The Development of 31-Tone Music
Volume 20 of the Collected Works of
Christiaan Huygens appeared in 1940, and
this meant the birth of 31-tone music.
This 20th volume contains the complete writings on the theory of music by
Huygens, the Dutch astronomer, physicist and mathematician, who was born in
1629 and died in 1695. His great knowledge of the theory of music and the wide
extent of his reading of the works of predecessors and contemporaries appear
from these writings.
Huygens rejected well-tempered tuning which was propagated in the 2nd half of
the 17th century and fully advocated maintaining the triad with the pure third.
The result of his investigations, arising from dissatisfaction with
well-tempered tuning, was the discovery of a Nouveau Cycle Harmonique,
which did not close after 12, the customary division of the octave, but after
He called the smallest interval in this 31-part system the diése, the fifth
part of a whole note. He showed that the division of the octave into 31 parts
had all the advantages of well-tempered tuning, particularly unlimited
possibilities of modulation, without the triad being marred in an unacceptable
way through the third that was tempered to far too great an extent. Huygens
stated himself that in his 31-tone system each diese contains all consonances
in a rising and falling tone and in all cases in an equally pure way.
A distinction is made in his system between the large (major) and small (minor)
half notes, being respectively diatonic and chromatic. the large half note
containing 3/5 and the small half note 2/5 of the 5-part whole note.
We can gain an impression of the extent of the expressive value of the
difference between large and small half notes if we consider the theme of Jan
P. Sweelinck's Chromatic Fantasy.
An important part in the new 31-tone cycle is the presence of the harmonic
seventh, the seventh in the series of natural notes. The sign with the
seventh note points to the lowering of 1/5 interval, one diese. This harmonic
seventh had already been described by the great Italian violin-teacher Giuseppe
Tartini, who found it on the violin in the series of harmonics.
He says of it in his Trattato di Musica: "Such a seventh is consonant,
not dissonant. Consequently, it needs no preparation, nor does it need to be
resolved. It can rise, it can fall and, if it is intonated purely, it can just
as well be held."
Tartini then expands on the possibilities of its use, both melodically and
The pleas made by Tartini for the use of the harmonic seventh and by Huygens
for the use of a division of the octave into 31 parts, in which the pure third
was present in addition to the seventh, did not find favour with composers of
their time of with those of about two and a half centuries later.
The seventh was not used, at least consciously, as a consonance, nor was
thought given to its use melodically or harmonically during the heyday of
chromaticism in the previous century. It must, of course, not be forgotten that
the development of keyboard instruments had not advanced enough technically for
the building of playable instruments with 31 notes in an octave.
The Dutch physicist, Prof. Adriaan D.
Fokker, took note of the views of Huygens on the theory of music at a time
when he could not practise his normal profession due to circumstances (World
War II, 1940-45).
In addition to Huygens' publications, he studied Tartini and the theoretical
works of Gioseffo Zarlino, Jean-Philippe Rameau and Leonhard Euler, the Swiss
mathematician, each in their own way being concerned with problems of tuning,
consonance of the pure third and seventh or with harmony. As late as 1961, Sir
Donald Tovey could be read in the Encyclopaedia Britannica as having said in
his article on harmony: "Harmony had not yet found a place for so simple a
natural phenomenon as the 7th note of the harmonic series". Willem Pijper
recognized that acceptance of the harmonic seventh and diese temperament which
went with it could mean a complete revolution in the field of music.
Round about Christmas 1941, Prof. Fokker and the head of the electronics section
of Philips' Gloeilampenfabriek, Dr. van der Pol, held lectures in the auditorium
of Teyler's Museum in Haarlem on the possibilities of diese temperament and the
use of new chords with the harmonic seventh. As a result of these lectures, the
expert in musical theory, Martin Lürsen, who was head teacher at the Royal
Conservatory in The Hague, became interested in the possibilities of their
use. He was later to become prize-winner in a competition for compositions using
dieses and sevenths, for his collection Modi antichi, Musiche nuove, in August
1945. He used in his collection the various modes, "genera musica", of Leonhard
Euler (1707-1783), to whom we have already referred. This inventive
mathematician was summoned by the Russian empress Catherine I from Basle to the
Russian capital. At the head of a laboratory for esperimental physics, he
studied, amongst other things, sound. He linked to this a musical theory which
appeared in 1739 under the title Tentamen novae theoriae musicae,
"Attempt at a new theory of music".
He was of the opinion that a reason has to be found for everything that happens,
including the fact that people find this music beautiful and that ugly. There
must be an answer to the question, he said, why one race enjoys a particular
music, while another race rejects that same music and does not find it enjoyable.
He ascribed this to a certain order which the ear can discover in the sounds.
The degree to which and the pleasure with which we recognize that order
determines the pleasure we derive from harmony and the succession of sounds. He
ascribed to this a certain measure of the order in an interval, physical order,
and opposed those who asserted that musical beauty is found merely in the
equality of intervals, the division of the octave into 12 equal steps, in place
of the simplicity of the physical succession.
His genera musica are consequently composed of combinations of pure
thirds, fifths and sevenths. There is found in his genera, for example, a scale
that had already been described by Tartini, the gipsy scale - C, D, E flat, F
sharp, G, A flat, B, C, the whole tone scale - A flat, B flat, C, D, E, F
sharp, A flat, which was used by Debussy amongst others, Bartók used both
scales together in his Saebadan suite, Hajeza.
Euler advocated use of the seventh consistently, but he encountered the
difficulty that keyboard instruments were not equipped to produce intervals
with the natural seventh. Of importance was his thought that the "complete"
chord, with the addition of the seventh, had to be a necessary starting-point.
Funds were made available by the Dutch "Sound Foundation", concerned with
acoustic problems, for the construction of a small
organ which produced Euler's genera. The pipes were tuned so purely that the intervals sounded free of beats.
It was for this small organ that the competition for compositions was arranged,
this leading, as we have said, to the awarding of the prize to Martin Lürsen's
collection. His 21 short compositions were not all intended for the organ,
although they could be performed on it. During the presentation of the prize in
August 1945, a section of the composition was performed by a string trio
consisting of Jos de Clerck, Jan Tegel and Karel van Leeuwen Boomkamp, being the
first 31-tone concert. Characteristic of the collection is the Chorale.
Well known Dutch artists, such as Willem Pijper and Eduard van Beinum, conductor
of the Concertgebouw Orchestra, were enthusiastic, although they were then still
of the opinion that the problem was merely of theoretical importance and that
there was still no need for a completely new terrain.
During the 1948 festival of the ICM a demonstration was held in Hilversum. The
harpsichordist Hans Philips demonstrated two differently tuned harpsichords: one
in well-tempered tuning, the other in tuning with pure thirds (Zarlino, Euler),
so-called meantone tuning. Surprise in international circles was great,
Alois Hába came to Teyler's Museum in
Haarlem in order to put his experiences with the quarter tone system to the
test on the small experimental organ. He stated on that occasion that his
quarter tones were not always equally large, but that he used smaller or larger
quarter tones depending on the harmonic requirement. He consequently arrived in
this way at tuning in fifth notes, diese tuning in 31 tones.
The next step was the building of an organ of normal compass with a keyboard
based on the division of the octave into 31 steps.
The organ-builders Pels & Zoon in 1950 placed in the Teyler Museum an
organ with a 31-tone keyboard,
designed by Prof. Fokker, and on which it was possible to play chords as well
as rapid passages.
Alongside this organ was placed one with a usual 12-note keyboard. By means of
a switching system it is always possible to choose on this auxiliary organ 12
of the available 31 notes, corresponding to the genera of Euler, including
meantone tuning. It is possible, without an exceptional playing technique, to
allow the new possibilities to be heard on this auxiliary organ.
The main organ consistently gives the division into 31 notes per octave and
demands a high degree of specialization, both mentally and technically.
One of the first composers who became interested in the great possibilities was
the native of Rotterdam, Jan van Dijk, a
pupil of Willem Pijper, who composed Pezzi per Organo trestunisoso for
the organ, being pieces for organ with or without obligato wind or strings.
The organist, Paul Chr. van Westering consecrated the organ on 8th September,
1950 at a concert at which, in addition to playing the auxiliary organ, he
improvized on the 31-note keyboard. He collected his improvisations together as
Six Inventions. The compositions by Jan van Dijk were performed for the
first time on 10th September, 1951.
This was followed by monthly concerts given on the auxiliary keyboard, by
several Dutch and foreign organists, Paul Chr. van Westering improvized from
time to time on the 31-note keyboard owing to the lack of specific literature.
The first major composition was written by
Henk Badings, being his Preludium
and Fuga 1952, which was performed on the large keyboard in November 1952
by Anton de Beer, who was the first person
to play the organ in its entirety, i.e. using independent part writing in the
pedals. This composition appeared in two versions, one being for ordinary
church organ. Anton de Beer later played both versions, on the 31-tone organ
and an ordinary organ. Records made by the AVRO broadcasting company were also
compared and discussed by the composer in a broadcast.
The seventh plays an important part, melodically and harmonically, in the
31-tone version of this composition. In a subsequent Preludium and Fuga
1954 by Badings, the style had evolved to a distinct 31-tone way of
writing. Comparisons with other versions in the 12-note style of writing are no
longer possible. This work is based on a virtuoso technique, and performance on
the organ demands a brilliant 31-tone technique. During these years the
notation of the dieses also developed to a degree of clarity which no longer
left anything to be desired and in which the notes can be clearly distinguished
from one another without the number of tines of the stave having to be
Apart from Badings, van Dijk and van Westering, the following have also
composed works for the organ: Anthon van der
Horst, Hans Kox, Arie de Klein (Prof.
Fokker), Anton de Beer, Peter Schat, the
American Joel Mandelbaum, the Swiss
Eugen Frischknecht, the Englishmen Richard Orton and
Alan Ridout and the Franco-Russian
All these have in the course of the years been performed by Anton de Beer
during the monthly concerts and have been provided with spoken annotations. In
1961 he compiled a two-part method for the 31-tone keyboard.
The organist Wim Dalm and his pupil Paul van Tongeren have likewise recently
mastered the technique of playing on the large keyboard. When playing,
spontaneous improvisations continue to play an important part.
Martin Lürsen won in his Modi antichi, Musicha nuove already thinking of
instruments other than the organ alone for performing 31-tone music. This also
applies to the majority of the composers concerned with 31-tone music. The
possibilities of the human voice were not neglected either, and the repertoire
of 31-tone music has consequently extended widely.
Henk Badings had, in addition to the works mentioned above composed other works
for the organ, 2 sonatas for 2 violins, a string quartet and a collection of
Contrasts for mixed a cappella choir.
As early as 1946, Jan van Dijk wrote exercises for string quartet, followed in
1949 by exercises for mixed choir, 2 choral works in 1952, one being for double
choir, and a concerto for trombone, violin and violoncello in 1961.
Jaap Geraedts composed studies for little and for two flutes,
Hans Kox has written four pieces for string quartet, pieces for violin, for
violin and baritone, and four pieces for two trumpets and trombone.
Ton de Leeuw wrote an electronic study in 1957, Alan Ridout a partita for cello
solo, a string trio, a sonata for two violins and a work for violin and
baritone. Joel Mandelbaum composed an opera in which 31-tone temperament is used
incidentally. Very recently, Hans Kox was commissioned to compose a sonata for
two violins while Henk Badings is just completing a concerto for two violins and
Outstanding musicians have devoted themselves to interpreting the instrumental
and vocal music.
The above-mentioned sonatas and pieces for two violins are being performed with
great regularity on concert-platforms in the Netherlands and elsewhere by the
man-and-wife duo Bouw Lemkes and Jeanne
Lemkes-Vos, who have in this way contributed much to spreading 31-tone
music. Their standard repertoire are amongst the two sonatas for two violins by
Henk Badings, while the sonata by Hans Kox was composed specially for them.
They have cultivated the technique for playing in pure intervals to a
particularly high degree. It may be mentioned that they play the music of the
old masters in their concerts in meantone tuning. The two sonatas by Badings
make high demands on the technique and ears of the performers, being amongst
the best that has been composed in the 31-tone sphere.
Bouw Lemkes has studied in detail the theoretical backgrounds of the 31-tone
technique, and their performances of 31-tone music are often coupled with a
clear and elucidating explanation. He has written a Course for Two Violins,
incorporating all the rich experience of orchestral and solo practice. He gives
lectures at the annual International Course for Conductors of the Netherlands
Radio Union on problems of intonation in general and about the possibilities of
and need for studying the 31-tone technique in particular.
The interests of 31-tone music are promoted by the Huygens-Fokker Foundation,
which was founded in 1960 under the name "careful listening" and which
propagates careful listening. The Foundation is supported by the government in
its activities. It is difficult to predict the direction that musical
development will take. It is evident that no field of musical research can be
neglected in a time which is characterized by changes and innovations without
parallel. The possibilities already presented by Huygens and Tartini deserve
every attention precisely now.
The possibilities offered by the use of the harmonic seventh are still on the
whole merely surveyed and have not yet been fully investigated. The melodic
possibilities of the small differences in pitch of the diesis have been
indicated, but not yet fully examined. It is clear that a field is lying fallow
for composers that do not throw overboard all the acquisitions from the past,
but who can work with an extension of the existing field. New tone rows are
available, new series, new sounds, possibilities of cadences and modulations, in
the widest sense of the word. The concept of consonant and dissonant is becoming
vaguer and vaguer, yet is finding new elements in the extension from 12 to 31.
If the seventh should become common property, the harmonic 11th and 13th will
come under consideration for use, they scarcely having been thought of yet.
The development of about 20 years of 31-tone music, the nuclei which have formed
in Germany around Prof. Vogel and Prof. Pfrogner, in England around composers
such as Ridout and Orton, in America around Joel Mandelbaum and Prof. Gerdine
point to a slow, but constantly rising interest on the part of experts of
standing who are not only becoming convinced of the need, but principally also
of the possibilities in the practice of music-making with 31 tones per octave.
Anton de Beer, Sonorum Speculum, 1965