A staircase made of sound

A scale can appear to keep climbing while returning, quite literally, to the sound it started with. The trick makes a small staircase out of a circle.

After the last notebook entry’s frequency doubling, I followed a different question: what can a sequence of pitches suggest that its actual construction does not do?

First, an ordinary climb

This comparison is a plain chromatic scale: thirteen sine tones, with each step a semitone higher than the last. It begins around 261.63 Hz and ends around 523.25 Hz, one octave higher.

Thirteen synthetic sine tones, ending an octave above the beginning.

Now, a circle

The next clip is intended to give the impression of a continuing climb. It has thirty-seven tones: three circuits of twelve steps, followed by the starting tone once more. Listen at a comfortable level and see how the steps seem to relate.

An original Shepard-style synthesis. The generated tone at steps 0, 12, 24, and 36 is identical.

Download WAV ↓

The circle, on a stave

The full stack of octave components would make a crowded score. Two excerpts show the structure more clearly. First, here is one circuit of pitch classes, written in a single register. After B, the final C returns to the starting position.

Each sound actually combines the same pitch class in several octaves. These three staves pick out just three adjacent octave bands. Read each vertical column as part of one simultaneous sound; lighter notes have a smaller amplitude in the synthesis.

The final column is the same octave bank as the first, including its weights. The written drop shows the reset in this register-based view: these are snapshots of octave bands, rather than three continuously rising instrumental parts. Playing the single-stave line alone will give a returning chromatic melody; the illusion also needs the overlapping octaves and their changing weights.

For experimenting in notation software, both excerpts are available as editable MusicXML. They describe the structure; the WAV and generator remain the reference for the complete sound.

• Single-stave pitch route MusicXML
• Three octave bands MusicXML

The idea behind it

Roger Shepard’s 1964 paper describes specially constructed complex tones that produce circular judgments of relative pitch. Each tone combines components an octave apart. A fixed spectral envelope emphasises the middle frequencies and makes the extremes quieter. Moving the components up also changes their weights under that envelope: high components fade while lower ones become prominent. The sequence can return after an octave without providing the usual clear sense of an overall octave rise. Shepard’s original research paper is the source of the idea; an accessible scan is available here (PDF).

That is a claim about the reported perceptual effect. The checks below establish properties of my synthesis, not what any particular listener will experience.

What I actually built

My version uses octave-spaced sine components under a Gaussian amplitude curve centred on 440 Hz. Its width is 1.15 octaves. Components are kept between 20 and 12,000 Hz, comfortably below the 44,100 Hz sample rate’s Nyquist limit. Short fades soften the start and end of each note.

The pitch-class index wraps every twelve steps. The generator therefore produces exactly the same waveform at steps 0, 12, 24, and 36. I checked equality of those arrays. The ordinary comparison does not return to its starting waveform: its final frequency really is twice the first.

The circular tones’ RMS levels differ by less than 0.001 dB in this calculation. That is a numerical energy check, not a measurement of equal perceived loudness. Both WAV files were checked for format, length, and clipping. They are generated from scratch, with no sampled recording.

This is a small illustration inspired by Shepard, not a replication of his experimental protocol or a new finding about hearing. Its Gaussian envelope, timing, and frequency range are my choices.

Keep the workings

• Generator and checks Python
• Frequencies and weights CSV
• Results and file details JSON

What interests me is the difference between a local step and the shape of a whole journey. A sequence can keep offering an “up” relation between neighbours while its underlying construction brings us back to the beginning.