# 31 equal temperament

Figure 1: 31-ET on the regular diatonic tuning continuum at P5= 696.77 cents, from (Milne et al. 2007).[1]

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 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).  Play  Each step represents a frequency ratio of 21/31, or 38.71 cents ( Play ).

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.

## History

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 and a third of a semitone. In 1666, Lemme Rossi first proposed an equal temperament of this order. Shortly thereafter, 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 51/4, 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's frequently used in concerts since it moved.

## Scale diagram

The following are the 31 notes in the scale:

 Interval (cents) 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 Note name A B A♯ B♭ A B C♭ B♯ C D C♯ D♭ C D E D♯ E♭ D E F♭ E♯ F G F♯ G♭ F G A G♯ A♭ G A Note (cents) 0 39 77 116 154 194 232 271 310 348 387 426 465 503 542 581 619 658 697 735 774 813 852 890 929 968 1006 1045 1084 1123 1161 1200

The five "double flat" notes and five "double sharp" notes may be replaced by half sharps and half flats, similar to the quarter tone system:

 Interval (cents) 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 Note name A A A♯ B♭ B B C♭ B♯ C C C♯ D♭ D D D D♯ E♭ E E F♭ E♯ F F F♯ G♭ G G G G♯ A♭ A A Note (cents) 0 39 77 116 154 194 232 271 310 348 387 426 465 503 542 581 619 658 697 735 774 813 852 890 929 968 1006 1045 1084 1123 1161 1200

## Interval size

Here are the sizes of some common intervals:

 interval name size (steps) size (cents) midi just ratio just (cents) midi error harmonic seventh 25 967.74 Play 7:4 968.83 Play −1.09 perfect fifth 18 696.77 Play 3:2 701.96 Play −5.19 greater septimal tritone 16 619.35 10:7 617.49 +1.87 lesser septimal tritone 15 580.65 Play 7:5 582.51 Play −1.86 undecimal tritone, 11th harmonic 14 541.94 Play 11:8 551.32 Play −9.38 perfect fourth 13 503.23 Play 4:3 498.04 Play +5.19 septimal narrow fourth 12 464.52 Play 21:16 470.78 play −6.26 tridecimal augmented third, and greater major third 12 464.52 Play 13:10 454.21 Play +10.31 septimal major third 11 425.81 Play 9:7 435.08 Play −9.27 undecimal major third 11 425.81 Play 14:11 417.51 Play +8.30 major third 10 387.10 Play 5:4 386.31 Play +0.79 tridecimal neutral third 9 348.39 Play 16:13 359.47 play −11.09 undecimal neutral third 9 348.39 Play 11:9 347.41 Play +0.98 minor third 8 309.68 Play 6:5 315.64 Play −5.96 septimal minor third 7 270.97 Play 7:6 266.87 Play +4.10 septimal whole tone 6 232.26 Play 8:7 231.17 Play +1.09 whole tone, major tone 5 193.55 Play 9:8 203.91 Play −10.36 whole tone, minor tone 5 193.55 Play 10:9 182.40 Play +11.15 greater undecimal neutral second 4 154.84 Play 11:10 165.00 −10.16 lesser undecimal neutral second 4 154.84 Play 12:11 150.64 Play +4.20 septimal diatonic semitone 3 116.13 Play 15:14 119.44 Play −3.31 diatonic semitone, just 3 116.13 Play 16:15 111.73 Play +4.40 septimal chromatic semitone 2 77.42 Play 21:20 84.47 Play −7.05 chromatic semitone, just 2 77.42 Play 25:24 70.67 Play +6.75 lesser diesis 1 38.71 Play 128:125 41.06 Play −2.35 undecimal diesis 1 38.71 Play 45:44 38.91 Play −0.20 septimal diesis 1 38.71 Play 49:48 35.70 Play +3.01
Circle of fifths in 31 equal temperament

The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, which have no approximate fits in 12 equal temperament and only poor fits in 19 equal temperament. The composer Joel Mandelbaum (born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.[2] The tuning has poor matches to both the 9:8 and 10:9 intervals (major and minor tone in just intonation); however, it has a good match for the average of the two. Practically it is very close to quarter-comma meantone.

This tuning can be considered a meantone temperament. It has the necessary property that a chain of its four fifths is equivalent to its major third (the syntonic comma 81:80 is tempered out), which also means that it contains a "meantone" that falls between the sizes of 10:9 and 9:8 as the combination of one of each of its chromatic and diatonic semitones.

## Chords of 31 equal temperament

Many chords of 31-ET are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad ( Play ), which might be written C-E-G, C-D-G or C-F-G, and the Orwell tetrad, which is C-E-F-B.

I-IV-V-I chord progression in 31 tone equal temperament.[3]  Play  Whereas in 12TET B is 11 steps, in 31-TET B is 28 steps.
Csubminor, Cminor, Cmaj, Csupermajor (topped by A) in 31 equal temperament

Usual chords like the major chord is 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).

Cmaj7 chord and Gminor chord, twice in 31 equal temperament, then twice in 12 equal temperament

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. This difference cannot be made in 12-ET.

## References

1. Milne, A., Sethares, W.A. and Plamondon, J., "Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
2. Keislar, Douglas. "Six American Composers on Nonstandard Tunnings: Easley Blackwood; John Eaton; Lou Harrison; Ben Johnston; Joel Mandelbaum; William Schottstaedt", Perspectives of New Music, Vol. 29, No. 1. (Winter, 1991), pp. 176-211.
3. Andrew Milne, William Sethares, and James Plamondon (2007). "Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum", p.29. Computer Music Journal, 31:4, pp.15–32, Winter 2007.