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.

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.

Sunday, December 21, 2014

Bullshit oscillating at 432hz

For the next topic, I will write a rebuttal to one very mistaken theory about music that is making the rounds on the internet, and that also has a number of adherents - even among real musicians.

This theory sometimes is coated in new age-inspired pseudoscientific terminology, and sometimes it is bundled with a conspiracy theory. A small minority seems not to hold any specific pseudoscientific stances, but appear to just go along with the bandwagon in order to utilize the additional traction for support the tuning standard in general that the new-agers and conspiracy nuts would . I will investigate the (in)validity of almost all the arguments presented.

432hz adherents believe that if we retuned the western scale so that A440 was replaced by A432, music would be more harmonious, and we'd be more happy and peaceful, and I really have to restrain myself not to say something silly like 'and Jesus would come back and lions and lambs would spoon together'.

This will, again, be thoroughly detailed. The point with it will be to show just how lacking of any rational ground to stand on a lot of the A432 claims are. I am also interested in debating this with adherents, so if you believe that A432 has weird mystical properties, please contact me and try to convince me differently. I will present your argumentation exactly as it is, and respond to it with serious arguments.

Saturday, December 6, 2014

On Context, Isaiah and the Reason for the Season

Language is maybe the most important tool we humans have at our disposal. Using it and its even more specialized variety 'meta-language', we have been able to develop maths, science and so on. However, sometimes, we do not see the forest for all the trees - we miss the context because we are blinded by words.

So, let us have a look at a prophecy where a lot of the discussion back and forth entirely misses the point of the text being discussed. The Christian idea that Jesus was born of a virgin has its roots in the Septuagint translation of the Old Testament, and the evangelists' use of one particular verse.

A lot of the further discussion of this topic basically is philological - does the term almah signify a young woman in general or does it signify a virgin? Arguments in favor of young woman seem way more solid, and even the Greek parthenos seems less clearly to have signified virgins than previously assumed, thus we cannot even be sure that the LXX translators intended the meaning 'virgins'. Further, it seems the texts of the LXX that are accessible to us are all from traditions maintained by Christian scribes, so that weakens the case in favour of virgin a bit as well.

However, there are good reasons even if almah meant virgin to understand the text as not at all being a prophecy about Jesus - and this even if we grant the assumption that God exists, can promise things he fulfills in the future, and also knows the future. These are not assumptions I believe to be true, but let us be magnanimous for a moment!
Therefore the Lord himself shall give you a sign; Behold, a virgin shall conceive, and bear a son, and shall call his name Immanuel.
What does any reader in the west when he comes across this verse in isolation think of? Alas, this blinds us to the meaning of the context in which it is found! So strong are our associations, that they blank our minds for a while, our reading comprehension fails.

I have even seen people rejecting the idea that this was written before Jesus was born on the grounds that 'since prophecy really does not exist, Isaiah cannot have predicted an alleged virgin birth, thus it must have been written after the idea of Jesus' virgin birth had taken hold in order to fabricate a prophecy'. This is not a common misunderstanding, but it still shows how relatively reasonable skeptics just fail to read it in context.

Of course, context is a magic word in other ways too - how often do not Christian apologists tell us we must read things in context, meanwhile trying to divert our attention from that very context? In this case, they do not even need to do that, though, since centuries of indoctrination has overruled our reading comprehension.

So, let us look a bit closer at what Isaiah actually says. I will not bother to go through the Hebrew text here, since the English is sufficient to illustrate the point I am making. This is partially based on the exegesis found here[1].
And it came to pass in the days of Ahaz the son of Jotham, the son of Uzziah, king of Judah, that Rezin the king of Syria, and Pekah the son of Remaliah, king of Israel, went up toward Jerusalem to war against it, but could not prevail against it.
And it was told the house of David, saying, Syria is confederate with Ephraim. And his heart was moved, and the heart of his people, as the trees of the wood are moved with the wind.
So, removing the extra verbiage, Ahaz, king of Judah, learned that Rezin and Pekah, kings of Syria and Israel, were in cahoots to attack him. He was distressed by this. It would seem the author inserts stuff from a later vantage point - "but could not prevail against it" seems to be an indication as to how the narrative will conclude as far as the military campaign goes.
Then said the LORD unto Isaiah, Go forth now to meet Ahaz, thou, and Shearjashub thy son, at the end of the conduit of the upper pool in the highway of the fuller's field; And say unto him, Take heed, and be quiet; fear not, neither be fainthearted for the two tails of these smoking firebrands, for the fierce anger of Rezin with Syria, and of the son of Remaliah. Because Syria, Ephraim, and the son of Remaliah, have taken evil counsel against thee, saying, Let us go up against Judah, and vex it, and let us make a breach therein for us, and set a king in the midst of it, even the son of Tabeal: Thus saith the Lord GOD, It shall not stand, neither shall it come to pass.
So, God tells Isaiah to talk to King Ahaz, to tell him not to fear. For the record, Ahaz was not a particularly righteous king. Next, God's promise to Ahaz is given in greater detail:
For the head of Syria is Damascus, and the head of Damascus is Rezin; and within threescore and five years shall Ephraim be broken, that it be not a people. And the head of Ephraim is Samaria, and the head of Samaria is Remaliah's son. If ye will not believe, surely ye shall not be established.
 The promise, quite clearly, is that Syria and Samaria will be crushed. But God has more to say, relating to this very promise about Syria and Samaria.
Moreover the LORD spake again unto Ahaz, saying,
Ask thee a sign of the LORD thy God; ask it either in the depth, or in the height above.
But Ahaz said, I will not ask, neither will I tempt the LORD.
So, God wants Ahaz to ask for a sign by which God can affirm this promise - a little miracle to show that he is up to his promise. Ahaz, however, scoffs at this.
And he said, Hear ye now, O house of David; Is it a small thing for you to weary men, but will ye weary my God also? Therefore the Lord himself shall give you a sign; Behold, a virgin shall conceive, and bear a son, and shall call his name Immanuel.
So, God himself picks the sign(, and as it turns out decides to renege on what he just told Ahaz). Here, of course, we run into the place where the virgin birth famously is promised. Looking closer at the Hebrew - provided that we grant that Almah is virgin - we find that it is not "a virgin", but "the virgin"; it is also preceded by the somewhat demonstrative הִנֵּה, and thus it seems even more likely the author of this narrative thought of it as the prophet actually pointing out one particular virgin. Of course, there's no oddity about pointing out that someone who currently is a virgin will conceive. She may no longer be a virgin when the conception occurs, obviously, but we may very well identify a person by what qualities they have in the present, regardless of their qualities at another time. However, even then we have good reasons to think it's not 'virgin' that is the intent of the text. Why should anyone think the sign God is giving to Ahaz is the birth of the Messiah? Nothing in the context this far indicates that such an interpretation is valid.
Butter and honey shall he eat, that he may know to refuse the evil, and choose the good.
For before the child shall know to refuse the evil, and choose the good, the land that thou abhorrest shall be forsaken of both her kings.
So, the sign by which God will affirm that he has the power to fulfill his promise, is that what he promises will be fulfilled before the boy reaches his teenage years. This is not a very useful sign - it is like "to prove to you that I can repay this loan within ten years, I will repay this loan within ten years". This is really an insult to Ahaz, God is giving a useless sign as a rhetorical device.

The promise is now turned pretty much upside down, with God promising terrible things to befall the kingdom of Judah. So, in case this prophecy really is about virgin births, we must have a virgin birth of the time-keeping boy in king Ahaz time and later on, we have Jesus. That is mighty odd, is it not? However, this means we have a clear idea what the text actually is saying, and the usual understanding that pretty much everyone has if asked what that particular verse means appears incongruous.

Even when reading the text by oneself, it is likely that one reads the bit about the conception and birth as though the prophet temporarily diverts from the affairs of the day to a prophecy centuries into the future and then jumps back to the main topic - as though he was incapable of keeping to a topic.

There still is literature that assumes Isaiah 7:14 is representative of Jewish beliefs about the Messiah, without having looked at any other evidence about the verse in question than its Christian interpretation. This is a pretty important fact regarding how people tend to think of Judaism.

Now, I am not just pointing out what Isaiah actually was saying, I am making a point that this is just one example of - associations that words or phrases have in our minds somehow trumping the actual meaning of the text or utterance, breaking our reading comprehension (or listening comprehension) temporarily. It is a problem lots of people run into, and sometimes makes people come up with ludicrous ways out of the mistaken ideas it causes them - such as the example I mentioned with a skeptic denying that Isaiah could have been written before the birth of Jesus.

In part I suspect this relates to how language works - our brains are pattern-matching algorithms, and a word, a phrase or even a whole poem can trigger associations. Often, these associations are not particularly complicated - but the same word may trigger slightly different associations for each one of us. However, the environment in which most native speakers of English have grown up make Behold, a virgin shall conceive trigger an association with several of Christianity's central tenets. Some other phrases and words may have a comparable power in distorting how we understand a text or utterance, and it is important that we realize that our minds are not immune to this.

[1] Rabbi Tovia Singer, Dual Prophecy and the Virgin Birth,

All other quotes from Isaiah 7, King James Version.