Pianotech

is really interesting.

Open the page in two tabs or windows in your browser.

Set one to sound 440

Set the other to 440 and use the > button to increase the frequency. Hear the beats. Go to the sine tab further along to change the upper frequency from sine to triangle. The 3rd harmonic comes in. 

At 506 for instance it's horrible but 513 gives an undertone starting to appear, and at 520 it's sounding nice although I don't see why it should. We can hear the undertone going lower as we get to 528 and when we get to 550 the undertone is in tune and, importantly, gets louder.

Now go right to the note selector and press Db/C# which will take you to all ET notes. Press C#5 and hear the undertone shift to be out of tune. That's our standard industry-wide tuning.

Then there's another phenomenon of importance and this relates to stretch in particular.

With the one page still generating A440 with the other go to A6. It's in tune.

Go to A7 and it's flat. Go to A#7 and it's sharp. Go to A4 then go to A#7. A#7 is more in tune then than A7. Go back to A4. A8 then is flat. A#8 is happier. 

So is top treble tuning really about inharmonicity? Is it more about our own ears and perception of pitch, nothing to do with the piano. This is why in the C6-C7 region I often feel that a tuning is flat even when one's tuned it properly to the 2nd partial and instead simply go for the maximum stretch setting on my machine - and why I feel that many ETD apps are simply too complicated and clever! Were I to tune the top octave properly to the 2nd partial of the lower octave, I'd regard it as flat. But that's where one can hear it audibly resonating rather than to the 4th partial of two octaves below, and duplex scales add muddle well to the equation.

Best wishes

David P