequency, probably by temporal coding. Amplifying a chord tone causes hearing both tone and its fundamental tone, though fundamental frequency has zero intensity.
properties: octave
Animals conditioned to respond to pitch respond almost equally to its above and below octaves.
properties: phase differences
People cannot hear phase differences, but hearing can use phase differences to locate sounds.
properties: rhythm
Hearing can recognize rhythms and rhythmic groups.
properties: timing
People perceive two sounds less than three milliseconds apart as the same sound.
processes: contrast
Hearing uses lateral inhibition to enhance contrast to distinguish sounds.
processes: damping
Later tones constrain basilar membrane. Lower-frequency later tones constrain basilar membrane more. If later tone is more than 1000 Hz lower than earlier tone, to hear first tone requires high loudness. If later tone is more than 300 Hz higher than earlier tone, to hear first tone requires moderate loudness.
processes: filtering
Hearing integrates over many neurons to filter frequencies to find their individual intensities. Hearing performs limited-resolution Fourier analysis on sound frequencies [Friedmann, 1979].
processes: important sounds
Important sounds use more neurons and synapses.
processes: memory
Previous sound experiences help distinguish current sound patterns.
brain
Because brain is viscous, sound cannot affect brain tissue.
For short sounds in noisy backgrounds, hearing can complete missing sounds or sharpen noisy sounds {continuity effect} {perceptual restoration effect}. Hearing does not fill in short silences with sounds, but sharpens temporal boundaries. Hearing does not know when it fills in.
Sound radiates in all directions from sources and reflects from various surfaces back to ears {echo perception}. Hearing can distinguish echoes from their source sounds. Hearing uses binaural signals to suppress echoes.
Body and head, including pinnae and ear canals, transmit and absorb different-frequency, different-elevation, and different-azimuth sounds differently {head-related transfer function}.
People can perceive sound frequency {pitch, sound}|.
frequency
People can hear ten frequency octaves, from 20 Hz to 20,000 Hz. Lowest frequencies, 20 Hz to 30 Hz, are also highest vibrations detectable by touch.
Shortest hair-cell hair lengths detect highest frequencies. High-frequency tones vibrate basilar-membrane stiff narrow end, far from oval window. Above 3000 Hz, higher hearing neurons respond to frequency, tone pattern, or intensity range.
Low-frequency tones activate all hair cells, with greater activity near oval window and its long-hair hair cells.
sensitivity
People are most sensitive at frequency 1800 Hz.
neuron firing
Maximum neuron firing rate is 800 Hz. After sound frequency and firing rate reach 800 Hz, firing rate drops abruptly, and more than one neuron carries sound-frequency information. After sound frequency and firing rate reach 1600 Hz, firing rate drops abruptly.
Auditory neurons have frequency {characteristic frequency} (CF) at which they are most sensitive. The characteristic frequency is at the maximum of the frequency-intensity spectrum (threshold tuning curve). For CF = 500 Hz at 0 dB, 1000 Hz is at 80 dB, and 200 Hz is at 50 dB. For CF = 1100 Hz at 5 dB, 1500 Hz is at 80 dB, and 500 Hz is at 50 dB. For CF = 2000 Hz at 5 dB, 3500 Hz is at 80 dB, and 500 Hz is at 80 dB. For CF = 3000 Hz at 5 dB, 3500 Hz is at 80 dB, 700 Hz to 2000 Hz is at 50 dB, and 500 Hz is at 80 dB. For CF = 8000 Hz at 5 dB, 9000 Hz is at 80 dB, 1000 Hz to 3000 Hz is at 60 dB, and 500 Hz is at 80 dB. For CF = 10000 Hz at 5 dB, 10500 Hz is at 80 dB, 5000 Hz is at 80 dB, 1000 Hz to 2000 Hz is at 60 dB, and 500 Hz is at 80 dB.
Auditory-nerve channels carry frequency-range {critical band} information.
For 100-Hz to 6000-Hz sound stimuli, basilar membrane has electric pulses, with same frequency and intensity, caused by potentials from all hair cells, that do not fatigue.
For 20-Hz to 900-Hz sound stimuli, auditory-neuron axons have electric pulses {microphonic electric pulse}, measured in cochlear nerve, with same frequency and intensity [Saul and Davis, 1932]. For 900-Hz to 1800-Hz sound stimuli, auditory-neuron axons have electric pulses with same frequency and one-half intensity. For 1800-Hz to 2700-Hz sound stimuli, auditory-neuron axons have electric pulses with same frequency and one-third intensity. For above-2700-Hz sound stimuli, auditory-neuron axons have electric pulses that do not correlate with frequency and intensity. Perhaps, auditory nerve uses summed potentials of microphonic-electric-pulse envelopes.
For below-500-Hz sound stimuli, auditory-neuron-axon signals have same frequency and phase {phase locking, hearing}.
Similar frequencies group together to make increasing loudness {recruitment, hearing}.
Tones that share one octave have perceivable sound features {tone chroma}.
Tone frequency determines low or high pitch {tone height}.
Noise or tones within two octaves of stimulus frequency can interfere with stimulus perception {critical band masking}. Pure tones mask high frequencies more than low frequencies, because higher frequencies activate smaller basilar-membrane regions. Complex tones mask low frequencies more than high frequencies, because lower frequencies have more energy than higher frequencies [Sobel and Tank, 1994].
Previous-tone {preceding tone} intensity-frequency spectrum affects neuron current-tone response.
Different later tone can decrease auditory-neuron firing rate {two-tone suppression}.
At each audible frequency, people have an intensity threshold {audibility curve}.
At each audible frequency, specific sound-pressure levels (SPL) cause people to hear equal loudness {equal loudness curve}.
At constant amplitude, auditory-neuron firing rate depends on frequency {isointensity curve}. For amplitude 20 dB at characteristic frequency, firing rate is 180 per second. For amplitude 20 dB at 500 Hz below or 500 Hz above characteristic frequency, firing rate is 50 per second. For amplitude 20 dB at 1300 Hz to 1400 Hz above characteristic frequency, auditory neurons have spontaneous firing rate.
At each frequency, people have a sound-intensity threshold {threshold tuning curve}.
Same-intensity-and-pitch sounds can have different harmonics {timbre, sound}|. Rapid timbre changes are difficult to perceive.
Clear tones {clarity, tone} have narrow frequency band. Unclear tones have wide frequency band.
Full tones {fullness, tone} have many frequency resonances. Shallow tones have few frequency resonances.
Shrill tones {shrillness} have higher frequencies. Dull tones have lower frequencies.
Sounds with many high-frequency components seem sharp or strident {stridency}. Tones with mostly low-frequency components seem dull or mellow {mellowness}.
People can hear sound energies as small as random air-molecule motions. {hearing, intensity} {sound intensity}. Because oval window is smaller than eardrum, sound pressure increases in middle ear. Middle-ear bones increase sound intensity by acting as levers that convert distance into force.
distortion
High sound intensities can strain materials past their elastic limit, so intensity and/or frequency change.
frequency
For same stimulus-input energy, low-frequency tones sound louder, and high-frequency tones sound quieter. Smaller hair-cell hairs have faster vibrations and smaller amplitudes.
maximum sound
Maximum sound is when physical ear structures have inelastic strain, which stretches surface tissues past point to which they can completely return.
pain
Maximum sound causes pain.
rate
For amplitude 40 dB to 80 dB at frequency between 2000 Hz below and 50 Hz above characteristic frequency, maximum firing rate is 280 per second {rate saturation, hearing}.
temporal integration
If sound has constant intensity for less than 100 ms, perceived loudness decreases {temporal integration, hearing}. If sound has constant intensity for 100 ms to 300 ms, perceived loudness increases. If sound has constant intensity for longer than 300 ms, perceived loudness is constant.
At loud-sound onset, stapedius and tensor tympani muscles contract {acoustic reflex}, to dampen stapes and eardrum vibration.
Tones can rise quickly or slowly from background noise level to maximum intensity {attack, hearing}| {onset, hearing}. Fast onset sounds aggressive. Slow onset sounds peaceful.
Tones can fall slowly or rapidly from maximum to background noise level {decay, hearing} {offset, hearing}.
Hearing perceives sound-source locations {source location} {sound location}, in space. Most space locations are silent. One space location can have several sound sources. Hearing determines sound location separately and independently of perceiving tones.
azimuth
Hearing can calculate angle to right or left, from straight-ahead to straight-behind, in horizontal plane.
elevation
Hearing can calculate height and angle above horizontal plane. People perceive lower frequencies as slightly lower than actual elevation. People perceive higher frequencies as slightly higher than actual elevation.
frequency and distance
Sound sources farther than 1000 meters have fewer high frequencies, because of air damping.
sound reflection and distance
Sound energy comes directly from sources and reflects from other surfaces. Close sounds have more direct energy than reflected energy. Far sounds have more reflected energy than direct energy. Reflected sounds have fewer high frequencies than direct sounds, because longer distances cause more air damping.
Hearing can separate complex sounds from one source into independent continuous sound streams {auditory stream segregation}.
Sound grouping has same Gestalt laws as visual grouping.
If one ear hears melody with large ascending and descending tone jumps, and other ear hears another melody with large ascending and descending tone jumps, people do not hear left-ear melody and right-ear melody but hear two melodies, different than either original melodies, that depend on alternating-tone proximities.
People separate sounds from multiple sources into independent continuous sound streams {auditory scene analysis} {source segregation}. Hearing separates sounds from different locations into independent continuous sound streams {spatial separation, hearing}.
Having two ears {binauralism} allows calculating time and amplitude differences between left-ear and right-ear sound streams from same space location.
Hearing can reject unwanted messages {focusing, hearing}, using binauralism to localize sounds.
The same sound reaches right and left ear at different intensity levels {interaural level difference} (ILD). Level difference can be as small as 1 dB. Intensity difference reflects stimulus distance, approaching or receding sounds, and body sound damping. Slight head movements are enough to eliminate direction ambiguity. Intensity differences due only to sound distance, or to approaching or receding sounds, are useful up to one or two meters. Beyond two meters, differences are too small to detect.
damping
Pinnae and head bones absorb sounds with frequencies higher than 1500 Hz, according to their frequency-related dampening function. Pinnae and head-bone damping differs on right and left, depending on source location, and hearing uses the intensity differences to determine space directions and distances beyond one or two meters.
brain
Lateral superior olive detects intensity-level differences between left-right ears and right-left ears, to make opponent systems. To find distance, two receptor outputs go to two different neurons, which both send to difference-finding neuron. Opposite-ear output goes to trapezoid-body medial nucleus, which lies beside pons lateral superior olive and inhibits same-ear lateral-superior-olive output. Interaural time difference and interaural level difference work together.
The same sound reaches right and left ear at different times {interaural time difference, hearing} (ITD), because distances from source location to ear differ, and ears have distance between them. Hearing can detect several microseconds of time difference. Slight head movements are enough to eliminate direction ambiguity. Interaural time difference uses frequencies lower than 1500 Hz, because they have no body damping.
Medial superior olive detects time differences between left-right ears and right-left ears, to make opponent systems. To find distances, two receptor outputs go to two different neurons, which both send to difference-finding neuron. Interaural time difference and interaural level difference work together.
In a cone {cone of confusion} {confusion cone} from head center into space, sounds have same intensity and timing, because ear timing differences (interaural time difference) and intensity differences (interaural level difference) are zero.
Electronic instruments {audiometer}| can test hearing.
Amplified auditory-nerve signals played through speakers sound same as stimulus sounds {microphone effect}.
People can study subjective sense qualities or psychological changes evoked by sound stimuli {psychoacoustics}.
Physical sound attributes directly relate to music attributes {music, hearing} {hearing, music}. Physical-sound frequency relates to music pitch. Music is mostly about frequency combinations. However, above 5000 Hz, musical pitch is lost. Physical-sound intensity relates to music loudness. Physical-sound duration relates to music rhythm. Physical=sound spectral complexity relates to music timbre.
However, frequency affects loudness. Intensity affects pitch. Tone frequency separation affects time-interval perception. Harmonic fluctuations, pitch changes, vibrato, and non-pitched-instrument starting noises {transient, sound} affect timbre. Timbre affects pitch.
emotion: chords
Chords typically convey similar feelings to people. Minor seventh is mournful. Major seventh is desire. Minor second is anguish. Humans experience tension in dissonance and repose in consonance.
emotion: pitch change
Music emotions mostly depend on relative pitch changes (not rhythm, timbre, or harmony).
emotion: key
Music keys have characteristic emotions. Composers typically repeat same keys and timbre, and composers have typical moods.
song: melody
Note sequences can rise, fall, or stay same. People can recognize melodies from several notes.
song: musical phrase
People perceive music phrase by phrase, because phrases have repeated often and because phrases take one breath. Children complete half-finished musical phrases using tones, rhythm, and harmony.
brain
No brain region is only for music. Music uses cognitive and language regions.
Musical pitch makes musical notes {tone, hearing}.
octave
Tones can be double or half other-tone frequencies. Octaves go from a note to similar higher note, such as middle-C at 256 Hz to high-C at 512 Hz. Hearing covers ten octaves: 20 Hz, 40 Hz, 80 Hz, 160 Hz, 320 Hz, 640 Hz, 1280 Hz, 2560 Hz, 5120 Hz, 10240 Hz, and 20480 Hz.
octave tones
Within one octave are 7 whole tones, 7 + 5 = 12 halftones, and 24 quartertones.
overtones
Tones two, four, eight, and so on, times fundamental frequency are fundamental-frequency overtones.
sharpness or flatness
Fully sharp tone has frequency one halftone higher than tone. Slightly sharp tone has frequency slightly higher than tone. Fully flat tone has frequency one halftone lower than tone. Slightly flat tone has frequency slightly lower than tone.
musical scales
Musical scales have tone-frequency ratios. Using ratios cancels units to make relative values that do not change when units change.
equal temperament scale
Pianos have musical tones separated by equal ratios. Octave has twelve equal-temperament halftones, with ratios from 2^(0/12) to 2^(12/12) of fundamental frequency. Frequency ratio of halftone to next-lower halftone, such as C# to C, is 2^(1/12) = 2^.08 = 1.06. Starting at middle-C, ratios of tones to middle-C are 2^0 = 1 for middle-C, 2^.08 = 1.06 for C#, 2^.17 = 1.13 for D, 2^.25 = 1.19 for D#, 2^.33 = 1.26 for E, 2^.42 = 1.34 for F, 2^.50 = 1.41 for F#, 2^.58 = 1.49 for G, 2^.67 = 1.59 for G#, 2^.75 = 1.68 for A, 2^.83 = 1.78 for A#, 2^.92 = 1.89 for B, and 2^1 = 2 for high-C. See Figure 1. F# is middle tone.
equal-temperament scale: frequencies
Using equal-temperament tuning and taking middle-C as 256 Hz, D has frequency 289 Hz. E has frequency 323 Hz. F has frequency 343 Hz. G has frequency 384 Hz. A has frequency 430 Hz. B has frequency 484 Hz. High-C has frequency 512 Hz. Low-C has frequency 128 Hz. Low-low-C has frequency 64 Hz. Lowest-C has frequency 32 Hz. High-high-C has frequency 1024 Hz. Higher Cs have frequencies 2048 Hz, 4096 Hz, 8192 Hz, and 16,384 Hz. From 32 Hz to 16,384 Hz covers nine octaves.
tone-ratio scale
Early instruments used scales with tones separated by small-integer ratios. Tones had different frequency ratios than other tones.
tone-ratio scale: all possible small-integer ratios
In one octave, the 45 possible frequency ratios with denominator less than 13 are: 3/2; 4/3, 5/3; 5/4, 7/4; 6/5, 7/5, 8/5, 9/5; 7/6, 11/6; 8/7, 9/7, 10/7, 11/7, 12/7, 13/7; 9/8, 11/8, 13/8, 15/8; 10/9, 11/9, 13/9, 14/9, 16/9, 17/9; 11/10, 13/10, 17/10, 19/10; 12/11, 13/11, 14/11, 15/11, 16/11, 17/11, 18/11, 19/11, 20/11, 21/11; 13/12, 17/12, 19/12, and 23/12.
tone-ratio scale: whole tones
In octaves, the seven whole tones are do, re, mi, fa, so, la, ti, and do, for C, D, E, F, G, A, and B. The seven tones are not evenly spaced by frequency ratio. Frequency ratios are D/C = 6/5, E/C = 5/4, F/C = 4/3, and G/C = 3/2. For example, C = 240 Hz, D = 288 Hz, E = 300 Hz, F = 320 Hz, and G = 360 Hz. C, D, E, F, and G, and G, A, B, C, and D, have same tone progression. Frequency ratios are A/G = 6/5, B/G = 5/4, C/G = 4/3, and D/G = 3/2. For example, G = 400 Hz, A = 480 Hz, B = 500 Hz, C = 532 Hz, and D = 600 Hz.
tone-ratio scale: halftones
Using C as fundamental, the twelve halftones have the following ratios, in increasing order. 1:1 = C. 17:16 = C#. 9:8 = D. 6:5 = D#. 5:4 = E. 4:3 = F. 7:5 = F#. 3:2 = G. 8:5 = G#. 5:3 = A. 7:4 or 16:9 or 9:5 = A#. 11:6 or 15:8 = B. 2:1 = C.
tone-ratio scale: quartertones
The 24 quartertones have the following ratios, in increasing order. 1:1 = 1.000. 33:32 = 1.031. 17:16 = 1.063, or 16/15 = 1.067. 13:12 = 1.083, 11:10 = 1.100, or 10/9 = 1.111. 9:8 = 1.125. 8:7 = 1.143, or 7:6 = 1.167. 6:5 = 1.200. 17:14 = 1.214, or 11/9 = 1.222. 5:4 = 1.250. 9:7 = 1.286. 4:3 = 1.333. 11:8 = 1.375. 7:5 = 1.400. 17:12 = 1.417, 10:7 = 1.429, or 13/9 = 1.444. 3:2 = 1.500. 14/9 = 1.556, 11:7 = 1.571, or 19:12 = 1.583. 8:5 = 1.600. 13:8 = 1.625. 5:3 = 1.667. 12:7 = 1.714, or 7:4 = 1.75. 16:9 = 1.778, or 9:5 = 1.800. 11:6 = 1.833, or 13:7 = 1.857. 15:8 = 1.875. 23:12 = 1.917. 2:1 = 2.000. Ratios within small percentage are not distinguishable.
tone intervals
Two tones have a number of tones between them. First interval has one tone, such as C. Minor second interval has two tones, such as C and D-flat, and covers one halftone. Major second interval has two tones, such as C and D, and covers two halftones. Minor third interval has three tones, such as C, D, and E-flat, and covers three halftones. Major third interval has three tones, such as C, D, and E, and covers four halftones. Minor fourth interval has four tones, such as C, D, E, and F, and covers five halftones. Major fourth interval has four tones, such as C, D, E, and F#, and covers six halftones. Minor fifth interval has five tones, such as C, D, E, F, and G-flat, and covers six halftones. Major fifth interval has five tones, such as C, D, E, F, and G, and covers seven halftones. Minor sixth interval has six tones, such as C, D, E, F, G, and A-flat, and covers eight halftones. Major sixth interval has six tones, such as C, D, E, F, G, and A, and covers nine halftones. Minor seventh interval has seven tones, such as C, D, E, F, G, A, and B-flat, and covers ten halftones. Major seventh interval has seven tones, such as C, D, E, F, G, A, and B, and covers eleven halftones. Eighth interval is octave, has eight tones, such as C, D, E, F, G, A, B, and high-C, and covers twelve halftones.
tone intervals: pairs
Tones have two related ratios. For example, D and middle-C, major second, have ratio 289/256 = 9/8, and D and high-C, minor seventh, have ratio 9/16, so high-C/D = 16/9. The ratios multiply to two: 9/8 * 16/9 = 2. E and middle-C, major third, have ratio 323/256 = 5/4, and E and high-C, minor sixth, have ratio 5/8, so high-C/E = 8/5. F and middle-C, minor fourth, have ratio 343/256 = 4/3, and F and high-C, major fifth, have ratio 2/3, so high-C/G = 3/2. G and middle-C, major fifth, have ratio 384/256 = 3/2, and G and high-C, minor fourth, have ratio 3/4, so high-C/G = 4/3. A and middle-C, major sixth, have ratio 430/256 = 5/3, and A and high-C, minor third, have ratio 5/6, so high-C/A = 6/5. B and middle-C, major seventh, have ratio 484/256 = 15/8, and B and high-C, minor second, have ratio 15/16, so high-C/B = 16/15.
The ratios always multiply to two. Tone-interval pairs together span one octave, twelve halftones. For example, first interval, with no halftones, and octave, with twelve halftones, fill one octave. Major fifth interval, with seven halftones, such as C to G, and minor fourth interval, with five halftones, such as G to high-C, fill one octave. Major sixth interval, with nine halftones, such as C to A, and minor third interval, with three halftones, such as A to high-C, fill one octave. Major seventh interval, with eleven halftones, such as C to B, and minor second interval, with one halftone, such as B to high-C, fill one octave. Minor fifth interval and major fourth interval fill one octave. Minor sixth interval and major third interval fill one octave. Minor seventh interval and major second interval fill one octave.
tone intervals: golden ratio
In music, ratio 2^0.67 = 1.59 ~ 1.618... is similar to major sixth to octave 1.67, octave to major fourth 1.6, and minor seventh to major second 1.59. Golden ratio and its inverse can make all music harmonics.
tone harmonics
Tones have harmonics {tone harmonics} that relate to tone-frequency ratios.
tone harmonics: consonance
Tone intervals can sound pleasingly consonant or less pleasingly dissonant. Octave tone intervals 2/1 have strongest harmonics. Octaves are most pleasing, because tones are similar. Tones separated by octaves sound similar.
Major fifth and minor fourth intervals are next most pleasing. Major-fifth 3/2 and minor-fourth 4/3 tone intervals have second strongest harmonics.
Major third 5/4 and minor sixth 8/5 intervals are halfway between consonant and dissonant. Minor third 6/5 and major sixth 5/3 intervals are halfway between consonant and dissonant.
Major fourth 7/6 and minor fifth 12/7 intervals are dissonant. Major second 8/7 and minor seventh 7/4 intervals are dissonant, or major second 9/8 and minor seventh 16/9 intervals are dissonant. Minor second 16/15 and major seventh 15/8 intervals are most dissonant.
Ratios with smallest integers in both numerator and denominator sound most pleasing to people and have consonance. Ratios with larger integers in both numerator and denominator sound less pleasing and have dissonance.
Three tones can also have consonance or dissonance, because three tones make three ratios. For example, C, E, and G have consonance, with ratios E/C = 5/4, G/E = 6/5, and G/C = 3/2.
Tone ratios in octaves higher or lower than middle octave have same consonance or dissonance as corresponding tone ratio in middle octave. For example, high-G and high-C have ratio 6/4 = 3/2, same as middle-G/middle-C.
Tone ratios between octave higher than middle octave and middle octave have similar consonance as corresponding tone ratio in middle octave. For example, high-G and middle-C have ratio 3/1. Dividing by two makes high-G one octave lower, and middle-G/middle-C has ratio 3/2.
tone harmonics: beat frequencies
Frequencies played together cause wave superposition. Wave superposition makes new beat frequencies, as second wave regularly emphasizes first-wave maxima. Therefore, beat frequency is lower than highest-frequency original wave.
If wave has frequency 1 Hz, and second wave has frequency 3 Hz, they add to make 1-Hz wave, 3-Hz wave, and 2-Hz wave, because every other 3-Hz wave receives boost from 1-Hz wave. Rising 1-Hz wave maximum coincides with first rising 3-Hz wave maximum and falling 1-Hz wave maximum coincides with third falling 3-Hz maximum, while first falling 3-Hz wave maximum, middle rising and falling 3-Hz maximum, and third rising 3-Hz maximum cancel.
If one wave has frequency 2 Hz, and second wave has frequency 3 Hz, they add to make 2-Hz wave, 3-Hz wave, and 1-Hz wave, because every third 3-Hz wave receives boost from 2-Hz wave. First rising 2-Hz wave maximum coincides with first rising 3-Hz wave maximum, while first falling 3-Hz wave maximum, middle rising and falling 3-Hz maximum, and third rising and falling 3-Hz maximum cancel.
Beat frequency is difference between wave frequencies: 3 Hz - 2 Hz = 1 Hz in previous example. Beat frequencies are real physical waves.
Small-integer frequency ratios have lower beat frequencies and reduce beat frequency interference with original frequencies. Two waves with small-integer frequency ratios superpose to have beat frequency that has small-integer ratios with original frequencies. Two waves with large-integer frequency ratios superpose to have beat frequency that has large-integer ratios with original frequencies.
Middle-C has frequency 256 Hz, and middle-G has frequency 384 Hz, with ratio G/C = 3/2. The waves add to make 384 Hz - 256 Hz = 128 Hz beat wave, with ratio C/beat = 2/1 and G/beat = 3/1.
Middle-C has frequency 256 Hz, and middle-E has frequency 323 Hz, with ratio E/C = 5/4. The waves add to make 323 Hz - 256 Hz = 67 Hz beat wave, with ratio C/beat = 4/1 and E/beat = 5/1.
Middle-C has frequency 256 Hz, and middle-D has frequency 289 Hz, with ratio D/C = 9/8. The waves add to make 289 Hz - 256 Hz = 33 Hz beat wave, with ratio C/beat = 8/1 and D/beat = 9/1.
Middle-C has frequency 256 Hz, and middle-A has frequency 430 Hz, with ratio A/C = 5/3. The waves add to make 430 Hz - 256 Hz = 174 Hz beat wave, with ratio C/beat = 3/2 and D/beat = 5/2.
Middle-C has frequency 256 Hz, and middle-B has frequency 484 Hz, with ratio B/C = 15/8. The waves add to make 484 Hz - 256 Hz = 228 Hz beat wave, with ratio C/beat = 9/8 and B/beat = 17/8.
Roger Shepard [1964] gradually increased or decreased all tones of a chord, keeping the tones separated by octaves. Pitch repeats when reaching the next octave, so tones rise or fall but do not keep rising or falling {Shepard tone} {Shepard scale}, an auditory illusion.
Brain recognizes music by rhythm or by intonation differences near main note {music, processing}. Brain analyzes auditory signals into tone sequences with pitches, durations, amplitudes, and timbres. First representation {grouping structure} segments sound sequence into motifs, phrases, and sections. Second representation {metrical structure} marks sequence with hierarchical arrangement of time points {beat}.
Brain can find phrasing symmetries {time-span reduction}, using grouping and metrics.
Brain can hierarchically arrange tension and relaxation waves {prolongational reduction}. In Western music, prolongational reduction has slowly increasing tension followed by rapid relaxation.
Brain-injured people can be unable to distinguish voices but can recognize other sound types {hearing, problems}. If they listen to speech recorded using different voices for different syllables, they cannot understand words.
Middle-ear bone or tendon damage decreases sound amplitude {conductive hearing loss}.
Infection causes middle-ear inflammation {otitis media}|, typically in children.
Middle-ear bones can grow abnormally {otosclerosis}|, affecting hearing.
Adverse conditions {ototoxic} can affect balance or hearing more than other systems.
Auditory-nerve or cochlea damage decreases loudness {sensorineural hearing loss}.
Perhaps, cochlea has band-pass filters {critical band theory}.
Perhaps, brain detects sounds by adding harmonic frequencies below 20 Hz, weighted by ratios {harmonic weighting}. 360 Hz uses 180/2, 120/3, 90/4, 72/5, 60/6, 51.4/7, 45/8, 40/9, 36/10, 32.7/11, 30/12, and so on. 720 Hz uses 360/2, 240/3, 180/4, 144/5, 120/6, 102.8/7, 90/8, 80/9, 72/10, 65.4/11, 60/12, 51.4/14, 45/16, 40/18, 36/20, 32.7/22, 30/24, and so on.
At frequencies above 900 Hz, brain detects stimulus frequency by cochlea-hair maximum-amplitude location {place coding} {place theory}, so pitch depends on activity distribution across nerve fibers.
At frequencies below 900 Hz, brain detects stimulus frequency by impulse timing {temporal theory} {temporal code}, because timing tracks frequency. Adjacent auditory neurons fire at same phase {phase locking, code} and frequency, because adjacent hair cells link and so push and pull at same time.
Perhaps, sound intensity depends on number of activated basilar-membrane sense cells and special high-threshold cells {threshold, hearing} [Wilson, 1971] [Wilson, 1975] [Wilson, 1998].
For frequencies less than 2400 Hz, frequency detection depends on cooperation between neuron groups firing in phase {volley theory} {volley code}. For frequencies less than 800 Hz, auditory-neuron subsets fire every cycle. For frequencies above 800 Hz and less than 1600 Hz, auditory-neuron subsets fire every other cycle. For frequencies above 1600 Hz and less than 2400 Hz, auditory-neuron subsets fire every third cycle {volley principle}. For example, three neurons firing at 600 Hz every third cycle can represent frequency of 1800 Hz.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225