For some reason, I woke up this morning wondering whether music could provide an analogy to help people understand higher-dimensional math.

I'm not very advanced in my mathematical studies. I enjoy watching the "Numberphile" channel Youtube, but I often find the concepts don't stick. One of the concepts that doesn't even seem to make contact is the idea of imagining the topology of objects in space that has more than three dimensions.

But now - bear with my ignorance - I'm thinking I might have the corner of something between my very confused fingertips. And it's basically an analogy from music to spatial dimensions.

Click to enlarge the illustration, if you like. I'm putting all the figures into one image, to save myself time and effort, but I'm going to talk about each image separately.

Consider Fig. 1. Suppose that this note, the D just one step above Middle C, represents a dimensionless dot on a one-dimensional line extending forever in both directions.

To keep it company, add the notes F-sharp and A (Figs. 1a and 1b), but only as separate dots on the same conceptual line - the line, say, of musical pitches extending upward and downward from the origin (say, Middle C), clear out of the range of human hearing. You could say the integer units on that line represent the tones of the equally-tempered 12-tone scale, repeating through a succession of octaves, like a Base-12 grid that has a heavier line at every 12th unit.

Moving on to a two-dimensional plane, suppose you plotted points D, F-sharp, and A on a graph with any arbitrary origin point (Middle C would do). In this instance, the points needn't be integer units or lie on a single line. When you transpose those notes up a half-step, as in Figures 2, 2a, and 2b, the new notes E-flat, G, and B-flat are interrelated in a way similar to the relationship between D, F-sharp, and A. The process of transposing the three notes, as a group, would be analogous to using three-dimensional math (with complex numbers, involving the "imaginary unit") to rotate a group of three points on a plane while preserving the angles between them.

But there is more you can do on a two-dimensional plane than plot a bunch of dimensionless dots and rotate them. I'm not yet up to delving into all the possibilities, but a blatantly obvious one is to draw line segments of various lengths, connecting pairs of dots on the plane. Likewise, when two notes sound together, consecutively or simultaneously, they form an interval. For example, D and F-sharp (Fig. 3) form the interval of a major third; D and A (Fig. 3a) form a perfect fifth; and F-sharp and A (Fig. 3b) form a minor third.

As with the single points, complex-number math can rotate line segments, individually or as a group, on the plane while preserving their lengths and the angles between them. Likewise, a set of musical intervals can be transposed, or as it were rotated, into similar intervals in a different key, like transposing the D/F-sharp/A intervals in Figs. 3, 3a, and 3b to the E-flat/G/B-flat intervals in Figs. 4, 4a, and 4b.

Another operation those Numberphile videos taught me you can do with a two-dimensional figure on a plane is to draw a plane inversion of the figure - and inversion, like rotation, can be done in higher-dimensional spaces too. I'm really straying into dangerous waters here, way over my head, but as I understand it, one can discover the reciprocal of a number (like turning a fraction upside down) by plotting a point and/or a circle around it in relation to the origin and radius of an arbitrary reference circle, then graphing an inversion of that point and/or circle's position and/or size, via a bunch of conditions that don't seem too far-fetched when someone who knows what he's talking about explains them.

After that no doubt compelling description, I am sure you'll agree this procedure bears some analogy to the process of transforming the musical intervals in Figs. 3, 3a, and 3b to their inversions in Fig. 5, 5a, and 5b. In the latter series of intervals, we find the D on top of the F-sharp instead of under it, changing a major third into a minor sixth; the D above rather than below the A, changing a perfect fifth into a perfect fourth; and the F-sharp above rather than below the A, changing the minor third into a major sixth. It is as if, in my imaginary universe governed by very shaky analogies, the reciprocal forms of the previous intervals were plotted in reference to an invisible, or rather inaudible, circle of musical inversion. Gads, that's terrible. Or maybe, if you catch what I'm throwing, it's brilliant.

Nah, it's probably just terrible.

But wait, I'm not done yet! There's still Fig. 6 to consider. When I first started cooking up this series of analogies, I was thinking of relating three-dimensional figures to the musical triad - a three-tone chord that, by a little note-shuffling, sort of like reducing fractions to their simplest form, can be distilled down to an interval of a third (like D to F-sharp) on top of another third (F-sharp to A), or a third (F-sharp) and a fifth (A) above the root tone (D). But now, it occurs to me the triad could also, and perhaps more aptly, illustrate the idea of a closed shape in the two-dimensional plane. Then again, when I moved from one dimension to two, I re-positioned the three notes as points on a plane, not on a straight line. So, perhaps I can be excused for using the same musical example two different ways.

So, yeah, in the 2-D plane, you can do the same operations with a closed figure as you could do with one or more line segments. I didn't bother to illustrate the rotation principle by transposing the D-major triad (Fig. 6) into an E-flat-major triad, but based on what I did with Figs. 3 and 4, I trust your imagination can get you there. You could do the same thing with Figs. 6a and 6b, the first and second inversions of the D-major triad - and inversion really is the musical term that applies here; I'm not just letting the planar-inversion analogy get away from me here. Each of these triad inversions could also be "rotated" into E-flat-major, or whatever key you want, sort of like using quaternion numbers (like complex numbers, only more so) to rotate a 3-D figure in space. 6a and 6b are still D major, even though a different tone in the original triad has been stuck at the bottom of the pile; but they have been transformed enough to give them a distinct sound.

Nevertheless, I think the triad could also be useful in an analogy to 3-D space, with a three-dimensional figure (sphere or otherwise) being transformed in some way, perhaps quite a dramatic way, when subjected to spherical inversion. But where I really wanted to go with this comes in Fig. 7, where I attempt to extend the analogy into a higher-dimensional space (in this case, 4-D space). If you accept a triad as comparable to a 3-D figure, where do you go from there? To start with the simplest possible answer, you could go to Fig. 7, which combines sequentially or, Fig. 7a, simultaneously, three similar triads - in this case, three major triads, D major, F-sharp major, and A major. It's as if you took the D-major triad from Fig. 6, rotated it to reveal a similar major triad rooted on each tone of the D-major chord, then brought all three major triads together into one figure - analogous to a tesseract or a 4-D supersphere.

Attempts to visualize 4-D figures using geometric imagery can never be quite precise, somewhat like the limitations of a 2-D picture representing a 3-D figure. But with music, it is possible to hear three different triads, each rooted on a different tone of the first triad, all at once. If you play them all together, but in such a way that you can hear each triad as a distinct identity within the superchord - say, by playing each triad in a separate register - you can actually hear it as a chord of chords. You can, to drive the analogy home, make the 4-D figure pop out of 3-D space - just as certain optical illusions can make a 3-D figure seem to pop out of a 2-D illustration.

Fig. 7b suggests one further level of sophistication. By musical analogy, we have already generated an optical illusion of a tesseract or supersphere, as it were, a very basic 3-D figure that seems to pop out into 4-D. But why stop at a supersphere? Why not go for another order of strangeness, and have each rotated sphere that pops out of the surface of the original sphere, be not only a rotation but also an inversion? Why not, indeed, include two different inversions - ah, but here the analogy is stretched past the breaking point, since I don't think math allows for more than one reciprocal of a given number. Music does, which is why I guess music beats math. It actually makes it possible, just conceivable, to "visualize" (in your mind's ear) a superfigure that has 3-D figures rotated three different ways, each a distinct inversion of the other two, and that pop out of each other.

In higher dimensions still, I guess you could be building musical tesseracts on each of the tones of all three chords in Figure 7b, rotated into the additional triads rooted on A-sharp (B-flat) and E, not to mention the other notes in those triads. And since your chord has more than three tones in it, you can also invert it in more distinct ways - sort of like how a four-note chord (like a dominant seventh chord, D/F-sharp/A/C) has three inversions, including the one with the 7th at the bottom; and a 9th chord has four inversions; etc. You could get super-crazy in your exploration of higher dimensions in harmony, though to our 3-D ears the resulting tone clusters may soon stop sound as if anything new had been added.

But in case I haven't sufficiently beaten up on math at the expense of music, I should point out that I've only touched on one aspect of musical complexity in the above analogies. Nowhere have I mentioned rhythm, melodic shape, tonal design, formal/dramatic structure, contrapuntal texture, number and type of movements, tone color, dynamics, lyrics, etc, etc. Counterpoint itself forks into such layers of complexity as number of voices, free imitation vs. canon vs. fugue, inversion and retrograde, augmentation and diminution, etc. One of those Numberphile videos shows evidence that origami (the traditional Japanese art of paper folding) is capable of doing harder math than Euclidean geometry (that stuff with a compass and a straight-edge). Perhaps it shouldn't be amazing that a soft subject like music rivals higher-dimensional topology for sheer complexity. Maybe this makes it good news that the Voyager Golden Record, now traveling through interstellar space, includes music by Beethoven, Bach, Mozart, and Stravinsky, among others in addition to greetings in 55 languages and a bunch of mathematical and scientific diagrams. If mankind's bright future ever depends on one thing on the Voyager spacecraft making an impression on E.T., my money is on the Brandenburg Concerto.

At least, saying so helps me feel good about being way better at music than at math. Fish out.

## Tuesday, December 5, 2017

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