Monday, December 22, 2014

Bullshit Oscillating at 432 Hz: A Primer on Acoustics

This is some prerequisite material to understand some of the relevant ideas which the A432-community utterly fail to grasp or account for.

Bullshit Oscillating at 432 Hz: A Primer on Acoustics

Sound consists of fast, relatively small oscillating changes in pressure (and for most hearing-related purposes, air is the medium in which these changes take place and travel). Air is rarefied and compressed due to the interaction of atoms - basically, they push each other out, and are pushed back in return. You have probably seen spectral diagrams of songs and sounds. These basically map relative pressure at some spatial point at any given moment onto the vertical axis, and time onto the horizontal axis.

Amplitude correlates with volume, and basically measures how greatly the atoms are offset.

Tones are a special kind of sounds – they are the subset that have regularly recurring peaks and troughs. That is, if the time it takes for the wave to go from one top to the next is the same for a lot of peaks, you are dealing with a tone. A complication exists, though: most things that produce regular waveforms of this kind, also produce other waveforms simultaneously! A string or organ pipe or glass of water that is agitated to produce a frequency f, also produces a set of other frequencies, called overtones or harmonics. In most musical instruments, these are integer multiples of f, where f signifies the frequency of whichever tone we are discussing at the time: 2f, 3f, 4f, ... The amplitude generally is lesser with each new note as we ascend this series, but exceptions exist. A simple example of that is the clarinet, where even frequency multiples are entirely omitted, thus leading to the situation where amp(odd number * f) > amp(even number * f), even if the odd number is way greater than the even number. Some instruments also may have other exceptions. One final set of exceptions is that not all instruments have exclusively integer multiples - most pianos have near-integer multipes, and bells can have really complicated multiples. Many percussive instruments are exceptions as well.

This will be relevant when looking at the misconceptions about scales and harmony that the A432-community labours under.

Sound travels at roughly 344 meters per second in air (subject to changes due to changes in temperature, dryness, etc). Inversely, the length between the peaks of the waveform for a tone at frequency x is 344./x meters. So, 344 hz in air would have the wave length of approximately one meter. However, assuming no wind, if the speaker were travelling along a straight line at 10 meters per second, a stationary listener in front of the speaker would hear 354 hz. The speed of the speaker is not added to the speed of sound – the speed of sound is entirely relative to the medium in which it travels. So, the number of wavepeaks that reach the listener will increase, as the distance between the wave peaks is reduced (or the opposite, if he is travelling the other way). This is known as the Doppler effect. The Doppler effect is nice in that it conserves intervals - if the speaker switched to playing a frequency that is y times 344hz, the listener would hear y times 354hz.

The formula is f' = f * (c + v)/c, where c = the speed of sound in the relevant medium, and v = velocity of the speaker. More generally, it is f' = f * (c + vs)/(c + vl), where vs and vare the speed of the speaker and the listener along the line. Calculating it if the movement vectors are not on a line is more complicated, but we will ignore that for now. Since we are dealing with a factor, overtones will be affected proportionally - ((f * (c + vs))/c) / ((f * 2(c + vs))/c)) = (f)/(f * 2) = r – overtones or sets of frequencies will be related by the same factor (not by the same difference in exact number of hertz).

It turns out our hearing is mostly logarithmic – we do not hear an absolute difference in hertz as a meaningful way of classifying how tones relate. 400hz and 450hz simultaneously sounds different from 300hz and 350hz simultaneously - but not just because the latter pair is lower! 400hz and 450hz simultaneously sounds as though the two notes relate in the same way that 300hz and 337.5hz do – the pairs share the same ratio, and therefore we hear these pairs as similar. This is also relevant when looking at the misconceptions and misinformation the A432 community spread about scales.

In other gasses, liquids or solids, sound travels at other speeds (and in solids, there evens exist two 'different kinds' of sound, travelling at different speeds - sheer waves and compression waves). Sound travelling in your body travels at another speed than sound travelling in the surrounding air – and this may further differ between your bones, your muscles, your skin, your intestines, etc.

If an orchestra is playing outdoors in A432 upwind from you, and the wind is six meters per second, you will hear it play in A440. If it is downwind from you, you will hear it play in roughly A424. We find this by dividing 440/432, then solving (c + vs)/c = 440/432, where c = 344, thus (344 + vs)/344 = 440/432. This is equivalent to 1 + vs/344 = 1 + 8/432   vs/344 = 8/432 ≡ vs/43 = 4/27 ≡ v= 162/27 = 6.

This will be relevant later on when looking into cymatics, a scientific method thoroughly misunderstood by the A432 community.

No comments:

Post a Comment