Music and the Missing Fundamental
By Marshall Chasin On March 27, 2012 ·
When it comes to music, is an extended low end better?
The energy spectrum of speech is from roughly the middle of the piano keyboard to a little higher than the highest note on the right side. The entire left side of the piano keyboard is unknown to speech and questions arise why we even have a left side of the keyboard. Cynical people may say that it#8217;s simply to give the left hand something to do.
Less cynical people may say that it gives the rhythm that is missing when the right hand is doing all the work. But do we need to hear the left hand side of the piano in order to #8220;hear#8221; the left hand side of the piano keyboard? How#8217;s that for sounding like a tautology?
To see where I#8217;m going with this, let#8217;s turn our attention to the telephone and the human voice. The pitch of the human voice is given by the fundamental frequency usually written as f0 or F0. For men, it#8217;s about 125 Hz, for women it#8217;s about an octave higher, and for children, it#8217;s still higher. The telephone however only is capable of transducing sounds between 340 Hz and 3400 Hz. If a telephone cannot transduce anything below 340 Hz, how can one know on the phone whether the speaker is a man, woman, or child? A frequency of 125 Hz simply cannot be transduced over the phone yet people rarely make mistakes when identifying whether someone is a male or a female.
This conundrum derives from an erroneous assumption that something needs to be heard in order to be perceived, and also exactly what that #8220;something#8221; is.
This conundrum is based on the missing fundamental- how can a man#8217;s pitch (of 125 Hz) be perceived if only the third or fourth harmonic is heard (harmonic energy that is within the band width of a telephone). The answer is that it is erroneous to assume that the pitch is the fundamental frequency, even though numerically it is the value of f0. It turns out that the pitch is the difference in frequency between any two successive fundamentals.
I know that this sounds like splitting hairs, but let#8217;s consider a man#8217;s voice (or a violin) where the fundamental frequency f0 is at 125 Hz (close to an octave below middle C). Like a violin, the human voice is considered to be a half wavelength resonator- the vocal chords are held rigidly at both ends of the larynx (or voice box). All half wavelength resonators have integer numbered harmonics of the fundamental frequency. If f0 is 125 Hz, then there are equally spaced harmonics at 250 Hz, 375 Hz, 500 Hz, 625 Hz, 750 Hz, #8230; Notice that the difference between any two successive harmonics is also 125 Hz and it#8217;s this difference that is the main element in perceiving the pitch of the sound- the difference has the same numerical value as f0. While f0 is not within the bandwidth of a telephone, the upper frequency harmonics are and their differences in frequency tell us about the more bass sounds that have missed the boat.
Hearing aids that can transduce only sounds above middle C then are not overly limited. It is true that the bandwidth does not go down to 50 Hz, but the harmonically related cousins of this missing 50 Hz are well within the bandwidth of the hearing aid. Enhancing the lower frequency regions of the hearing aid will serve to increase audibility of non-speech noises and increase the masking of the more important elements of speech. Hard of hearing consumers may feel that their own voice is echoey and reverberant.
Having a hearing aid bandwidth that has been optimized for speech is not all that different than one that can be optimal for music as well. The left hand side of the piano, like a child of the 19th century, is best seen but not heard#8230;
But wait a minute! When listening to music we are not just listening to the notes. We are feeling the bass; the vibro-tactile response; the overall #8220;feel#8221; of the music. And this is where real low frequency energy that originates from the left hand side of the piano keyboard rears its head.
Not all of this low frequency information is lost, even with a hearing aid that only amplifies from middle C and up. Low frequencies have long wavelengths and as such go through walls, doors, and even occluding ear molds of hearing aids. Most ear molds have a vent, and many are completely non-occluding so the lower frequency sounds still enter the ear, bypassing the hearing aid. Even a hearing aid with a limited frequency response will not really limit the sound that hearing aid wearers experience.
By Marshall Chasin On March 27, 2012 · 2 Comments
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Posted - 2012/05/12 : 20:09:03
