How did they figure 440 vibrations a second for the tone A in the old days?

Answer 1

I'm not sure what you mean by 'old days'. A was assigned 440Hz as a standard as recently as 1945, and at that time pitch standards included Tuning Forks, tuned bars, and instruments like the Stroboconn (produced by Conn's Experimental Laboratory in 1936.) Before that time, A was assigned the frequency 438Hz or 435Hz, and prior to that assignment A wasn't a specified frequency. (There are flutes, for example, from the 18th century with up to seven center-sections so the flutist could adjust his pitch to match the local 'standard' over a very large range!)

During the renaissance and early baroque periods, A was simultaneously maintained at two different pitches, which were about a minor third apart, depending on where the music was played, one pitch for church, another for chamber music.

Generally, we now use A=440 for renaissance music, A=415 for baroque music and A=392 for some french baroque music. Some HIP groups use even more pitch standards, either with different instruments for each pitch standard or with much retuning!

As for how frequency was determined, we have tuning forks from as far back as the 1600's, and pitch pipes from before that. Generally, a pipe cut to a given length will resonate at a given frequency regardless of its diameter or the device that makes the air in it vibrate (Flute/fipple or reed). So once a length of a pitch-standard pipe is determined, more can be made without regard to numerical cycles-per-second, and then tuned to one standard by adjustment until the beats go away. Likewise, tuning forks can be made and tuned to a standard. Like most standards in the "old days", someone decided what the standard would be, and everyone else was expected to match it.

The determination of A=435, A=438 and finally A=440 was made by increasingly-international committees in a time (the 19th and 20th centuries) when technology was up to measuring frequency with some considerable accuracy, and the advent of the electronic counter with an oven-controlled crystal clock made measuring frequency with high accuracy possible. It is worth noting that this technology is the same that is used for determining time accurately.