[hunter2_]

Further reading: https://my.ptg.org/HigherLogic/System/DownloadDocumentFile.a...

Notably:

> In equal temperament, all perfect fifths are “contracted”, while all perfect fourths as “expanded”. Minor thirds are contracted, while major sixths are expanded. Major thirds are expanded, while minor sixths are contracted. The piano tech must have knowledge of the approximate beat rates the intervals of equal temperament in the temperament octave: The beat rate of perfect fourths within the temperament octave may be about 1 beat per second. The beat rate of perfect fifths within the temperament octave may be about 1/2 beat per second. The beat rate of F3-A3 major third is about 7 beats per second and that of higher thirds are faster.

In my previous comment, my memory was a bit off when I said "about half the beat rate"... that's the difference in rate between fourths and fifths, apparently.

> Example: to check the tuning of D4 within the temperament octave, play A3-D4 and G3-D4. The fourth should beat faster than the fifth. If the fifth is too fast and the fourth too pure perhaps the D4 is flat; if the fourth and fifth beat at the same rate, perhaps the D4 is flat; if the fourth beats too fast and the fifth is too pure, perhaps the D4 is sharp.

> you can use more and more checks as one progresses through the sequence and tune each new note as a “best compromise” with all the previous notes, that is, each new note will not depend only on the last note tuned, so there will be more of a chance that errors will not accumulate in the later notes tuned.