Friday, December 26, 2014

Bullshit Oscillating at 432Hz: On Resonance

One concept often referred to by the 432hz enthusiasts is 'resonance'. Apparently, things in the world resonate "at 432hz" in ways that ... I dunno. Magic. They're not all that clear on what resonance actually is, nor do they want to clarify exactly what they thing it does to things, except it's clearly good and magical.

They think the 'universe' itself resonates at 432hz, but also that pretty much each of its component parts has that same magical resonance. Everything, of course, is vibrations, and so on. It's a cornucopia of vibrations, resonances and frequencies. What else is there to expect when new age kooks are involved? Sigh.

I previously talked about the speed of sound. (Which confusingly enough also is called 'c'. Thanks, science, was that the best letter you got?) This is a somewhat relevant part of resonance. If a system resonates at a frequency, this means it reinforces that frequency. A system may resonate at several different frequencies, and even simultaneously so. A frequency is reinforced if its wavelength in that material (say, a string) corresponds to the length of that string or a half or third or n:th part of its length.

A relevant example of just how dumb the A432hz claims are, is the claim that Stradivarius violins have exceptional resonance at A432hz. We will now look at why that claim is genuinely dumb.

Resonance in a violin depends on the speed of sound in the relevant kind of wood, the shape of the wooden parts, and a variety of other things. However, there are interesting complications in how resonance in violins works with regards to actual musical use.

Ever noticed how synth strings sound comparatively lifeless compared to the violin? In part, this is because violin resonance is not uniform. When you play a tone, say, A440, the string also produces harmonics. These are integer multiples of the fundamental frequency - you get something along the lines of A440, a880, e'1320, a'1760, c#''2200, e''2640, ... and each of these has its own amplitude. However, different frequencies resonate differently in the violin body. Thus, when you play A440 or you play B495 (with the harmonics b990, f#'1485, b'1980, d#'2475, ...) the relative amplitude of the harmonics will not be the same for B495 as they would have been for A440.

If you are mathematically inclined, you could best imagine what happens as a function along these lines:
a1sin(x) + a2sin(2x) + a3sin(3x) + ... + ahsin(hx), where all h are integers, and ah are values in the range [0, 1]. ah goes to zero as h goes to infinity. Essentially, the faster the oscillation of some overtone, the smaller the width that that oscillation imparts to the waveform. However, in the case of an acoustic instrument, this abstracts away the importance of the fact that ah is not the same for each hx! Thus, it'd be better to have
f(x)sin(x) + f(2x)sin(2x) + f(3x)sin(3x)  + ... + f(hx)sin(hx), where f(hx) gives the amplitude for that particular frequency, and f(x) is (most likely) a continuous function that goes to zero as x goes to infinity - but oscillates quite a bit on the way.

For people for whom maths is difficult to keep up with: the timbre of an instrument is the result of lots of waves, that interrelate in this way: in the time the lowest wave goes /\, the next-lowest goes /\/\. There's even a further one that goes /\/\/\ in the same time, and so on. However, the faster they go, the less high they go.

Some pictures! Let us pay no heed to the actual values along the x-axis now - the same "relative" situation will obtain for any note. We have several wave forms which if we were to separate them we'd obtain graphs like these describing them. The first few pictures below here are in the sequence sin(x), sin(2x), sin(3x), sin(4x), sin(5x):

We call the lowest frequency in a tone its 'fundamental', and that frequency is generally the frequency we will say the tone 'has'.


The second frequency is an octave above the first - notice how the number of peaks or troughs is twice that of the previous waveform.



An octave and a fifth above the fundamental, we have the third frequency - its troughs and peaks number thrice that of the fundamental.



Double octave, followed by major third over double octave:




These waves happen together, but their amplitudes are different. If we were to plot them all on the same curve, we'd get something like this (amplitudes subject to variation):

How high (and low) each wave goes is determined by the factor I previously labelled ah, so in this case a3 is 0.7, a4 is 0.5, etc. If we were to add together (sin(x) + sin(2x) + ... + sin(4x), we would obtain something like this:
If, however, we were to add together those given in the multiwave graph I just posted, we would obtain this:


If you keep adding more 'partials' to it or just alter the amplitude of any one partial wave, the wave form will slightly change but the pattern we have here will be recognizable there. However, in reality the faster waves will more often not reach 'as high' and 'as low' as the slower waves. The graph below illustrates another similar pattern:


The ear is surprisingly good at recognizing differences between different-shaped waves of these kinds - that is in part how we recognize trumpets from clarinets from violins from guitars, or even how we distinguish different vowels. Of course, if the difference is subtle enough, it will not necessarily be recognized at all.

Furthermore, our ear-brain interface is so used to waves being related by integer factors that if you were to hear a wave of this form: a2 * sin(2x) + a3 * sin(3x) + a4 * sin(4x) + ... your brain would fill in the missing sin(x) for you!

Now, when a violinist plays, he will often impart a vibrato - he will repeatedly continuously alter the frequency slightly over a certain range of frequencies. The resonances will also change, due to the aforementioned phenomenon – resonances differing for different frequencies and thus the shape will change. This is what makes the violin sound comparably more 'alive' than a synth tone. It seems good quality violins even have drastic changes in timbre over short ranges, and thus the shape of the wave that is produced at different fundamental frequencies. So, how does the physics of that work out?

Resonance is the result of standing waves and other similar things, and standing waves occur when the wave length of a tone is the same - or a divisor - of the length of the thing in which the vibration happens. Since the violin contains many lengths, a line in the violin body that happens to have such a length will start vibrating at such a frequency (and lines with approximately the same frequency may start vibrating too).

Look at the shape of the violin body. You may notice that it is not a circle or a sphere, but rather a shape with some complications to it. This means that depending on where in the wood or where in the air inside of the resonance chamber you draw a straight line, you'll have a different length - thus also a different set of frequencies resonating along that line. Since the wood does not have a perfectly identical density throughout, this may affect the resonance slightly at different frequencies.

So what if a Stradivarius violin resonates well at A432? It resonates well - and in different ways - throughout its entire range! And the variations in resonance are intentional! What of course makes the use of this pretend evidence even more interesting is that Stradivariuses have been proven not to sound 'superior' in double-blind tests: high quality modern violins, as well as high-quality antique violins of other skilled luthiers have been ranked the same in such tests. Simply put: if we believe that a musician is playing a Stradivarius, we trick our brain into thinking it sounds better than we would think if we knew he was playing a modern high-end violin. Certainly the Stradivarius violins are not bad, they're quite great instruments - but there is nothing magically perfect about them. It's interesting indeed that the A432hz enthusiasts are willing to use irrelevant, debunked and disproved reasoning, as well as name-dropping to bolster their case.

Furthermore, it is well known that violinists tend to use vibrato, a method wherein the pitch of the tone they are playing is periodically altered - basically it glides audibly between an upper and a lower pitch slightly off from the tone they are playing. The above variety in resonance makes this effect not only produce an alteration in pitch level, but also a slight alteration in timbre. This even further makes violins sound appealing to us, in a way that a single frequency's magical resonance properties wouldn't have any relevance to whatsoever.

What is more, there is a problem when the whole instrument resonates very well at some frequency. This is one of two phenomena that go by the name 'wolf tones'. Due to strong resonances when the whole instrument resonates, even nearby tones may cause an awkward, ugly sound. Jamie Buturff says:
We're stuck to 440Hz and are the whole day covered in this "not related" music! It is clear that we must return to the natural vote of 432 Hertz. A Stradivarius violin resonance is at 432Hz, it's built to do so. [Jamie Buturff, The Frequency of the Universe]
If Jamie Buturff were correct, A432 would sound like shit on that violin, since you'd end up having a lot of unwanted resonances and a strong spike in volume for that exact frequency! The A432 community are idiots who don't know the first thing about acoustics, yet pontificate about it as though they were experts.

Chances are, however, that they just claim that A432 is the main resonance of the Stradivarius violins, since this is a nice soundbite. I would even bet they just made it up.

Violins sound good not because a certain frequency resonates, but because of the complex interaction of resonance strengths for different overtones. A432-enthusiasts will never care about the actual physics of music, though, so can be dismissed as ignorant woo-peddlers.



4 comments:

  1. Just a teensy musicological quibble to your fundamentally excellent piece: The continual vibrato we hear today on string instruments is a relatively modern invention, popularized by Fritz Kreisler around the 'teens of the last century. The farther back you go in the history of recorded violin, the less vibrato - but the more portamento - you hear.

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  2. It's all related to the human voice and its registers.. Tuning A4=440Hz with 12-TET places the notes in the scale at a wonky high annoying resonance which is to far from the natural relaxed A4 at ~432Hz. I am not a new age believer or anything like that.. Just a musician and singer for 22 odd years. THe highest pitch for A4 is 438Hz.. Anything above that and the second passaggio gets to tough for heavy voices to operate safely.. You have to constant due exercises to get the balance correct which is not needed with a small 11.8 cents detune to 437Hz for exampel. The former European pitch at 435Hz was due to singer and physician recommendation. At 435Hz you have to do some transition at the second passaggio but you can still blend some lower resonance into the sound which creates a much more pleasent sound.. At 432Hz (-31.8 cents) then almost no transition have to be made at a certain passaggio note and the resonance of the singers voices on countless tracks I detuned -30 cetns (after getting used to the downtune) it that it sound much more honest and a true reflection of the person who is singing as the notes in the 12-TET scale is much closer to that persons natural relaxed talking pitch.. But sometimes you need a little lift and tension and hence why 435Hz is a good middle ground..

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  3. G.O.D. The Grand Orchestral Design

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