Why can we hear a sound with one ear (e.g. the left) when it is actually coming from the direction of the other (e.g. the right)?

• Indeed, if we expose the right ear to light, the left one will be in shadow. So is there no shadow for sound waves?
• There is, but a shadow is a property that depends closely on the wavelength of the waves considered. The closer the size of the obstacles gets to the wavelength, the more blurred the shadow becomes. When the size of the obstacle is comparable to the wavelength of the wave striking it, the shadow no longer exists, and the wave can propagate beyond the obstacle, going around it as if it did not exist at all. This phenomenon is called diffraction. You can see an example by following the instructions on the page diffraction by an obstacle.
• The width of the head (about twenty centimetres) is comparable to the wavelength of many audible sounds. These sounds can therefore go around the obstacle and reach the other ear.
• Of course, for some (high) frequencies the head is a real obstacle, so these frequencies will reach the ear "in shadow" strongly attenuated (they will hardly fail to arrive at all, because the two ears are not completely isolated from each other, and sound can also propagate through the skeleton).
• This means that the spectral content of a complex sound, that is, the intensity of the frequencies that make it up, will be slightly different at the two ears when it comes from one side rather than from the front!

What is the purpose of having two ears?

• The brain can very quickly analyse a great deal of information about an acoustic signal reaching the auditory system. With two ears it can also analyse the difference between two different "perceptions" of the same source. In particular:
  1. a sound coming from the right reaches the right ear slightly earlier than the left.
  2. The signal will be more intense on the right because the head acts as an obstacle for the left ear, especially at high frequencies.
  3. Moreover (see the previous question) the sound reaching the left ear will also have a slightly darker timbre than the one reaching the right ear.
• By processing these differences, the brain can determine information such as the distance and position of the sound source.

Having two ears is roughly the equivalent of having two eyes: it adds the third dimension to the sound field around us. For more details see the page on localisation of the sound source.

How far out of tune must a note be for us to notice?

It depends on how the sounds are compared.

• Sounds heard one after the other. The smallest perceptible difference in pitch between two pure tones is called the difference limen (or just noticeable difference). In the middle region of audible frequencies, at moderate volume, it is a few thousandths of the frequency, that is roughly between 3 and 10 cents. It is larger at low frequencies and smaller for trained listeners. For example, at middle C (262 Hz) 6 cents correspond to less than 1 Hz. It is a statistical threshold, which varies from person to person.
• Sounds heard together. When two notes sound at the same time the ear is much more sensitive, because small differences produce beats. Two sounds in unison that differ by 1 Hz produce one beat per second, which is clearly audible. The same applies to other intervals, thanks to beats between harmonics: in a fifth played with sustained tones even an error of 2 cents produces slow but audible beats (see the question on mistuned consonances).
• In real music the tolerance is much wider. The notes of instruments are not pure tones and vibrato makes the pitch oscillate by even tens of cents. Moreover, small differences in tuning are often perceived as differences in timbre rather than in intonation. A piano on which a few notes were tuned a few cents away from equal temperament would probably not sound "out of tune", only a little different. After all, not even a normal piano is tuned exactly to equal temperament: because of the stiffness of the strings, their harmonics are slightly higher than exact multiples of the fundamental. For this reason the tuner slightly "stretches" the octaves.

The effects of different scales and tunings can be heard on the page comparisons between different scales.

Why is an E "the same" E even in different octaves?

Two notes an octave apart have frequencies in the ratio 2:1, yet we call them by the same name and perceive them as the same note, higher or lower. There is no structure in the ear that resonates for each note: the cochlea analyses frequencies continuously, like a bank of very many resonators tuned to closely spaced frequencies (see physiology of the auditory system). Two notes an octave apart therefore excite different regions. The similarity has two main origins.

• The harmonics coincide. The harmonics of the higher note (2f, 4f, 6f...) all coincide with harmonics of the lower note. Played together, the two notes produce no beats and almost fuse into a single sound.
• The periodicity is compatible. A sound of frequency 2f repeats after a time 1/(2f). So it also repeats after twice that time, that is 1/f. The auditory system also derives pitch from the periodicity of the signal (see the missing fundamental). A note and its octave share this periodicity. Not by chance, "octave errors", that is mistaking a note for the same note an octave higher or lower, are frequent even among musicians.

For this reason the pitch of a sound is often described by two components. The first is tone height (higher or lower), which increases continuously with frequency. The second is chroma (the name of the note), which repeats at every octave. Illusions such as the Shepard scale are based on this distinction.

Octave equivalence therefore has a physical basis, but also a cultural component. It is recognised in almost all musical traditions, but it is not absolute: for example, many listeners prefer slightly "stretched" octaves (see the dialogue on pitch perception).

Why can one word in a recording sound higher than another, even though its fundamental frequency is lower?

A real example: in the spectrogram of the English sentence one of these nice new books, the word nice has a fundamental of about 290 Hz, while new has a fundamental of about 150 Hz, almost an octave below. Yet when listening, new may sound about a semitone above nice.

This is not a missing fundamental: in the spectrum of new the harmonics are spaced by about 150 Hz, so the fundamental is there. It is, however, much weaker than the second harmonic, which lies at about 305 Hz, almost a semitone above the fundamental of nice. Low frequencies are often attenuated by small microphones and loudspeakers, so the effect also depends on how the sound is recorded and played back.

A note and its octave have the same name (see the previous question), and the auditory system is more certain of the chroma than of the octave. When the fundamental is weak and the second harmonic dominates, it is easy to assign the sound a pitch an octave above the "physical" one. Comparing the voice with the notes of a piano then gives the right name (about a semitone above nice), but not necessarily the right octave. Indeed, listening to the same file on headphones, other listeners hear new as lower than nice.

Are there "lower harmonics", that is sounds at sub-multiple frequencies, generated by the ear?

In general, no. A linear system excited by a pure tone responds at the same frequency. A non-linear system, such as the ear at high volume, adds higher harmonics (multiples of the frequency) and, if there are two or more sounds, combination frequencies, that is sums and differences of the frequencies present (see Tartini's third sound). A difference frequency can even be lower than all the original sounds, but it is not a sub-multiple of any of them.

Sub-multiple frequencies (subharmonics) appear only in strongly non-linear systems and under special conditions, for example in certain voice sounds or in overdriven wind instruments. No mechanism of the ear is known that generates subharmonics of a pure tone. The "series of lower harmonics" (undertone series) found in some music theories is a theoretical construction, symmetrical to the harmonic series, with no physical counterpart in sounds.

The brain can, however, "add" a note lower than those present when it reconstructs the pitch of a complex sound: this is the case of the missing fundamental.

Do two instruments playing the same note at different distances from the listener seem to have different pitches?

No, if they are tuned alike. Propagation does not change the frequency: the sound of the distant instrument arrives later, but with the same frequency. The frequency changes only if the source or the listener is moving (see Doppler effect).

Other things change instead.

• Loudness: the distant instrument sounds weaker.
• Timbre: high frequencies are attenuated by the air and absorbed by surfaces more than low ones, so from a distance the sound is "darker". In a church or a large hall, moreover, the distant sound arrives mixed with many more reflections (see architectural acoustics).

These differences in timbre and loudness can sometimes make the pitch seem different, but only slightly: for pure tones perceived pitch depends slightly on loudness, while for the sounds of instruments the effect is negligible. If the two instruments are not perfectly in tune, you hear them beat wherever you are.