Cool musical illusion - tritone paradox

2026-07-06T00:00:00Z

Have you heard of “Shepard tones”? They are produced when you play a note in every octave at once, so the “pitch height” is ambiguous.

Well, if you play one of these shepard tones (just a single note, like a ) and you then play its tritone which is half an octave away, some people will hear the second note as higher, and some will hear it as lower.

Try it!

Which you hear seems to depend on your perception of the first note as high or low, which is connected with how you process pitch in language (and therefore is not something you can easily change). People tend to hear the same thing their mother hears, and there are regional tendencies as well (England vs California have opposite tendencies).

This would only be possible if everyone has a latent sense of absolute pitch!

If you arrange the pitches in a loop or circle, you generally hear up for half of the circle and down for the other half, with a little ambiguity in the middle where you might be able to hear it either way.

I find that if you listen to them too quickly, they all seem to be in the same direction. But if you pause between each one, or jump around, you can identify which tones you always hear strongly in one direction, and which ones you can hear both directions. Try it out! See if people around you hear things differently than you.

Tritone paradox piano

References and notes

This effect was first discovered by psychologist Diana Deutsch - read more and see references on the Wikipedia entry for Tritone paradox. Thanks to Joseph Sardin for his shepard tone generator which produced these sounds. To make the tones a little less harsh and make the illusion as salient as possible, I tried to center the octaves on the human vocal range, 80hz to 1000hz.

The Trivium: A Tool for Learning Anything

2021-04-03T00:00:00Z

Information today has become siloed. It’s a common belief that little, if any, expertise from one field of knowledge transfers over to other fields. But there is a forgotten tool that anyone can use to confidently approach new subjects and problems: the Trivium.

The ancients considered the liberal arts to be composed of seven parts. First was a foundation called the trivium, composed of grammar, logic, and rhetoric. Following that was the quadrivium, which was arithmetic, geometry, music, and astronomy (numbers, numbers in space, numbers in time, numbers in time and space). The trivium (literally, “three ways”), is a framework for learning.

Grammar

Grammar is about gaining knowledge: collecting information, without judgment or analysis. It answers the questions “Who, what, where, and when?” This is possibly the most important step, since (depending on what you study) essential information may not be readily available.

Logic

Logic is about gaining understanding. It answers “why?” This is the foundation for relating to the world. It has three components: filtration, correlation, and analysis. It places the information gathered into context, and eliminates inconsistency and resolves conflicting perspectives.

Rhetoric

Rhetoric is about acting wisely: the application of knowledge and understanding, put into correct action. It answers “how?”

Methodically applying this method provides a clear next step when encountering any unfamiliar topic. This is a powerful tool for methodically determining the best way to achieve your goals.

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