Pythagorean tuning

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Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most pleasing to the ear. Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 1800 B.C. in Babylonian texts.[1] It is the oldest way of tuning the 12-note chromatic scale.



Pythagorean tuning is based on a stack of intervals, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is considered to yield the same note. Starting from D for example (D-based tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down:

This succession of eleven factors 3:2 spans across a wide range of pitch. Since notes differing by a factor of 2 are given the same name, it is customary to divide or multiply the notes repeatedly to bring them all within one span of a factor 2.

For instance, the A is tuned such that the frequency ratio of A and D is 3:2 — if D is tuned to 288 Hz, then the A is tuned to 432 Hz. The E above A is also tuned in the ratio 3:2 — with the A at 432 Hz, this puts the E at 648 Hz, 9:4 above the original D. When describing tunings, it is usual to speak of all notes as being within an octave of each other, and as this E is over an octave above the original D, it is usual to halve its frequency to move it down an octave. Therefore, the E is tuned to 324 Hz, a 9:8 above the D. The B at 3:2 above that E is tuned to the ratio 27:16 and so on. Starting from the same point working the other way, also from D to G is tuned as 3:2. With D at 288 Hz, this arrives at G at 192 Hz, or, brought into the same octave, to 384 Hz.

When extending this tuning however, a problem arises: no stack of 3:2 intervals will fit exactly into any stack of factors 2. Thus, a longer stack, such as this (obtained by adding one more note to the stack shown above)

will be similar but not identical in size to a stack of 7 factors 2. More exactly, it will be about a quarter of a semitone larger (see Pythagorean comma). Thus, A and G, when brought in the basic octave, will not coincide as expected. The table below illustrates this, showing for each note in the basic octave the conventional name of the interval from D (the base note), the formula to compute its frequency ratio, its ratio, its size in cents, and the difference (ET-dif) in cents between its size and the size of the corresponding one in the equally tempered scale.

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