ERLEBNISLAND MATHEMATIK

The sound gyroscope

Figure 1: The sound gyroscope in Erlebnisland

Here you can see the exhibit “Tonkreisel” in the MATHEMATICS ADVENTURE LAND. The Tonkreisel is a musical instrument for one or more players. The playing surface consists of 15 floor sensors arranged along a spiral line on seven colored circle segments. When the sensors are pressed down with the ball of the foot, sung tone syllables (do, re, mi, fa, so, la, ti) sound at specific pitches. In the center of the instrument is a rotating cylinder whose 84 positions each encode different distributions of the tone syllables and pitches. The following explanations are intended as an invitation to explore the tonal gyroscope as a medium for playfully conveying both mathematical and music-theoretical insights. Small “compositions” for two, three and more players can be studied here and then tried out on the tonal gyroscope.

Music researchers have used mathematical tools to gain interesting insights into the structure of the diatonic scale and its modes. These include Eytan Agmon, Gerald Balzano, Norman Carey, John Clough, David Clampitt, Jack Douthett, Julian Hook and Eric Regener, among others. The exhibit is particularly inspired by the following paper: Jack Douthett et al. (2008): Filtered Point-Symmetry and Dynamical Voice-Leading. In: Music Theory and Mathematics: Chords, Collections, and Transformations. University of Rochester Press. This text explains how some of these insights were exploited in the construction of the tonal gyroscope. Involved in the realization of the Exponant were: Thomas Noll (idea and overall concept), Antje Werner and Robert Thiele (design), Firma Kluge (construction), Marije Baalman (sensor technology and programming), Jörg Garbers (programming), and the team of MATHEMATICS ADVENTURE LAND led by Michael Vogt (concept, construction, logistics). The solmization syllables were sung by students of the Hochschule für Kirchenmusik in Dresden.


First, some math: Seven out of Twelve

You can divide a circle into seven regular segments and you can divide a circle into twelve regular segments. However, you cannot do both at the same time. To be more precise: You cannot select seven corners from a regular dodecagon in such a way that they then form a regular heptagon. After all, the number 7 is not a divisor of the number 12. Even worse: The largest common divisor of 7 and 12 is 1. Nevertheless, there is a maximally regular selection of 7 from 12. It is as regular as possible. What looks like a “lazy compromise” at first glance is the key to an extremely interesting mathematical structure. In Figure 2, you can see how the maximum regular selection is created. On the outside, the circle is divided into seven segments and on the inside into twelve segments of equal size.

Figure 2: The selection 0, 2, 4, 6, 7, 9, 11 of 7 segments from the 12 segments 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 is maximally regular.

The concrete selection in Figure 2 depends on the position of the inner circle, which is thought to be movable, relative to the fixed outer circle. On the basis of the numbering 1 to 7 (outside) or 0 to 11 (inside) one can identify each segment, while the inner circle turns against the outer one. In the separating black circle line there are exactly seven slots, which are also regularly distributed. They are located in the center of each of the segments of seven. In each configuration, the color “flows” from the outer and invariable segments of seven through the slits into the respective passing segments of twelve. This determines the respective selection. There are 84 configurations in total. This can be seen from the counter in the center (see the following Figure 3).

Figure 3: Three of a total of 84 configurations of the circles that can be rotated against each other.

Each of the segments of twelve (white numbers from 0 to 11) moves past the red field of seven (with the black number 1) at some point. Meanwhile, exactly seven configurations result. And: 84=12\cdot 7.


The seven steps of the scale

The circular playing surface is divided into seven equally sized colored segments. These segments represent the seven steps of the diatonic scale.

In music theory, the steps are numbered 1, 2, 3, 4, 5, 6, 7. On the playing surface, it is the red field that bears the number 1. From there, the numbers increase counterclockwise around the circle: orange = 2, yellow = 3, and so on. (see figure 4 below)

Figure 4: The seven colored segments correspond to the seven steps of the scale

On each field there are ground sensors that trigger sounds when pressed. The sensor positions are connected to each other with a spiral line. This spiral begins and ends on the first step, i.e. the red field with the number 1. Only there are three sensors. All other step fields have two sensors. If you follow the spiral counterclockwise, the pitches of the sounds increase. Both the lowest and highest tones belong to the first step. The tones that each belong to the same step are octave related. They sound very similar to our perception, although they are far apart in pitch. If you turn the cylinder in the middle of the instrument, the pitches or tone syllables change in each case. Nevertheless, the lowest tone always sounds at the outer end of the spiral and the highest tone at the inner end of the spiral. Step 1 is therefore always the beginning and end of the scale.

The Latin interval designation Octava (i.e. the eighth) is based on the counting of the steps, which leads here exactly once around the circle. That the spiral leads twice around the circle allows a larger range of two octaves for making music.


Seven from twelve tones

The distribution of white and black keys on the usual piano keyboard is maximally regular (see the following figure 5, left). Black keys, of which there are only 5, are not adjacent to each other at any point, and the two places where two white keys are adjacent to each other follow each other — as regularly as possible.

Figure 5: Distribution of the white and black keys on a piano keyboard (left) and the identification of the 7 white keys as steps of the C major scale.

The colored piano keys (see Figure 5, right) relate the C major scale C-D-E-F-G-A-H-C’ to the configuration with the number 1 (left). Compared to the piano, the tonal gyroscope has the limitation that only 7 notes per octave register are playable in each individual configuration. So there are no black keys in the strict sense. However, by turning the cylinder, the tone meanings of the colored step fields can be changed so that those tones that a piano player must produce with black keys become so-called root tones. This means that these tones behave in the configuration in question in the same way as the tones C-D-E-F-G-A-H-C’ in configuration 1.

Figure 6
Figure 7

84 diatonic modes

The rotating cylinder in the center of the sound gyroscope is illuminated from the inside. Seven immobile vertical shadow stripes around the cylinder wall let the light fall only into the seven immobile bright stripes in between. The following figure 5 shows an unwinding of the cylinder wall (here still without consideration of the movable graphic).

Figure 8

If one divides the entire cylinder wall all around into 84 equally narrow vertical stripes, 5 of these 1/84 stripes are allotted to each shadow and 7 to each of the light-flooded areas.

This structure is also remotely reminiscent of a piano keyboard. However, at first glance it looks as if there are two black keys too many here. More precisely: Instead of 5 black keys with the width of 7 narrow stripes each, as one would expect, there are 7 shadows with the width of 5 narrow stripes each. Figure 9 shows how the already presented method for maximum-regular selection of seven segments out of twelve works with this shadow distribution.

Figure 9

Each of the digits 0, 1, …, 11 is selected exactly and appears accordingly in “White” if it lies within one of the seven colored segments. Otherwise it lies in a shadow and appears in “black”. It turns out that always in two shadows none of the digits appears. These shadows therefore remain empty and consequently do not represent any black keys. Each selected digit passes through exactly seven adjacent rotation positions until it enters a shadow again. These positions can be associated with the tone syllables fa, do, so, re, la, mi, ti. This can be followed in figure 10.

Figure 10

While in each individual position the tone syllables appear in the circle counterclockwise in the order do, re, mi, fa, so, la, ti, the tone syllables associated with a fixed digit run through the order fa, do, so, re, la, mi, ti from position to position. Why this is so can be seen from a mathematical consideration.


Tones and note names on the cylinder

To designate the twelve chromatic semitones, one uses in traditional music theory instead of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 note names, such as C, C#, D, D#, E, F, F#, G, G#, A, A#, H. However, one cannot limit oneself to the selection of these 12 names, because in most of the 84 positions then incorrect designations of the selected tones arise. In Figure 11, those names that dip into one of the seven shadows are replaced by other names before they emerge from the shadow again.

Figure 11

The red level 1 always contains names that begin with the letter C, i.e. …, Cx, C#, C, Cb, … (pronounced: …, Cisis, C sharp, C, C flat, …). The orange level 2 always contains names beginning with the letter D, i.e. …, Dx, D#, D, Db, Dbb, … (pronounced …, Disis, D sharp, D, D flat, Deses, …), etc. This representation does not show yet the connection of the note names with different tone letters among themselves, since they disappear in each case into the shadows, and/or emerge from there. The structure of the note names is two-dimensional and can be represented in adequate form on the movable cylinder shell. Here is a detail:

Figure 12: The structure of note names

Figure 13 shows how the network of note names moves against the unmoving vertical shadows. The diagonal shadows, which also move along with the vertical unmoving shadows, have the effect that the seven note names (for the notes just selected) belonging together in each of the 84 positions are in the illuminated parallelograms that lie next to each other at the same height all around the cylinder shell.

Figure 13

Whole tone steps and semitone steps

While the seven step segments of the circular playing surface are all the same size, the audible step intervals are of different sizes. There are five larger steps (whole tones) and two smaller ones (half tones). The latter correspond to those two shadows that do not represent black keys.

Figure 14: Whole tone and half tone steps

You can see that the colored lines for whole and semitone intervals (blue and red), are embedded in a whole network of such intervals, which connects all notes with each other. If you rotate the cylinder one position further (see the two graphs placed below each other in Figure 15), exactly two intervals change from the current selection. In the transition from C-ionic (C-D-E-F-G-A-H-C) to C-lydian (C-D-E-F#-G-A-H-C), the semitone E-F is replaced by the whole tone E-F#, and conversely, the whole tone F-G is replaced by the semitone F#-G.

Although here apparently only at one place is “operated around”, the step pattern remains nevertheless altogether. It is now shifted by four steps. The five intervals between the tone syllables do-re, re-mi, fa-so, so-la and la-ti are always whole tones and the two intervals mi-fa and ti-do are always semitones. As you can see on the left in Figure 15, the pattern do-re-mi-fa-so-la-ti reappears shifted by four steps. If you have wondered about this oddity long enough, you can make mathematical considerations for a better understanding.

Figure 15

Tuning gyroscope: three players form a rotating triangle

You cannot select three points in a regular heptagon that form a regular triangle, because 3 is not a divisor of 7. But there is a maximum regular arrangement, which is the basis for a “choreography” for three players here.

Figure 16 below shows the minimum change in position of a triangle that turns once in a circle in a total of 21 positions. Its three corners each point to those step fields on which the three players are currently standing and where they should also press down a floor sensor of their choice.

Each time the triangle changes position, only one corner crosses the border between two step fields. Consequently, only one player at a time has to move a step along the spiral. The other two players remain standing with their foot on the sensor that has just been depressed.

The result is a sequence of triads (and their inversions), which in music theory are called diatonic triads. The “mechanics” of the interplay of the three players is based on the same mathematical principle as already the selection of the seven diatonic tones from twelve chromatic tones, which is the basis of the construction of the tonal gyroscope.

Figure 16: The three players

Mnemonic: The movement is counterclockwise, (ascending melody steps) and the one who has two free fields in front of him always takes a step to the next sensor.

It gets interesting when you combine both movements: the progression of the three voices in a sequence of triads, as described here, and the alteration of the underlying diatonic modes by rotating the cylinder.


Tuning gyroscope: four players form a rotating square

You cannot select four points in a regular heptagon that form a regular quadrilateral (square), because 4 is not a divisor of 7. However, there is a maximum regular arrangement that forms the basis for a “choreography” for four players here.

Figure 17 shows the minimum change of position of a square that turns once in a circle in a total of 28 positions. Its four corners point to the step fields on which the four players are currently standing and where they should also press down a floor sensor of their choice.

At each change of position of the square only one corner crosses the border between two step fields. Consequently, only one player at a time has to move a step along the spiral. The other three players remain standing with their foot on the sensor that has just been depressed.

The result is a sequence of four tones, which in music theory are called diatonic seventh chords. The “mechanics” of the interplay of the four players is thus based on the same mathematical principle as the selection of the seven diatonic tones from twelve chromatic tones, which is the basis for the construction of the tonal gyroscope.

Figure 17: Choreography for four players

Mnemonic: The movement is clockwise (descending melody steps), and it is always the player who steps to the next sensor who has another player on the field directly behind him.

It gets interesting when you combine both movements: the progression of the four voices in a sequence of seventh chords, as described here, and the alteration of the underlying diatonic modes by rotating the cylinder.


Some more math: Once again 7 out of 12

If two numbers are prime divisors, that is, if they have no common divisor other than the number 1, then there are multiples of one number that are direct neighbors of certain multiples of the other number. The following two sequences of numbers are the multiples of the divisor-alien numbers 7 and 12:

Figure 18: Neighbors among the multiples of the numbers 7 and 12

Here, 35 and 36 are adjacent (35=5\cdot 7 and 36=3\cdot 12). Likewise 48 and 49 are adjacent (48=4\cdot 12 and 49=7\cdot 7).

But what does this fact have to do with the tone gyroscope? We first write the upper (arithmetic) sequence of multiples of 7 each as the sum of a multiple of 12 and a remainder: 0=0\cdot 12+0, 7=0\cdot 12+7, 14=1\cdot 12+2, 21=1\cdot 12+9, 28=2\cdot 12+4, 35=2\cdot 12+11, 42=3\cdot 12+6, 49=4\cdot 12+1. The sequence of the first seven remainders (0, 7, 2, 9, 4, 11, 6) is on the one hand an arithmetic sequence wrapped around the circle (or a regular 12-corner) and on the other hand we already know it as a maximal regular selection of 7 points from a regular 12-corner. So we have two alternative ways of obtaining the scale. A curiosity of the numbers 7 and 12? Or a general connection? Is any arithmetic sequence wrapped around a regular polygon maximally regular? We can easily do an experiment with step size (period) 1. The sequence (0, 1, 2, 3, 4, 5, 6) is undoubtedly an arithmetic sequence, but as vertices on the dodecagon (0, 1, …, 11) they all lie next to each other and are anything but regularly distributed. So you have to think a little bit about the relation between the step size and the number of tones. Given the step size 7 and the number 7 of tones involved, 7 times 7 equals 4\cdot 12+1. In fact, it turns out the following: exactly the above fact that 48 and 49 are neighbors is responsible for the fact that the arithmetic sequence of 7 elements is maximally regular at step size 7. We owe a proof of this assertion here. But musicians can realize a third circumstance that clarifies the proof idea: every diatonic interval (except the prime) exists in two specific sizes: minor and major second, minor and major third, pure and augmented fourth, diminished and pure fifth, minor sixth and major sixth, minor and major seventh. These “species” in turn appear in various multiplicities. There are four minor and three major thirds (or three minor and four major sixths). There are two minor and five major seconds (or five minor and two major sevenths). There are six perfect fourths and one augmented fourth (or one diminished fifth and six perfect fifths). Every decomposition of the total number 7 occurs: 7=3+4, 7=2+5 and 7=1+6. From this we can already conclude that the scale must be an arithmetic sequence. Because the decomposition 7=1+6 says that there are 6 equal intervals and these must necessarily form a connected chain.

The structure of the scale as an arithmetic sequence is known as the circle of fifths. For a better understanding of the mechanics of alteration (turning the cylinder), it is useful to look at this aspect again in more detail. Once again, we take the sequence (0, 7, 2, 9, 4, 11, 6) and add the number 7 to each element and again take the remainder when divided by 12, obtaining (7, 2, 9, 4, 11, 6, 1). This sequence can be rearranged to (7, 9, 11, 1, 2, 4, 6,) or also to (1, 2, 4, 6, 7, 9, 11) and proves in both cases also to be maximally regular and therefore belongs to other positions of the tone gyroscope. Since we add the number 7, which is also the step size within the sequence, we get the same remainders also if we omit the “0” at the beginning in (0, 7, 2, 9, 4, 11, 6) and add the next element of the arithmetic sequence at the end: the “1”. Although we add 7 to each element, the result differs from the original sequence only in one single number. After reordering, this is particularly clear: the sequence (0, 2, 4, 6, 7, 9, 11) becomes the sequence (1, 2, 4, 6, 7, 9, 11). In note names: (C, D, E, F#, G, A, H) becomes (C#, D, E, F#, G, A, H). This corresponds to the mode change from C Lydian to C# Locrian. Conversely, in (0, 7, 2, 9, 4, 11, 6), one can omit the “6” and instead add the number 5 in front equal to -1\cdot 12 + 7. The sequence (5, 0, 7, 2, 9, 4, 11) is reordered to (0, 2, 4, 5, 7, 9, 11). In note names (C, D, E, F, G, A, H). This corresponds to the mode change from C Lydian to C Ionic.

Figure 19

The step patterns of Lydian and Ionic are shown again below in comparison. The shift of the fourth step corresponds either to an exchange of whole tone and semitone at steps 3-4 and 4-5 or to a cyclic permutation of the whole pattern (shift by 4 steps to the right).

Figure 20: Step pattern of Lydian and Ionic

In order to appreciate this connection on the description level of the step patterns, one can take for illustration any word from seven letters and understand it as a cycle (i.e. after the last letter comes again the first). On the one hand, the letters can be permuted cyclically by so many positions (7 possibilities), on the other hand, two adjacent letters can be swapped (seven possibilities). For a word like dresden, none of these permutations lead to the same result, although the letters “d” and “e” appear twice.

Figure 21: “Dresden” as a cycle

This is different for the word aabaaab, which represents the step pattern of the Ionic mode. Here the letter “a” stands for whole tone and “b” for semitone. Here there are seven different cyclic permutations (see Figure 22). Only four of the seven permutations of adjacent letters actually yield different words. Two of these, in turn, agree with cyclic permutations: aabaaba (mixolydian) and aaabaab (lydian).

Figure 22: aabaaab as cycle

Such considerations belong to the field of algebraic combinatorics on words. For a more extensive online reading, see [1].

Text: Thomas Noll


Literature

[1] Clampitt, David and Thomas Noll (2010): “Modes, the Height-Width Duality, and Handschin’s Tone Character”, Music Theory Online, Volume 17, Number 1, March 2011.