The musical tuning system in common use in the last couple hundred years is known as Equal Temperament. It consists of twelve equally-spaced notes per octave (each a "half-step" apart). (The math behind it involves the "twelfth-root-of-two", because a note's frequency multiplied by that number twelve times yields a note having double the original frequency, which is an octave above.) This system is sometimes referred to as "12tET" (12 tone Equal Temperament).
Although many people may debate it, Just Intonation is a more mathematically correct and natural tuning system. Notes in this system are spaced such that the ratio of their frequencies are equal to simple integer ratios. There are an infinite number of notes in an octave, but only a few would be used in a given musical selection.
The 5-Limit Just Intonation tuning system is confined to ratios of integers which are products of 2, 3, and 5 (prime numbers up to 5). Increasing the limit to 7 would include the 7th harmonic, which sounds flat to Western ears.
Click here to jump right into some more of the mathematics involved.
(Note: The tuning examples use MIDI "pitch wheel" bending to fine tune the notes. You can check your computer's sound equipment with the Pitch Wheel Test before trying the examples.)
The following is the first few bars of the "Promenade" from Modest Petrovich Moussorgsky's "Pictures at an Exhibition" (MIDI format),
in equal temperament, and
in just intonation (5-limit).
If you're still having trouble hearing the differences, don't "fret" -- try ET and JI side-by-side.
Email me to let me know which you like better, and why. Thanks!
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