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Just (black) major and parallel minor triad, compared to its equal temperament (gray) approximations, within the chromatic circle. Pythagorean tuning has been attributed to both Pythagoras and Eratosthenes by later writers, but may have been analyzed by other early Greeks or other early cultures as well.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths [2] which are "pure" or perfect, with ratio :. This is chosen because it is the next harmonic of a vibrating string, after the octave (which is the ratio 2 : 1 {\displaystyle 2:1} ), and hence is the ...
Pythagorean tuning was a system of just intonation that tuned every note in a scale from a progression of pure perfect fifths. This was quite suitable for much of the harmonic practice until then ( See: Quartal harmony ), but in the Renaissance, musicians wished to make much more use of Tertian harmony .
A Pythagorean tuning is technically both a type of just intonation and a zero-comma meantone tuning, in which the frequency ratios of the notes are all derived from the number ratio 3:2. Using this approach for example, the 12 notes of the Western chromatic scale would be tuned to the following ratios: 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 729: ...
Comparison between tunings: Pythagorean, equal-tempered, quarter-comma meantone, and others.For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order or the cycle of fifths; as in each of these tunings except just intonation all fifths are of the same size, the tunings appear as straight lines, the slope indicating the relative tempering with respect to ...
For 12 tone equally-tempered tuning, the fifths have to be tempered by considerably less than a 1 / 4 comma (very close to a 1 / 11 syntonic comma, or a 1 / 12 Pythagorean comma), since they must form a perfect cycle, with no gap at the end, whereas 1 / 4 comma meantone tuning, as mentioned above, has a residual ...
A man tuning an upright piano. Piano tuning is the process of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the ...
In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning (frequency ratio 3 : 2 ); the result is 3 / 2 × [ 80 / 81 ] 1 / 4 = 4 √ 5 ≈ 1.49535, or a fifth of 696.578 cents. (The 12th power of that value is 125, whereas 7 octaves ...