Contributions in this column describe how smartphones and tablets can be used to analyze experimental data in physics experiments. This is possible in nearly all areas of physics—from classical mechanics, to thermodynamics, electrics, and radioactivity. While many examples already showed the analysis of acoustic experiments with mobile devices, no psychoacoustic phenomenon has been studied with the smartphone yet. Nevertheless, acoustic illusions are great starting points for an intrinsic motivation of learners. One great acoustic illusion, often found in science centers, is the so-called Shepard scale illusion. This paper shows how to use mobile devices to analyze and produce Shepard tones to get a deeper understanding of the acoustic concepts like tone, sound, pitch and frequency.

Contributions in this column describe how smartphones and tablets can be used to analyze experimental data in physics experiments. This is possible in nearly all areas of physics—from classical mechanics,1–3 to thermodynamics,4,5 electrics,6 and radioactivity.7 While many examples already showed the analysis of acoustic experiments with mobile devices,8,9 no psychoacoustic phenomenon has been studied with the smartphone yet. Nevertheless, acoustic illusions are great starting points for an intrinsic motivation of learners. One great acoustic illusion, often found in science centers, is the so-called Shepard scale illusion. This paper shows how to use mobile devices to analyze and produce Shepard tones to get a deeper understanding of the acoustic concepts like tone, sound, pitch and frequency.

A Shepard scale is a series of sounds that, when played in a loop, gives the impression of an endless rising scale. It is an auditory illusion, first described by the American psychologist R. Shepard10 in the 1960s. The individual sounds—the Shepard tones—are periodical signals, which consist of a set of frequencies: the fundamental f0 plus the harmonics fn that are powers of two or, in other words, octaves. The amplitude A as a function of time t is:

A(t)=A0(sin2πf0t)+A1(sin2πf1t)+A2(sin2πf2t)++An(sin2πfnt).
(1)

For a typical Shepard tone, the amplitudes Anof the high (e.g. n > 2 in Fig. 1) and low (e.g. n < 2 in Fig. 1) frequency harmonics are smaller. This leads to an ambiguity in the perception of the sounds pitch: the same sound can be perceived as low-pitched in the first round and as high-pitched in the second round.

Fig. 1.

Spectrogram of two full circles of the Shepard scale (tones played one after another). Left: phyphox experiment audio spectrum. Shown is the intensity of the frequencies (white = high, black = low) over the time. The record of the sounds 1 (red) and 5 (green) are highlighted. The spectrogram confirms that the sound of the buttons stays the same. Right: Screenshot of the app Shepard Illusion.

Fig. 1.

Spectrogram of two full circles of the Shepard scale (tones played one after another). Left: phyphox experiment audio spectrum. Shown is the intensity of the frequencies (white = high, black = low) over the time. The record of the sounds 1 (red) and 5 (green) are highlighted. The spectrogram confirms that the sound of the buttons stays the same. Right: Screenshot of the app Shepard Illusion.

Close modal

As Suits11 recently described, the distinction between pitch of a sound and frequency of a sound is important to understand this phenomenon. Many studies show that the perception of pitch is very individual and dependent on the geographical region.12,13

In this experiment, students can investigate the number of harmonics of a Shepard tone and compare their frequencies. Furthermore, it can be tested if the harmonic frequencies of a tone are multiples by the power of two of the fundamental:

f0=12f1=14f2==12fn.
(2)

Two mobile devices were used to analyze the Shepard scale illusion. The free Android app Shepard Illusion14 was used on one device to produce the sound—and to explore the illusion. Suits explains how to use audacity on a PC to produce the sounds. On the other device the free application (iOS and Android) phyphox15 was used to analyze the sounds.

To demonstrate and explore the Shepard illusion with the application in the first step, students can simply push the buttons (1-7, see Fig. 1) one after another clockwise to get the illusion of an endless rising scale—o r anticlockwise for an endless falling scale. In a second exploration, a stunning comparison between button 1 and 5 can be observed. First step: Playing buttons 1 up to 5 (1, 2, 3, 4, 5), and then alternately 1 and 5, suggests that the pitch of sound 5 is higher than of sound 1. Second step: Playing buttons 5 up to 1 (5, 6, 7, 1), and then alternately 1 and 5, suggests that the pitch of sound 5 is lower than that of sound 1. The illusion of the endless rising scale lies in the perception: The brain is fooled as the perception of pitch of the sound is constantly shifted.

To investigate this phenomenon, the second device is used to record and analyze the Shepard tones. For the analysis the app phyphox was used. It offers the experiment audio oscilloscope, which can be used to visualize the sound and show the periodicity of the Shepard tone. The phyphox experiment audio spectrum calculates the fast Fourier transform (FFT) and can be used to analyze the spectra of the Shepard tones (see Fig. 2).

Fig. 2.

Analysis of the waveform of the Shepard tone 1. Left: Screenshot of the app Shepard Illusion. Buttons 1 and 5 are highlighted. Right: The oscilloscopic representation reveals that the played notes are not based on a single sinusoidal wave but rather a composition of several frequencies. These can be identified by looking at the audio spectrum (FFT) of the sound. The harmonic frequency with the highest amplitude is n = 2.

Fig. 2.

Analysis of the waveform of the Shepard tone 1. Left: Screenshot of the app Shepard Illusion. Buttons 1 and 5 are highlighted. Right: The oscilloscopic representation reveals that the played notes are not based on a single sinusoidal wave but rather a composition of several frequencies. These can be identified by looking at the audio spectrum (FFT) of the sound. The harmonic frequency with the highest amplitude is n = 2.

Close modal

What do the Shepard tones look like on an oscilloscope? Are they really tones or rather a composition of different sinusoidal waves?

The analysis of the Shepard tones of the app Shepard Illusion confirms that the tones are a periodical signal that is composited of multiple frequencies (Fig. 2 shows exemplary sound 1). To analyze the components of the different Shepard tones, we looked at the fast Fourier transform (FFT) of the different tones. The phyphox experiment audio spectrum records the sound and calculates the FFT.

According to Eq. (2), the lowest detected frequency in the FFT of sound 1 (see Fig. 2) is its fundamental f0 (328 Hz). The other peaks in the FFT are nearly exact powers of two (656 Hz, 1312 Hz, 2648 Hz, 5273 Hz, 10546 Hz). This relation can be shown respectively for the sounds 2-7. For example, sound 2 has as its lowest frequency 375 Hz, which is a bit more than one whole step above the fundamental in sound 1. A look at the History tab of the experiment audio spectrum offers the possibility to record a spectrogram of the Shepard tones. By playing the Shepard tones one after another, a direct comparison is possible. The spectrogram in Fig. 1 shows the composition of the single Shepard tones as well as their correlation: One round (1-7) of sounds accords to an octave and is perceived as such.

The Shepard illusion is a great and stunning phenomenon. It combines the physics of acoustics, theory of music, and psychology of perception. The presented experiment offers various options to investigate the science behind this phenomenon with no additional material needed.

Shepard illusions had recently been used in many soundtracks for films,16 as the illusion of endless rising sounds is a possibility to create tension.

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