Frequency Transposition for Music: Revisited

Recently I wrote a blog about why frequency transposition (compression or shifting) should never be used with music. Although the reasoning was quite technical, it was based on the fact that in the higher frequency region, speech mostly has a “continuous spectrum” such as a band of noise or friction, much like what one would see with the consonants [s] or [f].  Transposing this band to a lower, presumably healthier frequency region, can be very helpful for speech intelligibility. The same cannot be said of music however.  Music, being a “line spectrum”,  is made up of a fundamental (or tonic) and then a series of well-defined harmonics that extend into the higher frequency region.  Changing the exact location of each of these harmonics (which define the instrument and the musical note) would have drastic effects on the quality of the music.  If a harmonic was at 4196 Hz (the top note on the piano keyboard), moving it down to 3000 Hz would destroy the musical quality.

But a client of mine (who reads this blog) pointed out that that was not entirely true: if the transposition was exactly an octave lower (by a factor of 2) then all of the harmonics would line up. Initially I thought that this was a great idea. In a violin, if a 4000 Hz harmonic was transposed downwards by exactly a factor of 2, then a 2000 Hz signal would be created. This would line up with an already existing harmonic at 2000 Hz and, perhaps with some auditory training, the hard of hearing person would learn to recognize that as a fuller harmonic range?

But let’s do some math to be sure…. I know that we are all excited to get started!

Let’s start with half wavelength instruments. These have integer multiples of the fundamental (or tonic) frequency.  Typically these include instruments where the vibrating column or body forces the standing waves to have nulls at either end.  The violin, piano, guitar, and voice immediately come to mind.  In each of these cases, the string is held tightly at either end of the instrument and can only vibrate in the center between these two end points.  The voice is like this as well; our larynx has a rather elastic set of vocal chords that are held tightly in place at the front of our larynx and at the back (where the arytenoids allow us to change our vocal quality).  If one were to examine the vibration of the vocal chords, or that of any stringed instrument, the vibration would be exactly ½ the length of a full wavelength. Hence the name of this type of resonator.  Half wavelength resonators are also found with “conical” musical instruments even though they are not “held tightly at either end of the instrument” like a violin or a guitar.  For reasons that are beyond this blog, conical instruments such as the oboe, bassoon, saxophone, tuba (and even the clarinet in the top register only), function as 1/2 wavelength resonators as well.

To make the math a bit easier, let’s stick with the standard audiometric frequencies- 250 Hz, 500 Hz, 1000 Hz, instead of the real musical notes, 262 Hz, 524 Hz, 1048 Hz … If a stringed instrument was played an octave above middle C or 500 Hz, there would be harmonics at 2 x 500 Hz, 3 x 500Hz, and so on.  The harmonics of 500 Hz, would be 1000 Hz, 1500 Hz, 2000 Hz, 2500 Hz, 3000 Hz, 3500 Hz, 4000 Hz, and so on.

If a 2:1 frequency transposition or frequency shifting was imposed, the 4000 Hz harmonic would be at 2000 Hz which would line up nicely with an already existing harmonic in that note. With sufficient auditory training, or perhaps with the benefit of time, a listener may be able to learn that there was a richer harmonic structure. But what would happen to the harmonic just above or below 4000 Hz?  Applying this same transposition 2:1 ratio, 3500 Hz would now become 1750 Hz.  There is no harmonic for this note at 1750 Hz and as such would be an “inharmonic”- with the resultant degradation of the musical quality.

So even a frequency compression ratio that reduces everything by one octave would not bring the listener into a “sweet spot” for music listening. Musical instruments that are ½ wavelength resonators such as all strings, vocal music, oboe, bassoon, flute, saxophone, tuba, and even the coronet (but not the trumpet) would not have their quality improved by any form of frequency transposition or shifting.

Well, what about the other category of musical instruments, the ¼ wavelength resonators? This comprises the clarinet (on the low register only), trumpet, trombone, and French horn.  Despite the flares of the ends of these brass instruments, they are not conical because the flare is an “exponential” horn- the diameter increase is not consistently greater as it would be with a cone instrument such as an oboe or tuba.  (If you don’t believe me trying placing a straight edge along-side the flare of the oboe or tuba- it is a linear constant flare.  This would not be the case with a trumpet or a trombone.)

Instruments that are quarter wavelength resonators have odd numbered multiples of the fundamental frequency or tonic. Using the same audiometric round number frequencies as above let’s do the calculations.

A 500 Hz fundamental frequency would have harmonics at 3 x 500 Hz, 5 x 500 Hz, 7 x 500 Hz, and so on for a harmonic series made up of 1500 Hz, 2500 Hz, 3500 Hz, 4500 Hz, and so on. (Incidentally this is the formant structure for the reduced vowel shwa that we have all learned about in speech sciences).

Imposing a 2:1 ratio frequency transposition or shifting of 4500 Hz, would bring it down to 2250 Hz which is not in the harmonic series. It is an “inharmonic” and like the ½ wavelength resonator instruments, would result in a degradation of the musical quality.

Well, what about a frequency compression of 3:1 for these instrument? While this would work for the first harmonic (like a 2:1 ratio would work for the first harmonic of  ½ wavelength resonator instruments), this would also not work for ¼ wavelength resonator instruments either.  The 4500 Hz harmonic would now be 4500/3 = 1500 Hz…. WAIT A MINUTE… 1500 Hz is an harmonic for this note.  Let’s try it with 3500 Hz…. 3500/3 = 1166 Hz which is not in the harmonic structure of a 500 Hz fundamental.

So, other than that one blip that perhaps a 3:1 frequency transposition ratio can work, this does not apply to all harmonics within this note.

So, further to my conclusion from the previous blog that frequency transposition or shifting cannot work for music and should never be used in a music program: the clinical approach should be just gain reduction in that offending damaged frequency region.  Frequency transposition or shifting of any degree will mathematically result in harmonics that are created where they are not supposed to be.  While this is not a rigorous mathematical proof, I doubt that there are any frequency transposition ratios that will not result in undesirable harmonics.  I am sure that my old math professors would have given me only a C- for this, but that’s still pretty good – nobody gets an A in Canadian educational systems!