On Marcel Duchamp’s Erratum Musical

La Mariée mise à nu par les célibataires, même. Erratum musical.
Page 1 Page 2
The history of this analysis
Other Duchamp works on manuscript paper
Realisations of Selections I-VIII (1979)
Using partitions to define a blurred line
Three Crashes
Erratic Lines

Duchamp's La Mariée mise à nu par les célibataires, même. Erratum musical  is concerned with time, and develops the particle-equivalent of his 3 Stoppages-étalon. The original Stoppages-étalon are both wave-like and timeless (stopped, frozen). The fundamental, dimensionless question is: Which path(s) between two points require the least energy if, as in modern physics, a straight line is not the answer? *
Page 1 of the Erratum musical is a record of Duchamp’s initial, developing ideas.
Page 2 contains “three standard stoppages” and the description of a device (The Apparatus). These are more than just artworks: The three stoppages are also instructions for changing the (space-time) relation between piano keys and the frequencies with which they are assocated. The Apparatus, while running, generates a probabilistic field. When it stops, the result is an ambiguous particle-stoppage that can be used to create a stoppage that is “a kind of new musical alphabet”.

An incidental note relates the analysis to Donald Hoffman’s Interface Theory of Perception.

* Curved space and time were, of course, burning issues at the beginning of the 20th century: Poincaré, Einstein, Cubism, Futurism... Nude descending a staircase...

La Mariée mise à nu par les célibataires, même. Erratum musical.
These pages, and all the extracts from them below: © Association Marcel Duchamp/VG Bild-Kunst, Bonn 2019
According to Serge Stauffer’s Duchamp: Gesammelte Schriften, the originals of these pages are currently at The Foundation for Contemporary Performance Arts, New York.

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:
Note that 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 predates these pages, 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 thinking aloud).
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” seems to 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 ordering 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 I-VIII. Perhaps he was thinking of investigating the matter further — with more wagons and a finer-grained selection of 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 explanation of each selection follows below:
February 2019: A score (Erratum Musical I-VIII) containing the above graphs transcribed as music notation can now be viewed and automatically performed using my Assistant Performer software. See “Unfinishable” below.

Page 1
Selection I
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.
Selection 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.

Selection II

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 sequences.
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.

Selection III
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!

Selection IV
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).

Selection V
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 happens now.
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 Stoppages-étalon come after 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 sorted 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).

Page 2
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 VIII, at 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.
Note that 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.

Selection VI
Sub-selection L contains the first 11 randomly chosen numbers in numerical order.
M 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 L only contains 11 numbers, not 12.

Selection VII

Sub-selection O contains the first 22 randomly chosen numbers in numerical order.
N 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.

Selection VIII

Sub-selection Q contains the first 10 randomly chosen numbers in numerical order.
P 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.

The Apparatus

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 instrument

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 in Duchamp: Gesammelte Schriften (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 F), allowing 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 field.
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:
  1. The number of values in each wagon is as equal as possible.
  2. 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:

Note that:
  • 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 account: Situations 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 Donald Hoffman’s introductions to his Interface Theory of Perception here (short) 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 complexity.
Note that:
  • 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!
    [added 31.01.2019]: See also Duchamp’s comments on chance, as reported in Serge Stauffer’s Duchamp: 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 above (which could also be rotated by 90°, 180° and 270°) — but these are blurred lines, rather than being continuous, bent lines.
Bent lines, that can be usd in the same way, can be created simply by interpolating values between the values in a single sub-selection (the 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 there to 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) (coloured red) in the following diagrams. Note that, 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 below.

Interpolating one pitch per piano key between the values of sub-selection L in Erratum musical VI:

Interpolating one pitch per piano key between the values of sub-selection O in Erratum musical VII:

Interpolating one pitch per piano key between the values of sub-selection Q in 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 is 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.

The history of this analysis
30th October 1979: Jacques Caumont and Jennifer Gough-Cooper (J&J) asked me to see what I could make of Duchamp’s Erratum Musical. I was spending a couple of days at their house in Normandy. When I left, I gave them all the resulting sketches, including the realisations of Selections I-VIII on manuscript paper. I made no attempt to analyse The Apparatus.
3rd August 1991: J&J were preparing Marcel Duchamp: Work and Life and the major Duchamp Retrospective at the Palazzo Grassi in Venice (September 1992 - June 1993). Jacques wrote to me including copies of my 8 scale realisations, saying that he thought Duchamp’s La Mariée... Erratum Musical was related to the 3 Stoppages, and that he would be interested in hearing my comments before putting the two pages in that context at the Palazzo Grassi.
27th September 1991: I sent J&J the detailed analysis that was the starting point for this one.
2010: The 1991 analysis is mentioned in Système D by Jaques Caumont and Françoise le Penven.
December 2018 - 23rd January 2019: This current version was prepared, and essentially completed. It is closely based on the 1991 analysis, but the analysis of The Apparatus and all the diagrams on graph paper are new.
24th - 27th January 2019: Jennifer (who is my cousin) visited Heinz and me in Cologne. 25th: drew the Erratic Lines (freehand). 26th: went to the exhibition "Duchamp: 100 Fragen. 100 Antworten." at the Staatsgalerie in Stuttgart. The curator, Dr. Susanne Kaufmann, gave us a private guided tour. (Many thanks!) Being previously ignorant of Serge Stauffer, I bought his Duchamp: Gesammelte Schriften
29th January 2019: Uploaded this web page for the first time.
30th - 31st January 2019: Added this and the following changes as a result of reading in Stauffer’s book:
  • Added the location of the originals of the Erratum Musical (under their first appearance above)
  • Added a note at the top of The Apparatus analysis, saying I think Stauffer was right that the Apparatus uses 85 (not 89) notes.
  • Extended point 1 in the incidental note to include Duchamp's remark about chance.
18th February 2019:
  1. Uploaded the following scores to my Assistant Performer:
  2. Added the Three Crashes appendix below.

Other Duchamp works on manuscript paper

  1. Erratum Musical (for Yvonne-Magdaleine-Marcel)
    © Association Marcel Duchamp/VG Bild-Kunst, Bonn 2019
    faire une em-prein-te mar-quer des traits u-ne fi-gure sur une sur-face im-pri-mer un sceau sur ci-re
    google translation:
    make an imprint mark features a figure on a surface print a seal on wax

    This work is much simpler than the La Mariée mise à nu par les célibataires, même. Erratum musical. The two works seem to be unrelated, except that both involve music and chance operations (“taking numbers out of a hat”).
    Here, each part has each of its 25 pitches once, which is what happens when one takes 25 different pitches out of a hat (which is what Duchamp said he did).
    Marcel's range is a fourth lower than Yvonne and Magdaleine’s.
    According to toutfait.com, this piece was written with Duchamp’s two sisters (who were both musicians) during a New Year’s visit in Rouen in 1913, and ”first performed publicly by the Dada artist Marguerite Buffet at the Manifestation of Dada on March 27, 1920“.
  2. avoir L’apprenti dans le soleil
    avoir l'apprenti dans le soleil
    © Association Marcel Duchamp/VG Bild-Kunst, Bonn 2019
    I think this is directly related to Einstein's prediction that light rays are bent by the Sun's gravitational field, and that there is therefore no such thing as a straight line.
    Bicycle wheels relate to the Sun and its rays.
    Musical manuscript paper is a chiffre for time, which Einstein also dilates, but which cannot be put directly on paper.
    Note that Duchamp’s first four readymades can all be related to the subject of bent space-time:
    1. The bicycle wheel should really be turning (in time). There would then be an inward force on its rim. Unfortunately, it is usually displayed in a stopped state “for conservation purposes”. The stationary wheel is a stoppage — a recording of the moment when it stopped.
    2. The bottle rack is a bent line in rotation, surrounded by bent lines (It is actually very like modern representations of the Big Bang.)
    3. The snow shovel is a bent plane.
    4. The urinal is a bent volume — with a few black holes.
      (Duchamp: "Problem: trace a straight line on Rodin’s The Kiss as seen from a sight” — Marcel Duchamp: Work and Life p.144)

Realisations of Selections I-VIII (1979)
These are scans of the copies sent to me in 1991. They are approximately at the original size.
The red markings are from 1991.
The pitch distribution in scale III here is different from the one in the main analysis above.
Also, this scale III contains an error: In the last line, the value 72 occurs twice. The first 72 should read 79, and be the D# above the written G#.

Using partitions to define a blurred line
In Erratum musical Selection III, Duchamp used an ordered sequence of partitions to define sub-selection E. Each successive partition had values that were in random order but in a higher range. This procedure blurs the line rather than bending it, so its not surprising that he dropped this idea in subsequent selections.
As can be seen in the following examples, the degree to which the line is blurred is related to the size of the partition. If there were 89 partitions, the line would be completely straight and sharp. If there was only one partition, the line would be diffused over the whole field (as in Selection I).
This process of blurring a line is very like the way in which modern black-and-white printers create grey-scale images by scattering random black dots inside a larger “pixel”. Interesting to note that stochastic processes can be used to add a second dimension to a fundamentally one-dimensional line...

Splines were originally long, flexible planks used to create curves in the shipbuilding industry.
See Wikipedia — (especially the History section). Also search for e.g. "spline drawing instruments".

Thinking that it would be more intuitive to have a keyboard in which key-frequencies always increase from left to right, I decided to draw the right-hand solutions above using the cubic spline tool in Paint.net (version 4.1.5). The procedure was as follows:
  1. Draw a line from a red point (1) to the red point three seqments to the right (red point 4). The new line has 4 control points. Control point 1 is now on red point 1, control point 4 is on red point 4, Control points 2 and 3 are somewhere in between.
  2. Drag control point 2 to red point 2, and control point 3 to red point 3.
  3. If there is already a tangent at red point 2, adjust control point 1 so that the new line’s tangent matches there. Control point 1's position can be freely chosen, providing that the tangents match.
  4. Adjust control point 4 so that the tangent at red point 3 is reasonable for the following segment, then draw the tangent. This required some trial and error, to ensure that the following segment never had to curve downwards.
  5. Copy the central section of the new line to the final spline sequence. Throw the outer two segments away.
  6. Repeat from step 1 for all the inner splines, then complete the outer splines (the first and last) with freely chosen end tangents.

Three Crashes
This is an original composition which can be viewed and automatically performed using my Assistant Performer software. It closely follows Duchamp’s instructions.
There are many references in the literature to the Three Crashes in relation to the Large Glass, but they seem to have been left in a conceptual state by Duchamp himself. I have tried to create something that is at least consistent with many of the cryptic descriptions and speculations...

In this realisation, each “Crash” consists of the data harvested from a single run of the (simulated) Apparatus. The Crashes are performed simultaneously, with each Crash being performed eleven times in succession, each time with a different internal order — as described above: (au gré) interchangeable.
To keep Crashes 1-3 distinguishable, they rotate around the listener at different positions: When a Crash is “in front”, it is faster and louder, and moves from right to left. When “behind”, it is slower and quieter, and moves from left to right. Alternate “repetitions” of a particular Crash are performed from top to bottom or bottom to top of the “scale”.
The different Wagons in each Crash are distinguished by their loudness, so that it is possible to follow them individually, especially when the Crash is at its fastest/loudest (front centre).
The best way to hear all these relations is to listen to the score on headphones, while following the position of the moving cursor.

Erratic Lines

www James Ingram 2019