31 equal temperament

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In music, 31 equal temperament, 31 ET, which can also be abbreviated 31 TET (31 tone ET) or 31 EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equally-proportioned steps (equal frequency ratios). Play Each step represents a frequency ratio of √2 , or 38.71 cents (Play). 31 EDO is often thought to be a very accurate approximation of quarter-comma meantone temperament. More generally, it is a regular diatonic tuning in which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On an isomorphic keyboard, the fingering of music composed in 31 EDO is precisely the same as it is in any other syntonic tuning (such as 12 EDO), so long as the notes are spelled properly—that is, with no assumption of enharmonicity. Division of the octave into 31 steps arose naturally out of Renaissance music theory; the lesser diesis‍ – the ratio of an octave to three major thirds, 128:125 or 41.06 cents – was approximately one-fifth of a tone or two-fifths of a semitone. In 1555, Nicola Vicentino proposed an extended-meantone tuning of 31 tones. In 1666, Lemme Rossi first proposed an equal temperament of this order. In 1691, having discovered it independently, scientist Christiaan Huygens wrote about it also. Since the standard system of tuning at that time was quarter-comma meantone, in which the fifth is tuned to √5 , the appeal of this method was immediate, as the fifth of 31 EDO, at 696.77 cents, is very slightly closer to a true 2:3 interval than the fifth of quarter-comma meantone.