Pythagorean Harmony

Monday 12/8/08

The Cult of Fractions

Pythagoras was a famous Greek philosopher and scientist who lived about 2500 years ago. He started a cult-like religion based on the idea that the true nature of the world is expressed in numbers. Specifically, they believed that rational numbers - fractions and mixed numbers - could be used to describe everything that exists.

One story about Pythagoras tells about how he worked to figure out how to find the length of the diagonal of a square based on the length of its sides. He was quite upset when one of his disciples, Hippasos, discovered that the length of the diagonal could not be written as a fraction of the length of the side.







Pythagoras first got interested in music when he was walking past a forge and heard that the sounds of the blacksmiths' hammers sounded good together. He talked to the blacksmiths and found out that this was because the anvils they were using were scaled down copies of each other: one full size, one half size, and one two thirds size. He got an idea that maybe a simple fraction relationship in the size of two instruments is what makes them sound good together. This is the idea we are going to experiment with today.

Creating related notes

We are going to start with a full straw as the basis of our musical scale, and cut other straws to be simple fractions of it. Then we will play these shorter straws together with the full straw to see how they sound.
  1. Cut one straw to be 1/2 of a full straw.

    a) The musical interval between the full straw and this is called an             .

    b) When played together, these two notes sound                                     .

  2. Cut one straw to be 2/3 of a full straw.

    a) The musical interval between the full straw and this is called a             .

    b) When played together, these two notes sound                                     .

  3. Cut one straw to be 3/4 of a full straw.

    a) The musical interval between the full straw and this is called a             .

    b) When played together, these two notes sound                                     .

  4. Cut one straw to be the diagonal of a square whose sides are your two half-straws.
    (It may be easier to mark the length of a half-straw on the sides of your paper)

    a) The musical interval between the full straw and this is called a             .

    b) When played together, these two notes sound                                     .

Creating a musical scale

The full musical scale is made by adding in other notes that relate to each other by simple fractions. Many simple songs can be played using just the first five notes in a musical scale:

C:

D:

E:

F:

G:

A famous example of a song whose tune uses only these five notes is "Ode to Joy" from Beethoven's Ninth Symphony, one of the most famous pieces of choral music of all time:

Freude, Schöner Götterfunken,
Tochter aus Elysium,
Wir betreten feuer-trunken,
Himmlische, dein Heiligtum!

Deine Zauber binden wieder,
Was die Mode streng geteilt;
Alle Menschen werden Brüder,
Wo dein sanfter Flügel weilt.
Joy, beautiful spark of God,
daughter of heaven,
with hearts aflame we approach,
heavenly one, your shrine.

Your magic power binds again
what strong custom had divided.
All people become brothers
where your soft wings fly.
E E F G G F E D
C C D E E D D
E E F G G F E D
C C D E D C C

D D E C D E F E C
D E F E D C D (G)
E E F G G F E D
C C D E D C C