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Comment by a3n | Hacker News Reader

> the BBC required their orchestras to tune to 440 Hz instead of 439 Hz because 439 is a prime number, and the corresponding frequency is hard to generate electronically.

I followed, until this. Earlier he debunks the significance of 432 per second (including that it's a sum of four consecutive primes), because a second is an arbitrary length of time. But now he says that 439 per second is difficult to generate electronically, because it's prime.

I'm not an EE. I'm willing to believe, but could a knowledgeable someone help us out here?

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From other sources it doesn't seem clear if this actually was considered in the decision, but it describes how they got the tone:

> The B.B.C. tuning-note is derived from an oscillator controlled by a piezo-electric crystal that vibrates with a frequency of one million Hz. This is reduced to a frequency of 1,000 Hz by electronic dividers; it is then multiplied eleven times and divided by twenty-five, so producing the required frequency of 440 Hz. As 439 Hz is a prime number a frequency of 439 Hz could not be broadcast by such means as this (http://www.wam.hr/sadrzaj/us/Cavanagh_440Hz.pdf)

Since 1000 Hz also was (is?) a broadcasted test tone, it makes sense that it is much easier to get perfectly matching 440 and 1000 Hz from the same high-frequency source than it is to get 439 and 1000 Hz. A more precise way of describing the problem is then that it's hard to generate since it doesn't share any divisors with other desired test frequencies.

If they only required 439 Hz they could have just used a slightly differently tuned source oscillator, yes.

I wonder why they decided it was better to design and build an 11x multiplier and a 25x divider rather than cut a 440 kHz crystal and run it with a copy of the 1000x divider electronics they already had.

Because 1MHz crystals are common off-the-shelf items, 440kHz are not?

In order for that explanation to make sense, one has to assume that 11x multipliers and 25x dividers were common off-the-shelf items during the time period in question.

If the choice is between a custom crystal and two custom boards, the custom crystal is going to be the far less painful choice.

I would bet that the BBC was using the 1 MHz crystal and the associated division/multiplication circuitry for other frequency control purposes, not just the musical test tone. 1 MHz is smack-dab in the middle of the medium wave band usually used for AM broadcasting, and the BBC has always broadcast on many different frequencies at the same time.

Having all the broadcast frequencies controlled by a single reference crystal would have some obvious advantages (and a few disadvantages too, of course).

Arbitrary dividers and multipliers are fairly easy to design in a circuit. Certainly far easier than cutting a custom crystal.

The world is so ... physical. :) Thanks.

To common ways to produce precise oscillating signals are tuned crystals or RC oscilators. In both cases, you're looking at common parts (i.e. standardised to x Hz), and then multiplying or dividing by integer multiples to get to the frequency you want. With a prime number, you'd need a component specifically tuned to that or a multiple of it.

I found a plausible explanation here:

http://www.network54.com/Forum/27140/message/1296951241/More....

tl;dr: The BBC reference pitch was generated by dividing and multiplying a 1MHz crystal oscillator. A440 was achievable by this means; A439 was not.

You raise a good point. a second is an arbitrary length, so nothing universally significant about 439 Hz or 432 Hz or etc.

However, when you are trying to build a device that oscillates at x Hz, there are engineering-ly sigificant values. For example (as already mentioned) a 1kHz oscillator might be easier to come by. As an additional example, the AC power in the UK is at 50 Hz, a factor of 440 Hz. So there is a free 50 Hz reference available everywhere, that can be multiplied (with a PLL, which is more robust to, say, temperature differences, than say, building a 50 Hz or 440 Hz reference yourself).

(50 Hz is just an example. AC line frequency probably is not stable enough in the short term to be a useful pitch reference)

AC line frequency has been recognized as a preferred reference for pitch. This is what made Hammond electric clocks the first reliable AC powered timekeepers.

Later, Hammond famously produced electro-mechanical organs referenced to line frequency.

Well, what do you expect from somebody simply following in the creative footsteps of Leonardo DaVinci & Thos. Edison?

Interesting.

I don't think it's a particularly stable reference in the short term. It's true that the power companies adjust it to hit exactly the nominal frequency over a period of many hours, but not at any one instant.

In fact: https://en.wikipedia.org/wiki/Utility_frequency#Time_error_c...

439 being a prime would be an issue if you were up multiplying a lower frequency to generate it, for example using a PLL. However it is such a slow signal that it would be generated by dividing down a higher frequency. For example you can divide it down from an 18 MHz clock.

So, nope his claim doesn't make sense.

439 from 18M requires a divisor of 41, which does sound trivial. Meanwhile 440 from 1M requires only dividing by 5 and 2 when first multiplied by 11, which might have been significantly simpler than dividing by 41.

I wonder if there is some easy way of figuring out the simplest base number-multiplier-divisor chains for getting certain numbers?

Dividing a frequency by a large prime, like 41, is somewhat difficult without digital electronics. Smaller factors are easier.

I imagine historic gearmakers for watches and clocks would have worked out such a system.

PLL's have been around since the 1920's. They could easily divide by n. The actual value of n didn't matter (within some bounds).

Would it not just have been created natively using an appropriate osciallator, given that the reference was from 1939?

A truly decent audio oscillator was only available after 1939, when Hewlett & Packard issued their first instrument.

In a 1985 letter, Dave [Packard] described Hewlett's audio oscillator as "the foundation on which Hewlett-Packard Company was able to grow into the largest manufacturer of electronic instruments in the world, the keystone that allowed four and one-half decades of major contributions to electronic measurement technology and equipment."

Maybe but it'd have been significantly more complex, here they could take a MHz piezo-electric crystal and tack on a few electronic multipliers and dividers (/1000 *11 /25) to get the desired frequency.

Quartz oscillators are usually made in at leas 10th of kHz range, for whatever reason. Second most stable oscillators are LC and would require large coils for lower frequencies.

It would still have been divided down. There aren't many "Hz" oscillators. They're all usually in KHz/MHz range.

Perhaps he meant to say the BBC made that choice for that reason. Not that they're necessarily right to do so.

Or perhaps seconds are relevant in the way sound was broadcast.

And that doesn't even make sense.

An instrument will go ever so slightly out of tune by the time the orchestra is done playing. Temperature and humidity play a large role in how an instrument is tuned.

Sibling comments have the explanation, but good catch. It so happens 439 Hertz = 1000 Diddly (a just-now-invented unit defined as "cycles per 2.277904328 seconds")

Maybe if your "electronics" are vacuum tubes and relays.

In the digital domain it's not an issue at all.

Electronic =/= digital.

It is electronic gating to bring down a 1khz signal. Division and multiplication to a prime number is pretty hard...

But there's nothing special about the 1k signal either, because a second is arbitrary. The original signal might have been 878 Hz instead (or equivalently 1000 cycles / 1.139 seconds which is no more or less privileged than 1000 cycles / 1.000 seconds) and then the gating to 439 is easy.

What do you think they were using in 1939.

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