Timbre comes from the overtones stacked on the fundamental — and the overtones come from the shape of the space the instrument vibrates in. Slide the dimension to hear a string become a drum, a solid, and the fractals in between. Switch between a box and a ball to hear how much the shape matters once the dimension is fixed.
Grey lines are the pure harmonic series (integer multiples of the fundamental). Coloured stems are the actual modes of a 1.000-dimensional box. They line up completely only at d = 1; above that the stems drift off the grid and the tone stops sounding harmonic.
At d = 1 every partial shares a period, so the trace repeats exactly. Off the integers nothing divides evenly and the wave never quite closes — that restlessness is the inharmonic timbre.
An idealised instrument’s overtones are the eigenfrequencies of the Laplacian on its body. For a d-dimensional box with clamped edges the modes are indexed by integers n1…nd ≥ 1 with
f ∝ √(n12 + n22 + … + nd2)
At d = 1 that is just 1, 2, 3, 4… — the harmonic series, which is why strings and pipes sound pitched and sweet. At d = 2 it becomes the ragged √(m2+n2) ladder of a drumhead, and by d = 4 or 5 so many modes crowd into the first couple of octaves that the tone turns into a gong or a wash of noise.
A non-integer number of indices makes no sense, so the fractional case is built from mode density instead. Weyl’s law says the number of modes below a frequency grows as N(f) ∝ fd, so the n-th partial sits at fn ∝ n1/d. That exponent is precisely what spectral dimension means for a fractal: the Sierpiński gasket has ds = 2 log 3 / log 5 ≈ 1.365, so a gasket-shaped drum really does grow its modes more slowly than a membrane but faster than a string.
So for d between two integers k and k+1, each partial is placed by geometric interpolation between the two integer lattices, using the blend weight w = (k+1)(d−k)/d. That particular weight is what makes the asymptotic density exponent (1−w)/k + w/(k+1) equal 1/d — a plain linear blend in d would drift off the target (at d = 1.5 it would land near 1.33). At whole numbers the weight collapses to 0 or 1, so you hear the true box spectrum, degeneracies and all; in between, the partials slide smoothly and monotonically from one lattice to the next. Amplitudes fall off with frequency (the brightness control) and grow with each mode’s multiplicity.
A ball is a different eigenproblem, and a more elegant one. Separating variables gives modes f ∝ jν,s, the s-th zero of the Bessel function of order ν = l + d/2 − 1, where l is the angular order. The dimension now sits inside the order of a Bessel function, and the number of angular modes of degree l continues to fractional d through gamma functions. So nothing has to be interpolated: d enters analytically, exactly as it does in Stillinger’s formulation of fractional-dimensional space. Slide the dimension with Ball selected and you are hearing a directly computed spectrum, not a blend.
Does the shape matter? Less than you would guess for the overall character, and more than you would guess in the detail. Weyl’s law makes mode density depend on dimension and volume but not shape, so both resonators grow their modes at the same rate and travel the same arc from sweet to clangorous. Measured over the partials being sounded, the two agree on effective dimension to within a few percent. But the individual partials land in different places: a square drum’s third partial is exactly 2× the fundamental while a round drum’s is 2.136×, a difference of 114 cents — more than a semitone. That gap widens with dimension, to a typical 218 cents by 4D and 378 cents by 5D. At d = 1 the two coincide precisely, because a 1D box and a 1D ball are the same interval.
One extra piece of physics matters for the ball. Rotational symmetry gives its degeneracies no upper bound — a 4-ball has (l+1)2 modes at a single frequency — so left unchecked the high angular orders would swamp everything. But a mode of order ν and wavenumber k turns over at radius ν/k and decays exponentially inside that, so a strike at 0.7 of the radius barely couples to modes that hug the rim. Including that factor is what makes the ball’s spectrum converge to something independent of how many modes we bother to compute.
Dimension five is where the bookkeeping starts to bite. Degeneracies grow as fd−2, so past about d = 3.2 the brightest rolloff the slider asks for would leave the partials getting louder with frequency; it is floored to keep a downward tilt. And the fundamental — one lone mode among thousands by 5D, outranked in loudness by dozens of degenerate groups — is always sounded, or the note you picked would drop out of the mix and the pitch would jump as you crossed the slider past 4.
Honest caveats. The 1/d density is an asymptotic law, and the first few dozen modes sit well short of that limit: the negative boundary term in N(f) = Cdfd − c fd−1 + … means any dimension measured down here is approximate, and a ball’s surface term differs from a box’s, which is most of the residual disagreement between them. Real fractal drums also have log-periodic gaps and localised modes that neither construction reproduces. And the amplitude law is a perceptual choice, not physics: true modal amplitudes depend on how and where the body is struck. Treat the in-between settings as a faithful sketch of fractal spectra, not a solved eigenproblem.