Harmonic Eigenspace: A Web-based Application for Navigating and Composing Microtonal Harmony

1. Introduction

• The application pairs a navigable, sonified dissonance volume with a Modal Studio for forming progressions and generating modal-interchange substitutions in 31- and 53-TET.The Modal Studio’s scale editor supports interval gradations and chord types absent from 12-TET, including many of the 152 catalogued tetrads.Source: §1 · pp. 2–3
• The Eigenspace locates tetrad types by their roughness-based dissonance profiles, with three root-relative interval ratios defining chord quality as a transposition-invariant direction.Local minima in the dissonance field serve as landmarks for measuring distances to chord types.Source: §1 · p. 2
• The Eigenspace places 12-, 31-, and 53-TET chord types as discrete objects in a psychoacoustically grounded coordinate system without borrowing principles from another tuning system.The paper’s technical contribution is the web application, while its theoretical contribution is the transposition-invariant Eigenspace; the listening study examines responses to chords and progressions.Source: §1 · pp. 2–3
• The paper cautions that perceived chord dissonance varies with cultural context and timbre, and that some traditions do not organize pitch through chord qualities.It therefore presents the Modal Studio as a way to explore 31- and 53-TET progressions in a more idiomatic context.Source: §1 · p. 2
• The paper’s research questions ask whether chord quality can be represented in a psychoacoustic coordinate system and whether 53-TET modal-interchange progressions sustain perceived coherence and musicality.The second question also asks how ratings vary with the tonic chord’s proximity to familiar 12-TET qualities.Source: §1 · p. 3

2.1. Geometric and Spatial Models of Harmony in 12-TET

• Existing geometric models represent harmony through pitch-class relations, chord proximity, or voice-leading; the paper instead constructs a psychoacoustic coordinate space for tetrad types across 12-, 31-, and 53-TET.The review notes that earlier spatial models often derive their geometry from music-theoretic axioms rather than perceptual measurement.Source: §2.1 · p. 4

2.2. Interactive Systems for Harmonic Exploration

• Interactive systems offer navigable harmonic layouts or data-driven chord continuations, while the reviewed sequential generation systems remain bound to 12-TET.The review describes Harmony Space as a two-dimensional grid for exploring tonal structure and ChordRipple as a corpus-based continuation recommender.Source: §2.2 · p. 4

2.3. Cultural Practices of Microtonal Music

• Microtonal practice spans varied musical traditions, instruments, and tuning systems, including Partch’s 43-tone scale and 53-TET’s role in modern Turkish makam theory.The review also notes the cost barrier to wider adoption of the Lumatone, which supports 31-TET and 53-TET mappings.Source: §2.3 · p. 5

2.4. Microtonality in Computation and Practice

• Earlier computational work includes a discontinued multi-temperament composition system and a dataset of microtonal MIDI and audio; the paper addresses the gap in real-time interactive tools grounded in psychoacoustic dissonance.The review notes that 53-TET closely approximates just-intonation intervals, including the perfect fifth and major third.Source: §2.4 · p. 6

3.1. Sensory dissonance for pairs of sinusoids

• The model rescales frequency separation by a frequency-dependent critical bandwidth and weights pairwise roughness by the quieter sinusoid’s loudness.The scaling accounts for critical bandwidth growing with frequency; the parameters are fitted to experimental data.Source: §3.1 · p. 7

3.2. Dissonance curves for complex tones

• With six harmonic partials, dissonance minima occur at unison, octave, and just ratios 6:5, 5:4, 4:3, 3:2, and 5:3, but not at 8:5 or 7:4.Partial coincidences create the minima; the six-partial spectrum lacks the eighth and seventh partials needed for coincidences at 8:5 and 7:4.Source: §3.2 · p. 8

3.3. The dissonance surface for triads

• For triads, pairwise dissonances form a two-dimensional surface over root-relative ratios, with unique triads restricted to s ≤ r and minima in consonant valleys.The 12-TET major, minor, augmented, and diminished triads map near, but not exactly at, local minima.Source: §3.3 · pp. 8–10

3.4. The Harmonic Eigenspace

• For tetrads, total dissonance sums interactions across all six voice pairs, and its level sets define the dissonance volume.The three upper voices are represented by ratios α, β, and γ relative to the root.Source: §3.4 · p. 11
• Chord quality is an invariant direction represented by (α, β, γ), while transposition changes the scale factor along that ray.Every chord’s transpositions lie on one ray through the origin.Source: §3.4 · p. 12
• Local-minimum locations are register-invariant, and the 12-, 31-, and 53-TET chord vocabularies occupy discrete lattices in the same volume.The 12-TET grid is a coarser subset of the 53-TET grid; distinct neighboring dominant-seventh versions can be compared on the same psychoacoustic basis.Source: §3.4 · pp. 12–13

4. Harmonic Extension Principle

• Modal interchange extends 12-TET reharmonisation by drawing on the five third and seventh gradations in 31-TET and ten in 53-TET, creating substitutions without 12-TET equivalents.Modes built from non-standard Holdrian comma distributions also generate chord types absent from the diatonic vocabulary, and interchange among them expands substitution possibilities.Source: §4 · p. 13, §4.2 · p. 15

4.1. Chord Quality Extensions

• 31-TET combines five third and five seventh gradations into 25 chord types, while 53-TET combines ten of each into a 100-chord main grid.The 53-TET vocabulary also includes 40 half-diminished chords, 10 augmented chords, and two suspended chords; together the three temperaments contain 192 chords.Source: §4.1 · p. 13, Table 1 · p. 14, Table 2 · p. 14

4.2. Modal Interchange in 53-TET

• In parallel interchange, chords are borrowed from a mode sharing the same root; in relative interchange, the root is reinterpreted as a degree of another key.For a C Ionian progression, the Aeolian parallel substitutions are A♭maj7, Dm7♭5, and Gm7; the relative substitutions are Cm7, Fm7, and B♭7.Source: §4.2 · p. 15, Figure 5 · p. 15

5. The Application

• The Eigenspace combines a sonified dissonance volume for comparing 12-, 31-, and 53-TET chord vocabularies with a Modal Studio for developing scales, progressions, and voicing.The Eigenspace visualisation uses interactive nodes and chord positions; nodes can be filtered by dissonance.Source: §5 · p. 16, §5.1 · pp. 16–17, Figure 6 · p. 17

5.1. The Harmonic Eigenspace

• The Eigenspace displays chord positions against local-minimum landmarks, which align with just-intonation intervals and let nearby tempered chords be read as variants of the same harmonic region.The 12-TET major seventh lies very close to its just-intonation minimum, near a stable consonance valley; the web app uses 78 of the 100 retained minima.Source: §5.1 · pp. 16–17, Table 5 · p. 17

5.2. Modal Studio

• The Modal Studio lets users reshape a 53-TET scale, automatically derive its seven modes and chord qualities, and place a progression beside parallel and relative substitutions for audition.Selecting a grid cell opens its chord in the Voicing Editor, where notes can be repositioned across a five-octave spiral display.Source: §5.2 · pp. 18–20, Figure 8 · p. 19, Figure 9 · p. 20

6.1. Part 1: Chord Dissonance Ratings

• The 31 participants rated 19 distinct isolated 53-TET seventh chords on a 1–10 dissonance scale after one duplicated stimulus was excluded.For model comparison, the Eigenspace dissonance values were linearly rescaled to the rating range; this affine rescaling does not affect correlation analysis.Source: §6 · p. 21, §6.1 · p. 22
• Mean dissonance ratings ranged from 2.52 for m7 to 7.52 for smsm7, while Consonant chords had the lowest category mean and Neutral chords the highest.The category means were 3.37 for Consonant, 4.89 for Intermediate, 5.06 for Dominant, 5.61 for Diminished, 6.16 for Less Consonant, and 6.40 for Neutral.Source: §6.1 · p. 22, Figure 11 · p. 22

6.2. Part 2: Chord Progressions Ratings

• Participants rated ten 53-TET progressions on coherence, musicality, dissonance, and novelty, using 1–10 scales.Each progression began and ended on the same tonic and contained five to seven seventh chords; progressions were grouped by tonic type.Source: §6.2 · p. 23, Figure 12 · p. 23, Table 7 · p. 24

7.1. The Harmonic Eigenspace

• The Eigenspace places chord quality on a transposition-invariant coordinate direction, letting 12-, 31-, and 53-TET chord types coexist and be compared within one psychoacoustic volume.Its 100 local minima partition the volume into basins around just-intonation sonorities; dissonance values also distinguish tempered chords within a basin.Source: §7.1 · pp. 24–25

7.2.1. Part 1 - Chord Dissonance

• Per-chord Eigenspace and perceived dissonance were positively associated (Spearman ρ = .481, p = .037; Pearson r = .569, p = .011; n = 19 chords).Neutral chords were rated more dissonant than their Eigenspace value, while diminished chords were rated less dissonant; agreement was highest for familiar chord qualities. The correlation intervals were wide, and the study cannot distinguish cultural, timbral, or presentation-order effects.Source: §7.2.1 · pp. 25–26

7.2.2. Part 2 - Chord Progressions

• Progressions averaged coherence M = 6.86 and musicality M = 6.43; Neutral-tonic Progression 4 scored musicality M = 4.23 versus M = 8.00 for Dominant-tonic Progression 5.Neutral-tonic progressions received the lowest musicality ratings, while major- and dominant-tonic progressions received the highest; the ratings are exploratory descriptions of first-encounter tolerance. Novelty and dissonance were perceptually entangled for Neutral progressions.Source: §7.2.2 · pp. 26–27

7.3. Limitations and Future Work

• The listening study’s results remain exploratory: Part 2 was reference-free and first-encounter, and correlations were estimated over 19 chords with wide confidence intervals.Prior microtonal experience was not included as a covariate, and the dissonance ratings depended on two chosen reference chords.Source: §7.3 · p. 27
• The present Eigenspace is computed for harmonic timbres; inharmonic spectra would produce different minima, basins, and chord vocabularies.The authors propose a family of timbre-specific Eigenspaces; extending the interval gradation to fifths and using the coordinates for machine-learning models are planned directions.Source: §7.3 · pp. 27–28

• The authors report no financial or commercial relationships that could be construed as a potential conflict of interest.Source: Disclosure statement · p. 28
• Claude Opus 4.6 was used for proofreading, while the authors retained responsibility for the paper’s intellectual content, research design, analysis, and interpretation.Source: Disclosure statement · p. 28

• The funding information is marked “Hidden during the review process.”Source: Funding · p. 28

Frequency Spectrum Visualisation

• The frequency visualisation shows a clicked tetrad’s deviation from 12-TET, with root, α, β, and γ coded white, red, green, and blue.Each axis has a ×2 control to move its note up one octave; the root can also be changed by clicking a piano key or pressing Q–I.Source: Appendix A · pp. 29–30, Figure A1 · p. 30

Audio Engine

• The audio engine offers sine, triangle, sawtooth, and square waveforms, integrated reverb, and an envelope graph for attack, decay, and release timing.The envelope controls operate within a 2-second range.Source: Appendix A · p. 30, Figure A2 · p. 31
• The Chord Memory grid stores nodes of any type as sequences or progressions for interactive playback with the application as a MIDI controller in a DAW.The grid can also be linked to a Launchpad Pro.Source: Appendix A · p. 30, Figure A3 · p. 31