The Boltzmann distribution assigns each state a probability that falls off exponentially with its energy. Put musical dissonance on the energy axis, and the rule becomes a composer: the root and its consonances are low-energy and likely, the tritone is high-energy and rare, and a single Temperature sets how strict that preference is.
Where it comes from
A law about heat, repurposed as a law about taste.
In statistical mechanics, the Boltzmann distribution describes how a system in contact with a heat bath at temperature T populates its available states. States with lower energy are occupied more often. The probability of a state with energy \(E_i\) is
- \(E_i\)
- The energy of state i. In Boltzmann, the dissonance of a pitch against the root.
- \(T\)
- Temperature. It sets the width of the distribution, not a single value.
- \(Z\)
- The partition function. It sums the weight of every state so the probabilities add to one.
- \(e^{-E_i/T}\)
- The Boltzmann factor. Energy enters as a negative exponential, so high energy means low odds.
Nothing in the equation knows it is about gas molecules. It is a general statement: given energies and a temperature, here is the most probable way to spread your bets. Replace "energy" with any cost you want a system to avoid, and the same exponential gives you a tunable preference. Boltzmann replaces energy with dissonance.
The energy axis
Energy is dissonance against the root.
To turn the distribution into a musical instrument, every pitch needs an energy. Boltzmann measures each pitch class by its interval distance from the root. The root sits at the bottom of the well. The perfect fifth is the lowest non-root consonance. Thirds and sixths sit in the middle. The minor second and the tritone sit at the top, the least stable intervals against a tonic.
This is a deliberate, opinionated map. It is not the only way to score consonance, but it captures the ordering most listeners share: a phrase that spends time on the root and fifth sounds settled, and a phrase that keeps landing on the tritone sounds tense. Because the energies are an array you can reshape with the Scale and Root controls, the landscape is the part of the instrument you design before you ever touch Temperature.
The Scale you choose removes some pitches from the set entirely and re-weights the rest. The custom scale ring lets you draw your own energies by hand, raising one degree and lowering another, so the same Temperature produces a different center of gravity.
The control parameter
One knob runs the whole field.
Temperature is the single parameter that decides how sharply the distribution favors low energy. As \(T \to 0\), the Boltzmann factor collapses onto the lowest-energy state, so the generator returns to the root again and again. As \(T \to \infty\), every exponential approaches one, the distribution flattens, and every pitch becomes equally likely. Between those limits the field reorganizes continuously.
Two readouts make the state legible. Entropy measures how spread out the distribution is, in nats:
Entropy is near zero when one pitch dominates and reaches its maximum, \(\ln N\), when all N pitches are equally likely. Order is the complement, \(1 - S/S_{\max}\), scaled to a percentage: high when the phrase is predictable and tonal, low when it is exploratory. These are not decorations. They are the same quantities that describe order and disorder in a physical system, read directly off the pitch distribution you are hearing.
Interactive lab
Move the temperature. Watch the distribution melt.
Each bar is one pitch class, colored by energy: teal at the consonant low end, amber at the dissonant high end. Bar height is probability on a fixed 0-to-1 scale. Drag Temperature toward zero and the root dominates; warm it and the fifth is the first consonance to rise. Drag it higher and the bars level out; every pitch comes into play and Order falls toward zero. This is the exact distribution the sequencer samples from, redrawn as you turn the knob.
Drawing notes
Two ways to walk the same distribution.
Knowing the distribution is half the job. The other half is drawing notes from it, and the way you draw changes the melodic shape even when the statistics are identical.
Independent sampling draws each note straight from the distribution, with no memory of the last one. The long-run histogram is exactly P(E), but consecutive notes can leap anywhere, so the line is angular and unpredictable.
Metropolis sampling reaches the same distribution by a random walk. It proposes a small step from the current pitch, then accepts it with probability
A move to lower energy is always accepted; a move to higher energy is accepted only sometimes, more often when Temperature is high. With symmetric neighbor proposals, run this walk and the visited pitches still follow the Boltzmann distribution, but they arrive by neighbors instead of leaps. The result is a melodic contour: the line wanders, dwells near the root, and occasionally climbs into tension before settling. Same odds, different motion. Both samplers are honest; the choice is whether you want a scatter or a path.
From law to instrument
How the distribution becomes a sequencer.
Boltzmann wraps this engine in performance controls. The distribution decides pitch; a separate layer decides time. A Euclidean rhythm skeleton spreads hits as evenly as possible across the step count, and a barrier can gate a step the way an activation energy gates a chemical reaction, opening as the system heats.
The endless stream is only useful if you can keep a phrase you like, so LOCK re-seeds the generator at every loop top. That quenches the stochastic process into a repeatable orbit: a phrase that plays the same way each pass. MUTATE then redraws a tunable fraction of the steps from the live distribution, taking the loop from bit-exact to molten without leaving the energy landscape. SEED names the phrase, and the dice draws a new one.
Hold a chord and ARP rebuilds the whole landscape from the notes under your fingers: the lowest held note becomes the ground state, and the others take their energies from their intervals against it. The same law that runs the free generator now runs the arpeggiator, so the temperature gesture means the same thing in both.
Practical reading: design the landscape first with Scale and Root, set how strict it is with Temperature, choose Independent for scatter or Metropolis for line, then LOCK a phrase and use MUTATE to keep it alive.
References
Source material.
- Boltzmann, L. (1877). "Uber die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Warmetheorie und der Wahrscheinlichkeitsrechnung." Wiener Berichte 76, 373-435.
- Gibbs, J. W. (1902). Elementary Principles in Statistical Mechanics. Yale University Press.
- Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). "Equation of state calculations by fast computing machines." Journal of Chemical Physics 21(6), 1087-1092.
- Hastings, W. K. (1970). "Monte Carlo sampling methods using Markov chains and their applications." Biometrika 57(1), 97-109.
- Toussaint, G. (2005). "The Euclidean algorithm generates traditional musical rhythms." Proceedings of BRIDGES: Mathematical Connections in Art, Music, and Science, 47-56.