AlexS wrote:But what do you mean by "pure, just intonation"?

This has to do with musical notes whose frequencies are related by simple numeric ratios.

It all begins with Overtones, or Harmonics. A "musical" tone (like a plucked string) sounds different than an "unmusical" tone (like hammering on an old drain-pipe). The reason is that the different frequencies contained in the sound tend to be simple multiples of one another... a Fundamental (bass pitch), and a second harmonic at twice that frequency (which sounds an octave higher), and a third harmonic at three times the fundamental frequency (and which sounds an octave plus a fifth higher), and so forth.

The most important interval is the Octave... say from one "C" to the next "C" on the keyboard. These notes share the same name, and in some way sound like the same note (except for one being higher than the other).

The next most important interval is the Fifth. Take the third harmonic, and lower it by an octave (to bring it back into range). Frequency was x3, and divide by two, so the Fifth is a frequency ratio of 3 to 2, or 3/2. The interval from "C" to the next "G" on the keyboard.

Fourth harmonic is the octave of the octave. Fifth harmonic sounds at two octaves and a third. Lower by two octaves to bring it back into range. Frequency was x5, and divide by four (two twice), so this ratio is 5 to 4, or 5/4. This is called a Major Third, the interval from "C" to the next "E" on the keyboard.

From here the C-major scale can be defined... from C up a pure major third to E, and up a pure fifth to G. Then from G up a pure major third to B (3/2 x 5/4 = 15/8), and up a pure fifth to D (which will thus be 3/2 x 3/2 = 9/4, but we can lower it by an octave to bring it back into range, getting 9/8 as the interval from "C" to the next "D" on the keyboard. And from C down a pure fifth to low F (which will thus be 1 divided by 3/2 = 2/3, but we can raise it by an octave to bring it back into range, getting 2 x 2/3 = 4/3 as a Pure Fourth, the interval from "C" to the next "F" on the keyboard. And from there up a pure major third to A. 4/3 x 5/4 = 5/3 as a pure major sixth, the interval from "C" to the next "A" on the keyboard.

The C-Major Scale in Just Intonation (all harmonically pure intervals, NO BEAT on any interval):

C 1

D 9/8

E 5/4

F 4/3

G 3/2

A 5/3

B 15/8

C 2

The Ratios:

D to C is 9/8 to 1, called Major Whole Tone

E to D is 5/4 div by 9/8 = 5/4 x 8/9 = 10/9, called Minor Whole Tone

F to E is 4/3 div by 5/4 = 4/3 x 4/5 = 16/15, called Diatonic Semitone

G to F is 3/2 div by 4/3 = 3/2 x 3/4 = 9/8, again Major Whole Tone

A to G is 5/3 div by 3/2 = 5/3 x 2/3 = 10/9, again Minor Whole Tone

B to A is 15/8 div by 5/3 = 15/8 x 3/5 = 9/8, again Major Whole Tone

C to B is 2 div by 15/8 = 2 x 8/15 = 16/15, again Diatonic Semitone.

None of these intervals match any interval in 12-tone Equal Temperament, except for the Octave. And on a Piano with stretched octaves, that one fails too.

There are many other tunings and temperaments of interest... Pythagorean, the various Mean Tones, and the various Well-Tempered varieties. But the Just Intonation is where they all take origin.