Significant original text and figures not included in the article in The Mathematical Intelligencer

Geometric Construction of Pythagorean and Just Musical Scales and Commas

Not unexpectedly, the notes are found in the same order of succession as in the circle of fifths. This is a circle with the 12 notes arranged after each other according to an interval of a fifth. The procedure above corresponds to what Helmholtz described, [Helmholtz, 1954, p. 278]:
Pythagoras constructed the whole diatonic scale from the following series of Fifths: F–C–G–D–A–E–B.

Geometrical Construction

We start with two equilateral triangles combined in a six-pointed regular geometric star as shown in Fig. 7. The line from the lower left to the upper right corner gives the triangle highlighted in the figure. The horizontal base, the hypotenuse, of this triangle is the frequency axis with 0 frequency to the left at the origin, O, the note C normalized to frequency 1 in the middle and C2=2 to the right. From elementary geometry and trigonometry one will find that the point marked F corresponds to 4/3 and the point marked G corresponds to 3/2.

Figure 7: Six-pointed regular star with linear frequency axis O–C2 and relative frequencies for C, F, and G marked.

Tetractys of the decad

The topic of this article has been the geometrical construction of the Pythagorean and just scales and in particular the Pythagorean comma, the syntonic comma, and the schisma. Such constructions have roots back to the Pythagoreans who recognized a link between musical scales and geometry. Pythagoras is also well known for the statement that ”all things are number”. A central figure connected with this statement is called the tetractys of the decad (tetra = four, decad = ten). The tetractys is a triangular figure with ten points in four rows. Starting from the top, the rows have a single point, two, three and four points respectively as shown in Fig. 8. The most obvious connection between the tetractys and music is the presence of the ratios of the perfect fourth (4:3), the perfect fifth (3:2), and the octave (2:1) which are so fundamental to both the Pythagorean and the just scales. We find it intriguing that all ten tetractys points appear in Fig. 7, and that in particular six of the points (marked with solid circles) are central to the construction of the scale based on that figure. One cannot but wonder whether Pythagoras or the Pythagoreans may have seen a deeper connection between the tetractys and the geometrical constructions of
this article.

Figure 8: The tetractys of the decad is a triangular figure with ten points
arranged in four rows of one, two, three, and four points (Image: Wikipedia, User: Jossifresco).

Pythagorean Scale

Incidentally, the parallel key to C major, A minor starting with A0, now appears on the frequency axis (see Fig. 2 in main article published 16 March 2023 in TMIN).


[Helmholtz, 1954] Helmholtz, H. v. (1954). On the sensations of tone (AJ Ellis, Trans.). New York: Dover.(Original work published 1877).