"Circle" can produce the feeling of infinity. Making good use of mathematical methods can cleverly create such an illusion.

The pitch class itself has the characteristic of a cycle. Counting a note up or down with 12 semitones means an octave higher or an octave lower. They sound similar. When we sing, they are given the same pitch name. The mathematical expression of a higher octave or a lower octave is that the frequency is twice or half.

The pitch class circle. [1]

Using the cyclic characteristic of the pitch class circle, an infinite ascending and descending effect can be produced. There is no highest pitch, nor the lowest pitch. According to the description in Diana Deutsch's book, components of several octaves are added up. A fixed, bell-shaped envelope in the frequency domain is to give the mid-range sound the highest amplitude, while the amplitude in high and low frequencies gradually decreases [2]. My own experiment is to design a band-pass filter directly in the frequency domain. The design procedure is as follows.

Use 10 octave harmonics to do additive synthesis and then process by a band-pass filter. From the frequency domain's perspective, when the tone is ascending, the low-frequency harmonics fade in, while the high-frequency harmonics fade out. When the tone is descending, the low-frequency harmonics fade out and the high-frequency harmonics fade in.

Ascending Shepard Scale

Descending Shepard Scale

Ascending Risset Glide

Descending Risset Glide

This example is associated with the famous illusion "Penrose stairs".

Penrose stairs. [3]

[1] Deutsch, Diana. Musical Illusions and Phantom Words (p. 64). Oxford University Press. (2019)

[2] Same as [1].

[3] __https://commons.wikimedia.org/wiki/File:Impossible_staircase.svg__