1. Pythagorean Scale (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.
It is impossible to achieve exactly, but
it is possible to achieve approximately
If octave has 12 semitones, then 7 octaves have 12x7 semitones.
If perfect 5th has 7 semitones, 12 perfect 5th have 7x12 semitones.

2. 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/octave!
7 semitones/ perfect 5th!

3. 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)

4. More about the circle of fifth

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):
Chromatic semitone
Diatonic 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

7. Mean-tone temperament
Pythagorean 3d are out of tune (E/C is 81/64~ instead 5/4=1.25)
Alterations of the Pythagorean scale have been developed
An attempt to alter the Pythagorean scale by flattening the third so that the major and minor third correspond to the just intervals
It, however, leads to the sharps and flats being more out of tune
C D E F G A B C
C D-1/2δ E-δ F+1/4δ G-1/4δ A-3/4δ B-5/4δ C
Syntonic comma: δ = (81/64)/(5/4) =
Advantage: 3d and 6th sound much better
Disadvantage: the 5th and 4th are no longer perfect intervals,
deteriorates as more sharps and flats are added

8. Pythagorean temperament
Comparing scales
The meantone temperament, the fifth is 3.5 cents short of the equal temperament, which leads to a short of 3.5x12=42 cents when the circle of fifths is completed
The Pythagorean fifth is 2 cents greater leading to 24 cents overlap
The circle of fifth:
Equal temperament
Mean tone temperament
Pythagorean temperament

9. Scale of just intonation
The just scale is based on the major triad, a group of three notes that sound particularly harmonious, where the first two notes are a major 3rd apart and the last two are a minor 3rd apart, with frequency ratios of 4:5:6
λf:λi ff:fi example # of half steps
4:5 5:4 major third (C,E) or (Ab,C) 4
5:6 6:5 minor third (C,Eb) or (A,C) 3
All seven notes of the just scale can be obtained by letting these three triads consist of notes with the frequency ratios 4:5:6
If C:E:G is 4:5:6, it also means 1:5/4:3/2
G:B:D => B = 5/4 * 3/2 = 15/8 and D = 3/2 * 3/2 = 9/4.
To drop the D to the same octave it becomes 9/8
F:A:C => F = 2 ÷3/2 = 4/3 and A = 2 ÷6/5 = 5/3 (this is in opposite direction)
C D E F G A B C
/ / / / / /
9/ / / / / / /15

10. C D E F G A B C
/ / / / / /
9/ / / / / / /15
We have three intervals: 9/8, 10/9 and 16/15 called the major whole tone, the minor whole tone, and the semitone respectively
Disadvantage:
Five fifths are perfect but the other are not perfect. D:A is imperfect. And the same is true of the fourths.
When you do the sharps and flats, enharmonic notes (G# and Ab) are not equal as they can be derived by various triad combinations

11. Any tuning system that uses integers to represent the ratios for all intervals is called Just Intonation.  Just scales can include, but are not limited to, the use of the pure tones.  More than likely, a scale is derived from the use of one or more of the pure tone ratios (as in the Pythagorean Scale). 
Just intonation systems are developed around one particular note, the root.  The other notes can be determined systematically (as in the Pythagorean Scale) or decided upon by choice.  Any scale created will be considered Just as long as the ratios are in integer form.  All the notes in the scale are individually determined from the root or a pre-established note in the scale. 
Just tuning depends on the scale one is using.  Since all the notes in the scale are related to each other, and (more importantly) to the root of the scale, the notes will seem to be in tune as long as one stays in the same key.  However, if one modulates into another key in the same system, there will be some problems because the ratios between the notes and the new root will be different from those of the previous root.

12. Modulation of one tone by another
(nonlinear effect)
One frequency is considerably smaller then the other
1) Amplitude modulation
2) Frequency modulation