Wednesday, January 14, 2015

Bullshit Oscillating at 432hz: Pythagoras and Numerological Nonsense

Quite a few A432hz enthusiasts claim that only in A432hz are all the tones of the scale integers.

This is a truth with quite a bit of modification. Some of the advocates of this claim also claim that you can make your music A432hz by tuning it down using software. There are instructions all around the internet, especially on how to use Audacity to achieve such a detuning.

However, what will happen if you use Audacity to those ends is the following set of pitches:
A = 432
Bb = 457.688056763
B = 484.90360487
C = 513.737473681

Of course, the actual pitches will vary a bit from these idealized values, as singers deviate from their exact pitch both intentionally (to make their melody or harmony bit more expressive or more 'in tune' in quite a different sense from the one proposed by the A432hz enthusiasts) or by mistake. The same goes for free pitch instruments such as trombones or violins. Guitarists often have badly intonated guitars, so their pitches may also deviate quite a bit, and intentional and unintentional string bends add to the deviation there. Hammond organs have a peculiar tuning of their own that approximates regular tuning but is ever so slightly off, etc. The organ-builder may have missed by a hundredth of a millimetre the exact length a certain pipe should have been, and thus the tuning may be ever so slightly off, thus making the A:WHATEVER come out as A:WHATEVER±a bit.

Thus, the table of tunings for a small bunch of pitches provided there is only a sort of idealized average. Electronic music might get pretty close, though.

Why do the A432hz people believe that tuning to A432hz gives integer frequencies to most of or indeed to the whole scale? Many of them favor Pythagorean tuning, which is not just a question of readjusting the tuning of the reference pitch, it is a question of calculating the other pitches in other ways relative to the reference pitch. (Which requires tuning every pitch on your instrument differently, individually.)
Tuning down a song with audacity does not achieve that result. However, if you were to build your own instrument in such a way that it does have Pythagorean tuning, the idea regarding the integers will be slightly true, and I will explain why in a bit.

Why they believe that this is only achievable with A432hz is a bit less easy to understand. I have no idea, to be honest. I guess they just don't understand evidence-based thinking?

So, why does A432hz give integer frequencies with Pythagorean tuning? It's not entirely true, but it is true for the keys of C major/A minorA. Pythagorean tuning consists of tuning a bunch of new intervals by repeatedly tuning a new one up a perfect fifth, and a new one on top of that down a perfect fourth, and repeating that pair of operations (at some points, two perfect fourths will need to be added in sequence for this formulation to work, however). The untempered perfect fifth is a ratio between two frequencies, exactly 3/2. So, 100hz and 150hz are a perfect fifth apart. 

We start by C256, and we immediately obtain G384. We now multiply that by 3/4 (the perfect fourth is 3/4 downwards, 4/3 upwards. Notice that 3/2 * 4/3 = 2) and get 288. We go on to obtain 432 hz, and from there we still add 324hz, 486hz, and 364.5. So, in the [256, 512]-range we have one non-integer, but since octaves correspond to doubling a frequency, that problem does disappear in the 512-1024 range, as well as even in the 432-864 range (which is of interest if we focus on A).

However, why should we construct scales using this method? This method was indeed known to the ancient Greeks but so were other methods, such as those described by Archytas, for instance. It does give very nice fifths, but it sacrifices the consonance of the major and minor thirds significantly. Unlike equal temperament, you either end up with infinitely many pitches or a wolf interval.

Pythagoras* allegedly discovered that having two things that differ by simple ratios - 2/1 or 3/2 and such - produces consonant intervals. Examples given in Pythagorean literature consist of anvils whose masses differ by such a ratio, strings of the same dimensions weighted down by weights differing by such a ratio, etc. Weird enough, the examples given in the early Pythagorean descriptions don't work - they simply do not produce results that correspond to the perfect fifth.

The piece here below that is indented might not interest all readers. It has to do with number theory and pythagorean tuning.
3/2 does produce a very consonant interval. The method I gave above is basically the same as stacking 3/2s on top of each other, and sometimes reducing them by octaves (dividing by powers of two) to keep them in the same octave - 1/1 - 3/2 - 9/4 (9/8) - 27/8 (27/16) - 81/16 (81/64) - 243/32 (243/128), etc.
Now, we need to look a bit at factorization to understand why A432 / C256 have these results.
432 factors to 2 * 2 * 2 * 2 * 3 * 3 * 3. 256 factors to 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2. When multiplying, we simply concatenate the strings of factors. When dividing, we remove some shared factors:
555 / 27 = (3 * 5 * 37) / (3 * 3 * 3) = (5 * 37) / (3 * 3). 55 * 231 = (5 * 11) * (3 * 7 * 11) = (3 * 5 * 7 * 11 * 11)
This does not give us beautiful and easily comprehensible numbers, but this way of illustrating multiplication and divsion may illustrate why certain things work the way they do.
2 * 2 * 2 * 2 * 3 * 3 * 3 can obviously be divided by 3 exactly three times without yielding a non-integer. Multiplying 256 by 3/2 will be doable up to eight times until we've depleted the twos from the factorization. Since we basically alternate between adding a 3 and removing a 2 (by multiplying by 3/2), and adding a 3 and removing two 2s (when multiplying by 3/4), we can basically calculate how long it'll take to deplete the 2s - we're removing an average of one and a half per iteration, and thus we run out on the sixth iteration, which explains why the seventh tone is off by half from an integer.
By 432, we have already depleted a few 2s - we have four left. Thus, if we want to build an A major scale (which is a sequence of five leaps of fifths and one leap of 3/4 down from the starting point, including the notes at both ends, this giving us seven notes) we will deplete our 2s before getting all the way:
A: 432, E:  648, B: 486, F#: 729, C#: 546.75, G#: 820.125, D: 576

Pythagorean tuning takes one very consonant interval, and reiterates it to build a full scale. It is a useful musical scale, and probably the tuning that most medieval European music was composed in. However, it has certain issues that make almost all music composed since the renaissance fit less well with it:

  • its thirds give rather dissonant chords.
  • it does not form a 'cycle', it forms a 'spiral', alternatively 'it requires an infinite number of notes (or it breaks somewhere)
The first problem is the result of how dissonance works. We recall that the C major chord consists of C,E and G. We know that G is very consonant and therefore ignore that for now. We instead look at E, which is 648hz. This E is at 81/64 the frequency of C. We notice that a very nearby ratio, 80/64 = 5/4 looks fairly simple in comparison. We produce tables of overtones of the relevant notes:









The slightly lower E at 5/4 in fact has less dissonance, due to the overtones coinciding perfectly every fourth/fifth overtone for the pair C+E' , whereas the 81/64 overtones nearly never coincide and slightly more often also reach into the dissonant 'critical bandwidth'. This is actually entirely audible as well, we can compare the effect of these chords:

Listen Music Files - Embed Audio Files - Pythagorean vs Just Intonati...

Since music is fundamentally a subjective thing, some may prefer the first chord, some may prefer the second. Personally, I find them useful for different purposes - however, the chord you get on your average guitar is a good enough approximation of both for most purposes.

Turns out the first chord type does not really 'resolve' as well as the second - if you end a song on it, there'll be the kind of feeling lingering that 'hey, this song (or part of a song) hasn't come to a halt yet'. That might be a nice effect at times - but it's not what most classical or even pop music goes for. In medieval music, this kind of chord was not used as a consonance, but a dissonance that had to be resolved, either to a perfect fifth or a perfect fifth and an octave (so, in the key of C, C+G, or C+G+c). For more information on this, see Margo Schulter's monumental website on Pythagorean tuning and medieval harmony.

Thus, if you were to somehow magically retune all the works from basically the renaissance onwards up to this day to a Pythagorean tuning, you'd end up with a lot of songs whose chord progressions do not really work as their composers have intended any longer. But who cares for the intent the composers expressed in their compositions when you have a bunch of new age gurus telling you what to do?

As for the cycle thing, we need to look at the concept of modulation and the circle of fifths. In European music, the ideas of chord progressions and of modulation both have been of quite some importance for some time now. A chord progression is a sequence of chords, and chords are sets of three (or more) notes. Most musicians do not think of the progression Am Dm Am E as fundamentally different from the progression Abm Dbm Abm Eb. They may differ in how easy or difficult they are to play on a given instrument, but essentially they have the same internal structure - in isolation, they sound very similar. This can be achieved in both equal temperament and Pythagorean tuning. But, whereas this is possible when using any note as a starting point in equal temperament, it only is possible for a limited number of starting points in Pythagorean temperament (or, you end up with an infinite number of tones you have to work with). We want a system where if a chord is the Nth chord in one key, it has the same function relative to its key as the Nth chord of the other keys. (Of course, we could probably tolerate just a few keys for which it does not hold true, but it adds complications.)

In more modern European music, it happens that the key is changed during the work. Sometimes, this even happens repeatedly. Thus, we want a large set of workable keys between which we can switch.

Further, sometimes, performers' ranges are sufficiently wide for a given work, but the absolute reach does not go sufficiently high or low. In those cases, it is convenient for musicians to adjust the piece of music so that the range of the singer (or other performer) now corresponds to the adjusted piece's demands. Our voices aren't all created the same, so flexibility in this way is very useful.

As I mentioned, Pythagorean temperament does, to some extent, satisfy these demands. However.
It necessarily contains some breaking point. We've built our scale by adding new tones that are perfect fifths apart, and we notice that the distance to the tone from which we started never is an interval we have seen before. Either we stop somewhere, or we go on forever. If we go on forever, we end up with notes whose names would be monstrosities along the lines of C######## (which would be about a fifth of a semitone sharper than G).

The twelfth tone we add is 312/2(19). As it happens, this is fairly close to 1/1. So we ignore it altogether and close our cycle there, letting the error fall on the last tone. (We could go on, and let the error fall elsewhere, but this is as convenient an ending point as we get - we don't end up with dozens of named notes, nor do we end up with a bunch of notes that are very close to each other.) Thus, the last fifth we have is of the form (3/2) / (531441/524288) - which is a very ugly interval - the first note is a perfect fifth, the other is the wolf as it would be if it were tuned to C:
This error will also be present in any interval that "spans" this fifth. Different chords will sound drastically different, and transposing a song from one key to another may turn a chord in the song from consonant to dissonant or vice versa, ruining the song's structure altogether. (A given chord, say, 'G', will constantly be the same, of course, but what we're interesting in is retaining the structure of the scale such that, say, the chord built from the second tone of any given major key will sound sufficiently similar to every other such chord, so that we can say that it has the same 'function' relative to its key as the other 'second chords'.)

Equal temperament solves this by distributing the error given previously over each fifth, having each fifth just slightly off. The difference is barely perceivable.
Further, the fact that we've now reduced the fifth ever so slightly, adds up to four times more of a reduction of the major third - which nudges it closer to the very consonant major third in the first sound clip. And we end up with a system where each key can be used. The sample compares a pythagorean and an equal temperament fifth C-G, C-G'; the second half leaves out the lower part of the intervals so we can just compare the pitch of the two Gs. The difference is tiny.

Now, I've gone on for quite a bit here about tuning. I don't particularly believe that equal temperament is superior in any musical sense than other tunings, but it has many advantages that explain why it is used. I am fairly convinced, however, that most repertoire since the renaissance on to this day would not work very well in Pythagorean renditions.

Because of a thing in arithmetics - viz. the n:th root of an integer will always be irrational unless that integer is another integer to the n:th power - all the frequencies we obtain, except at most one, will be irrational. Regardless if we tune to A432hz or A440hz (or any other hz whatsoever).

However, our dear A432hz enthusiasts have of course done their maths and picked their tuning system so as to have integers all the way. Yet, they do it wrong. Let's compare some different A432hz tuning tables:
     1, **234*56A440/12tetA432/12tet



Notice how the different sources do not agree on their pitches? Some of these intervals vary by as much as 22/21 (f=704 and f=672). In part this is because they use entirely different approaches to building their scale - the high 704 is not pythagorean at all despite the claims by the source, it's a way more esoteric interval (11/8). By arbitrarily picking our frequencies in such a manner, I can build an integer-only tuning based on A440, viz. A440, Bb466, B494, C523, C#554, D587, ... and this can be done for any arbitrary starting point in that region. The errors introduced for any interval by rounding the frequency to an integer number of hertz will be just slightly wider than the error of the perfect fifth in the 12-tone equal system. And that is of course an idealized error - singers, brass players, violinists, cellists, and even guitarists will regularly be further off.

Of course, ultimately, the second's length has been arbitrarily decided; we could have divided the day into ten equal hours and each hour into 100 equal units and each of those into another 100 equal units, and a tone at 432hz would now be described as ~373.248alternahz. The division of the day into 24*60*60 is arbitrary. In a world with alternahz instead of hz, other frequencies would be integers. Integer hz frequencies have no magical properties despite the dumb beliefs A432hz enthusiasts have regarding this.

But as I might have said before, if you don't like that priests, ministers, imams or rabbis tell you what music to listen to, you can always listen to new age gurus instead - they even have rituals that make your music 'spiritually permissible' (because what else does reducing its audio quality by an ever so slight amount of resampling artefacts in Audacity amount to, but a superstitious ritual - and unlike rituals by older, more well-established religions, this at least has the veneer of technology to it - but who am I kidding, it's really slightly worse than e-mailing a dozen 'hail Mary' into the digital void). Why turn to evidence-based reason when gurus make stuff so much easier? And the added anxiety from believing that Nazis have made the music you hear in the radio increase aggressiveness among your peers is certainly good for your health as well as well.

By further telling you to prefer Pythagorean tuning over other tuning methods, they're essentially imposing a certain music theory on you - one in which modulation is limited, one in which chord resolutions are much more restrictive, one in which the useful keys are much fewer and you end up having to buy new, expensive guitars because your Gibson or Martin or Taylor or Fender simply cannot be tuned in a Pythagorean fashion*. You get less, but at such a steep expense, who can refuse?

* Pythagorean guitars require complicated frets that are damn expensive to manufacture. You'll end up with one along the lines of the guitar neck pictured in this post. Those are not cheap, I can tell you. But of course, if a new age guru tells us to buy them, who are we to refuse? Who are we, indeed, to refuse?

A) A natural minor and C major contain the same seven notes. These are, when ordered as a series of fifths, F-C-G-D-A-E-B. Ordered as a regular scale they are C D E F G A B (c). (Or A B C D E F G). Think of F-C-... as though each "-" signifies "..., which equals 4/3 or 2/3 of ...", so F, which equals 4/3 of C, which equals 2/3 of G, ...

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