For a fairly thorough, basic explanation of just and even (or “tempered” or “equal tempered”) intonation see the following site (look for the What is… link): http://www.justintonation.net/</a](http://www.justintonation.net/">http://www.justintonation.net/</a)>
Equal temperament is a Western tonal concept (12 tones per octave), whereas much World Music makes more thorough use of the entire range of possibilities. Indian classical music, or even the Byzantine Chant used in (the non-Slavic jurisdictions of) the Eastern Orthodox Church make use of a greater number of tonal divisions. For example, Indian music is made up of 12 tones with a number of microtones, for a total of 22 microtones per octave. Byzantine music limited this somewhat early on, but still uses a broader palette than traditional Western music.
Most non-stringed “band” instruments common to western music are tuned using equal tempered intonation. While string players now attempt to play w/ just intonation. If I had to play my saxophone with classical string players I’d have some work to do to make myself in tune with them.
The following link goes into a little more technical detail: http://www.phy.mtu.edu/~suits/scales.html</a](http://www.phy.mtu.edu/~suits/scales.html">http://www.phy.mtu.edu/~suits/scales.html</a)>
I’ll include a portion of this doc here:
The table below shows the frequency ratios for notes tuned in the Just and Equal temperament scales. For the equal temperament scale, the frequency of each note in the chromatic scale is related to the frequency of the notes next to it by a factor of the twelfth root of 2 (1.0594630944…). For the Just scale, the notes are related to the fundamental by rational numbers and the semitones are not equally spaced. The most pleasing sounds to the ear are usually combinations of notes related by ratios of small integers, such as the fifth (3/2) or third (5/4). The Just scale is constructed based on the octave and an attempt to have as many of these “nice” intervals as possible. In contrast, one can create scales in other ways, such as a scale based on the fifth only.
Interval Ratio to Fundamental Ratio to Fundamental
Just Scale Equal Temperament
Unison 1.0000 1.00000
Minor Second 25/24 = 1.0417 1.05946
Major Second 9/8 = 1.1250 1.12246
Minor Third 6/5 = 1.2000 1.18921
Major Third 5/4 = 1.2500 1.25992
Fourth 4/3 = 1.3333 1.33483
Diminished Fifth 45/32 = 1.4063 1.41421
Fifth 3/2 = 1.5000 1.49831
Minor Sixth 8/5 = 1.6000 1.58740
Major Sixth 5/3 = 1.6667 1.68179
Minor Seventh 9/5 = 1.8000 1.78180
Major Seventh 15/8 = 1.8750 1.88775
Octave 2.0000 2.0000
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> You will note that the most "pleasing" musical intervals above are those which have a frequency ratio of relatively small integers. Some authors have slightly different ratios for some of these intervals, and the Just scale actually defines more notes than we usually use. For example, the "augmented fourth" and "diminished fifth," which are assumed to be the same in the table, are actually not the same.
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> The set of 12 notes above (plus all notes related by octaves) form the chromatic scale. The Pentatonic (5-note) scales are formed using a subset of five of these notes. The common western scales include seven of these notes, and Chords are formed using combinations of these notes.
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> As an example, the chart below shows the frequencies of the notes (in Hz) for C Major, starting on middle C (C4), for just and equal temperament. For the purposes of this chart, it is assumed that C4 = 261.63 Hz is used for both (this gives A4 = 440 Hz for the equal tempered scale).
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> ```text
Note Just Scale Equal Temperament Difference
C4 261.63 261.63 0
C4# 272.54 277.18 +4.64
D4 294.33 293.66 -0.67
E4b 313.96 311.13 -2.84
E4 327.03 329.63 +2.60
F4 348.83 349.23 +0.40
F4# 367.92 369.99 +2.07
G4 392.44 392.00 -0.44
A4b 418.60 415.30 -3.30
A4 436.05 440.00 +3.94
B4b 470.93 466.16 -4.77
B4 490.55 493.88 +3.33
C5 523.25 523.25 0
Since your ear can easily hear a difference of less than 1 Hz for sustained notes, differences of several Hz can be quite significant!
More info than you wanted?!
Ephrem Cristóir