31 equal temperament

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In music, 31 equal temperament (also known as tricesimoprimal), which can be abbreviated 31-TET, 31-EDO, 31-ET, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equal frequency ratios). Each step represents a frequency ratio of 2^1/31, or 38.71 cents ( Play (help·info)).

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. On this basis, Nicola Vicentino produced a 31-step keyboard instrument, the Archicembalo, in 1555, but it was not until 1666 that Lemme Rossi first proposed an equal temperament of this order. Shortly thereafter, having discovered it independently, famed 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 5^1/4, the appeal of this method is 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, which was an advanced insight for its time. 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.

Scale diagram

The following are 21 of the 31 notes in the scale:

The remaining 10 notes can be added with, for example, five "double flat" notes and five "double sharp" notes, or by half sharp and half flats, similar to the quarter tone system.

Interval size

Here are the sizes of some common intervals:

interval name	size (steps)	size (cents)	midi	just ratio	just (cents)	midi	error
major sixth	23	890.32	Play (help·info)	5:3	884.36	Play (help·info)	+5.96
perfect fifth	18	696.77	Play (help·info)	3:2	701.96	Play (help·info)	−5.19
septimal tritone	15	580.65	Play (help·info)	7:5	582.51	Play (help·info)	−1.86
undecimal tritone, 11th harmonic	14	541.94	Play (help·info)	11:8	551.32	Play (help·info)	−9.38
perfect fourth	13	503.23	Play (help·info)	4:3	498.04	Play (help·info)	+5.19
tridecimal major third	12	464.52	Play (help·info)	13:10	454.21	Play (help·info)	+10.31
undecimal major third	11	425.81	Play (help·info)	14:11	417.51	Play (help·info)	+8.30
septimal major third	11	425.81	Play (help·info)	9:7	435.08	Play (help·info)	−9.27
major third	10	387.10	Play (help·info)	5:4	386.31	Play (help·info)	+0.79
undecimal neutral third	9	348.39	Play (help·info)	11:9	347.41	Play (help·info)	+0.98
minor third	8	309.68	Play (help·info)	6:5	315.64	Play (help·info)	−5.96
septimal minor third	7	270.97	Play (help·info)	7:6	266.87	Play (help·info)	+4.10
septimal whole tone	6	232.26	Play (help·info)	8:7	231.17	Play (help·info)	+1.09
whole tone, major tone	5	193.55	Play (help·info)	9:8	203.91	Play (help·info)	−10.36
whole tone, minor tone	5	193.55	Play (help·info)	10:9	182.40	Play (help·info)	+11.15
lesser undecimal neutral second	4	154.84	Play (help·info)	12:11	150.64	Play (help·info)	+4.20
septimal diatonic semitone	3	116.13	Play (help·info)	15:14	119.44	Play (help·info)	−3.31
diatonic semitone, just	3	116.13	Play (help·info)	16:15	111.73	Play (help·info)	+4.40
chromatic semitone, just	2	77.42	Play (help·info)	25:24	70.67	Play (help·info)	+6.75
undecimal diesis	1	38.71	Play (help·info)	45:44	38.91	Play (help·info)	−0.20
septimal diesis	1	38.71	Play (help·info)	49:48	35.70	Play (help·info)	+3.01

The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, ratios which do not even have approximate fits within the 12 equal temperament and which have only a poor fit with the 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.^[1] It should be noted, however, that this tuning does not distinguish between the septimal major third and the (14:11) ratio, neither of which is matched particularly well in this tuning.

This tuning can be considered a meantone temperament. It has the necessary property that a chain of its four fifths are 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.

Tempering

One property of 31-et is that it equates to the unison, or tempers out, the syntonic comma of 81:80. It can therefore be considered a meantone temperament. It also tempers the 5-limit intervals 393216:390625, known as the Würschmidt comma after music theorist José Würschmidt, and 2109375:2097152, known as the semicomma.

In addition, it also tempers out 126:125, the septimal semicomma or starling comma. Because it tempers out both 81:80 and 126:125, it supports septimal meantone temperament. It also tempers out 1029:1024, the gamelan residue, and 1728:1715, the Orwell comma. Consequently it supports a wide variety of linear temperaments.

31-et also tempers out 99:98.

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 (help·info)), which might be written C-E-G, and the Orwell tetrad, which is C-E-F-B.

I-IV-V-I chord progression in 31 tone equal temperament.^[2] Play (help·info) Whereas in 12TET B♮ is 11 steps, in 31-TET B♮ is 28 steps.

References

1. ^ http://links.jstor.org/sici?sici=0031-6016%28199124%2929%3A1%3C176%3ASACONT%3E2.0.CO%3B2-G Six American Composers on Nonstandard Tunnings: Douglas Keislar; 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.
2. ^ 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

• de Beer, Anton, The Development of 31-tone Music^{[dead link]}
• Fokker, Adriaan Daniël, Equal Temperament and the Thirty-one-keyed organ^{[dead link]}
• Rapoport, Paul, About 31-tone Equal Temperament^{[dead link]}
• Terpstra, Siemen, Toward a Theory of Meantone (and 31-et) Harmony^{[dead link]}
• Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice