## 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.

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

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).

### References

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

## Geometric Construction of Pythagorean and Just Musical Scales and Commas

Open access link til artikkelen i The Mathematical Intelligencer finner du her (gratis):

Etter mange års forskning har jeg endelig fått publisert min første internasjonale vitenskapelige artikkel med meget god hjelp av professor i fysikk ved Universitet i Oslo, Sverre Holm, og dr. ing. Alv I. Aarskog.
Artikkelen ble publisert i journalen The Mathematical Intelligencer den 16. mars 2023 innunder et av verdens største akademiske forlag, Springer Nature.
Tittelen er «Geometric Construction of Pythagorean and Just Musical Scales and Commas».

Kort sammendrag:
Den pytagoreiske toneskalaen er knyttet til matematikk og basert på heltallsforhold. Det nye er at artikkelen demonstrerer at denne skalaen kan konstrueres geometrisk i en 30-60-90 graders trekant! I tillegg visualiseres at de fem hevede og fem senkede halvtonetrinnene, som befinner seg på de samme fem svarte tangentene på et piano, er forskjellige. Denne forskjellen kalles det pytagoreiske komma (531441/524288).
På samme måte konstrueres også den renstemte toneskalaen geometrisk. Tonene C, D, F og G er felles for begge toneskalaene, men de tre tonene E, A og H er forskjellige og skiller den pytagoreiske fra den renstemte toneskalaen.
Forskjellen mellom disse tre tonene visualiseres ved det syntoniske komma (81/80).
Videre vises den lille forskjellen mellom de pytagoreiske og syntoniske kommaene som er det såkalte skisma (32805/32768).

Alle disse begrepene er velkjente innenfor musikkteorien. Men selv om musikkteorien har slike kjente, matematiske intervaller, har det gjennom historien ikke vært opplagt å kunne konstruere dem geometrisk. Noen har forsøkt, (se referanselisten), men jeg er glad for at vi har klart å kunne konstruere og visualisere alle de kjente musikkbegrepene på en konsekvent og helhetlig måte i artikkelen.