Introducing The Humming of the Strings

Hello! I’m sorry that I haven’t been around much in the last few months, which have been rather busy for me. I’m hoping to continue more regular posting now, though, starting with the announcement of a new project: Point at Infinity‘s first spin-off blog, The Humming of the Strings.

Loyal readers will remember a few posts here on Point at Infinity about music. We had a post about the Shepard tone, an aural illusion that seems to perpetually rise in pitch, two posts about the Risset rhythm, the Shepard tone’s rhythmic cousin, and a post about Euclidean rhythms and their surprising connections with nuclear physics, computer graphics, and the Hebrew calendar. These were among the most enjoyable posts for me to write, and some of them were among the favorites of readers as well. Mathematics and music are two passions of mine, and there is a vast field of ideas to explore here, many of which don’t necessarily fit here on Point at Infinity. This is why I’m starting The Humming of the Strings, to provide a place for more in-depth exploration of musical topics that might be out of place here.

There is another, more technical reason for this blog. Over the last couple of years, I’ve sporadically been playing around with SuperCollider, an audio synthesis platform and programming language. It is a wonderful tool for exploring the connections between math and music, and I plan on using it to create music and audio samples for The Humming of the Strings, and on sharing what I learn in this process with the readers. I’m hoping that the details of the computer programming will be of interest to some readers, and I will try to make it as unintrusive as possible for those readers not interested in it. (I’m anticipating that most posts will consist of a non-technical body section followed by a technical appendix with details of the code.)

I hope you will join me over at The Humming of the Strings. I will likely primarily be posting there rather than here in the near future, though there may be some new content on Point at Infinity from time to time. Thanks for reading, and Happy New Year!

Advertisements

L’escalier du Diable

Welcome one, welcome all to the Point at Infinity sideshow, where today we present a tantalizing and diabolical selection of musical and mathematical curiosities. Just watch your step; these stairs can be a bit tricky.


A few months ago, you may recall, we published two posts about the Shepard tone and the Risset rhythm, aural illusions in which a tone or rhythm seems to perpetually rise or fall in pitch or in tempo but is actually repeating the same pattern over and over again, the musical equivalents of Penrose stairs.

372px-Impossible_staircase.svg
Penrose stairs. (Image in the public domain.)

To accompany the posts we created some sound samples so the readers could hear the illusions themselves. A couple of weeks ago, one of these samples was used in an internet radio program on audio paradoxes released by Eat This Radio, paired with some work of Jean-Claude Risset. The entire program is really excellent, ranging from a piece by J.S. Bach to mid-twentieth century audio experiments to modern electronic music, and I encourage all of you to listen to it.


One of the pieces in the radio program is a piano étude written by György Ligeti in the late twentieth century. The étude is named L’escalier du diable, or The Devil’s Staircase, and its repeated ascents of the keyboard have a striking resonance with the never-ending ascent of the Shepard tone.


The Devil’s Staircase is also the colloquial name given to a particular mathematical function introduced by Georg Cantor in the 1880s. It is a function defined on the set of real numbers between 0 and 1 and taking values in the same interval, and it has some quite curious properties. Before we discuss it, let’s take a look at (an approximation to) the graph of the function.

CantorEscalier.svg
Graph of the Devil’s Staircase. By Theon, CC BY-SA 3.0.

To appreciate the strangeness of this function, let us recall some definitions regarding functions of real numbers. Very roughly speaking, a function is called continuous if it has no sudden jumps, or if its graph can be drawn without lifting the pencil from the page. Continuous functions satisfy a number of nice properties, such as the intermediate value theorem.

The derivative of a function at a given point of its domain, if it exists, measures the rate of change of the function at that point. If the x-axis measures time and the y-axis measures the position of an object along some one-dimensional track, then the derivative can be thought of as the velocity of that object. If a function is differentiable at a point (i.e., if its derivative exists there) then it must be continuous at that point, but the converse is not necessarily true. (For example, if the graph of a function has a sharp corner at a point, then the function cannot be differentiable there.)

Let’s think about what it means for a function to have a derivative of 0 at a point. It means that, at that point, the rate of change of the function has vanished. It means that, if we zoom in sufficiently close to that point, the function should look like a constant function. Its graph should look like a horizontal line. What would it mean for a function to have a derivative of 0 almost everywhere? (Here “almost everywhere” is a technical term (which I’m not going to define) and not just me being vague.) One might think that this must imply that the function is a constant function. At almost every point in its domain, the rate of change of the function is 0, so how can the value of the function change?

One will quickly discover that this is not quite right. Consider the function defined on the real numbers whose value is 0 at all negative numbers and 1 at all non-negative numbers.

step

This function has derivative 0 everywhere except at 0 itself, and yet it increases from 0 to 1. It does this quite easily by being discontinuous at 0, which, in hindsight, seems sort of like cheating. So what if we also require our function to be continuous? Now we need more exotic examples, and this is where the Devil’s Staircase comes in, for the Devil’s Staircase is a continuous function, it is differentiable almost everywhere, it has a  derivative of 0 wherever its derivative is defined, and yet it still manages to increase from 0 to 1. Wild!

What is the Devil’s Staircase exactly? I’ll give two different definitions. The first proceeds via an iterative construction. Start with the function f_0(x) = x. Its graph, between 0 and 1, is simply a straight line segment increasing from (0,0) to (1,1). Now, look at the midpoint of this increasing line segment, and draw a horizontal line segment centered there whose length is 1/3 of the horizontal line of the original increasing segment. Now connect the ends of this line segment via straight lines to (0,0) and (1,1). This new curve is the graph of a function that we call f_1. It consists of two increasing line segments with one horizontal line segment between them. Now repeat the process that took us from f_0 to f_1 on each of these increasing line segments, and let f_2 be the function whose graph is the result. Continue in this manner, constructing f_n for every natural number n.

Cantor_function_sequence
First three steps of the iterative construction of the Devil’s Staircase. (Image in the public domain.)

It turns out that, as n goes to infinity, the sequence of functions \langle f_n \mid n \in \mathbb{N} \rangle converges (uniformly) to a single function. This function is the Devil’s Staircase.

A more direct but also more opaque definition is as follows: Given a real number x between 0 and 1, first express x in base 3 (i.e., using only 0s, 1s, and 2s). If this base 3 representation contains a 1, then replace every digit after the first 1 with a 0. Next, replace all 2s with 1s. The result has only 0s and 1s, so we can interpret it as a binary (i.e., base 2) number, and we let f(x) be this value. Then the function f defined in this manner is the Devil’s Staircase. Play around with this definition, and you might get a feel for what it’s doing.


And now, on our way out, some musical addenda. An encore, if you will. First, after making the Risset rhythms for the aforementioned post, I did some further coding and wrote a little program that can take any short audio snippet and make a Risset rhythm out of it. Here’s an example, first accelerating and then decelerating, using a bit from a Schubert piano trio.

You may recognize the sample from the soundtrack to Barry Lyndon.

Finally, I can’t help but include here one of my favorite pieces by Ligeti, Poema sinfónico para 100 Metrónomos.


Cover image: Devil’s Staircase Wilderness, Oregon, USA

Hippasus and the Infinite Descent

Today, dear Reader, we bring you a story. A story of Mathematics and Music, of Reason and Passion, of Drama and Irony. It is the story of Hippasus of Metapontum, of his remarkable life and his equally remarkable death. Before we begin, a note of warning. Not everything presented here is true, but all of it is meaningful.


Ippaso_di_Metaponto
Hippasus of Metapontum

A Greek philosopher and mathematician from the 5th century BCE, Hippasus was a follower of Pythagoras. The Pythagoreans believed in the transmigration of souls, subscribed to the belief that All is Number (where “Number” is, of course, “Whole Number”), made great strides in the study of musical harmony, and eschewed the eating of beans. Our hero was a particularly illustrious Pythagorean. He performed experiments linking the sizes of metal discs to the tones they emit upon being struck, developed a theory of the musical scale and a theory of proportions, and showed how to inscribe a regular dodecahedron in a sphere. The regular dodecahedron is a twelve-sided solid whose faces are regular pentagons, shapes which were dear to the Pythagoreans and central to our story. The pentagram, the five-sided star formed by extending the sides of a regular pentagon, and whose tips themselves form a regular pentagon, was a religious symbol of the Pythagoreans and a mark of recognition amongst themselves.

pentagram_coin
Ancient Greek coin with a pentagram

A corollary to the Pythagorean doctrine that All is Number was a belief, at the time, that any two lengths are commensurable, i.e., that given any two lengths, they are both whole number multiples of some fixed smaller length. In modern language, this amounts to the assertion that, given any two lengths, their ratio is a rational number, i.e., can be expressed as the ratio of whole numbers. It is only fitting that the first evidence to the contrary would come from the pentagram.

It happened when Hippasus was stargazing. He saw five stars forming a perfect regular pentagon, and inside this regular pentagon he formed a pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram. An infinite web of similar triangles was woven through his mind and, in a flash of insight, he realized something terrible: the lengths of one side of the regular pentagon and one side of the pentagram found inside it are incommensurable.

first_diagram
The lengths of one side of a regular pentagon and one side of the inscribed pentagram are incommensurable. In other words, if s denotes the length of the black line segment AB and t denotes the length of the blue line segment AC, then t/s (and therefore also s/t) is irrational.

The next day, he told his fellow Pythagoreans of his discovery, and they were horrified. They could not have this knowledge, which struck at the core of their belief system, getting out into the wider world. So they took Hippasus far out to sea, and they threw him overboard.


Infinite descent is a proof technique that morally dates back to the ancient Greeks but really came into its own in the work of Pierre de Fermat in the 17th century. The idea behind it is simple and immediately appealing. Suppose we want to prove that there are no positive integers satisfying a particular property. One way to prove this would be to show that, given any positive integer n satisfying this property, we could always find a smaller positive integer n' satisfying the same property. Repeating the argument with n' in place of n, we could find a still smaller positive integer n'' satisfying the same property. Continuing, we would construct a decreasing sequence of positive integers, n > n' > n'' > n''' > \ldots , the aforementioned “infinite descent”. But of course there can be no infinite decreasing sequences of positive integers (if a decreasing sequence starts with n, it can of course have at most n elements). One thus reaches a contradiction and concludes that there are no positive integers with the given property.

Fermat made great use of the method of infinite descent in his work on number theory. One particularly striking application came in the proof of a special case of his famous Last Theorem: there are no positive integers a,b,c such that a^4 + b^4 = c^4. Fermat showed that, if a_0,b_0,c_0 are positive integers such that a_0^4 + b_0^4 = c_0^4, then we can construct positive integers a_1, b_1, c_1 such that a_1^4 + b_1^4 = c_1^4 and c_1 < c_0. One can then continue, with a_1, b_1, c_1 in place of a_0, b_0, c_0, and obtain a_2, b_2, c_2 with a_2^4 + b_2^4 = c_2^4 and c_2 < c_1. This of course leads to the infinite descent c_0 > c_1 > c_2 > \ldots and a contradiction.

We will use the method of infinite descent to prove the result of Hippasus mentioned above. This will not be exactly the way the ancient Greeks would have presented the proof, but it is very similar in spirit, and we make no apologies for the anachronism. Let’s get started.


Suppose, for the sake of an eventual contradiction, that we have a regular pentagon whose side length is commensurable with the side length of the inscribed pentagram. Let s denote the side length of the pentagon and t denote the side length of the pentagram (i.e., the diagonal of the pentagon). In modern language, our assumption is that \frac{t}{s} is rational, i.e., that there are positive integers p and q such that \frac{t}{s} = \frac{p}{q}. By scaling the pentagon, we can in fact assume that t = p and s = q. So let this be our starting assumption: there is a regular pentagon whose side length s and diagonal length t are both positive integers.

Let us label the vertices of the pentagon by the letters A,B,C,D,E. The center of the pentagram forms another regular pentagon, whose vertices we shall call a,b,c,d,e. This is shown in the diagram below. Note that s is the length of the line segment connecting A and B (we will denote this length by |AB|), and t is |AC|. Let s' denote the side length of the inner pentagon, i.e., |ab|, and let t' denote the length of the diagonal of the inner pentagon, i.e., |ac|.

proof_diagram
Figure 1

We now make use of the wealth of congruent triangles in the diagram. We first observe that, whenever a pentagram is inscribed into a regular pentagon, the diagonals that form the pentagram exactly trisect the angles of the pentagon. (Exercise: Prove this!) Therefore, the triangle formed by A, E and D is congruent to that formed by A, e, and D, on account of their sharing a side and having the same angles on either end of that side. In particular, we have |Ae| = |AE| = s.

Next, consider the triangle formed by A, b, and d. By our observation at the start of the previous paragraph, the angle at b in this triangle has the same measure as the angle at A. But then this triangle is isosceles, so we have |Ad| = |db| = t'. Of course, we clearly have |Ad| = |Bd| = |Be| = |Ce| =…, so all of these lengths are equal to t'.

proof_diagram
Figure 1 again

Let’s now see what we have. Consider first |AC|. By definition, we have |AC| = t. But also |AC| = |Ae| + |Ce|, and we saw previously that |Ae| = s and |Ce| = t'. All together, we have t = |AC| = |Ae| + |Ce| = s + t', or t' = t-s.

Next, consider |Ae|. We have already seen that |Ae| = s. But we also have |Ae| = |Ad| + |de|. We saw previously that |Ad| = t', and by definition we have |de| = s'. All together, we have s = |Ae| = |Ad| + |de| = t' + s', or s' = s - t'. Since we already know that t' = t-s, this yields s' = 2s - t.

This might not seem like much, but we’re actually almost done now! The key observation is that, since s and t are integers, and since s' = 2s - t and t' = t-s, it follows that s' and t' are also (positive) integers. We started with a regular pentagon whose side length s and diagonal length t were integers and produced a smaller pentagon whose side length s' and diagonal length t' are also integers. But now we can continue this process, producing smaller and smaller regular pentagons and producing infinite descending sequences of positive integers, s > s' > s'' > \ldots and t > t' > t'' > \ldots.

pentagons
The infinite descent

This is a contradiction, and we have thus shown that \frac{t}{s} is irrational. But what is this ratio exactly? Well, it turns out to be a fascinating number, easily #2 on the list of Most Famous Ratios of All Time: \phi, a.k.a. the golden ratio! But that is a story for another time…


As a short addendum to today’s story, here’s a little-known fact about the end of Hippasus’ life. It turns out that, after being thrown overboard by his fellow Pythagoreans, he has not and will never in fact reach the seabed! For to get there, he would first have to travel halfway down, and then he would have to travel half of the remaining distance, and then half of the still remaining distance, and so on and so forth, completing an endless sequence of tasks, and thus he remains to this date in the midst of an infinite descent of his own.


Notes:

(1) \sqrt{2} is more commonly cited as being the first irrational number discovered by the Pythagoreans, and it is almost always the first number proven to be irrational in classrooms today. However, the proof of the irrationality of \sqrt{2} possessed by the Greeks is not the simple number-theoretic proof used today and is in fact a rather complex elaboration of the proof of the incommensurability of the side and diagonal lengths of a regular pentagon. Given this fact and the centrality of the pentagram in Pythagorean intellectual life, some scholars have suggested that perhaps this was in fact the first proof of the existence of incommensurability and that the proof of the irrationality of \sqrt{2} came later. We have adopted this hypothesis for the purpose of our story today.

(2) This story is derived from legends that significantly post-date the death of Hippasus. It seems unlikely actually to have happened as presented here.

(3) This post was in part inspired by the episode “Drowned at Sea”, from the excellent podcast, Hi-Phi Nation, by philosopher Barry Lam. Check it out!


Cover image: “Rainstorm Over the Sea” by John Constable

Beats by Bjorklund

You gotta keep ’em separated.

-The Offspring, “Come Out And Play”

(Friendly note to the reader before we begin: If you’re looking for music to listen to while reading this post, just scroll to the bottom…)

The Spallation Neutron Source (SNS), at Oak Ridge National Laboratory in Tennessee, provides the world’s most intense pulsed neutron beams for scientific research. By accelerating proton pulses and directing them towards a mercury target, the laboratory ejects neutrons from the target. By measuring the energies and angles of these scattered neutrons, scientists can uncover information about the magnetic and molecular structure of various materials.

Pulses at the SNS are grouped into super-cycles, each 10 seconds long and consisting of 600 pulses. Commonly, experimenters will want to perform a particular action on a fixed number of pulses (k, say) during each super-cycle and will want to have the executions of the actions spaced as evenly as possible across time. If k evenly divides 600, then the solution to this problem is trivial: just perform the action on pulses 0, \frac{600}{k}, \frac{2 \cdot 600}{k}, \frac{3 \cdot 600}{k}, \ldots, \frac{(k-1) \cdot 600}{k}. If k does not evenly divide 600, though, then the solution is not as obvious.

This is of course a special case of a general problem: Given cycles of length n, how can k events be spaced as evenly as possible across a single cycle? (Naturally, we should have k \leq n here.) As seen above, the problem is trivial if k evenly divides n, so we shall assume this is not the case. In fact, we shall assume that k and n are relatively prime, i.e., that they have no common positive divisors other than 1. We will also assume that k > 1, since it is also trivial to evenly distribute one event across a cycle.

This exact problem was tackled by E. Bjorklund, motivated by timing questions at the SNS. In this paper, he presents an algorithm for generating solutions. For ease of notation, let us denote a cycle by a sequence of 0s and 1s, with a 1 representing an event taking place and a 0 representing an event not taking place. For definiteness, we will always start the sequence with a 1, but, because we are dealing with cyclic phenomena, we will consider two “rotations” of the same base sequence to be the same. For example, we consider 10010 and 10100 to be the same, since the second is just the first “rotated” (and wrapped around) three spaces to the right.

We will not give a formal account of Bjorklund’s algorithm here, but we will illustrate it with an example that we hope will indicate how to carry it out in general. Suppose we are given the specific problem of evenly spacing eight events in a cycle of length thirteen. Our final sequence will thus have eight 1s and five 0s. We start with thirteen individual sequences, each of length one, with all of the 1s first and then all of the 0s:

[1] [1] [1] [1] [1] [1] [1] [1] [0] [0] [0] [0] [0]

You will notice that we have two different sequences in this list ([1] and [0], here). This will be true for every step of the algorithm. In the next step, we take all instances of the second sequence ([0], here) and append them in turn to instances of the first sequence. Since there are five instances of the second sequence, five instances of the first sequence will become longer, and we obtain:

[10] [10] [10] [10] [10] [1] [1] [1]

We again have two different sequences ([10] and [1]). And we again take all instances of the second sequence and append them in turn to instances of the first sequence, obtaining:

[101] [101] [101] [10] [10]

Repeating this process again yields:

[10110][10110][101]

Technically, since there is only one instance of the second sequence in this stage, we could stop here and just append all of the sequences in order to obtain the correct solution, but let us continue for illustration’s sake. Next, we get:

[10110101][10110]

And, finally:

[1011010110110}

(which is just a rotation of what we would have gotten two stages earlier: [1011010110101]). Thus, the optimal way of evenly distributing eight events in a single cycle of length 13 is given by the sequence 1011010110110. A similar process will yield the correct answer in any other specific instance of this problem.

(The astute reader might have noticed that this algorithm bears a resemblance to Euclid’s famous algorithm for calculating the greatest common divisor of two positive integers. This is not a coincidence.)


You might be slightly disturbed by my lack of rigor at this point. Indeed, there are two ways in which I am slightly cheating the reader. First, I have not really said what I mean by having events be spaced “as evenly as possible.” Intuitively, 1011010110110 is more even than 111111110000, or 1101101101100, or, if one thinks about it, 1011011011010, but how do we know it is “as even as possible,” and what do we even mean by this? Second, even if I had given a rigorous definition of “evenness,” I have not proven that Bjorklund’s algorithm does in fact deliver the optimal result (again, this should be intuitively plausible, but should perhaps not be blindly accepted). I will do neither of these things here, but will instead direct the interested reader to the excellent paper, “The distance geometry of music,” by Demaine, Gomez-Martin, Meijer, Rappaport, Taslakian, Toussaint, Winograd, and Wood, from which much of this post was derived.


Unsurprisingly, the problem of evenly distributing k events or items into cycles of length n does not show up only in nuclear physics. Let us briefly look at a couple of other interesting appearances.

Computer graphics: Suppose you want to depict a straight line on a computer screen with a slope of \frac{k}{n}. Computer screens consist of grids of pixels, so, unless the line is horizontal or vertical, it cannot be drawn perfectly straight. Here is an example of a segment of a line with slope -\frac{5}{11}:

832px-Bresenham.svg
A line with slope -5/11, superimposed over a pixel approximation. Created by Crotalus horridus, Public Domain

It should be clear that, at least as long as k < n, the problem of drawing this line consists of choosing k out of every n horizontal spaces after which to move your line one space vertically. To make your line as straight as possible, this distribution should be as even as possible. Indeed, the situation depicted above can be represented by the sequence 10101001010, where 1 indicates a vertical shift and 0 represents the absence of a vertical shift. This is precisely the result one gets (after a rotation) if one applies Bjorklund’s algorithm to the numbers 5 and 11.

Leap year distribution: The problem of reconciling the fact that one revolution of the earth around the sun does not take an exact integer number of days with the desire to have years that last an exact integer number of days is one that affects every calendar design. It is handled particularly interestingly in the Hebrew calendar. The Hebrew calendar is a lunisolar calendar, in which each month lasts one full cycle of the moon, but in which adjustments are made so that each month occurs at roughly the same stage of the solar year during each cycle. The Hebrew calendar reconciles these two contradictory demands by having twelve months in a typical year but, during certain “leap years,” having one of these months (namely, Adar) repeated to yield a year of thirteen months. It turns out that, to keep the Hebrew calendar very close to being in sync with the solar cycles, one needs to have seven out of every nineteen years be leap years. It is also natural to want these leap years to be spaced as evenly as possible. And, indeed, the distribution of these seven leap years in nineteen-year cycles in the Hebrew calendar is given precisely by a rotation of the result of Bjorklund’s algorithm applied to the numbers seven and nineteen.


Perhaps the most interesting appearances of the problem of evenly distributing events in a cycle, though, are in music. If one thinks of a musical measure consisting of n beats as being a cycle of length n, of a 1 as being a note being played, and of a 0 as being the absence of a note being played, then our strings of 1s and 0s can be thought of as musical rhythms. And it turns out that applications of Bjorklund’s algorithm, when interpreted in this way, yield a startling variety of musical rhythms used in practice around the world. We end this post by presenting a few prominent examples.

In what follows, we notationally replace 1s with ‘x’s and 0s with dots. This is more conducive to following along with musical rhythms, as it is visually easier to get lost in strings of 1s and 0s than in strings of ‘x’s and dots. An ‘x’ thus denotes a note being played and a dot denotes the absence of a note being played (or, often, an ‘x’ denotes an accented beat and a dot denotes an unaccented beat).  Given numbers k < n, we denote by E(k,n) the result of Bjorklund’s algorithm applied to k and n. The sequence that follows is the rotation of this result that is actually heard in the piece of music. As before, we restrict ourselves to the interesting cases in which k and n are relatively prime and k > 1. (Of course, the standard waltz rhythm [x .  . ], for example, would be E(1,3), and the standard backbeat rhythm [. x . x ] would be a rotation of E(2,4), but these are sort of boring in this context.)

There’s some good music here. Enjoy!

E(2,5) – (x . . x . ): “Take Five” by The Dave Brubeck Quartet.

E(3,7) – (x . x . x . . ): “Money” by Pink Floyd.

E(3,8) – (x . . x . . x . ): This is the famous tresillo rhythm, which can be heard throughout Latin American and sub-Saharan African music as well as in such songs as “Hound Dog,” “Despacito,” and the song we feature here, “Intro” by The xx.

E(5,8) – (x . x x . x x . ): This is the cinquillo rhythm, an elaboration of the tresillo that is also ubiquitous in Latin American and sub-Saharan African music. It also features prominently in “Come Out and Play” by The Offspring.

E(4,9) – (x . x . x . x . . ): This is our second track from the classic Time Out album, which was inspired by musical styles (especially time signatures) observed during travels in Europe and Asia. Brubeck saw this particular beat being played by Turkish street musicians and then used it for  “Blue Rondo à la Turk” by The Dave Brubeck Quartet.

E(4,11) – (x . . x . . x . . x . ): “Outside Now” by Frank Zappa.

E(5,11) – (x . x . x . x . x . .): “Promenade” from Pictures at an Exhibition by Modest Mussorgsky.

E(7,12) – (x . x . x x . x . x . x): This is one of the most important rhythms in sub-Saharan Africa, often used as a bell pattern. It is also a common beat in the Palo music of the Caribbean. It can be heard played by the claves player in this performance by Los Calderones, which I have been listening to non-stop for the last hour. (Note also that, if one re-interprets this sequence in a harmonic rather than rhythmic setting, in which each step of the cycle represents a half-step in pitch, an ‘x’ represents the inclusion of that note in a scale, and a dot represents the exclusion of that note from a scale, then this sequence (this exact rotation, in fact) precisely corresponds to the major diatonic scale, the most common musical scale in Western music!)

E(4,13) – (x . . x . . x . . x . . . ): This rhythm can be heard in the instrumental sections of “Golden Brown” by The Stranglers.

E(5,16) – (x . . x . . x . . . x . . x . . ): This rhythm forms the basis for much bossa nova music and, when significantly slowed down, “Pyramid Song” by Radiohead.


Cover image: Detail from “Wall Drawing #260” by Sol LeWitt

Infinite Acceleration: Risset Rhythms

In our most recent post, we took a look at and a listen to Shepard tones and their cousins, Shepard-Risset glissandos, which are tones or sequences of tones that create the illusion of perpetually rising (or falling) pitch. The illusion is created by overlaying a number of tones, separated by octaves, rising in unison. The volumes gradually increase from low pitch to middle pitch and gradually decrease from middle pitch to high pitch, leading to a fairly seamless continuous tone.

The same idea can be applied, mutatis mutandis, to percussive loops instead of tones, and to speed instead of pitch, thus creating the illusion of a rhythmic track that is perpetually speeding up (or slowing down). (The mechanism is exactly the same as that of the Shepard tone, so rather than provide an explanation here, I will simply refer the reader to the previous post.) Such a rhythm is known as a Risset rhythm.

I coded up some very basic examples on Supercollider. Here’s an accelerating Risset rhythm:

And a decelerating Risset rhythm:

Here’s a more complex Risset rhythm:

And, finally, a piece of electronic music employing Risset rhythms: “Calculus,” by Stretta.

 

Infinite Ascent: Shepard Tones

Have you ever been watching a movie and noticed that the musical score was seeming, impossibly, to be perpetually rising, ratcheting up the intensity of the film more and more? Or perhaps it seemed to be perpetually falling, creating a deeper and deeper sense of doom onscreen? If so, it is likely that this effect was achieved using a Shepard tone, a way of simulating an unbounded auditory ascent (or descent) in a bounded range.

To understand how Shepard tones work, let’s look at a simplified implementation of one. We will have three musical voices (middle, low, and high), with an octave between successive voices. The voices then start to move, in unison, and always an octave apart, up through a single octave, over, say, five seconds. As they go, though, they also change their volumes: the middle voice stays at full volume the whole time, the low voice gradually increases from zero volume to full volume, and the high voice gradually decreases from full volume to zero volume. The result will simply sound like a tone rising through an octave, and it can be represented visually as follows.

shepard_1

This by itself is nothing special, though. The trick of the Shepard tone is that this pattern is then repeated over, and over, and over again. Each repetition of the pattern sounds like a tone ascending an octave, but, because of the volume modulation, successive patterns are aurally glued together: the low voice from one cycle leads seamlessly to the middle voice of the next, the middle voice from one cycle leads seamlessly to the high voice of the next, and the high voice simply fades away. The result sounds like a perpetually increasing tone.

shepard_2

Note the similarity to the visual barber pole illusion, in which a rotating pole causes stripes to appear to be perpetually rising. Also, this whole story can be turned upside down, which will lead to a perpetually falling tone.

Let’s hear some Shepard tones in action! Now, in practice, using only three voices does not create a particularly convincing illusion, so, to make these sounds, I used nine voices, spread across nine octaves. Also, linearly varying the volume, as in the above visualization, seems to make it more noticeable when voices enter or fade away, so I used something more like a bell curve.

(Technical notes: These Shepard tones were created in Supercollider, using modified code written by Eli Fieldsteel, from whose YouTube tutorials I have learned a great deal of what I know about Supercollider. Also, I used a formant oscillator instead of the more traditional sine oscillator.)

First, a simple ascending Shepard tone:

The effect becomes more convincing, and the tone more interesting, if multiple Shepard tones are played simultaneously at a fixed interval. Here, we have two ascending Shepard tones separated by a tritone, a.k.a. the devil’s interval, a.k.a. half an octave:

Next, three descending Shepard tones, arranged in a minor triad:

Finally, two Shepard tones, with one ascending and the other descending:


The origins of the Shepard tone lie with Roger Shepard, a 20th-century American cognitive scientist, as a sequence of discrete notes. The continuous Shepard scale, or Shepard-Risset glissando, which our code approximates, was introduced by French composer Jean-Claude Risset, who perhaps most notably used it in his Computer Suite from Little Boy from 1968.

More recently, it has prominently been deployed by Christopher Nolan and Hans Zimmer, as the basis for the Batpod sound in The Dark Knight and in the Dunkirk soundtrack.


Cover image: M.C. Escher, Waterfall

The Book of Sand

“It can’t be, yet it is,” the Bible peddler said, his voice little more than a whisper. “The number of pages in this book is literally infinite. No page is the first page; no page is the last. I don’t know why they’re numbered in this arbitrary way, but perhaps it’s to give one to understand that the terms of an infinite series can be numbered any way whatever.”

Then, as though thinking out loud, he went on.

“If space is infinite, we were anywhere, at any point in space. If time is infinite, we are at any point in time.”

His musings irritated me.

-Jorge Luis Borges, “The Book of Sand”

It’s a busy week for me, so just a short post today and an exhortation to take a few minutes to read Borges’ story, “The Book of Sand.” You can find a nice version of it here.

We will no doubt return to this and other of Borges’ fictions later. “The Book of Sand” will prompt reflection on our place in a potentially infinite universe. Attempts to determine the order type of its pages will lead naturally to the investigation of the fascinating topic of dense linear orders.

For today, though, we will just appreciate the story.

Oh, and then we will go over here and play around with Michel van der Aa’s lovely multi-tiered music video experiment based on “The Book of Sand” and other Borges stories. Join us!