Lionel and Roger Penrose devised an impossible staircase by distorting perspective. The most famous example of this is M.C.Escher's lithograph "Ascending and Descending"

The people in M.C. Escher's lithograph, "Ascending and Descending", seem to climb up and up a set of stairs, but keep returning to the same level where they started. This is an optical illusion. Could there be a similar musical illusion? Could someone design a musical scale that always seems to be going up in pitch but also goes all the way around to the beginning? Someone could, and did.

Diana Deutsch conducts research on the perception and memory of sounds. Some of this has inspired her to develop musical illusions, tricking people's ears in the same way artist M.C. Escher tricked people's eyes.

Diana Deutsch and her colleagues have created so many musical illusions, they have been able to release two CDs filled with them

Deutsch, a psychologist from the University of California San Diego says that musical pitch can be thought of as linear, going from low to high, like the keys of a piano. On the other hand, she says, "pitch also has a circular component which is acknowledged in note names, so as you go up a keyboard you go C, C sharp, D, D sharp, E and so on then you get to A, A sharp, B then you get to C again and continue C sharp, D, D sharp, E and so on, so you go round and round a circle at the same time."

And every time the notes start repeating again, like from one C to the next, we have gone up an octave in pitch.

Note names can be ordered in a circle like the one pictured here. This circularity was what inspired the illusion

Building upon research by Roger Shepard and Jean-Claude Risset, Deutsch found a way to build a "staircase" of musical tones that seems to both go up without end, and just like in Escher's illusion, they appear to never go up to another octave. By changing the harmonics of each tone, Deutsch had in effect, turned a linear musical scale into a circle.

She explains that tones made by natural instruments are composed of many tones, called the harmonics or overtones. Each overtone has a frequency that is a multiple of a fixed base frequency, and so they can be numbered according to that: 1 for the base one, and then 2, 3, 4, 5, and 6. "So if you take the odd-numbered harmonics," Deutsch explains, "numbers 1, 3, and 5, and reduce them in loudness so that you end up with harmonics 2, 4, and 6, you end up perceiving the same note but an octave higher, and that forms the basis of this illusion." (Listen to example)

Deutsch started with the G sharp that one would find in the middle of a piano keyboard, and as she descended note-by-note, she would lower the volume of the odd-numbered harmonics, until she got to A, which now sounded like it was an octave higher — just above the G sharp she started with — like in the example above.

After modifying the volume of the odd-numbered harmonics, Deutsch had a set of twelve tones. She played all the possible pairs of tones to her students and asked them to write down, for each pair, whether they were going up or down in pitch. They sounded like this. (Listen to example)

The student listeners confirmed that changing the harmonics effectively created the musical illusion of a never-ending scale, even though it circled around to its starting point at the end of every octave.

"So when these tones are played just going up," says Deutsch, "note by note, you can hear what sounds like a scale that goes up and up and up and never returns to the beginning, or it can appear to be going down and down and down all the time and never returning to the beginning.

Listen to circular scale going up.

Listen to circular scale going down.

The scales appear to always be going up or down, and we can tell, if we play the first tones and the last tones, that we really never left the octave. (listen) — beginning and end of circular scale going up

Deutsch also developed two gliding scales, where the notes go up or down smoothly. These seemed to have a psychological impact on listeners. She observes "people have been saying that listening to the ascending glide makes them feel sort of uplifted." A descending scale had a different effect. "I've played this to groups of students and I've found," Deutsch says, "after they've been listening to this descending pattern that they seem to be kind of nodding, and they have their eyes closed, and you can tell that they're feeling themselves sort of going down and down and down as though into a bottomless pit." She says she would like to explore these effects on people's moods more closely.

Listen to glissando scale up.

Listen to glissando scale down.

Diana Deutsch says that coming up with illusions is fascinating and fun, but she also believes that her research helps scientists understand how the brain processes musical pitch. Her new results were presented for the first time August 1 at the annual meeting of the Society for Music Perception and Cognition, in Montreal.

The psychologist says she wants to continue exploring how the human mind understands music and perhaps, in the process, find new and fun ways to keep tricking our ears.