31 Тонды Тең Белгілі Жүйе

In music, a microtonal tuning system

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 equal sized steps (equal frequency ratios). Each step represents a frequency ratio of , or 38.71 cents
31 ET is a very good 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 ET is precisely the same as it is in any other syntonic tuning (such as 12 ET), 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 a fifth of a tone or a third 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 , the appeal of this method was immediate, as the fifth of 31 ET, at 696.77 cents, is only 0.19 cent wider than the fifth of quarter comma meantone. Huygens not only realized this, he went farther and noted that 31 ET provides an excellent approximation of septimal, or 7 limit harmony. In the twentieth century, physicist, music theorist and composer Adriaan Fokker, after reading Huygens's work, led a revival of interest in this system of tuning which led to a number of compositions, particularly by Dutch composers. Fokker designed the Fokker organ, a 31 tone equal tempered organ, which was installed in Teyler's Museum in Haarlem in 1951 and moved to Muziekgebouw aan 't IJ in 2010 where it has been frequently used in concerts since it moved.

Many chords of 31 ET are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad , which might be written C–E–G, C–D–G or C–F–G, and the Orwell tetrad, which is C–E–F–B. Usual chords like the major chord are rendered nicely in 31 ET because the third and the fifth are very well approximated. Also, it is possible to play subminor chords (where the first third is subminor) and supermajor chords (where the first third is supermajor). It is also possible to render nicely the harmonic seventh chord. For example on C with C–E–G–A. The seventh here is different from stacking a fifth and a minor third, which instead yields B to make a dominant seventh. This difference cannot be made in 12 ET.