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In tunings such as 1:1, 9:8, 5:4, 3:2, 7:4, 2:1, all the pitches are chosen from the harmonic series (divided by powers of 2 to reduce them to the same octave), so all the intervals are related to each other by simple numeric ratios. Pythagorean tuning Prelude No. 1, C major, BWV 846, from the Well-Tempered Clavier by Johann Sebastian Bach.
N ≈ 3.516015 is the square of the smallest positive solution to cos(x)cosh(x) = −1, [8] which arises from the boundary conditions of the prong’s cantilevered structure. L is the length of the prongs, (m) E is the Young's modulus (elastic modulus or stiffness) of the material the fork is made from, (Pa or N/m 2 or kg/(ms 2))
For pianos, pins are typically square with a slight taper. There are three standard sizes known as No. 1, No. 2 and No. 3, for pins up to 6.5mm, for pins 6.5mm to 7.25mm, and for pins larger than 7.25mm. No. 2 is the most common. Wrenches are supplied with an eight-point star. [1] Some early keyboard instruments have oblong-shaped tuning pins.
By definition, in Pythagorean tuning 11 perfect fifths (P5 in the table) have a size of approximately 701.955 cents (700+ε cents, where ε ≈ 1.955 cents). Since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of 700 − 11ε cents, which is about 678.495 cents (the wolf ...
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency. [2]
The just or Pythagorean perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a grad, the twelfth root of the Pythagorean comma (/). The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three ( 3 13 {\textstyle {\sqrt[{13}]{3}}} ).
The value of 5 1 ⁄ 8 ·35 1 ⁄ 3 is very close to 4, which is why a 7-limit interval 6144:6125 (which is the difference between the 5-limit diesis 128:125 and the septimal diesis 49:48), equal to 5.362 cents, appears very close to the quarter-comma ( 81 / 80 ) 1 ⁄ 4 of 5.377 cents.
In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios). Play ⓘ Each step represents a frequency ratio of 2 1/41 , or 29.27 cents ( Play ⓘ ), an interval close in size to the septimal comma .
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