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While desoldering useful components off of old computer hardware, I found quite a large number of 14.31818 MHz crystals.

This seemed odd to me. Why use such an irregular frequency with a very nontrivial conversion to human time units?

At first I thought that it must be a multiple of another frequency with a certain dedicated use (such as 44.1 kHz commonly used as an audio sampling frequency), but my guessing only led to two numbers pretty close to it: 1/7*10⁸ Hz and π/22*10⁸ Hz, both to about 2‰, and I can't seem to deduce what any of these would be useful for.

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It is exactly 4× the NTSC color-burst frequency of 3.579545 MHz. Since it is (well, used to be) used in huge quantities in commercial color TV sets, it is both commonly available, and particularly useful when you want to generate a signal to be displayed on such a TV.

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    \$\begingroup\$ Yes, that's reasonable. So in the parts which do not output a video signal, the only reason is availability? The answer does also still beg the question of why that particular frequency was chosen for the NTSC standard itself. \$\endgroup\$ – FlashCactus Jul 31 '16 at 14:15
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    \$\begingroup\$ The exact frequency for the NTSC color burst is 30 frames/sec x 525 lines/frame x 455/2 cycles/line / 1.001 (a correction factor that avoids a problem) = 3579545.4545... cycles/second. I don't remember the precise reason the correction factor is needed, but you can probably look it up. I think it had something to do with having the field rate (59.94 Hz) NOT precisely locked to the power line frequency. \$\endgroup\$ – Dave Tweed Jul 31 '16 at 14:21
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    \$\begingroup\$ For anybody else looking, the factor of 1.001 looks like it is present to prevent the sound carrier and video carrier from having similar frequencies, which could cause video artifacts. en.wikipedia.org/wiki/NTSC#Color_encoding \$\endgroup\$ – Eldritch Cheese Jul 31 '16 at 15:00
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    \$\begingroup\$ +1: Decades ago (and with a European bias) we referred to NTSC as 'never twice same colour'. \$\endgroup\$ – copper.hat Aug 1 '16 at 16:30
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    \$\begingroup\$ @copper.hat: Sure, and PAL = "Pay for Added Luxury" (re: cost of delay line) and SECAM = "System Essentially Contrary to American Method". \$\endgroup\$ – Dave Tweed Aug 1 '16 at 16:43

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