# Pythagorean Intervals

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Related KMR-pages:

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The basic pythagorean tuning intervals:

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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:

/////// 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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