This paper presents a web-based application for navigating and composing microtonal harmony, built on the Harmonic Eigenspace, a four-dimensional psychoacoustically grounded space in which tetrad chord types are located by their spectral dissonance profiles, computed with Sethares's roughness/dissonance model. The coordinate system is transposition-invariant: a coordinate triple (α, \b{eta}, γ) locates the three upper notes in relation to the root, so a chord quality corresponds to a direction in the space, the invariant ray along which transposition acts, while the root frequency sets the scale. The dissonance field over these coordinates can be computed at any register; the locations of its local minima are register-invariant, as they arise from partial-coincidence ratio conditions. The dissonance volume contains 100 local minima that align with just-intonation intervals and act as landmarks, organising the space into basins around the most consonant tetrads. We embed tetrads from three tonal equal temperaments as discrete lattices within this continuous volume. The application presents this space through two components: the Harmonic Eigenspace as a navigable 3D visualisation of the dissonance volume in which all nodes are playable, and a Modal Studio that extends modal interchange logic to the ten-gradation interval vocabulary of 53-TET. The application also functions as a MIDI controller with MIDI Polyphonic Expression support, usable in any digital audio workstation that supports this format. A listening study with 31 participants used both scenes of the application: listeners first rated isolated 53-TET chords alongside chords familiar from Western practice, such as the maj7 and the m7; they then rated chord progressions composed in the Modal Studio, measuring their acceptance or rejection of microtonal progressions heard for the first time.
Figures & tables
Figure 1: Sensory dissonance DF(α) over one octave for a harmonic timbre with six partials at integer multiples of a 500 Hz fundamental and loudnesses 0.88k , the configuration of Sethares (2005) ; the curve is normalised so that the unison is 0 and the maximum is 1. Orange markers: local minima, located at the just-intonation ratios 6:5 , 5:4 , 4:3 , 3:2 , and 5:3 (partial coincidences); no minimum arises at 8:5 or 7:4 , which would require the eighth and seventh partial. Black diamonds: the 12-TET steps, with dissonance values unison 0.00 , m2 0.83 , M2 0.40 , m3 0.33 , M3 0.33 , P4 0.18 , tritone 0.34 , P5 0.08 , m6 0.33 , M6 0.18 , m7 0.20 , M7 0.38 , octave 0.00 . The perfect fourth and fifth fall close on their just minima; the thirds and the major sixth lie on the shoulders slightly off theirs; the minor second, the tritone, and the major seventh lie in regions of similar dissonance.
Figure 2: Two-dimensional dissonance surface DF(r,s) for triads, evaluated over all frequency ratio pairs (r,s)∈[1,2]×[1,2] . Blue regions are consonant valleys; red ridges are dissonant peaks. Numbered markers indicate local-minima. Diamond markers show the positions of the major, minor, augmented, and diminished triads of 12-TET.
Figure 3: Three-dimensional dissonance surface DF(r,s) for triads. Consonant valleys contain the local minima (white markers), the most consonant triads of the surface. Coloured markers show the positions of the major, minor, augmented, and diminished triads of 12-TET.
Figure 4: The dissonance volume D(α,β,γ) rendered over the tetrahedral domain 1≤α≤β≤γ≤2 (the ordering removes permutational duplicates), colour-coded from consonant (blue) to dissonant (red). The volume appears as a cloud of regions because the field is continuous: every point is a sounding four-note combination, defined by its three interval ratios above the root and belonging to no particular tuning system, and the application can audition any of them. Dissonant regions concentrate where adjacent voices come close together; the blue pockets are the consonance regions whose deepest points are the local minima. The placement of the 12-, 31-, and 53-TET chord vocabularies inside this continuum is shown in Figures 6 and 7 .
SM
M
n
m
sm
SM
SMSM7
SMM7
SMn7
SMm7
SMsm7
M
MSM7
Maj7
Mn7
Dom7
Msm7
n
nSM7
nM7
nn7
nm7
nsm7
m
mSM7
mM7
mn7
min7
msm7
sm
smSM7
smM7
smn7
smm7
smsm7
Table 1: 31-TET chord names (28 chords). The 5×5 grid varies the third (rows) and seventh (columns) quality; the fifth is fixed at step 18. Three additional chords with different fifths are also considered: (ø7, o7, M+7)
SM
ˆ M
maj
vM
N
n
ˆ m
m
vm
sm
SM
SMSM7
SM ˆ M7
SMmaj7
SMvM7
SMN7
SMn7
SM ˆ m7
SMm7
SMvm7
SMsm7
ˆ M
ˆ MSM7
ˆ M ˆ M7
ˆ Mmaj7
ˆ MvM7
ˆ MN7
ˆ Mn7
ˆ M ˆ m7
ˆ Mm7
ˆ Mvm7
ˆ Msm7
M
MSM7
M ˆ M7
Mmaj7
MvM7
MN7
Mn7
M ˆ m7
Mm7
Mvm7
Msm7
vM
vMSM7
vM ˆ M7
vMmaj7
vMvM7
vMN7
vMn7
vM ˆ m7
vMm7
vMvm7
vMsm7
N
NSM7
N ˆ M7
Nmaj7
NvM7
NN7
Nn7
N ˆ m7
Nm7
Nvm7
Nsm7
n
nSM7
n ˆ M7
nmaj7
nvM7
nN7
nn7
n ˆ m7
nm7
nvm7
nsm7
Table 2: 53-TET main grid chord names (100 chords). The third quality (rows) and seventh quality (columns) are combined as {third}{seventh}7 ; the fifth is fixed at step 31.
n
ˆ m
m
vm
sm
ˆ m
ø ˆ mn7
ø ˆ m ˆ m7
ø ˆ mm7
ø ˆ mvm7
ø ˆ msm7
m
ømn7
øm ˆ m7
ømm7
ømvm7
ømsm7
vm
øvmn7
øvm ˆ m7
øvmm7
øvmvm7
øvmsm7
sm
øsmn7
øsm ˆ m7
øsmm7
øsmvm7
øsmsm7
Table 3: 53-TET half-diminished chord names (40 chords). The third quality (rows) and seventh quality (columns) are combined as ø{third}{seventh}7 ; the fifth is fixed at step 26.
M+SM7
M+ ˆ M7
M+maj7
M+vM7
M+N7
Table 4: 53-TET augmented chord names (10 chords). The third is fixed at step 18 and the fifth at step 35; only the seventh varies.
Figure 5: Parallel and relative Aeolian interchanges applied to a I vi ii V I progression in C major. The parallel interchange replaces chords with their C Aeolian counterparts; the relative interchange borrows from E ♭ major, where C functions as the 6th degree.
Note
Axis
Ratio
Frequency
Root (A3)
—
1.0000
220.0 Hz
Major 3rd (C ♯ 4)
α=24/12
1.2599
277.2 Hz
Perfect 5th (E4)
β=27/12
1.4983
329.6 Hz
Major 7th (G ♯ 4)
γ=211/12
1.8877
415.3 Hz
Table 5: 12-TET maj7 chord example at 220 Hz.
Figure 6: Overview of the zone-nodes with dissonance colour-coding. The 12-, 31-, and 53-TET chords are embedded within the space. A consonance canal is visible along the region where the fifth is perfect; diminished and augmented chords fall into adjacent higher-dissonance regions.
Figure 7: The 12-, 31-, and 53-TET chord positions with zone-nodes hidden, showing the temperament lattices and their dissonance colour-coding in isolation.
Figure 8: The Scale Editor. A circular visualisation of the 53-TET octave with twelve draggable nodes representing scale degrees. Repositioning a node redefines the interval pattern and triggers automatic derivation of all seven diatonic modes and their chord qualities.
Figure 9: The Grid. An 8 × 8 chord-progression sequencer. The first row holds the user’s primary progression; the remaining rows are populated automatically with parallel and relative modal substitutions.
Figure 10: The Voicing Editor. A concentric spiral display spanning five octaves in which individual chord tones can be repositioned to craft specific distributions across registers.
#
Chord
Category
α
β
γ
Dchord
Closest LM
Distance
1
vMvM7
Consonant
1.2490
1.4999
1.8734
11.855
LM-1
0.0016
2
ˆ m ˆ m7
Consonant
1.2009
1.4999
1.8013
11.587
LM-3
0.0015
3
SMSM7
Intermediate
1.2990
1.4999
1.9484
13.556
LM-61
0.0805
4
SMm7
Intermediate
1.2990
1.4999
1.7779
14.932
LM-63
0.0352
5
maj7
Intermediate
1.2654
1.4999
1.8981
13.584
LM-1
0.0280
6
m7
Intermediate
1.1853
1.4999
1.7779
13.287
LM-3
0.0271
Table 6: Euclidean distance from each listening test chord to its closest local minimum in ( α , β , γ ) space. Chords are grouped by perceptual category.
Figure 11: Part 1: Perceived dissonance ratings for 19 isolated 53-TET chords, ordered by Eigenspace dissonance. The boxes show the interquartile range with the median line. Red dots indicate the Eigenspace dissonance value, scaled to the listener-rating range. Box colours indicate chord category: Consonant (light blue), Intermediate (pale blue), Neutral (grey), Dominant (orange), Less Consonant (light-orange), Diminished (red)
Figure 12: Part 2: Rating distributions per progression across four perceptual dimensions. Progressions are grouped by tonic type: Major (orange), Neutral (purple), Dominant (grey), Minor (blue). Dashed lines indicate medians.
#
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Major tonic ( vMvM7 )
1
CvMvM7[I]
G#vMvM7/C[VI]
C#vMvM7[II]
F ˆ m ˆ m7[IV]
D#vMm7[III]
CvMvM7[I]
3
CvMvM7[I]
C#vMvM7/F[II]
F#vMvM7[V]
Bb ˆ m ˆ m7[VII]
G#vMm7[VI]
CvMvM7[I]
10
CvMvM7[I]
D#vMvM7/G[III]
G#vMvM7[VI]
C ˆ m ˆ m7[I]
ˆ BbvMm7[VII]
CvMvM7[I]
Neutral tonic ( NN7 )
2
CNN7[I]
CNN7/vvE[I]
ˆˆ Fø ˆ m7[IV]
vvAnN7[VI]
ˆ Døvm7[II]
ˆ Gn ˆ m7[V]
Table 7: Chord progressions used in Part 2 of the listening test, grouped by tonic type.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 13: Frequency spectrum visualisation showing the deviation from standard 12-TET positions on a piano-roll reference.
Figure 14: Audio engine settings. Waveform selection (sine, triangle, sawtooth, square), reverb control, and an envelope graph for attack, sustain, and release timing.
Figure 15: The Chord Memory grid is used to store data from nodes of any type. It serves as a chord progression or sequence to play with an external audio workstation.
This paper deals with the introduction of "geometric harmony", a discipline that explicitly addresses the spectral characteristics of musical gamut. The framework of Western music, from Renaissance to the present, represents sound in terms of "pitch"-as is evident from its five-line staff notation system-and employs the fundamental frequency as its representative value, 440 Hz, etc. In this paper, by taking the timbres of specific individual instruments as elements and examining the Wasserstein distance between two voices, and Wasserstein deviations between three or more voices, we demonstrate that it is possible to expand the system whilst retaining the entire framework of conventional music theory. At the same time, as an example of practical utility in ensemble playing, we provide a detailed account of the two-voices affinity of the "Throat G" on clarinet, a note known for its fragility in ensemble contexts.
Yusei Tamura, Shigekazu Ishihara, Ken Ito
The University of Tokyo 7-3-1 Hongo, Bunkyo Ward, Tokyo, JAPAN · Hiroshima International University 555-36 Kurose Gakuen-dai, Higashi-Hiroshima, Hiroshima JAPAN
Conventional music representations describe acoustic energy over time and frequency but do not explicitly expose relations among simultaneous frequency components. We introduce the Dissonance Spectrum (DS), a nonnegative time--frequency representation that applies a tolerance-based rational pitch-relation kernel with logarithmic harmonic distance to a constant-Q spectrum and attributes aggregate pairwise interactions back to individual frequency bins. Controlled music-theory tests show strong ordinal agreement for intervals, harmonic-function connections, and church modes, and weaker but significant agreement across diverse chord voicings. DS is then encoded by a lightweight parallel branch whose zero-initialized residual projection preserves the baseline function at initialization. Across six paired training seeds in open-ended music question answering and categorical and dimensional music emotion recognition, DS obtains the highest mean on every reported endpoint relative to the unchanged baseline, a parameter-matched Gaussian-input branch, and an architecture-matched magnitude-CQT branch. These results support DS as an interpretable, complementary representation, while listener-specific perception and broader task coverage remain open problems.
Tianle Wang, Xinyi Tong, Liangke Zhao +7
Beijing Institute for General Artificial Intelligence, Beijing, China · Central Conservatory of Music, Beijing, China · Peking University, Beijing, China
Several tonal pitch spaces and computational models have been proposed to analyze tonal structure in Western tonal music, many of them grounded in principles from music theory and used to support tonal analysis with important implications for tonal tension. In parallel, data-driven methods such as skip-gram have been used to learn chord embeddings from symbolic corpora, but their ability to recover tonal structure and its relation to tonal tension remains underexplored. In this work, we investigate how skip-gram chord embeddings reflect tonal structure and whether they provide a useful basis for analyzing structural aspects of tonal tension. Using chord sequences with and without transposition-based augmentation, we evaluate the learned spaces from geometric, functional, and tension-related perspectives. We show that augmented embeddings exhibit strong transposition equivariance, recover a clear circle-of-fifths structure, and support interpretable shifts between key-related regions of the learned space. We then derive embedding-based measures from chord-to-key distance and contextual chord-distance relations, and show that they capture meaningful aspects of tonal tension structure through correspondence with matched tonal measures and moderate alignment with human tension profiles. Across analyses, transposition-based augmentation generally improves the stability, tonal coherence, and interpretability of the learned space.
Maral Ebrahimzadeh, Gilberto Bernardes, Sebastian Stober
Otto von Guericke University Magdeburg, Artificial Intelligence Lab, Magdeburg, Germany · INESC TEC, Faculty of Engineering, University of Porto, Portugal