Real-Time Fractional Fourier Sound Synthesis and Processing
Esteban Gutiérrez1, Behzad Haki1, Carlos Sing Long4, Frederic Font1, Xavier Serra1, and Rodrigo Cádiz2,3,
1 Department of Information and Communications Technologies, Universitat Pompeu Fabra
2 Music Institute, Pontificia Universidad Católica de Chile
3 Department of Electrical Engineering, Pontificia Universidad Católica de Chile
4 Instituto de Ingeniería Matemática y Computacional, Pontificia Universidad Católica de Chile
This webpage provides supplementary materials for our paper "Real-Time Fractional Fourier Sound Synthesis and Processing", submitted to the Journal of the Audio Engineering Society (JAES).
5. Sound Examples
This section presents a set of sound examples illustrating the methods introduced in Section 5. To highlight both the real-time capabilities of our implementation and the geometric interpretability of the FRFT, all examples except the final one were recorded as videos using our real-time Max implementation and are accompanied by spectrograms.
5.1. α–Synthesis and Processing Examples
The following examples illustrate the α-synthesis and processing techniques.
5.1.1. Vanilla Synthesis Examples
A sinusoid at 440~Hz was synthesized using all combinations of the following parameters: window sizes of 2048 and 65536 samples; hop sizes equal to the full window, half, and one quarter; and α in {0,0.01,0.1,0.4}. This configuration provides a broad parameter space to examine the effect of the RFRFT on pure tones. The experiment was repeated using a square wave at the same frequency.
Large windows clearly reveal the chirp-like transformations induced by the double rotation property. With smaller windows, the chirps become sufficiently steep to be perceived as novel sound textures, occasionally producing emergent pitch sensations. Varying the hop size enables further control over the resulting textures.
If you want to run quick synthesis experiments, use the following interactive widget.
5.1.2. Synthesis + Parameter manipulation example
Here, the order of the FRFT is dynamically modulated while overlap-and-add parameters remain fixed. Sinusoids and square waves are again used as inputs, with α controlled in real time via an LFO. Even with simple source signals, this approach yields a wide range of continuously evolving textures, demonstrating sophisticated control possibilities.
5.1.3. Minimal Processing Examples
Two audio excerpts, a percussed clarinet and a staccato string ensemble texture, were α-processed using the same parameter combinations as in Example 5.3.1.
Partials in the input signals are transformed into frequency-modulated tones, while noisier components remain noise-like but acquire tonal coloration due to excited chirps in the α-domain. Window and hop size again play a significant role in shaping the resulting textures.
If you want to run quick processing experiments, use the following interactive widget (click on the folder icon to upload your own sounds).
5.1.4. Processing + Parameter Manipulation
A collection of percussive, synthetic pitched, and acoustic pitched sounds was processed using the RFRFT with order modulated in real time by an LFO. Percussive sounds retain sharp transients for small windows while resonant components are transformed into audible chirps. Complex pitched sounds become collections of non-harmonically related chirps whose structure evolves over time, producing metallic and time-varying timbres.
5.2. α–Filters and Convolution Examples
These examples illustrate the α-filtering and convolution methods
5.2.1. Vanilla Filters
White noise was α-filtered using various band-pass filters to explore perceptual characteristics of different regions in the α-domain. Parameters included window sizes of 32768 and 2048 samples; α in {0,0.1,0.5}; hop sizes equal to a fourth, a half, and a full window; and manually controlled filter kernels. The experiment was repeated using a 440~Hz sinusoid.
The resulting bands correspond not to fixed frequencies but to chirps whose modulation depends on α and window size. For the sinusoid, prominent energy bands appear across the α-domain for all orders, with markedly different sonic characteristics as α varies.
5.2.2. Filters + Parameter Manipulation
A string ensemble and a sung voice were processed using α-filters with time-varying α. Small windows produce heavily distorted yet input-coherent signals, reminiscent of bit crushing or fuzz. The spectral color of this distortion is controllable via α-domain band selection and can introduce metallic qualities. With larger windows, α low-pass filtering yields rhythmic amplitude-modulation effects while preserving intelligibility.
5.2.3. Vanilla Convolutions
Several sounds including a sinusoid, a drum pattern, and a sung voice, were α-convolved with square waves using fixed parameters. Window and hop sizes matched those in Example 5.3.1, with α in {0,0.01,0.1}.
Large windows smooth transients, reducing rhythmic clarity in the drum pattern. Smaller windows preserve transients and introduce a bit-crushed quality. The sinusoid yields synthetic pitched sounds, while the sung voice produces textures similar to those obtained through α-filtering.
5.2.4. Convolution + Parameter Manipulation
Pairs of sounds, including a sinusoid with a square wave and a sung voice with itself, were α-convolved while modulating α in real time. The sinusoid–square wave pair produces frequency-modulated pitched noise, while the self-convolved voice generates structured chirp systems that retain many perceptual qualities of the original signal for large windows and small α.
5.3. α–Ring Modulation Examples
These examples demonstrate α-ring modulation.
5.3.1. Vanilla Ring Modulation
A triangular wave and an acoustic guitar excerpt were used as carriers, with a triangular wave as modulator. Carrier parameters matched those of Examples 5.3.1. The modulator order α_m was defined relative to the carrier order α_c as α_m in {α_c,2α_c,1-α_c,-α_c}.
Using α_m=α_c or 2α_c generally produces similar outcomes. For small α, results resemble standard ring modulation. Larger α values yield stronger frequency modulation for α_m=1-α_c, while α_m=-α_c produces interference patterns perceived as rhythmic structures, consistent with the multiplication of oppositely directed chirps.
5.3.2. Ring Modulation + Parameter Manipulation
Using the same sound material as in the previous example, α was modulated in real time via an LFO. Time-varying α enables continuous navigation between the behaviors observed in the static case, making the perceptual impact of α modulation particularly salient.
5.4. Other Examples
5.4.1. Feedback Structures
A sinusoid is processed within a feedback loop in which the output of the α-processing stage is reinjected into its input. This configuration produces chaotic, evolving, and non-repeating sound textures.
5.4.2. Rhythmic Patterns
Sub-audible frequencies are used in α-synthesis with α close to 1. These components are transformed into broadband chirps. With small windows, this results in complex rhythmic patterns composed of short, irregular resonances.
Acknowledgements
This research was supported by ANID Fondecyt Regular Grant #1230926, ANID Anillo ATE220041, Government of Chile, and the project "IA y Música: Cátedra en Inteligencia Artificial y Música (TSI-100929-2023-1)" funded by the "Secretaría de Estado de Digitalización e Inteligencia Artificial and the Unión Europea-Next Generation EU". We would also like to thank Diego Vera for his contributions to an early version of this project during his undergraduate research.