Figuring out: Shepard Tone

The Shepard Tone is an interesting audio illusion that creates the impression of an always rising or falling pitch that really doesn’t get anywhere. Despite feeling like always going up or down, is always stuck in an eternal auditory fractal. How is this possible? And could it be useful for sound design?

The Penrose Stairs is a nice visual equivalent. Depending on perspective, it looks like they are always going up or down but we are really just going in circles.

The Penrose Stairs is a nice visual equivalent. Depending on perspective, it looks like they are always going up or down but we are really just going in circles.

History & Working Principles

Roger Shepard described this idea in his 1964 paper “Circularity in Judgements of Relative Pitch”. The original concept was conceived in a musical context with pitches jumping in discrete steps, aka notes.

He basically stated that to create an apparently always rising melody, we would need to create a circular or looping pattern consisting of sets of ascending notes that are faded in and out in an specific timing.

So we would start with just one ascending scale of notes. This scale will get to the end of the instrument range pretty soon so the trick is to sneak in a new set of notes doing the same thing but fading them in slowly as we fade out the previous set.

If we do this with the proper timing, we get the feeling of an eternally ascending scale.

You can see that working in the following example:

As you can see, the key to get this effect is to use volume to mask the replacement of older octaves with newer octaves that will always start from a lower tone, giving the illusion of an ascending tone overall, despite the average pitch staying constant.

Here is the same concept written in MIDI in Pro Tools. Volume is indicated by the velocity bars below. The lower green notes are rising in volume, while the blue higher ones are fading out. The central red notes stay at the same volume. Again, the average pitch is always the same, but the dynamic changes in the 3 scales provide the illusion of eternal ascension.

Later on, Jean-Claude Risset created a continuous version, called the Shepard-Risset glissando where the pitch glides without discrete jumps, making the overall effect more convincing and seamless.

In this case, the principles stay the same, there is always a new octave gradually fading in to replace the octave that will gradually fade out. This version can be much more useful for Sound Design although to achieve it we need to use instruments or synths that are able to glide through different pitch values in a smooth way.

Risset also tried to apply to concept to rhythm, layering different versions of a beat at proportional tempos (30, 60, 120 & 240 for example) and fading them in and out to create the illusion of an always ascending rhythm. Check out this examples created by music researcher Dan Stowel, below you can see how one of them looks in the spectrogram. Notice the upwards pattern and how the different versions fade in and out.

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Building our own designs

Now that we have a good idea of how these effects work, let’s see if we can get creative and build a Shepard-Risset tone that could be useful in a sound design context.

I first tried using Native Instrument’s Form since it is a sample based synthesizer where you can use any sample as a source. I used this tutorial as a starting point.

Basically, you trigger several octaves at once and use two LFOs, one to control the pitch so it’s always rising and a second to control the oscillator level so it rises and then falls. Also, I adjusted the general envelope so sounds have a long attack and release just so they blend together as they come and go. This is the result just using a sine wave:

It basically works but the overlapping is a bit noticeable. I then tweaked the timing and started to play with different sounds:

Form only gives you 30 minutes to demo the plugin so I decided to use the limited time to go for one of the most obvious applications of the Shepard tone: en engine ramping up or down. Here are some of the ones I came up with. Keep in mind that the advantage of generating these is that you have an infinite amount of acceleration and deceleration which can be very handy for later editing.

All of them are quite obvious, you can tell where the sound is re-starting and new octaves are fading in. I think you could fix this playing with the volume values (although I don’t know if an LFO is the most confortable way of doing this) or maybe using more octaves.

Lastly, this last example is interesting because on top of the Shepard effect, I was changing the length of the sample to enhance the feeling of acceleration: as the sample gets shorter, the engine feels to be going faster. I tried to play around with the plugin, kind of driving in real time. This could also have interesting applications for video-games.

After this, my demo expired and I felt I didn’t have enough time to improve the effect and play around with the settings. So I looked for an alternative and after some failed experiments I found “Endless Series”, an specialized Shepard plugin by Oli Larkin.

It offers two synthesiser modes plus four audio processing modes so you can create Shepard tones from scratch or using an audio sample as a base.

There are also a nice amount of variables you can tweak to customise the result. So let’s hear some of the tones I got from this plugin.

First, here are some examples just using the synthesizer built into the plugin. You can create a discrete or a continuous (glissando) tone (Example 1). In the case of a discrete or stepped tone, you can use several different musical scales. A chromatic scale will give you the classical Shepard feel (Example 2) but you can also play with other more exotic ones. In example 3 below I tried creating a dreamy, impressionist whole tone one. Is cool, but that last one doesn’t have much of the going up in pitch feeling.

The plugin also works nicely if you want to create engines. Here are a couple of examples.

As for the audio processing mode, there are different effects that you can apply. The simplest mode, am input, just applies the Shepard processing to the sample as far as I can tell. It works in an strange way with tonal content, the Shepard effect is not very pronounced and it adds a descending tone for some reason which doesn’t help. Here is this mode applied to just a sine wave:

This same mode goes nuts with more noisier content. Here is another example using an engine sound. As you can hear, the am input mode introduces a lot of noise and artefacts. I tried playing around with the settings and using other source material but I could not make it sound clean. I don’t know if I’m missing something.

But there are two other modes that can give better results. There is a flanger and a phaser setting. As you can hear, they sound much cleaner, although the effect is quite mild in the case of the phaser. I just wish there was a way to have a “sheparded” sound as clean as this but without the flanger effect on top.

In summary, I feel that I didn’t find the perfect “Shepard Machine” but I’m sure that there are other options out there. I was also thinking that there is probably no plugin that can do everything perfectly (like sample based and synth based and musical options, etc…) so maybe an array of different plugins may be needed for different purposes.

Use in media

Shepard tones have been used in several music and film projects, sometimes in a subtle way, other times quite explicitly.

In music, they can give a very trippy and psychedelic feel (see Pink Floyd’s “Echoes” below) or they can be used to create rising tension (Used extensively in movies like “Dunkirk” or “Flight”). As some of my examples from before showed, they can also be used to create fantasy or sci-fi vehicle engines. In “The Dark Knight”, Nolan wanted the batpod to feel like an unstoppable force that doesn’t even shift gears which sounds like a perfect use case for the Shepard tone.

But probably my favourite example, and maybe this is just nostalgia, is in Super Mario 64, which features an endless staircase that you need to overcome to get to the final boss. The game gives you the illusion of an eternal ascend but you are just running on a “virtual treadmill” and getting nowhere. Analogously, the music is using a Shepard tone to achieve the same effect, an apparent ascension that is really just circular. A great example of a Shepard tone used in an interactive environment.