The Just Scale
Just intonation, also known as pythagorean tuning, is the oldest form of musical tuning. In this system, the 7 diatonic notes of the major scale are measured in their relation to the fundamental note, which serves as the tonic of the scale. The following diagram graphically illustrates how these ratios are used to derive the scale:
If you would like a more detailed explanation of the just scale, click here.
The just scale represents absolute harmonic perfection. The ratios used assure that all musical consances will produce no beat frequency. The scale has one tremendous disadvantage though. Because of the ratios used, this tuning system can only apply to one key signature at a time. This means that an instrument tuned to a just C scale (all white notes on the piano), will be out of tune if it attempts to play a D scale. This is shown graphically below:
The top scale is a C scale, the middle a D, and the bottom a G scale with octave displacements. Most notes are not too far apart (after all, these are closely related scales), but there are visible discrepencies about the placements of As, Es, and Cs. These discrepencies become worse as more and more transpositions are applied.
So, the just tuning system will produce georgeous consonances, but will become uglier and uglier as music moves to new key signatures. This means that, when using just tuning, music can only be performed in one scale, eliminating exciting musical elements such as transpositions, modulations of all kinds, and all 20th Century theory. Read about the tempered scale to hear how this was solved.