beyond 88

Yesterday Om pointed out that it was piano day - the 88th day of the year honoring the 88 keys of most pianos.  I read his post a day late, but there's still hope for another celebration with the same name. An obvious choice might involve the Bösendorfer grand where options include 92 and 97 keys in addition to the normal 88.  Music has been written for these beautiful instruments, but why stop there? 

 

Press a key on the piano and a hammer strikes one or more strings which begin to vibrate producing a sound.  The frequency of the vibration depends on the length of the string with longer strings vibrating at lower frequencies.¹  The left most key causes the longest string to oscillate 27 times per second - 27 Hertz (Hz) - while the right most key produces a 4,186 Hz frequency.  In between the keys are divided into 7 octaves, each composed of 12 keys.  Each octave doubles the frequency so the C above middle C has a frequency twice as high. The notes are evenly divided so each key has a frequency 2^1/12 times - just under six percent higher than the preceding key. 

 

Now we're ready.

 

Let's say you wanted a piano with the range of human hearing - usually given as 20 to 20,000 Hz in young children.   Going from 4,186 to 20,000 Hz, after a bit of arithmetic, requires 27 extra keys on the right.  Five keys on the left will get you to 20 Hz..   One hundred and twenty keys and some long arms would be sufficient to entertain young children to the limit of their range.

 

But what about dogs?  Canine hearing goes up to about 40,000 Hz.   All  we need to do is add an octave, or 12 keys, to cover their range.   Other mammals can hear even higher pitches.  Bats top out around 160,000 Hz so we add two more octaves giving us a 156 key instrument.  We don't have to go much higher as the atmosphere strongly attenuates higher frequencies. A bat can produce a very loud noise, but even their sensitive ears are only good for a 20 meter range.²

 

We've left out elephants which go down to about 12 Hz - that works out to 9 keys on the left. We've grown the beastmaster piano to 165 keys.   I won't cover whales because pianos don't work underwater. 

 

The game can go on.  There are any number of natural sounds of lower frequency (infrasound), but we'll leave it at that..  

 

Well - almost.  There are very faint sounds in the universe - ripples that move compress and expand space-time itself.  They come across a wide range of frequencies, but exquisitely sensitive instrumentation is required for some of the loudest - like colliding black holes where the energy that goes into the resulting gravity wave can be the equivalent of several solar masses.   It turns out the frequency of the first one detected was a chirp that went from about 35 Hz up to about 250 Hz (close to middle C).  You can play the signal as audio.

So much information there, but it may not be pleasing so let's end with one of the most demanding pieces for the human voice - in this case a coloratura soprano.  The Queen of the Night in Mozart’s  Die Zauberflöte. 

I'll leave you to find some good piano music today.  Perhaps Om's list is a place to start. 

_________

¹ It also depends on the mass of the string, its diameter and tension.  Additionally a rich set of other frequencies are produced giving the louder central frequency richness and character.  There's a good deal of art in the design and construction of a fine piano and none of the sound the same. 

² It's a feature.  Longer range would confuse them with signals from other bats, plus they tend to work a few meters from their prey.

 

Comments

[Gregg Vesonder]

wonderful, just simply wonderful. 'Tis wonderful to know a physicist with a bit of quirkiness. But then again are there any other kind. Thanks so much, Steve. --Gregg

[Jheri]

This is SO much fun!!