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Difference between 12 just perfect fifths and seven octaves. Difference between three Pythagorean ditones (major thirds) and one octave. A just perfect fifth has a frequency ratio of 3:2. It is used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to a given initial note, the frequency of any other note.
In Pythagorean tuning all notes are tuned as a number of perfect fifths (701.96 cents play ⓘ).The major third above C, E, is considered four fifths above C. This causes the Pythagorean major third, E + (407.82 cents play ⓘ), to differ from the just major third, E ♮ (386.31 cents play ⓘ): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma play ⓘ).
Using justly tuned octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). Pythagorean tuning uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds.
The term perfect has also been used as a synonym of just, to distinguish intervals tuned to ratios of small integers from those that are "tempered" or "imperfect" in various other tuning systems, such as equal temperament. [6] [7] The perfect unison has a pitch ratio 1:1, the perfect octave 2:1, the perfect fourth 4:3, and the perfect fifth 3:2.
Euler's Tonnetz. The Tonnetz originally appeared in Leonhard Euler's 1739 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae.Euler's Tonnetz, pictured at left, shows the triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a ...
"The Pythagorean system would appear to be ideal because of the purity of the fifths, but some consider other intervals, particularly the major third, to be so badly out of tune that major chords [may be considered] a dissonance." [2] The Pythagorean scale is any scale which can be constructed from only pure perfect fifths (3:2) and octaves (2: ...
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In the formulas, x = 4 √ 5 = 5 1 ⁄ 4 is the size of the tempered perfect fifth, and the ratios x : 1 or 1 : x represent an ascending or descending tempered perfect fifth (i.e. an increase or decrease in frequency by x ), while 2 : 1 or 1 : 2 represent an ascending or descending octave.