Kosmófono

A generative instrument for the sonification of binary black hole systems.

00 Gravitational Waves 01 Kosmófono 02 Controls 03 Two Clocks

Versión en español

00 Gravitational Waves

Spacetime waves that travel through the Universe.

According to the theory of General Relativity, gravity is a manifestation of the curvature of spacetime. Certain cosmic events, such as the collision of black holes, can create curvature waves that travel through the Universe: these are gravitational waves. Their existence was predicted by Albert Einstein in 1916, and they were first measured in 2015.

MASSES CURVE SPACETIME TWO MASSES IN MOTION GENERATE GRAVITATIONAL WAVES

Consider a binary black hole. As they orbit each other, the binary emits gravitational waves whose frequency is proportional to the frequency at which they orbit: the faster they spin, the higher the frequency of the emitted wave. During its evolution, the orbit shrinks and the pair spins faster and faster, so that the gravitational wave rises in frequency and amplitude.

As it emits gravitational waves, the orbit shrinks. The frequency and amplitude of the wave increase until the merger. merger

As it emits gravitational waves, the orbit shrinks. The frequency and amplitude of the wave increase until the merger.

An experimental feat

At 09:50:45 on September 14, 2015, the LIGO interferometer measured that the length of its 4 km mirrors oscillated with an amplitude of 10e-19 meters. This was the first detection of a gravitational wave. The signal was consistent with the collision of two black holes of 36 and 29 times the mass of the Sun.

The orbit

The frequency of the emitted wave is twice the orbital frequency:

f_GW = 2 f_orb = (1/π) √(GM/a³)

where G is Newton's constant, M is the mass of the pair, and a is the distance between the black holes. For example, a binary emitting at 440 Hz orbits at 220 Hz, that is, it makes 220 turns per second.

Maximum frequency

Binaries with smaller masses can get closer to each other. Therefore, they can have higher orbital frequencies and emit waves at a higher frequency. We can approximate the maximum emitted frequency by estimating the innermost stable circular orbit before the merger:

f_fus = 4396 Hz / (M/M⊙)

where M⊙ is the mass of the Sun.

Frequency curve

Considering how the separation of the binaries decreases over time, the emitted frequency as a function of time is:

f(t) = f₀ (1 − t/t_c)^(−3/8)

where f₀ = (1/π) √(GM/a₀³) is the initial frequency and t_c = 5c⁵a₀⁴ / (64G³M³) is the time remaining until the merger, with a₀ the initial separation and c the speed of light.

01 Kosmófono

Binary black holes: the basis of an instrument for generative music and visuals.

Kosmófono is a digital audiovisual instrument where users can create, in real time, pairs of black holes, see their orbits, and hear the gravitational waves that emanate from them. Users can vary at will the physical parameters that define the binaries, becoming the creator of a unique audiovisual Universe.

1CONTROLS M mass per hole a initial separation e eccentricity z redshift G Newton's constant h Planck's constant EQUATIONS initial f f₀ = (1/π) √(GM/a³) final f f_fus = c³ / (6^(3/2) π G M) the curve f(t) = f₀ (1 − 64G³M³t / (5c⁵a⁴))^(−3/8) merger time t_fus = 5c⁵a⁴ / (64 G³M³) evaporation t_evap = 5120π G²M³ / (ħ c⁴) 2PREVIEW the pair about to be created frequency as a function of time initial f 48 Hz final f 2142 Hz merger t 25.0 s evaporation t 1.8·10⁶⁸ years

When a binary is generated, an acoustic signal is activated whose frequency follows the time evolution of the gravitational wave frequencies. Kosmófono is an instrument for generative music and visuals, where the piece is generated algorithmically from the binary's initial parameters and the rules of evolution of gravitational physics.

3QUANTIZATION 55 110 220 440 880 1760 Hz frequency as a function of time continuous quantized attack each instant is rounded to the nearest degree of the chosen scale NOTES note pitch onset length Fa#1 46.2 Hz 0 s 1 s G#1 51.9 Hz 1 s 4.72 s La1 55 Hz 5.72 s 3.98 s Si1 61.7 Hz 9.69 s 4.06 s Do#2 69.3 Hz 13.75 s 2.32 s Re2 73.4 Hz 16.07 s 1.84 s Mi2 82.4 Hz 17.91 s 1.88 s Fa#2 92.5 Hz 19.79 s 1.38 s G#2 103.8 Hz 21.17 s 1 s La2 110 Hz 22.17 s 1 s Si2 123.5 Hz 23.17 s 1 s Mi3 164.8 Hz 24.17 s 823 ms Do#4 (random) 277.2 Hz 24.99 s merger The note changes if at least one second passes between notes. The merger note is random, representing the theoretical uncertainties of that moment. 4ORBIT Fa#1 0 s Do#2 13.75 s Mi3 24.17 s the orbit shrinks over time and the emitted note rises

Black holes emit waves whose frequency varies continuously; in Kosmófono, those frequencies are quantized within a musical scale of your choice. While a circular orbit emits a single frequency, elliptical orbits emit a series of harmonics of the orbital frequency.

ORBIT EMITTED HARMONICS (POWER) e = 0.9 e = 0.6 e = 0.3 e = 0 1 2 5 10 20 50 100 nth harmonic of the orbital frequency
THE CHORD OVER TIME 27 55 110 220 440 880 1760 Hz 0 s 0.5 s 1 s 1.5 s 2 s There are five voices at birth and only two at the merger due to the circularization of the orbit

Merger

When the black holes collide and merge, the equations used by Kosmófono are not a good approximation to describe the waves that emanate from there, nor are there formulas that describe the process. To know exactly what happens at that moment, it is necessary to run expensive non-perturbative gravity simulations. To account for this uncertainty in the merger mechanism, the note that sounds at this moment is chosen randomly.

Chords

In Kosmófono, the harmonics are not sonified as timbre, but as chords. The chord is made up of the 3 most intense harmonics.

As the black holes approach each other, the orbits circularize, and the harmonic series cleans itself up until it is left as a single line. Therefore, for elliptical orbits both the fundamental frequency and the emitted chord change over time.

Scales

You can choose between different scales to quantize the frequencies emitted by the black holes. Currently you can choose between the major, minor, major pentatonic, minor pentatonic or hirajoshi scales, for any of the 12 notes of the chromatic scale. Likewise, the note change is only triggered if there is at least one second between one note and the next, so there can be jumps of several notes.

02 The Controls

How to control the binaries.

The range of the controls is designed to contain binaries that emit in the audible range (from 20 to 20,000 Hz) and that, at least, emit for the order of minutes across that whole frequency space. Certain combinations result in inaudible waves (infrasound or ultrasound), as well as binaries that emit for hundreds or thousands of days.

101 102 103 104 105 10−4 10−3 10−2 10−1 100 M⊙ f_GW [Hz] 1 s 1 min 1 h 1 day 1 yr a₀ = 1 km a₀ = 10 km a₀ = 100 km a₀ = 700 km cutoff: merger INSTRUMENT WINDOW lines of equal a₀: same initial separation isochrones: time remaining until merger instrument range: 20 Hz to 20 kHz, 10⁻³ to 5 M⊙ total mass Merger times and frequencies swept for binaries of different mass. The range of the controls is the region where the frequencies fall within the audible range and last, at least, on the order of minutes.
lines of equal a₀: same initial separation
isochrones: time remaining until merger
instrument range: 20 Hz to 20 kHz, between 10⁻³ and 5 M⊙ of total mass

Merger times and frequencies swept for binaries of different mass. The range of the controls is the region where the frequencies fall within the audible range and last, at least, on the order of minutes.

In this plot, each pair created moves horizontally, from left to right. The initial separation (a) defines the initial frequency, and therefore the left end of the path in frequency space. As the orbits shrink, the binary moves to the right, until it reaches its maximum emission frequency f_fus at the moment of merger.

Let us look in a bit more detail at what each parameter of the binaries does.

M 5·10⁻⁴ – 2.5 M⊙ logarithmic

Mass

The mass of each black hole raises the starting note (the greater the mass, the greater the speed of the pair), and lowers the final note (the greater the mass, the larger the radius of the last orbit). Both effects squeeze the path in frequency space, so more mass means fewer octaves and an earlier merger.

a 2.2 – 552 km logarithmic

Initial separation

The distance at which it is born determines the initial emission frequency, since f is proportional to a^(−3/2). Moving the binaries apart lengthens the path, resulting in a longer time to collision. The initial separation cannot be smaller than the radius of each black hole (in that case the merger time is 0).

e 0 – 0.99 linear

Eccentricity

How far the orbit departs from a circle. While the circular orbit emits a single frequency, the elliptical one emits a series of harmonics of the orbital frequency. In Kosmófono, the harmonic series is represented as a chord, since the timbre is determined by the instrument that sounds. The emission of gravitational waves circularizes the orbit, and the harmonic series cleans itself up.

z 0 – 1100 octaves

Redshift

Distance between the pair of black holes and the observer. Since the Universe is expanding, the wavelength and time are stretched between the moment of emission and the moment of observation, resulting in a cosmological Doppler effect. Because of this effect, the observed (heard) frequency decreases by a factor (1+z), as does the amplitude of the wave.

G G/30 – 3.3 G log, signed

Newton's constant

In our Universe, this parameter is a constant. Here we allow ourselves to vary it. The parameter scales the gravitational force between the pair: it raises the frequency as √G and shortens the duration as G⁻³. We also allow negative values of it. In this case, the pair repels instead of attracting: the orbit grows and the frequencies decrease over time.

h 0, y 10⁻⁶ – 10⁶ log, plus zero

Planck's constant

In our Universe, this parameter is also a constant. Varying it changes the Hawking evaporation time, a quantum effect stating that black holes evaporate over time. If the merger time is greater than the evaporation time, the binary evaporates before colliding. The classical limit, with no evaporation, corresponds to h=0.

03 Two Clocks

Two clocks: visuals and rhythm.

Binaries that emit in the audible range move at such a high speed that the human eye cannot follow their motion. To give a sense of motion and of correspondence between orbital frequency and emitted frequency, Kosmófono has a slow clock mode, which proportionally reduces the speed to a few turns per second.

REAL CLOCK
Circular orbit, proportional clock
Circular
Eccentric orbit, proportional clock
Eccentric
SLOW CLOCK
Circular orbit, slow clock
Circular
Eccentric orbit, slow clock
Eccentric

The slow clock also opens up the spectrum of possible frequency curves. As Johannes Kepler showed, in eccentric orbits bodies move faster when they are close to each other, and slower when they are far apart. This means we can define an instantaneous orbital frequency, which is proportional to the instantaneous speed of the pair. If we attribute the emission frequency to the instantaneous frequency, then the emitted frequency curves are no longer strictly increasing (or decreasing in the case of antigravity). With the slow clock, the frequency curves will show an oscillating character.

1 rising 0 1 circular orbit 2 chord 0 1 eccentric orbit 3 falling 0 1 circular orbit G<0 4 chord falling 0 1 eccentric orbit G<0 5 rising oscillation 0 1 eccentric orbit slow clock 6 falling oscillation 0 1 eccentric orbit slow clock, G<0

Different clocks give different visuals; in addition, the orbit itself can be drawn in several ways, independent of the clock. Real orbits stack each complete turn as a closed curve. Linear orbits connect points frame by frame without interpolating. The sweep draws lines between the position of the black holes and their initial positions. Color lines are like linear orbits but assigning a color to the emitted frequency (red/low, violet/high).

Real orbits
Real orbits, e = 0e = 0
Real orbits, e greater than 0e > 0
Linear orbits
Linear orbits, slow clock, newly bornSlow
Linear orbits, fast clock, large separationFast
Sweep
Color lines, fast clock, near the mergerFast
Color lines, slow clock, near the mergerSlow

Kosmófono

Reference: Kosmófono: a generative instrument for the sonification of binary black hole systems (in preparation).

Vicente Atal.

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