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31 Equal Temperament

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31 Equal Temperament

Figure 1: 31-ET on the syntonic temperament’s 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). About this sound Play   Each step represents a frequency ratio of 21/31, or 38.71 cents (About this sound Play  ).

31-ET is a very good approximation of quarter-comma meantone temperament. More generally, it is a tuning of the syntonic temperament 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.

Contents

  • History 1
  • Scale diagram 2
  • Interval size 3
  • Chords of 31 equal temperament 4
  • References 5
  • External links 6

History

Division of the 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 Bdouble flat A B Adouble sharp B C B C Ddouble flat C D Cdouble sharp D Edouble flat D E Ddouble sharp E F E F Gdouble flat F G Fdouble sharp G Adouble flat G A Gdouble sharp 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

Sometimes the five "double flat" notes and five "double sharp" notes are 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 Ahalf sharp A B Bhalf flat B C B C Chalf sharp C D Dhalf flat D Dhalf sharp D E Ehalf flat E F E F Fhalf sharp F G Ghalf flat G Ghalf sharp G A Ahalf flat 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 About this sound Play   7:4 968.83 About this sound Play   −1.09
perfect fifth 18 696.77 About this sound Play   3:2 701.96 About this sound Play   −5.19
greater septimal tritone 16 619.35 10:7 617.49 +1.87
lesser septimal tritone 15 580.65 About this sound Play   7:5 582.51 About this sound Play   −1.86
undecimal tritone, 11th harmonic 14 541.94 About this sound Play   11:8 551.32 About this sound Play   −9.38
perfect fourth 13 503.23 About this sound Play   4:3 498.04 About this sound Play   +5.19
septimal narrow fourth 12 464.52 About this sound Play   21:16 470.78 About this sound play   −6.26
tridecimal augmented third, and greater major third 12 464.52 About this sound Play   13:10 454.21 About this sound Play   +10.31
septimal major third 11 425.81 About this sound Play   9:7 435.08 About this sound Play   −9.27
undecimal major third 11 425.81 About this sound Play   14:11 417.51 About this sound Play   +8.30
major third 10 387.10 About this sound Play   5:4 386.31 About this sound Play   +0.79
tridecimal neutral third 9 348.39 About this sound Play   16:13 359.47 About this sound play   −11.09
undecimal neutral third 9 348.39 About this sound Play   11:9 347.41 About this sound Play   +0.98
minor third 8 309.68 About this sound Play   6:5 315.64 About this sound Play   −5.96
septimal minor third 7 270.97 About this sound Play   7:6 266.87 About this sound Play   +4.10
septimal whole tone 6 232.26 About this sound Play   8:7 231.17 About this sound Play   +1.09
whole tone, major tone 5 193.55 About this sound Play   9:8 203.91 About this sound Play   −10.36
whole tone, minor tone 5 193.55 About this sound Play   10:9 182.40 About this sound Play   +11.15
greater undecimal neutral second 4 154.84 About this sound Play   11:10 165.00 −10.16
lesser undecimal neutral second 4 154.84 About this sound Play   12:11 150.64 About this sound Play   +4.20
septimal diatonic semitone 3 116.13 About this sound Play   15:14 119.44 About this sound Play   −3.31
diatonic semitone, just 3 116.13 About this sound Play   16:15 111.73 About this sound Play   +4.40
septimal chromatic semitone 2 77.42 About this sound Play   21:20 84.47 About this sound Play   −7.05
chromatic semitone, just 2 77.42 About this sound Play   25:24 70.67 About this sound Play   +6.75
lesser diesis 1 38.71 About this sound Play   128:125 41.06 About this sound Play   −2.35
undecimal diesis 1 38.71 About this sound Play   45:44 38.91 About this sound Play   −0.20
septimal diesis 1 38.71 About this sound Play   49:48 35.70 About this sound 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 (About this sound Play  ), which might be written C-Ehalf flat-G, C-Ddouble sharp-G or C-Fdouble flat-G, and the Orwell tetrad, which is C-E-Fdouble sharp-Bdouble flat.

I-IV-V-I chord progression in 31 tone equal temperament.[3] About this sound 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.

External links

  • The Huygens Fokker foundation for micro-tonal music, in Dutch and English
  • Equal Temperament and the Thirty-one-keyed organFokker, Adriaan Daniël,
  • About 31-tone Equal TemperamentRapoport, Paul,
  • Toward a Theory of Meantone (and 31-et) HarmonyTerpstra, Siemen,
  • Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice
  • M. Khramov, “Approximation to 7-Limit Just Intonation in a Scale of 31EDO,” Proceedings of the FRSM-2009 International Symposium Frontiers of Research on Speech and Music, pp. 73–82, ABV IIITM, Gwalior, 2009.
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