The way a piano tuner tunes to equal temperament is by listening to beats or the sound of the interval getting louder and softer in a steady rhythm. Playing an octave should have no beats.

To put this into musical terms as opposed to mathematical terms, playing an octave should be like playing 1/8th notes with quarter notes where every other 1/8th note is at exactly the same time as a quarter note (people can pick nits here - but this is the idea). In contrast, fifths should have beats with equal temperament. A just fifth is like playing 3 against 2 with both 3 and 2 starting at the same time. With equal temperament, the 3 is a little bit slow.

Back to tuning, when

playing a fifth, a piano tuner is listening for a beat approximately every second. Piano tuners will also listen to major thirds which will have a faster rate of about 4 or 5 beats per second.

To connect the sound back to the math, if one has a fundamental at 440 hz, a just fifth above that would be 660 hz whereas an equally tempered fifth is a little flat at 659.25 hz. The result is that the fifth gets behind one oscillation every 1.3 seconds which is the interval at which you'll hear a beat. A just major third above 440 hz would be 550 hz whereas an equally tempered major third will be a little sharp at 554.36 hz. The result is that the equally tempered third will get ahead 4.36 oscillations every second which means the tuner will hear 4+ beats per second.

When you hear a piano tuner at work, they loudly play these intervals over and over, listening for beats because they can't judge if they're getting exactly 4.36 beats per second but they can judge if they made too much of an adjustment one place, causing problems in another place on the scale. If all fifths were just instead of being a little flat, octaves would be sharp. So that's why they keep listening to octaves.

You should be able to hear beats on a tuned piano - even an electronic one. Compare playing an octave with a fifth and listen for a slow pulse to the fifth.

 

The way a piano tuner tunes to equal temperament is by listening to beats or the sound of the interval getting louder and softer in a steady rhythm. Playing an octave should have no beats.

When playing a fifth, a piano tuner is listening for a beat approximately every second. Piano tuners will also listen to major thirds which will have a faster rate of about 4 or 5 beats per second.

To connect the sound back to the math, if one has a fundamental at 440 hz, a just fifth above that would be 660 hz whereas an equally tempered fifth is a little flat at 659.25 hz. The result is that the fifth gets behind one oscillation every 1.3 seconds which is the interval at which you'll hear a beat. A just major third above 440 hz would be 550 hz whereas an equally tempered major third will be a little sharp at 554.36 hz. The result is that the equally tempered third will get ahead 4.36 oscillations every second which means the tuner will hear 4+ beats per second.

When you hear a piano tuner at work, they loudly play these intervals over and over, listening for beats because they can't judge if they're getting exactly 4.36 beats per second but they can judge if they made too much of an adjustment one place, causing problems in another place on the scale. If all fifths were just instead of being a little flat, octaves would be sharp. So that's why they keep listening to octaves.

Piano tuners tune the equalized temperament in the middle of the keyboard and extend this to higher and lower octaves by tuning the octave. As a result, there is not a lot of variation in the rate of the beats for the subset of notes in the middle of the keyboard tuned by listening to beats.

Equal temperament has been achieved for 400+ years by listening to beats, not by devices unavailable until recently. See https://en.wikipedia.org/wiki/Piano_tuning under the heading of Temperament.

You should be able to hear beats on a tuned piano - even an electronic one. Compare playing an octave with a fifth and listen for a slow pulse to the fifth.