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
Method
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.
Source: §6 · p. 5Takeaways & Limitations
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.Source: §1 · p. 1
- The effect operates at the boundary between timbre and tonality, leaving the source audible while imparting chordal coloration.Source: §1 · p. 1
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.Source: §1 · p. 1
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.Source: §2 · p. 1
- 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.Source: §2 · p. 1
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.Source: §3 · p. 1
Log-frequency structure
- In log-frequency coordinates, octave transposition becomes translation, making the total response approximately periodic with period one octave.Source: §3 · p. 2
- Each octave contains K = |C| peaks fixed by the chord’s pitch classes, unlike a comb filter’s equally spaced linear-Hz peaks.Source: §3 · p. 2
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.Source: §3 · p. 2
- 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.Source: §3 · p. 2
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.Source: §3 · p. 2
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.Source: §4.1 · p. 2, §4.2 · pp. 3–4
- 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.Source: §4.1 · p. 2, §4.2 · p. 3
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.Source: §4.2 · p. 3
- 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.Source: §4.1 · p. 2, §4.2 · p. 3
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.Source: §4.3 · p. 4, §4.2 · p. 4
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.Source: §5.1 · p. 4, Figure 1 · p. 5
- 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.Source: §5.1 · p. 4
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.Source: §5.2 · p. 4
- 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.Source: §5.2 · p. 4
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.Source: §6 · p. 5
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.Source: §6 · p. 5
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.Source: §6 · p. 5