53 equal temperament

In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps (equal frequency ratios).  Play  Each step represents a frequency ratio of 2^1/53, or 22.6415 cents ( Play ), an interval sometimes called the Holdrian comma.

53-TET is a tuning of equal temperament in which the tempered perfect fifth is 701.89 cents wide, as shown in Figure 1.

Theoretical interest in this division goes back to antiquity. Ching Fang (78–37 BC), a Chinese music theorist, observed that a series of 53 just fifths (${\displaystyle (3/2)^{53}}$) is very nearly equal to 31 octaves (${\displaystyle (2/1)^{31}}$). He calculated this difference with six-digit accuracy to be ${\displaystyle 177147/176776}$. Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c. 1620–1687), who calculated this value precisely as ${\displaystyle (3^{53}/2^{84}=19383245667680019896796723/19342813113834066795298816)}$, which is known as Mercator's comma. Mercator's comma is of such small value to begin with (≈ 3.615 cents), but 53 equal temperament flattens each fifth by only 1/53 of that comma (≈ 0.0682 cent ≈ 1/315 syntonic comma ≈ 1/344 pythagorean comma). Thus, 53 equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning.

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