Tuning and the Least Common Multiple (LCM)

Musicians enjoy espousing the benefits and links between  Music and Math/Science. We could go back to Pythagoras and discuss his ideas on ratios and how there is a structure to what are considered perfect intervals – there are volumes of text on his findings from physics to cosmology. Let us look at a reversed structure of these relationships.

If you were to take a frequency and double it, you would find the octave. Triple it and you find the octave + fifth. Quadruple is the double octave. 5x adds a major third, 6x a minor third, 7x another major third. All this adds up to being a major second shy of a triple octave.

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This is called the “Chord of Nature”

Now, lets create the “Inverted Chord of Nature”

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Stacking this from the bottom; minor third, minor third, major third, fourth, fifth, octave.

The top note is what this is checking for intonation. This is where our 5th grade math can be applied. The least common multiple of the first 6 notes is that seventh note, but only when properly tuned. Even non-musicians can hear slight deviations from being properly tuned using this technique. The reason it works is because all the notes below point to the top in one of their multiples.

The bottom note generates the 1:7 ratio, the next is 1:6, 1:5, 1:4, 1:3, and 1:2. When you consider each of the lower notes as a Fundemental, 1:1 with themselves, but each rationally discordant with one another, the listener is forced to audiate a specific frequency in their mind.

Here it is in practice:

If you’re trying this out on your piano and it doesn’t seem to work, try it in a higher register, bass strings have what’s referred to as “Inharmonicity” which creates irrational partials simultaneously with the rational ones.

If it still doesn’t work, try contacting a Registered Piano Technician who can set it right.

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