31 equal temperament
This articleneeds additional citations forverification.(July 2024) |
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 astricesimoprimal,is thetemperedscale derived by dividing theoctaveinto 31 equal-sized steps (equal frequency ratios). Each step represents afrequencyratio of31√2,or 38.71cents( ).
31-ET is a very good approximation ofquarter-comma meantonetemperament. More generally, it is aregular diatonic tuningin which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On anisomorphic keyboard,the fingering of music composed in 31-ET is precisely the same as it is in any other syntonic tuning (such as12-ET), so long as the notes are spelled properly—that is, with no assumption ofenharmonicity.
History and use
[edit]Division of theoctaveinto 31 steps arose naturally out of Renaissancemusic theory;the lesserdiesis — the ratio of an octave to three major thirds, 128:125 or 41.06 cents — was approximately one-fifth of atoneor two-fifths of asemitone.In 1555,Nicola Vicentinoproposed an extended-meantone tuning of 31 tones. In 1666,Lemme Rossifirst proposed an equal temperament of this order. In 1691, having discovered it independently, scientistChristiaan Huygenswrote about it also.[2]Since the standard system oftuningat that time wasquarter-comma meantone,in which the fifth is tuned to4√5,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, or7-limitharmony. In the twentieth century, physicist, music theorist and composerAdriaan 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 inTeyler's MuseuminHaarlemin 1951 and moved toMuziekgebouw aan 't IJin 2010 where it has been frequently used in concerts since it moved.
Interval size
[edit]Here are the sizes of some common intervals:
interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
---|---|---|---|---|---|---|---|
octave | 31 | 1200 | 2:1 | 1200 | 0 | ||
minor seventh | 26 | 1006.45 | 9:5 | 1017.60 | −11.15 | ||
small justminor seventh | 26 | 1006.45 | 16:9 | 996.09 | +10.36 | ||
harmonic seventh,subminor seventh | 25 | 967.74 | 7:4 | 968.83 | − | 1.09||
minor sixth | 21 | 812.90 | 8:5 | 813.69 | − | 0.78||
perfect fifth | 18 | 696.77 | 3:2 | 701.96 | − | 5.19||
greaterseptimal tritone,diminished fifth | 16 | 619.35 | 10:7 | 617.49 | + | 1.87||
lesserseptimal tritone,augmented fourth | 15 | 580.65 | 7:5 | 582.51 | − | 1.86||
undecimaltritone,half augmented fourth, 11thharmonic | 14 | 541.94 | 11:8 | 551.32 | − | 9.38||
perfect fourth | 13 | 503.23 | 4:3 | 498.04 | + | 5.19||
septimal narrow fourth, half diminished fourth | 12 | 464.52 | 21:16 | 470.78 | − | 6.26||
tridecimal augmented third, and greater major third | 12 | 464.52 | 13:10 | 454.21 | +10.31 | ||
septimal major third | 11 | 425.81 | 9:7 | 435.08 | − | 9.27||
diminished fourth | 11 | 425.81 | 32:25 | 427.37 | − | 1.56||
undecimal major third | 11 | 425.81 | 14:11 | 417.51 | + | 8.30||
major third | 10 | 387.10 | 5:4 | 386.31 | + | 0.79||
tridecimal neutral third | 9 | 348.39 | 16:13 | 359.47 | −11.09 | ||
undecimalneutral third | 9 | 348.39 | 11:9 | 347.41 | + | 0.98||
minor third | 8 | 309.68 | 6:5 | 315.64 | − | 5.96||
septimal minor third | 7 | 270.97 | 7:6 | 266.87 | + | 4.10||
septimal whole tone | 6 | 232.26 | 8:7 | 231.17 | + | 1.09||
whole tone,major tone | 5 | 193.55 | 9:8 | 203.91 | −10.36 | ||
whole tone, major second | 5 | 193.55 | 28:25 | 196.20 | − | 2.65||
whole tone,minor tone | 5 | 193.55 | 10:9 | 182.40 | +11.15 | ||
greater undecimalneutral second | 4 | 154.84 | 11:10 | 165.00 | −10.16 | ||
lesser undecimal neutral second | 4 | 154.84 | 12:11 | 150.64 | + | 4.20||
septimal diatonic semitone | 3 | 116.13 | 15:14 | 119.44 | − | 3.31||
diatonic semitone,minor second | 3 | 116.13 | 16:15 | 111.73 | + | 4.40||
septimal chromatic semitone | 2 | 77.42 | 21:20 | 84.47 | − | 7.05||
chromatic semitone,augmented unison | 2 | 77.42 | 25:24 | 70.67 | + | 6.75||
lesser diesis | 1 | 38.71 | 128:125 | 41.06 | − | 2.35||
undecimaldiesis | 1 | 38.71 | 45:44 | 38.91 | − | 0.20||
septimal diesis | 1 | 38.71 | 49:48 | 35.70 | + | 3.01
The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, which have no approximate fits in12 equal temperamentand only poor fits in19 equal temperament.The composerJoel Mandelbaum(born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.[3]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 theaverageof the two. Practically it is very close to quarter-comma meantone.
This tuning can be considered ameantone temperament.It has the necessary property that a chain of its four fifths is equivalent to its major third (thesyntonic comma81:80 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.
Scale diagram
[edit]The following are the 31 notes in the scale:
The five "double flat" notes and five "double sharp" notes may be replaced by half sharps and half flats, similar to thequarter tonesystem:
Key signature | Scale | Number of sharps |
Key signature | Scale | Number of flats | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
C major | C | D | E | F | G | A | B | 0 | ||||||||||
G major | G | A | B | C | D | E | F♯ | 1 | ||||||||||
D major | D | E | F♯ | G | A | B | C♯ | 2 | ||||||||||
A major | A | B | C♯ | D | E | F♯ | G♯ | 3 | ||||||||||
E major | E | F♯ | G♯ | A | B | C♯ | D♯ | 4 | ||||||||||
B major | B | C♯ | D♯ | E | F♯ | G♯ | A♯ | 5 | ||||||||||
F♯major | F♯ | G♯ | A♯ | B | C♯ | D♯ | E♯ | 6 | ||||||||||
C♯major | C♯ | D♯ | E♯ | F♯ | G♯ | A♯ | B♯ | 7 | ||||||||||
G♯major | G♯ | A♯ | B♯ | C♯ | D♯ | E♯ | F | 8 | ||||||||||
D♯major | D♯ | E♯ | F | G♯ | A♯ | B♯ | C | 9 | ||||||||||
A♯major | A♯ | B♯ | C | D♯ | E♯ | F | G | 10 | Cmajor | C | D | E | F | G | A | B | 21 | |
E♯major | E♯ | F | G | A♯ | B♯ | C | D | 11 | Gmajor | G | A | B | C | D | E | F | 20 | |
B♯major | B♯ | C | D | E♯ | F | G | A | 12 | Dmajor | D | E | F | G | A | B | C | 19 | |
Fmajor | F | G | A | B♯ | C | D | E | 13 | Amajor | A | B | C | D | E | F | G | 18 | |
Cmajor | C | D | E | F | G | A | B | 14 | Emajor | E | F | G | A | B | C | D | 17 | |
Gmajor | G | A | B | C | D | E | F | 15 | Bmajor | B | C | D | E | F | G | A | 16 | |
Dmajor | D | E | F | G | A | B | C | 16 | Fmajor | F | G | A | B | C | D | E | 15 | |
Amajor | A | B | C | D | E | F | G | 17 | Cmajor | C | D | E | F | G | A | B | 14 | |
Emajor | E | F | G | A | B | C | D | 18 | Gmajor | G | A | B | C | D | E | F♭ | 13 | |
Bmajor | B | C | D | E | F | G | A | 19 | Dmajor | D | E | F♭ | G | A | B | C♭ | 12 | |
Fmajor | F | G | A | B | C | D | E | 20 | Amajor | A | B | C♭ | D | E | F♭ | G♭ | 11 | |
Cmajor | C | D | E | F | G | A | B | 21 | Emajor | E | F♭ | G♭ | A | B | C♭ | D♭ | 10 | |
Bmajor | B | C♭ | D♭ | E | F♭ | G♭ | A♭ | 9 | ||||||||||
F♭major | F♭ | G♭ | A♭ | B | C♭ | D♭ | E♭ | 8 | ||||||||||
C♭major | C♭ | D♭ | E♭ | F♭ | G♭ | A♭ | B♭ | 7 | ||||||||||
G♭major | G♭ | A♭ | B♭ | C♭ | D♭ | E♭ | F | 6 | ||||||||||
D♭major | D♭ | E♭ | F | G♭ | A♭ | B♭ | C | 5 | ||||||||||
A♭major | A♭ | B♭ | C | D♭ | E♭ | F | G | 4 | ||||||||||
E♭major | E♭ | F | G | A♭ | B♭ | C | D | 3 | ||||||||||
B♭major | B♭ | C | D | E♭ | F | G | A | 2 | ||||||||||
F major | F | G | A | B♭ | C | D | E | 1 | ||||||||||
C major | C | D | E | F | G | A | B | 0 |
Comparison between 1/4-comma meantone and 31-ET (values in cents, rounded to 2 decimals) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
C | C♯ | D♭ | D | D♯ | E♭ | E | E♯ | F | F♯ | G♭ | G | G♯ | A♭ | A | A♯ | B♭ | B | C♭ | C | |||||||||||
1/4 comma: | 0.00 | 76.05 | 117.11 | 193.16 | 269.21 | 310.26 | 386.31 | 462.36 | 503.42 | 579.47 | 620.53 | 696.58 | 772.63 | 813.69 | 889.74 | 965.78 | 1006.84 | 1082.89 | 1123.95 | 1200.00 | ||||||||||
31-ET: | 0.00 | 77.42 | 116.13 | 193.55 | 270.97 | 309.68 | 387.10 | 464.52 | 503.23 | 580.65 | 619.35 | 696.77 | 774.19 | 812.90 | 890.32 | 967.74 | 1006.45 | 1083.87 | 1122.58 | 1200.00 |
Chords of 31 equal temperament
[edit]Many chords of 31-ET are discussed in the article onseptimal meantone temperament.Chords not discussed there include theneutral thirdstriad ( ), which might be written C–E–G, C–D–G or C–F–G, and theOrwelltetrad, 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 issubminor) and supermajor chords (where the first third issupermajor).
It is also possible to render nicely theharmonic 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 adominant seventh.This difference cannot be made in12-ET.
See also
[edit]- Archicembalo,alternate keyboard instrument with 36 keys per octave that was sometimes tuned as 31TET.
References
[edit]- ^abMilne, 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, pp. 15–32.
- ^Monzo, Joe (2005)."Equal-Temperament".Tonalsoft Encyclopedia of Microtonal Music Theory.Joe Monzo.Retrieved28 February2019.
- ^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.JSTOR833076
External links
[edit]- The Huygens Fokker foundation for micro-tonal music, in Dutch and English
- Fokker, Adriaan Daniël,Equal Temperament and the Thirty-one-keyed organ
- Rapoport, Paul,About 31-tone Equal Temperament
- Terpstra, Siemen,Toward a Theory of Meantone (and 31-et) Harmony
- 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.
- 31 Tone Equal Temperament