How long is the Universal Keyboard?

 kw: book reviews, nonfiction, frequency, vibration

Once in a while I encounter a book with a wholly fresh idea, an unexpected view of a subject. Such a book is What the Ear Hears (and Doesn't): Inside the Extraordinary Everyday World of Frequency, by composer and musician Richard Mainwaring.

Frequency is related to everything physical. In the 1970's I worked on the campus of the California Institute of Technology (CalTech), and sometimes I would arrive early and read in the campus library. I was on the top floor, the tenth, one day, when the building began to shake. The elevator shafts began a regular Click-Click, a little longer than a second apart. Alarmed, I found a phone and called campus security. They said, "Oh, Dr. Richter is on the roof shaking the building with his earthquake machine." The machine was a strong variable-speed motor turning an off-center weight. I learned that the building is full of strain and motion sensors, and Dr. Richter (of the Richter Scale) did such experiments from time to time, to learn the resonant frequencies of the structure and how it responded to various frequencies.

Something less planned but no less alarming happened in South Korea when a tall building began to shake, rather badly. This building is much taller than the CalTech library. The "culprit" was not an earthquake, but a Tae Bo class of 23 people, stomping and taking lunges in unison, on the twelfth floor. Their workout was on the resonant frequency of the building. Mainwaring tells us that building engineers installed equipment to dampen and counter resonance in the future, making the building safe for fitness classes in the future.

For the frequency of anything to persist, it must stimulate a resonant system. For example, a tuning fork, when struck on something like the edge of your hand, oscillates at a characteristic frequency, such as the "standard A" at 440 Hz that is used to tune up an orchestra or any modern musical instrument. If you consider a tuning fork's reaction in detail, it is stimulated by the strike with a wide range of frequencies, but those that it cannot resonate with vanish quickly, leaving the frequency it is calibrated to.

In order to make all the frequencies he writes about relevant to readers of the book, the author attaches them to popular musical compositions, saying things like, "…the Techno Mart tower of Seoul gets vertically excited whenever it hears the opening G of Beethoven's Fifth Symphony (eleven octaves lower though…)". To unpack this: That "G" is 196 Hz (Hz, or Hertz, designates cycles per second, using lots fewer letters). Eleven octaves is the 11th power of two, or 2,048; 196/1,024 = 0.19 Hz, or a period of 5.2 seconds.

Early in the book the Infinite Keyboard is introduced, starting with the standard piano keyboard. A piano can play 88 notes, and at 12 notes per chromatic octave, that comes to seven-and-a-third octaves. The highest note on a piano is C8, with a frequency of 4,186 Hz, and the lowest C on the keyboard is C1, with a frequency of 32.7 Hz. These have a ratio of 128:1 because each octave, from C to C, encompasses a doubling of frequency, or a halving if counting downwards. The lower 1/3 of an octave ends on a low A (A0), with a frequency of 27.5 Hz. This is just a little above the lowest tone that most people are able to hear, 20 Hz, a "note" midway between F#0 (19.44 Hz) and G0 (20.6 Hz).

However, people in their teens and younger can hear many sounds higher than C8; the limit is about 20,000 Hz (AKA 20 kHz), which is a D# more than four octaves above C8. How long would a keyboard need to be to play that note? Each octave has a width of 165 mm (6.5 inches). That D# would then be 2.25x165 = 371 mm or about 14.6 inches further. At that point, it gets hard to reach both ends! This ignores the different widths of the white and black keys on the keyboard. The author does the math in the background so the reader doesn't have to.

Once that is set up, a survey of (almost) the entire realm of frequency begins with the author's pick for the lowest note in the universe, a slow oscillation of a galaxy-sized plasma jet with a period of 18.5 million years. He reports the frequency as 0.000000000000002 Hz. I dickered around with the figures and refined that a little: 0.0000000000000017 Hz. This is a B, 57 octaves, plus one note, below middle C. How far is that on the Infinite Keyboard? 9.4 meters, or almost 31 feet to the left. This shows the power of the logarithmic transformation between frequency and music notation.

At this point, let us touch on scientific notation. Very large and very near-zero quantities are cumbersome when written out. Thus, that low tone is better represented as 1.7E-15 Hz. "E-15" means "ten to the minus 15th power", which puts 14 zeroes in front of the "1", locating it in the 15th decimal place. More familiar numbers such as a million, or 1.0E+6 or one-billionth, or 1.0E-9, ought to make this comprehensible.

So this keyboard is getting long. It won't fit in your living room any more. Fast-forward to near the end of the book. The very low note was discovered by a space telescope called Chandra using gamma rays. These are some of the highest frequencies in the universe. X-rays and gamma rays are not usually reported in terms of frequency, but energy per photon. Photons of light have enough energy to stimulate the dyes in our cone cells, but not enough to do damage. Visible light photon energies range from 1.77 eV to 3.1 eV, while UVC, the ultraviolet rays that cause skin damage, have energies in the 4-5 eV range. The unit eV is "electron-volt"; 1 eV is the energy of an electron accelerated by a one-volt difference between two electrodes.

The X-rays that are used by your dentist or bone doctor have energies in the range 10,000 to 100,000 eV. They are called "penetrating radiation" because they'll easily go through flesh but not through bones or teeth. Gamma rays are much more energetic, in the millions to billions of eV. The most energetic gamma ray detected had an energy of 1.4 peta-eV, or 1.4 quadrillion eV. That's 1.4E+15 eV…about a million billion times as energetic as visible light. What frequency might that be? There is a proportionality constant that cranks out the frequency as 3.4E+29 Hz. On the Infinite Keyboard, that's a C# just over 90 octaves above middle C, at a distance of nearly 14.9 meters, or 48'-9". Put this together with the distance below middle C of the low-low note, and we find we need a total keyboard length of 24¼ meters, or just over 79½ feet. Hm! Eighty feet. That's all we need of an "infinite" keyboard to encompass all the frequencies ever detected.

That got me thinking: How long would the keyboard need to be to encompass all possible frequencies that "fit" in our universe? I would call it a Universal Keyboard!

• The lowest possible note is one that began at the Big Bang and has just completed one cycle, with a period of 13.8 billion years, or 4.35E+17 seconds. It is a "note" between E and F, 66.6 octaves to the left of middle C.
• The highest possible note is one with a period equal to the Planck time, or a frequency of 2.95E+42 Hz. It lies between C and C#, just over 133 octaves to the right of middle C.

The sum of these octave ranges is 199.7, and thus the full length of the Universal Keyboard is 32.95 meters or 108 feet. Building lots in my neighborhood are 100 feet wide. This keyboard would have to be placed on a diagonal to fit in the lot, and you'd need a big, big house with a basement rumpus room at least 75 x 80 feet in size, or more to have room to play the highest and lowest notes.

But beware: as you approach the highest notes, there is danger. Actually playing the top note would force a new Big Bang, and create a new universe. Your basement room would not survive, nor would the rest of known creation.

I've gone far beyond the author's scope. There's plenty to do with "only" 80 feet of keyboard: all of known science. I haven't touched on most things he gets into. I'll leave that up to you. I didn't recognize nearly any of the musical pieces the author uses to anchor our understanding of this note or that note, but that's OK. You might…read this book! Great fun.