Wednesday 15 March 2017

stop the world, i want to tune up

Last month NASA announced the discovery of a bunch of planets in the nearby star system TRAPPIST-1. The planets got a lot of notice because so many of them were in the "habitable zone" of the star (i.e. the bit where you'd get liquid water), and have roughly the same mass as Earth, which makes them prime candidates for space colonization.

But that's not the bit that caught my eye. From the article I linked:

The orbital periods of the innermost six worlds range from 1.5 days to 12.4 days; the outermost planet, known as TRAPPIST-1h, is thought to complete one lap in about 20 days. (Spitzer spotted just one transit by TRAPPIST-1h, so its orbital path is not well-known.)

The six inner planets are in near-resonance, meaning their orbital periods are related to each other by a ratio of two small integers.

For example, the ratio between Trappist 1e's "year" (at 6.1 days) and 1d's same (4.05) is 3:2, the same ratio applies between 1f and 1e. This sort of resonance is not massively unusual in celestial bodies. Ganymede, Europa and Io have 2:! resonances (Ganymede's period is 7.15 days, twice Europa's 3.55 days, which is twice Io's 1.77 days).  Bode's law is sort of a version of it: the idea that positions of the planets in our own solar system followed a similar mathematical pattern was devised in the 18th century and correctly predicted the positions of Uranus and Ceres. But somehow, the way it was being phrased reminded me of something else.

And that would be harmonies. That 3:2 is also the ratio of a perfect fifth. Other common intervals also very basic ratios: a major third corresponds to 5:4, a minor third is about 6:5. I'm not the first person to have spotted this, of course. That would be Pythagoras or someone.

But I think it's possible I might have been the first person to wonder what chord that solar system was playing. We can pose the question what colour is the universe, so why shouldn't I ask whether TRAPPIST-1 is emitting some weird jazz chord or what? And can I play it?

I'll show my working. The first thing is to establish what notes the various planets are playing. We know their periods (in days), converting that to period in seconds is easy (multiply by 86400). Converting that to frequency is also easy (take the reciprocal), but then you end up with unhelpful numbers like 0.00000286 Hz (for Trappist 1d). To convert this into the type of number we can compare against a pitch chart, we need to multiply it up a bit.

Octaves are a ratio too. A 2:1 ratio. If 440Hz is an A (as it is in standard tuning), then so is 220, and so is 110, etc etc etc.  So we can simply multiply our frequency by 2 enough times to get it to the human hearing range.  (Or rather, multiplying it by two to the power of the number of octaves we want to shift by: if we want to take it up three octaves, multiply by 23=8, if we want to take it up ten octaves, multiply by 210=1024. In this case if we multiply it by 226, we get things that are nearly but do not quite correspond to notes in our standard tuning. If we multiply by 226.03 we get them pretty much bang on.

Planet Period (days) Frequency (mHz) Adjusted freq (Hz) Note
1b 1.51 0.00766 525.2 C (523.3)
1c 2.42 0.00478 327.8 E (329.6)
1d 4.05 0.00286 195.8 G (196.0)
1e 6.10 0.00190 130.0 C (130.8)
1f 9.20 0.00126 86.2 F (87.3)
1g 12.35 0.00094 64.21 C (65.4)

This would appear to be a Cadd4. The root is C (lowest note, plenty of other Cs), and the fifth of C (G) is also present, and so is a major third E. But so is F, which is a perfect fourth away up from C - hence the chord name, C additionally with a 4th.  I can play that on a guitar, although not quite in that inversion,  I don't think.

Anyway, that's those planets. They are recognisable chord, and it's only a little bit out of tune (the .03 I added to to the octaves corresponds to A being at 431 Hz in that solar system.) What about this solar system. Are we a chord?

Planet Period (days) Frequency (µHz) Adjusted freq (Hz) Note
Venus 224.7 0.0515 451.75 between A (440) and A# (466.2)
Earth 365.256 0.0316 277.9 C# (277.2)
Mars 686.961 0.0168 147.7 D (146.83)
Ceres 1681.63 0.00688 60.36 B (61.74)
Jupiter 4,332.59 0.00267 93.71 F# (92.50)

I stuck Venus, Earth, Mars, Ceres (the biggest asteroid and the one whose discovery was thought to confirm Bode's law) and Jupiter into the spreadsheet as well. I have to shift more octaves, of course, because of the vastly longer years. Earth comes out at 272Hz if you shift it 33 octaves up. It's a little bit lower than a C#. If you nudge by that 0.03 again, it turns out to be a C# pretty much bang on. And... Mars is D, Ceres is B, Jupiter is F#. Venus resists, being between A and A#, but Venus is funny anyway. It rotates backwards, for one. So let's just exclude it.  Maybe it's a string bend.

I have less of an idea what chord that would be.  It's a Bm (B, D, F#), but with a C# in it as well? Bmadd2? Is that a thing? But it's not the chord we're playing here that matters. It's the tuning! Our solar system, like TRAPPIST-1, clearly has a natural tuning of A431. We can assume that since two completely different star systems (representing 100% of the sample) both match, it is some kind of universal physical constant, right?

And that is not the tuning most of us use. No wonder there's so much war and conflict and stuff these days. We're not in harmony with the universe!

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