Source-linked AI summary
Evoking Harmony via Convolution
Michael Gogins
TL;DR
The paper asks how chordal pitch content can be evoked from arbitrary source sounds without separating it from their timbre or introducing conspicuous artifacts. It constructs a chord-specific, octave-spanning convolution response with windowed sinusoidal grains and a dry Dirac component, then examines its filtering relationships and musical use. The resulting coloration can remain subordinate to source timbre or become a clearer harmonic shadow, with parameter choices shaping the outcome.
Problem
The paper addresses the divide between timbre-oriented electroacoustic sound and tonal practice while seeking to evoke chordal pitch content from broadband or mixed sources.
Method
The method convolves a source with a normalized impulse response containing loudness-balanced, chord-specific windowed sinusoidal grains across octaves plus a Dirac component.
Results
The effect works best as chordal coloration subordinate to source timbre, while wetter settings and pitched sources can produce a clearer harmonic shadow or more abstract textures.
Takeaways & Limitations
Harmony convolution provides a musical boundary technique in which source timbre remains audible while parameters control the strength and character of harmonic sensation.
Abstract
from arXiv · showhide
I show how to evoke the pitch-class content of a chord from an arbitrary source sound by convolving the source with an impulse response whose grains are one windowed sinusoid per pitch-class, across each octave of hearing range; while, at the same time, minimizing artifacts. A Csound user-defined opcode, chord_convolver, mixes a dry Dirac component into that response, and applies partitioned convolution once. I contrast the effect with a linear-frequency comb filter and with a generic constant-Q resonator bank, and I demonstrate musical use on a twilight field recording alongside the ruins of Chateau de Lagarde.
1 Introduction
The paper introduces harmony convolution, which evokes the pitch classes of a chosen chord from varied source sounds while preserving their timbral basis. It distinguishes this chord-specific, logarithmic approach from related filtering methods.
- Contribution: Harmony convolution evokes a chosen chord’s pitch classes by convolving varied source signals with octave-spanning sinusoidal grains.The grains occupy only the chord’s pitch classes, with one windowed sinusoid per pitch class across the hearing range.
- Contribution: The effect operates at the boundary between timbre and tonality, leaving the source audible while imparting chordal coloration.
- Relation to other methods: Its chord-specific logarithmic structure differs from a linear-frequency comb filter and from generic constant-Q or wavelet representations.Comb-filter teeth are periodic in linear Hz, while generic constant-Q tilings are not restricted to chord pitch classes.
2 Motivation
The paper addresses a historical split between electroacoustic practice centered on timbre and tonal practice centered on pitch relations. Its algorithm deliberately works between these domains by retaining the source while adding chordal sensation.
- Motivation: The paper responds to a practical disjunction between non-tonal electroacoustic work and tonal practice focused on abstract pitch relations.This split is situated in the roots of computer music during the 1950s and 1960s.
- Motivation: Its algorithm bridges timbre and tonality by leaving the source audible while imparting faint or strong chords or chord progressions.The strength of the harmonic impression depends on parameters and mix.
3 Mathematics
The method constructs a normalized, chord-specific impulse response from octave-related windowed grains and a Dirac component. Its frequency-dependent durations create a logarithmic constant-Q structure while controlling time–frequency and transient-smearing tradeoffs.
- Impulse-response construction: A chord-specific impulse response sums loudness-balanced causal grains and a Dirac component, then applies joint ℓ2 normalization.Each grain is a half-cosine-windowed sinusoid associated with a pitch class and octave.
- Log-frequency structure: In log-frequency coordinates, octave transposition becomes translation, making the total response approximately periodic with period one octave.
- Log-frequency structure: Each octave contains K = |C| peaks fixed by the chord’s pitch classes, unlike a comb filter’s equally spaced linear-Hz peaks.
- Time–frequency tradeoff: A shorter grain improves time resolution but worsens frequency resolution, whereas a longer grain produces the opposite tradeoff.The grain duration is therefore treated as a musical parameter.
- Time–frequency tradeoff: For T = 30 ms, the measured −3 dB bandwidth is 1.68 semitones above 440 Hz, while the capped low-frequency bandwidth reaches 14.6 semitones at 55 Hz.Above 440 Hz the design maintains constant Q; below the pivot, bandwidth remains constant in Hz and widens musically toward the bass.
- Artifact control: Causal, capped grains keep the harmonic response within the source duration, leaving under 3 % of each period’s energy outside the dry source interval in the stated impulse-train test.Uncapped or symmetrically windowed grains displace a third or more under the same test.
4 Method
The method constructs a chord-specific impulse response from pitch-class grains, balances their energy and loudness, and applies the response through one partitioned convolution. Its logarithmic pitch lattice distinguishes it from generic constant-Q banks and linear-frequency comb filters.
- Impulse-response construction: chord_convolver builds one impulse response from selected pitch-class grains and applies partitioned convolution once via ftconv.The response also includes a Dirac component and is jointly normalized.
- Impulse-response construction: Selected MIDI frequencies receive causal half-cosine grains, inverse A-weighting, and compensation-dependent amplitude scaling before synthesis.Frequencies are restricted to selected pitch-classes between 20 Hz and 0.95 of Nyquist.
- Levelling octaves: Equal-energy levelling uses A(f) ∝T(f)^-1/2 because grain energy, rather than amplitude or window integral, determines broadband partial contribution.A constant factor preserves the meaning of the impulse and Dirac gains.
- Levelling octaves: Inverse A-weighting then converts equal power per partial into approximately equal loudness per partial, while optional critical-band compensation attenuates low octaves by 4–6 dB.The recordings use equal loudness per partial without this additional low-octave attenuation.
- Comparison: A chord-centered constant-Q bank shares the logarithmic lattice but uses ongoing filtering, whereas a comb filter places equally spaced peaks in linear Hz.The comparison therefore distinguishes both spectral spacing and finite-impulse-response dynamics.
5 Results
The experiments compare dry, comb-filtered, constant-Q, and CM7 chord-convolved sources, then demonstrate chord convolution on a twilight field recording. The effects range from subtle coloration to clearly audible harmonic treatments.
- Comparison spectrograms: Figure 1 compares dry, comb, sparse constant-Q, and CM7 chord-convolver treatments of a click and white noise.The segments are loudness-matched within each source group to within 1.1 dB A-weighted.
- Comparison spectrograms: The comparison shows linear-Hz spacing for the comb, unrestricted log spacing for constant-Q, and a sparser chordal lattice for the convolver.Equal-energy normalization keeps the convolver’s chordal pattern articulated toward the top of the audible range.
- Sound walk: The sound walk processes footsteps, voices, cars, wind, bells, and jackdaws recorded near the ruins of Chateau de Lagarde under score-controlled chord-convolver parameters.Controls include onset, duration, fades, grain duration, wet and dry gains, compensation, and pitch-classes.
- Sound walk: Musical examples include chord progression, subtly treated bells, strongly treated bells with following voices, and “Bonjour” with following voices.The effects can be subtle or obvious depending on the treatment.
6 Discussion
Evoking harmony occupies a boundary between timbre and tonality through a logarithmic, chord-specific treatment that preserves the source while adding harmonic coloration. Levelling choices are both technical and compositional, with equal-energy normalization used for the recordings.
- Scope and identity: Evoking harmony differs from comb filtering and generic constant-Q analysis because its lattice is logarithmic and chord-specific.The preferred realization combines causal short-to-medium grains with a dry Dirac in one impulse response.
- Musical behavior: The treatment works best when harmonic sensation remains subordinate to the source timbre, while wetter settings and pitched sources can produce a clearer harmonic shadow or more abstract textures.The passage identifies source balance and wetness as audible determinants of the result.
- Levelling as composition: Equalizing amplitude, energy, or window integral creates roughly 17 dB of spectral tilt across the lattice, making levelling a compositional as well as technical choice.The recordings use equal energy so each pitch-class is equally present.