1. Pythagorean Scale Most consonant intervals:
Pythagoras (born about 580 B.C.)
Most consonant intervals:
1:1 – unison
2:1 – octave – 12 semitones
3:2 – perfect 5th – 7 semitones
4:3 – perfect 4th – 5 semitones
The Pythagorean system is an attempt to build a complete chromatic scale from only two (three) of the pure tones: the octave and the perfect fifth/forth
The goal was to close circle (Circle of fifths), i.e. to end up with the same note as started. Unfortunately, it is impossible.
If octave has 12 semitones, then 7 octaves have 12x7 semitones.
If perfect 5th has 7 semitones, 12 perfect 5th have 7x12 semitones.

2. How to construct a convenient scale?
Questions:
How to construct a convenient scale?
How to set up an interval between notes?
What is known:
Perfect fifth plus perfect forth is an octave
Perfect fourth up is the same as octave up and then perfect fifth down
Perfect fifth up is the same as octave up and then perfect fourth down
So we can use octave and perfect fifth to construct perfect fourth
12 perfect 5th are approximately equal to 7 octaves
12 semitones! 7 notes!
12 perfect 5th are approximately equal to 7octaves

3. 1
Perfect fifth (C:G)
Perfect fourth (C:F)
Octave (C’/C)

4. Frequency ratios of notes in Pythagorean scale
C F G C
/ /
9/8
C D E F G A B C
/ / / / / /
9/ / / / / / /243
Circle of fifth
C G D A E B
F# C# G# D# A# E# (F)

5. Pythagorean scale (continues)
Advantage: good for perfect 5th and 4th.
Disadvantage: poor for 3d (E/C is 81/64~ instead 5/4=1.25)
Syntonic comma: (81/64)/(5/4) =
Games with numbers
Tone:
Diatonic semitone (C/B, F/E):
Chromatic semitone (F#/F):
Diatonic semitone
Chromatic semitone
Pythagorean coma =

6. Pythagorean temperament
Comparing scales
The Pythagorean fifth is 2 cents greater leading to 24 cents overlap
The circle of fifth:
Equal temperament
Pythagorean temperament