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Why do pianos end on an a?

This is because they both have no sharps or flats. This means that if we play the key of A minor or C major on a piano or other keyboard instrument, we would usually play only white keys.

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When the piano was invented it did not have 88 keys and did not start on A. As composers such as Beethoven starting composing music that demanded a wider range of available notes, piano makers of the day responded by building piano's with an expanded range. The precursor of the piano was the harpsichord which was not the first keyboard (the organ was invented prior to the harpsichord). Very early keyboard instruments did not have black keys and therefore were not chromatic. The early harpsichords and the pianos that evolved from them only spanned four octaves and started on a C. This is probably due to the first keyboards having only white keys so they worked in the key of C or A minor.

Here is a quote from this article Fascinating Facts ... pianos ... 88 keys?

According to piano historian and registered piano technician Stephen H. Brady, medieval stringed instruments originally included only the white keys of the modern keyboard, with the raised black keys added gradually: “The first fully chromatic keyboards [including all the white and black keys] are believed to have appeared in the fourteenth century.” Before there were 88 key pianos the 85 key piano starting on A became the norm. From the same article quoted above: Michael Moore, of Steinway & Sons, theorizes that it was a combination of artistic expression and capitalism that gave rise to the 88- key piano. Great composers such as Mozart were demanding instruments capable of expressing the range of the music they were creating. Other composers piggybacked on the expanded range provided by the bigger, “modern” pianos. Piano makers knew they would have a competitive advantage if they could manufacture bigger and better instruments for ambitious composers, and great changes were in store between 1790 and 1890, as Stephen Brady explains: By the end of the eighteenth century, toward the end of Mozart’s career and near the beginning of Beethoven’s, piano keyboards had reached six full octaves, and a keyboard compass of six and a half octaves was not uncommon in early nineteenth-century grands. For much of the middle to late nineteenth century, seven full octaves (from lowest A to highest A) was the norm. Okay - so now we know why the piano keyboard grew larger over time.

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EDIT: And Wheat Williams did a great job (in his new answer) explaining how the technology of piano building evolved to make this expansion possible in order to meet the demand & desire for more notes.

But why did the piano builders all decide to start making the first key on the keyboard an A? Was it arbitrary? Was it a result of limitations of human hearing or engineering? Or was there some logic applied to the decision to start with A?

None of them are around to tell us for absolute certain - and whoever they told when they were alive apparently did not write it down. So all I can do is offer a very logical reason why starting with A became the accepted norm. When the concept of assigning letter names to identify musical notes was presented in the first century, it only made sense to start with A as the name of the first note. This naming of notes using letters predates the building of many instruments. Eventually the note that was identified by the letter name A became common to the tuning of most (if not all) stringed instruments. Thus the A note became the reference note for tuning of all modern instruments in an orchestra. In the early days of keyboard instruments, there was no standardized universal tuning reference that was adhered to. An organ in one church may have been tuned to a different pitch than one across the street. Along about the same time in history that piano builders started expanding the range of their pianos, there came a coordinated effort to standardize musical pitch. From a Wikipedia article on "Concert Pitch" - the French government passed a law on February 16, 1859, which set the A above middle C at 435 Hz.[2] This was the first attempt to standardize pitch on such a scale, and was known as the diapason normal. It became quite a popular pitch standard outside France as well, and has also been known at various times as French pitch, continental pitch or international pitch Piano tuners of the day (in the absence of modern digital tuners) used tuning forks which were tuned to A.

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They started with the A above middle C (which is what the tuning fork was tuned to sound) and tuned the other A notes using octaves. The 85 key piano became the standard before the modern 88 key piano did. There are still some old 85 key pianos around. The 85 key piano that was adopted by the piano makers of the day as the standard configuration spanned seven octaves and started on A0 and the last key was A7. This is very logical from a practical point of view given that the piano tuners of that time started with a tuning fork tuned to A = 435 Hz (this was later changed and now is A=440 Hz) and then tuned the other A's by ear using octaves. So it stands to reason that if we were going to build a piano with an extended range that satisfied the composers of the day and wanted to span 7 octaves (close to the limits of humans ability to distinguish notes) and if we knew that to tune that piano we would be starting with a tuning fork tuned to A above middle C and work from there - it only makes sense that we would go as far as we could in both directions to cover most of the effective range for human hearing. And to do that it makes the most sense to proceed in octaves for the sake of simplifying the tuning process. If we start in the middle on A and expand from there in both directions, we end up with a keyboard that starts and ends on A. So there you have it! Later the 3 keys were added to the end (to end on C) perhaps to complete the C major scale on the last octave.

music.stackexchange.com - Why is the lowest note on the piano an A? - Music Stack Exchange