Pythagorean Intervals

This page is a sub-page of our page on Music.

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Related KMR-pages:

The Logarithmic Piano

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The basic pythagorean tuning intervals:

Skärmavbild 2018-02-04 kl. 23.30.12

Skärmavbild 2018-02-04 kl. 23.32.06

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According to legend, the pythagoreans used to explore the structure of numbers by arranging pebbles in regular patterns that became known as “figures”. In the rectangular pattern below they discovered the basic pythagorean tuning intervals:

\, \begin{matrix} & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ 1 & & \blacksquare & 2/1 & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \square & \square & \blacksquare & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 2 & & \blacksquare & \blacksquare & 3/2 & \blacksquare & \blacksquare & \blacksquare & \square & \square & \blacksquare & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 3 & & \blacksquare & \blacksquare & \blacksquare & 4/3 & \blacksquare & \blacksquare & \square & \square & \blacksquare & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 4 & & \blacksquare & \blacksquare & \blacksquare & \blacksquare & 5/4 & \blacksquare & \square & \square & \blacksquare & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 5 & & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & 6/5 & \square & \square & \blacksquare & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 6 & & \square & \square & \square & \square & \square & \square & \square & \square & \blacksquare & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 7 & & \square & \square & \square & \square & \square & \square & \square & \square & \blacksquare & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 8 & & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & 9/8 & \blacksquare & \square & \square & \square & \square & \square & \blacksquare \\ 9 & & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & 10/9 & \square & \square & \square & \square & \square & \blacksquare \\ 10 & & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \blacksquare \\ 11 & & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \blacksquare \\ 12 & & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \blacksquare \\ 13 & & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \blacksquare \\ 14 & & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \square & \blacksquare \\ 15 & & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & \blacksquare & 16/15 \end{matrix} \,

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From Wikipedia:

Skärmavbild 2018-02-04 kl. 23.34.22

/////// OBS: The magic triangles described below are NOT related to pythagorean intervals.

Magic triangle:

\; \;\; 1 \; = \; 1 \; = \; 1 \, \times \, 1 \, \times \, 1 \;
\;\;\; 3 \; + \;\; 5 \; = \; 8 \; = \; 2 \, \times \, 2 \, \times \, 2 \;
\;\;\; 7 \; + \;\; 9 \;\, + \, 11 \; = \; 27 \; = \; 3 \, \times \, 3 \, \times \, 3 \;
\; 13 \; + \; 15 \; + \, 17 \; + \; 19 \; = \; 64 \; = \; 4 \, \times \, 4 \, \times \, 4 \;
\; 21 \; + \; 23 \; + \, 25 \; + \; 27 \; + \; 29 \; = \; 125 \; = \; 5 \, \times \, 5 \, \times \, 5 \;
\; 31 \; + \; 33 \; + \, 35 \; + \; 37 \; + \; 39 \; + 41 \; = \; 216 \; = \; 6 \, \times \, 6 \, \times \, 6 \;
\; \cdots \;

Moreover, we have:

\; 1 \; = \; 1^2 \;
\; 1 \; + \; 3 \; = 4 \; = \; 2^2 \;
\; 1 \; + \; 3 \; + \; 5 \; = 9 \; = \; 3^2 \;
\; 1 \; + \; 3 \; + \; 5 \; + \; 7 \; = 16 \; = \; 4^2 \;
\; 1 \; + \; 3 \; + \; 5 \; + \; 7 \; + \; 9 \; = 25 \; = \; 5^2 \;
\; 1 \; + \; 3 \; + \; 5 \; + \; 7 \; + \; 9 \; + \; 11 \; = 36 \; = \; 6^2 \;
\; \cdots \;

Both these triangles can be continued indefinitely.

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