There seems to be no single plan for
Selections I-VIII, though they are related to one another: Duchamp seems to
be working towards a way of creating semi-randomness. His first attempt (I) is abortive, but by V he
has found a way to create an erratic, corpuscular line, equivalent to the wave-like 3 Stoppages
I think that VI, VII and VIII, at the top of page two, are 3 standard stoppages, albeit in an unfinished
state. The letters A-Q are sub-selections from a gamut of 85 notes. Selections I-VIII each contain each pitch once.
My reasons for interpreting each Roman numeral I-VIII as a single line are that:
- He is trying to create “une sorte d’alphabet musical nouveaux”. Alphabets
are single sequences of letters — not polyphonic.
- His comment that “l’ordere de succession est (au gré) interchangeable” makes
sense in that, when the “period” consists of 2 or 3 “periodes fragmentées”,
the order in which the numbers are taken from the sub-selections can be varied: for example, III
has three “periodes fragmentées”:
which can be ordered
the realisations below, I generally distribute each sub-selection evenly along the finished line.
This makes for a relatively even curvature along the line, but is not essential.
I’m assuming that Duchamp is using the word period in the musical sense (as in “an
8-bar period”). In other words as an abstract, limited sequence unrelated to absolute
units of time (hours, minutes, seconds). In music, such periods can vary considerably
depending on how they are performed.
- A close parallel to the idea of the 3 Stoppages-étalon becomes evident.
Duchamp started (in I) by writing two clefs on manuscript paper, and then
decided to use numbers instead. The Erratum musical for Yvonne-Magdaleine-Marcel
so it looks as if he originally intended to use musical notation as in the earlier work, but
decided against that because numbers are easier to notate and manage.
I think its unlikely that he used these numbers for
anything - otherwise there would be some signs on the score indicating which of the notes he had used
(crossings out etc.).
The instructions at the end of page 2 (see The Apparatus
below) are written
in three clearly separate paragraphs
which were probably written after he had completed VIII (otherwise the spacing on the page would
have been different). The first paragraph is a simple explanation of what the numbers represent. The
second paragraph is the beginning of an attempt to be more precise about how these numbers could be used
to realize a peformance (he’s
I think the third text, with the sketch for an apparatus,
may have been added much later (days or weeks)
— he may have added the title on page 1 at the same time (similar handwriting).
The “appareil enregistrant automatiquement les periodes musicales fragmentées
me to be an attempt to embody the principles he’d been developing in I-VIII in an object —
as a visual artist he wanted things, not just concepts. Interestingly its an object that is
intended to be operated in time
Strangely, the idea of
the contents of each wagon before its use, is not mentioned in connection with the appareil
(perhaps, by then it was too obvious), and there are more wagons than are used in any of the selections
he was thinking of investigating the matter further — with more wagons and a finer-grained selection
pitches (1/4 tones).
It is easiest to visualize numbers if they are plotted on graph paper. Here is a summary of my
interpretation of the eight selections, showing that they converge on nearly straight lines. A more detailed
of each selection follows below:
February 2019: A score (Erratum Musical I-VIII
) containing the above graphs transcribed
notation can now be viewed and automatically performed using my Assistant Performer
software. See “Unfinishable” below
Here, B contains the values 10 and 35 together with 28 other numbers selected randomly from the 85.
Selection A contains the remaining 35 numbers in random order.
A is followed
by B. I followed his instructions, putting the values 10 and 35 at the end of B.
This strategy introduces the idea of partitioning the total selection, but is as yet no better than
simply selecting all the values in random order.
These numbers are in the left-right order of my 1979 scale I
Here, the first 25 numbers were selected at random for D, then the remaining 60 were selected in
random order for C. Note that both the C and D numbers are evenly spaced, indicating that they were
written in the order they were selected. He left himself plenty of space for the 60 C-numbers. The
fat “barline” at the end of C indicates that it is a single sub-sequence, not two separate
Duchamp may have originally intended C
to be followed by D (as in I A and B), but have noticed that since their
components are in random
order, C and D could just as well be simultaneous.
Note that the curly bracket at the start of the
system is a correction.
These numbers are in the left-right order of my 1979 scale II
This is where things begin to get interesting! Here, there are three sub-selections, with G
containing just 6 and 60. (G’s order may or may not have been deliberately ascending).
F contains the next fourteen numbers deliberately arranged in ascending order. This can be seen from
their spatial arrangement. He wrote
the numbers as they were chosen, at positions that would allow him to put the following numbers in order.
The spacing has nothing to do with either vertical alignments or absolute time.
The idea of creating randomly ascending “periodes fragmentées
” is further improved
in E. Here he made nine partitions (the beginning of the idea for wagons) arranged in ascending order
and then filled them randomly. The first partition contains the numbers 1 to 9, the second the numbers
10-19 and so on, the last partition containing 80-85.
It is possible that the idea for partitions came as a result of his experience with F, since they
arise when numbers are written on paper in positions that are estimated in relation to their value.
Note that E effectively consists of a non-random sequence of random sequences — like A followed
by B in Sequence I, except that each subsequence is in a range higher than its predecessor. This gives
a marked direction to the sub-sequence (E) as a whole, without individual notes in the sub-sequence
being necessarily higher than their predecessors (as in F). E and F certainly show the direction in
which his thoughts are moving.
Even if he was not thinking in terms of the Stoppages-étalon
at the beginning, by now he is!
He reverts to two sub-selections (G is rather ineffectual in III), but the general principle remains
the same. He seems, within the partitions in H to have tried to compromise between randomness and
straightness: The partition for 10-19 is completely straight (ascending), and has the typical
spacing for a straight selection — note that he put the 10 at the end of the 1-9 partition.
Partitions 1-9, 50-59 and 60-69 are straight except for one value (in 1-9 its the first value, in
50-59 and 60-69 its the last value).
Partitions 20-29 and 50-59 are straight except for two values (in 20-29 the first and last values,
in 50-59 the eighth and last values).
Partitions 70-79 and 80-85 have been amalgamated (note that the divider has been crossed out).
Possibly, he was trying out the use of smaller ascending partitions:
He may have mentally divided the 70s block into two straight segments plus one value...
or possibly these are simply random values within the 70s. In any case, the straightness within most
of the partitions is very marked, and I think it is indisputable that in IV he was somehow using
the straightness (first used in III F) one level lower (at the level of the partitions).
If all the partitions in IV had been completely straight, sub-selection H would have been a
single chromatic scale from 1-85 except for the values in sub-selection I. This is what I think
I think both J and K are in ascending order,
and that J did not have to be written out because it is nearly a chromatic scale.
Selection V is crooked because the values in K have effectively been moved from their standard positions.
Here, he effectively starts with a straight line, and then proceeds to bend it. (Did the
this Erratum musical
?) Notice that although K
is ascending, it has not got the typical spacing for a straight sequence. He may have just kept and
the number tokens before writing out their values, or written their values on a separate piece of
paper. He may have already started using a separate page for intermediary results in order to cope with
the complexities of sub-selection IV H (note the spacing there).
Selections I-V on page 1 of this Erratum musical
are the record of a closed train of thought
aimed at finding a way to generate erratic lines of discrete values. Selections VI, VII and
the top of page 2, are three examples that use the process he envisaged for V, without making any
further conceptual changes. In effect, they are three standard, quantized stoppages.
as with all the other selections except III, the three selections VI, VII and VIII each consist of two
sub-selections (périodes fragmentaires
The sketch for The Apparatus
prescribes 5 or 6.
I think the first two texts below VIII refer to Selections I-VIII, and that The Apparatus is a later,
incomplete addition (à developer
) that should be treated separately. Note the different handwriting.
contains the first 11 randomly chosen numbers in numerical order.
contains the remaining 74 numbers (from 85) in numerical order.
Duchamp seems to have changed his mind here. A number between 8 and 20 has been deleted, so that
only contains 11 numbers, not 12.
contains the first 22 randomly chosen numbers in numerical order.
contains the remaining 63 numbers in numerical order.
The number 65, to the left of the O
staff, could be the number of numbers in
a (randomly selected) pre-selection, from which he selected the first 22.
contains the first 10 randomly chosen numbers in numerical order.
contains the remaining 75 numbers in numerical order.
The number 40, to the left of the Q
staff, could be the number of numbers in
a (randomly selected) pre-selection, from which he selected the first 10.
The following translations
from the French were taken from
Marcel Duchamp: Work and Life
by Jennifer Gough-Cooper and Jacques Caumont.
Each n° indicates a note; an ordinary piano contains about 89 notes; each n° is the number in
order starting from the left.
Unfinishable; for a designated musical instrument (player piano, mechanical organs or other new
instruments for which the virtuoso intermediary is suppressed); the order of succession is (to
taste) interchangeable; the time which separates each Roman numeral will probably be constant (?)
but it may vary from one performance to another; a very useless performance in any case.
An apparatus automatically recording fragmented musical periods
Vase containing the 89 notes (or more: 1/4 tone) figures among n° on each ball.
Opening A letting the balls drop into a series of little wagons B.C.D.E.F. etc
Wagons B, C, D, E, F, going at a variable speed, each one receiving one or several balls
When the vase is empty: the period in 89 notes (so many) is inscribed and can be performed by a designated
another vase - another period - the result from the equivalence of the periods and their comparison
a kind of new musical alphabet allowing model descriptions. (to be developed).
The following analysis was completed before discovering Serge Stauffer’s German translation of the Erratum Musical
(pp 108-109). Stauffer says he thinks the apparatus uses 85
notes, and that 89 is an old misreading of Duchamp’s handwriting.
I think Stauffer is probably right but that the difference does not affect the following argument.
Duchamp is proposing that a performable (temporal) sequence should be made from 5 sub-selections (in
wagons B, C, D, E and
the sub-selections to contain different numbers of values.
He does not say how
the values should be combined.
Notice that the content of each wagon (whose values are random as to number and the order in which
they are observed) can be used to define an erratic path between points (1,1) and (89,89).
Until they are known, all that can be said about the values is that they exist in a probabilistic
The characteristics of this field can be changed by
- Ignoring the time at which the wagons are evaluated (by plotting each wagon’s content
on the same graph). Note that another vase is another period, so a period
is defined by all the wagons.
- Ignoring the time at which the values inside the wagons are evaluated (by always putting them
in ascending order — as in sub-selections L, M, N, O, P, Q)
For the following diagrams, I have simulated an instance of the stopped Apparatus, by selecting
the numbers 1-89 (one instance of each, as Duchamp says) under the following extra conditions:
- The number of values in each wagon is as equal as possible.
- Each wagon’s values are ordered numerically and distributed as equally as possible horizontally.
These extra conditions keep the envelope around the 5 sub-selections as narrow (least blurred) as possible,
so as to make this discussion easier to follow. Less stringent conditions would not affect
my conclusions, since in the end all
possible wagon-paths lead from (1,1) to (89,89).
There are three unambiguous ways to join these dots:
- Indicating that two dots are in sequence does not have to be done with a straight line. There
is, in fact, no single “correct”
way to do this! (see below)
- The paths
joining (1,1) to (89,89) in graph 3 are much shorter than the black paths in graphs 1 and 2.
- The direct diagonal between (1,1) and (89,89) can be created by collapsing graph 1 (horizontally) and
graph 2 (vertically).
Any of the coloured paths can be collapsed either horizontally or vertically onto the same direct
diagonal, but using fewer dots.
- There are other ways to define dot sequences
(absolute proximity, various diagonals etc.) but they all involve ambiguity unless the colour is taken into
will always arise
in which there can be more than one dot that
fulfils the “next dot in sequence” requirement.
A propensity for joining the dots is, I think, related to optical illusions — see
introductions to his Interface Theory of Perception
and here (longer)
I agree very much with Hoffman, that
our perceptions are probably an interface that shields us from ultimate, physical reality: we throw away
all the information
we can’t cope with and then mentally join the dots. Space and time are a brain strategy for reducing
Even though random numbers can be used while modelling otherwise inscrutable information,
that does not mean that we have understood the inscrutable information! We all agree on and use the results of scientific
experiments (quantum physics is highly successful), so its probably not “random” in the same way!
Gott würfelt nicht!
See also Duchamp’s comments on chance, as reported in Serge Stauffer’s
Gesammelte Schriften (page 97, footnote 92): ”Ihr Zufall ist nicht der gleiche wie mein Zufall, genau so wie Ihr Wurfelwurf selten der gleiche sein wird wie der meine.“ Stauffer says that the original is in "Tomkins 1965, seite 58" — which I have been unable to find. My back-translation would be: “Your chance is not the same as my chance, in the same way as your throw of the dice will seldom be the same as mine.” — if anyone knows where the original quote can be found, please let me know.
Stauffer’s Anmerkungen to this footnote lead eventually to Poincaré and Jouffret...
When listening to music or speech, we automatically chunk the incoming information (a continuous sound wave) so as to create points
(symbols) that we then connect to each other to create higher levels of meaning. We seem to need the flexibility to ignore
or create some of these levels. Chunking and joining the dots are opposite strategies.
It would be possible to create Duchamp’s “kind of new musical alphabet allowing model descriptions
from “the equivalence of the periods and their comparison
by using all the wagons simultaneously, as in graphs 1 and 2
(which could also be rotated by 90°, 180° and 270°) — but these are blurred lines, rather
than being continuous, bent lines.
lines, that can be usd in the same way, can be created simply by interpolating values
between the values in a single
sub-selection in a single wagon). This can be done by drawing either straight or curved lines between
the values. However, straight lines are not the best solution if we are looking for minimum energy paths,
because the line segments that meet at a knot have no common tangent, so an energy impulse
would be required
change the direction.
A curve that is often used in engineering is the cubic spline — as in the following diagrams.
Duchamp's Selections VI, VII and VIII were, I think, intended as “standard stoppages”, so I have
used the points from sub-selections VI (L)
, VII (O)
and VIII (Q)
in the following diagrams.
in contrast to Duchamp's procedures, this results in microtonal pitches
on the y-axis. It would, of course,
be possible to round the values to the nearest semitone or quarter-tone etc...
The diagrams on the left were constructed using data from a
cubic spline interpolation utility
Those on the right were drawn freely such that the slope
is always positive, using the cubic spline tool in Paint.net
(version 4.1.5) — see Splines
one pitch per piano key between the values of sub-selection L
Erratum musical VI
Interpolating one pitch per piano key between the values of sub-selection O
Erratum musical VII
Interpolating one pitch per piano key between the values of sub-selection Q
Erratum musical VIII
There are lots of ways to draw smoothly curved lines through a particular series of random points. All
that is necessary is that there should be a common tangent between
the curves that meet at each point. I imagine that
it would even be possible to bend a light ray through the points, given an appropriate gravitational
field — an idea that relates closely to the 3 Stoppages étalon.
All the possible paths through a
particular series of knots have the same fixed points at the knots, but
they are increasingly
different in-between. It is possible to think of this
blurring between the knots as a kind of wave of uncertainty that collapses when a particular curve
chosen. The act of choice fixes a moment in time. A
Stoppage is thus the record of a moment. It "collapses the wave function" that is all the possible
configurations of Duchamp's string as it falls to the ground.
If the method chosen to create the “kind of new musical alphabet allowing model descriptions” is to select just one of the wagons from the stopped Erratum Musical Apparatus, then we have to think of the Apparatus as containing a probability field even when it has stopped.