Symmetric Structure and Motivic Transformation in Melodies, Phrases, and Themes

Alfredo González-Espinoza

Figure 1: Internal symmetry classifies an object independently of its identity. Left: a snowflake is left unchanged by rotations in steps of

60^{\circ}

and by reflections (one mirror

\sigma

shown), 24 operations in all, its point group

D_{6h}

inherited from the hexagonal ice lattice. Middle: the water molecule

\text{H}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}\text{O}

is left unchanged by a rotation (the

C_{2}

axis shown) and two reflection planes, its point group

C_{2v}

. This classification holds whatever atoms occupy the vertices. Right: the motif

(60,64,67,64,60)

(MIDI notation) reads the same forward and backward, so it is left unchanged by retrograde (

F

) and belongs to the musical counterpart of that symmetry class. The set of operations fixing an object (its stabilizer ) sorts objects by internal symmetry, and objects sharing a class share a structural profile: spectroscopic selection rules in chemistry, transformation invariances in music. Within one group, the more operations fix an object, the fewer distinct forms it has (the orbit–stabilizer theorem).

Figure 2: Dihedral groups used in this work. Solid arcs are rotations, dashed lines reflection axes, dotted arcs the swaps they induce. (a)

D_{12}

on the chromatic circle:

r

transposes by a semitone,

f

inverts about the

0

–

6

axis, and

|D_{12}|=24

. (b)

D_{m}

at

m=5

:

R

shifts positions cyclically,

F

reverses them (retrograde). For even

m

the axis passes between vertices, and

|D_{m}|=2m

.

Figure 3: Windows and orbits. (a) A sequence

S

is read through a sliding window of length

m

(here

m=4

), giving

N=L-m+1

overlapping windows. (b)

G

partitions

\mathbb{Z}_{12}^{m}

into orbits, each with area proportional to

|\text{

\mathchar 525

}|=|G|/|\operatorname{Stab}|

. Arrows map windows to the canonical representative

\hat{o}

. Asymmetric motifs (

\operatorname{Stab}=\{e\}

) fill large orbits, while the palindrome

\hat{o}_{2}

is fixed by

F

, as

\text{H}{\vphantom{\text{X}}}_{\smash[t]{\text{2}}}\text{O}

is fixed by its

C_{2}

axis. (c) One orbit at

m=3

, each window labeled by the element that maps it to

\hat{o}=(0,0,1)

.

Regime	$H_{\text{macro}}$	$\langle H_{\text{intra-orbit}}\rangle$	$H_{\text{micro}}$
Strict ostinato a	$0$	$0$	$0$
Strict canon b	$0$	up to $S$ of that orbit	matches intra
Maximum motif variety c	up to $\log_{2}N$	$0$	matches macro

Table 1: Partition of the entropy budget under three compositional regimes.

Table 2: Window stabilizer types and the local shape each detects. A window is symmetric if it is fixed by some reflection

\rho

of the

D_{m}

position cycle (its central axis is only the simplest case) or by a rotation; conditions hold up to a transposition

k

. The Sketch column gives an illustrative example of one representative pitch–position contour per type, all drawn in a common frame (dots are successive notes). By analogy with molecular point groups, the asymmetric class corresponds to

C_{1}

and the retrograde class to a

C_{2v}

-type mirror. At

m=3

a single axis pairs one couple of positions, so repeated-note and neighbor figures (

[x,x,y]

,

[x,y,x]

) are retrograde-type; at

m=5

two couples must agree.

Figure 4: The eight example objects as pitch–position contours: four synthetic controls (ostinato, chromatic, symmetry images, random) and four tonal melodies (Twinkle, Ode to Joy, Frère Jacques, Happy Birthday).

Control	$H_{\text{orbit}}$ (m3 / m5)	$\eta$ (m3 / m5)	Expected behavior
Ostinato [0,4,7]	$0.00$ / $1.58$	$0.22$ / $0.00$	crystal endpoint; window-dependence
Chromatic 0:11	$0.00$ / $0.00$	$0.58$ / $0.52$	orbit entropy $\neq$ pitch entropy
Symmetry images (R, I, T, RI)	$1.99$ / $3.06$	$0.39$ / $0.20$	one idea, many surface forms
Random 0:11 , $n{=}500$	$4.06$ / $8.38$	$0.69$ / $0.07$	motivic-vocabulary ceiling

Table 3: Synthetic controls at

m=3

and

m=5

(MLE estimator, ceilings

\log_{2}K(3)=4.25

,

\log_{2}K(5)=10.12

bits).

Figure 5: (a) Motivic-entropy spectrum:

H_{\text{orbit}}\to 0

when a melodic line stays in one orbit (crystal, ostinato, strict canon) and

H_{\text{orbit}}\to\log_{2}K(m)

at orbital equipartition. The gap to the ceiling is the motivic compressibility. (b) Measured

H_{\text{orbit}}

: synthetic controls span the interval. Annotated melodies are shown for placement only, since their short length undersamples the distribution.

Figure 6: (a)

H_{\text{orbit}}

as a function of window size

m

for the eight examples. (b) The same values against the deployment ratio

\eta

, one path per example from

m=3

to

m=6

(marker size grows with

m

). The period-3 ostinato returns

H_{\text{orbit}}=0

only when

m

is commensurate with its period (

m=3,6

). Random pitch tracks the ceiling while its

\eta

falls from

0.69

toward

0

.

Data set	Sequence	Number	$L$	Measures
Solo keyboard corpus (KDF)	movement	852	322	$H_{\text{orbit}}$ , $\Delta H$ , $\eta$ , symmetry types, relations
Prelude & Fugue set	prelude, fugue	108	289	$\eta$ , $\Delta\eta$ , $\Delta H$ , $H_{\text{stab}}$ , relations
MTC-ANN 2.0.1	phrase ( $4\leq L\leq 16$ )	1,890	9	exact symmetry, $O/E$
Essen	phrase ( $4\leq L\leq 16$ )	44,765	8	exact symmetry, $O/E$
MTD	theme (all $L$ )	2,067	18	exact symmetry, $O/E$

Table 4: Data sets.

L

: median length of the analyzed sequence

S

in notes.

Period	Dates	Composers	Pieces
Baroque	before 1750	14	213
Classical	1750–1819	10	213
Romantic	1820–1889	14	213
20th century	from 1890	11	213

Table 5: Solo keyboard corpus by period (Schubert and Hummel placed in the Classical period); lengths and composer shares in Table S6 .

Gap	Structure it reflects	$m=3$	$m=4$	$m=5$	$m=6$
$\log_{2}K(m)-H_{\text{orbit}}^{\text{shuffle}}$	pitch bias (key, scale, range)	0.26	0.51	1.14	3.05
$H_{\text{orbit}}^{\text{shuffle}}-H_{\text{orbit}}^{\text{IAAFT}}$	linear correlation (spectrum)	0.00	0.04	0.07	0.08
$H_{\text{orbit}}^{\text{IAAFT}}-H_{\text{orbit}}^{\text{Markov-1}}$	one-step transitions	0.38	0.60	0.89	0.69
$H_{\text{orbit}}^{\text{Markov-1}}-H_{\text{orbit}}^{\text{real}}$	beyond one-step transitions	0.12	0.57	1.21	1.91

Table 6: Measured gaps between consecutive levels of Eq. ( 13 ): median over the 852 pieces, in bits. At large

m

the first row also contains the sampling ceiling (a line of

N

windows cannot exceed

\log_{2}N

bits).

Figure 7: Real

H_{\text{orbit}}

(black) against the shuffle (gray), IAAFT (blue), and Markov-1 (yellow) nulls (bands: central

95\%

of 199 surrogates; dashed: median) for one typical piece per period, the piece whose IAAFT deficit at

m=4

equals its period’s median: Froberger, Capriccio 3; Mozart, Fugue K. 375g; Liszt, Grand Étude 4; Scriabin, Étude Op. 8 No. 6.

Figure 8: The three measures across periods and forms,

m=3

–

6

. (a) Relative deficit of

H_{\text{orbit}}

against IAAFT by period (vocabulary). (b) Deployment ratio

\eta

by form family (deployment). (c) Symmetry-type entropy

H_{\text{stab}}

by period (mix of symmetry types).

213

pieces per period (Bar. Baroque, Cla. Classical, Rom. Romantic, 20th: 20th century);

508

labeled pieces by form (dance

65

, cyclic

211

, character

167

, contrapuntal

35

, free

30

). Character: single-movement pieces (prelude, nocturne, étude, impromptu, ballade, intermezzo); cyclic: sonata, suite, partita movements. Boxes: interquartile range and median; dashed: mean. Separation tests for every measure and grouping are in Table S8 .

Figure 9: Structure and deployment in the Prelude & Fugue set,

m=4

. (a)

H_{\text{orbit}}

, (b)

H_{\text{stab}}

(six symmetry classes), (c)

\eta

and (d)

\Delta\eta

(difference from the mean of 199 IAAFT surrogates), preludes (left) and fugues (right) by composer (

n=24

,

6

,

24

pieces per form). Jonckheere–Terpstra trend across composers for fugues, Holm-adjusted over

m=4

–

6

:

p=1.0

(a),

0.014

(b),

0.002

(c),

0.035

(d).

Figure 10: Symmetry types by period,

m=3

–

6

. (a) Share of windows in each symmetry class, computed per piece, averaged within each period, and stacked to one (Bar. Baroque, Cla. Classical, Rom. Romantic, 20th century). (b) Distinct symmetry subgroups found as pieces are added, by period. Lines and bands show the mean and

95\%

range of the rarefaction curve, diamonds and horizontal dashed lines the estimated total (Chao2) with its interval, and the vertical dashed line the common sample of

213

pieces.

Figure 11: Relations between recurring windows,

m=4

, pairs of asymmetric windows. (a) Share of each relation type relative to the within-orbit shuffle, which keeps the orbit of every window and the forms each orbit takes (filled, with

90\%

intervals that resample composers), and relative to shuffled, IAAFT, and Markov-1 surrogates (open). T transposition, I inversion, R retrograde, RI retrograde-inversion. (b) Transposition intervals relative to the within-orbit shuffle in the keyboard corpus and in the fugues of the Prelude & Fugue set, fifth or fourth (

k=5,7

), step (

2,10

), third or sixth (

3,4,8,9

), semitone (

1,11

), and tritone (

6

). Dashed lines mark

O/E=1

.

Figure 12: Composers relative to the within-orbit shuffle,

m=4

, pairs of asymmetric windows. (a) Transposition and (b) retrograde-inversion for composers with at least

10

pieces, in the same rows (ordered by transposition), shrunk toward the period mean (filled,

\pm 1

s.d.), with the unshrunk mean (open) and the period mean (dotted). Transposition varies more between composers than between periods (composer

13\%

and period

0.1\%

of the variance), whereas retrograde-inversion varies between periods (trend

\rho=-0.24

). Romantic composers coincide after shrinkage because their between-composer variance is estimated as zero.

Figure 13: Exact symmetry of whole sequences by length bin. (a) Share of symmetric segments (bars) and the share expected from each segment’s own notes in random order (black tick). (b) Enrichment

O/E=\sum O_{L}/\sum E_{L}

within each bin, with exact Poisson

95\%

interval [ 74 ] . The dotted line marks

O/E=1

. Bins with fewer than ten segments, fewer than

0.1

expected or no observed symmetric segments are omitted from (b). Per-length estimates are in Figure S18 .

$m$	3	4	5	6	7	8
$K(m)$	19	163	1,110	10,962	107,509	1,126,566

Table S1: Total orbit count

K(m)

for

G=D_{12}\times D_{m}

.

Motif	Non-identity elements of $\operatorname{Stab}$	$\|\operatorname{Stab}\|$	Family	Class	Forms
$(0,4,7)$	none	1	trivial	asymmetric	144
$(0,1,2)$	$fr^{10}F$	2	$\mathbb{Z}_{2}$	retrograde-inversion	72
$(0,6,0)$	$F$ , $f$ , $fF$	4	$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$	multiple	36
$(0,4,8)$	$r^{4}R$ , $r^{8}R^{2}$ , $fFR^{2}$ , $fr^{4}F$ , $fr^{8}FR$	6	$D_{3}$	multiple	24

Table S2: Stabilizers of four motifs at

m=3

(

|G|=144

), with the number of surface forms

|G|/|\operatorname{Stab}|

. Elements are written

f^{a}r^{k}F^{b}R^{j}

(inversion, transposition, retrograde, cyclic shift).

Figure S1: Retrograde-inversion returns a flat–rise–flat shape or contour to itself. Retrograde flips the contour in time (dashed vertical axis), inversion flips it in pitch (dashed horizontal axis), and transposition moves it back into place. The three operations together amount to a half-turn rotation of the contour, so any contour that looks the same rotated by

180^{\circ}

is fixed by retrograde-inversion. Every constant-step sequence (chromatic when step is one) is such a contour.

Reading	Windows equal when	Group	$K$ at $m=3,4,5,6$
block ( $m$ -gram)	identical	$\{e\}$	1,728; 20,736; 248,832; 2,985,984
interval $m$ -gram	equal up to transposition	$\mathbb{Z}_{12}$	144; 1,728; 20,736; 248,832
orbit	equal up to T, I, R, cyclic shift	$D_{12}\times D_{m}$	19; 163; 1,110; 10,962
permutation	same rank order (ties kept)	order-preserving maps	13; 75; 541; 4,683

Table S3: Readings of the windows of a line and their numbers of states

K

.

Figure S2: Orbit entropy among other entropy measures, keyboard corpus. (a) Median entropy normalized by

\log_{2}K

(bands: interquartile range), (b) NSB correction to the plug-in estimate, and (c) partial rank correlation with

H_{\text{orbit}}

(length removed), for windows of

m=3

–

6

notes.

Measure	Periods $p$ ( $\rho$ )	Forms $p$	Variance	Fugue $p$	Below / above (%)
block	1.00 ( $+0.13$ )	0.026	0.02 / 0.31 / 0.02	0.021	98 / 0
interval	1.00 ( $+0.11$ )	0.437	0.02 / 0.28 / 0.01	0.526	99 / 0
orbit	0.187 ( $+0.16$ ); 0.001 at $m=3$	1.00	0.04 / 0.22 / 0.01	0.864	97 / 0
permutation	1.00 ( $+0.03$ )	0.144; 0.040 at $m=3$	0.00 / 0.18 / 0.01	0.376	91 / 1
rate	1.00 ( $+0.02$ )	0.001	0.00 / 0.34 / 0.02	0.080	96 / 0
$\eta$	1.00 ( $-0.04$ )	0.001	0.00 / 0.19 / 0.02	0.002	27 / 42

Table S4: Entropy readings compared at

m=4

, keyboard corpus (Prelude & Fugue set for the fugue trend). Periods: Kruskal–Wallis with composers permuted between periods (Holm over

m

; values at

m=3

where they differ); ordered trend as Spearman

\rho

. Forms: Kruskal–Wallis with labels permuted within composers. Variance: adjusted

R^{2}

of ranks for period, composer beyond period, and form beyond composer. Fugue trend: Jonckheere–Terpstra, Bach to Shostakovich. Last column: share of pieces significantly below / above their IAAFT surrogates after Benjamini–Hochberg control. Bold:

p<0.05

. All values for

m=3

–

6

in Data/entropy_comparison_stats.csv ; permutations are independent of Table S8 , so

p

may differ in the third decimal.

Baroque	Bach 24, Handel 22, D. Scarlatti 22, Couperin 21, d’Anglebert 20, Dandrieu 20, Froberger 17, Corrette 12, Jollage 12, Jacquet de La Guerre 11, Rameau 11, Krebs 9, Mattheson 8, Telemann 4
Classical	Beethoven 30, Mozart 30, Haydn 28, Schubert 23, A.-L. Couperin 22, Cramer 20, Clementi 18, L. Mozart 17, Hummel 14, Soler 11
Romantic	Chopin 20, Alkan 19, Raff 19, Schumann 19, Brahms 18, Fauré 18, Liszt 18, Tchaikovsky 17, Saint-Saëns 13, Albéniz 12, Dvořák 11, Grieg 11, Mendelssohn 10, Gade 8
20th century	Busoni 30, Debussy 30, Scriabin 30, Joplin 20, Pozzoli 17, Ravel 17, Shostakovich 17, Satie 16, Bartók 13, Godowsky 13, Janáček 10

Table S5: Composers (pieces).

Period	Dates	Composers	Pieces	Works	$L$ , median (IQR)	Top
Baroque	before 1750	14	213	208	301 (186–577)	11%
Classical	1750–1819	10	213	193	322 (184–584)	14%
Romantic	1820–1889	14	213	209	317 (173–561)	9%
20th century	from 1890	11	213	212	327 (179–563)	14%
Total		49	852	822	322 (179–571)

Table S6: Solo keyboard corpus by period. “Top” is the share of the period’s pieces by its most represented composer; Schubert and Hummel are placed in the Classical period.

Question	Data	Test	Multiplicity	Effect size
Lower $H_{\text{orbit}}$ than nulls (§ 3.1 )	piece	rank of $H_{\text{orbit}}$ among 199 surrogates (two-sided); sign test across pieces	BH over pieces	$\Delta H$ , $\Delta H/\overline{H}_{\text{null}}$
Periods (§ 3.2 )	piece	KW; period labels permuted between composers (9,999); ordered trend as Spearman $\rho$ , same permutation; measures $H_{\text{orbit}}$ , $\eta$ , $H_{\text{stab}}$	Holm over $m$	$\varepsilon^{2}$ ; $\delta$ with composer-bootstrap 90% CI
Forms (§ 3.2 )	labeled piece	KW; form labels permuted within composers (9,999); $\eta$ also as residual on $\log L$ ; $\eta$ and $\Delta\eta$ against IAAFT	Holm over $m$	$\varepsilon^{2}$ ; $\delta$
Where measures vary (§ 3.2 )	piece	adjusted $R^{2}$ of ranks on period, composer, form (descriptive)	—	variance share
Composers, form fixed (§ 3.3 )	piece	one-sided JT (50,000 permutations); exact Wilcoxon signed-rank for fugue $-$ prelude pairs	Holm over $m=4$ – $6$ per measure	$\delta$ with 90% stratified-bootstrap CI
Symmetry types (§ 3.4 )	piece	class share ranked among 199 surrogates per null; sign test; period trend as Spearman $\rho$ with periods permuted between composers	BH over pieces	$O/E$ of class shares

Table S7: Procedure behind each result. KW: Kruskal–Wallis; JT: Jonckheere–Terpstra; BH: Benjamini–Hochberg;

\delta

: Cliff’s delta.

Figure S3: Window stabilizer-type profiles of the annotated melodies and controls at

m=3

and

m=5

. At

m=3

short palindromes dominate; at

m=5

the types separate return figures (retrograde), scalar or mirror motion (RI) and directed subjects (asymmetric).

Figure S4: Relative deficit

(H_{\text{orbit}}-\langle H_{\text{null}}\rangle)/\langle H_{\text{null}}\rangle

of the 852 corpus pieces under shuffle, IAAFT, and Markov-1 surrogates (199 each),

m=3

–

6

; labels give the share of pieces below their null.

		Periods			Forms
Measure	$m$	$p$	$\varepsilon^{2}$	trend $\rho$ ( $p$ )	$p$	$\varepsilon^{2}$
$H_{\text{orbit}}$ deficit vs IAAFT	3	0.003	0.06	+0.19 (0.010)	1.000	0.00
	4	1.000	0.02	+0.10 (0.599)	1.000	0.01
	5	1.000	0.01	+0.09 (0.599)	1.000	0.05
	6	1.000	0.00	+0.05 (0.599)	0.175	0.10
$H_{\text{orbit}}$	3	0.002	0.09	+0.22 (0.008)	1.000	0.00

Table S8: Separation of periods and forms for each measure. Periods: Kruskal–Wallis with period labels permuted between composers (Holm over

m

),

\varepsilon^{2}

, and the ordered trend Baroque

\to

20th century as Spearman

\rho

with composers permuted as blocks (Holm

p

in parentheses). Forms: Kruskal–Wallis with form labels permuted within composers (Holm over

m

),

\varepsilon^{2}

. Bold:

p<0.05

. Pairwise

\delta

in Data/cluster_tests.csv . † Not retained with :complexity lines (Section S5.3 ).

Measure	$m$	period	composer	composer $\|$ period	form	form $\|$ composer
$H_{\text{orbit}}$ deficit	3	0.06	0.17	0.10	0.00	0.02
	4	0.02	0.22	0.21	0.01	0.00
	5	0.01	0.26	0.25	0.05	-0.00
	6	0.00	0.30	0.30	0.10	0.00
$H_{\text{orbit}}$	3	0.09	0.21	0.13	0.00	0.02
	4	0.04	0.26	0.22	0.01	0.01

Table S9: Where the measures vary: share of rank variance explained (adjusted

R^{2}

of a least-squares fit on ranks) by period, composer, composer beyond period, form (labeled pieces), and form beyond composer. Descriptive; composers are nested in periods.

$m$	$\eta$	$\overline{\eta}_{\text{null}}$	$\Delta\eta$	above / below (%)	$\rho(\eta)$	$\rho(\Delta\eta)$
3	0.501	0.576	-0.071	4 / 80	+0.76	-0.38
4	0.284	0.265	+0.016	42 / 27	+0.76	-0.15
5	0.170	0.096	+0.063	76 / 3	+0.66	+0.35
6	0.107	0.044	+0.058	81 / 2	+0.58	+0.54

Table S10: Deployment ratio against

199

IAAFT surrogates per piece: medians of the real

\eta

, the null mean, and

\Delta\eta

; percentage of pieces significantly above / below the null after Benjamini–Hochberg control; Spearman

\rho

of

\eta

and

\Delta\eta

with

\log L

.

Period	composers with $\geq 2$ forms	dances	$m=3$	$m=4$	$m=5$	$m=6$
Baroque	5	19	+0.36 (0.240)	+0.36 (0.180)	+0.43 (0.212)	+0.32 (0.098)
Classical	8	23	+0.72 ( $<0.001$ )	+0.47 (0.011)	+0.41 (0.035)	+0.17 (0.119)
Romantic	10	12	+0.05 (0.657)	+0.16 (0.672)	+0.20 (0.541)	+0.23 (0.655)
20th century	5	11	+0.35 (0.314)	+0.56 (0.043)	+0.62 (0.015)	+0.67 (0.009)

Table S11: Deployment ratio

\eta

by form within each period: Cliff’s

\delta

of the other labeled forms against dances (positive: dances lower), with the

p

-value of a Kruskal–Wallis test over all forms with form labels permuted within composers (9,999 permutations). Bold:

p<0.05

.

Figure S5: Panels complementing Figure 8 ,

m=3

–

6

. (a)

\eta

by period (

213

pieces each). (b) Relative deficit of

H_{\text{orbit}}

against IAAFT and (c)

H_{\text{stab}}

by form family (

508

labeled pieces). Tests in Table S8 .

Test	Result
Fugue trend, $p$	$H_{\text{orbit}}$ 1.0; $\Delta H$ 1.0; $H_{\text{stab}}$ 0.014 ; $\eta$ 0.002 ; $\Delta\eta$ 0.035
$\delta$ of fugue $\Delta\eta$	Bach $\to$ Shostakovich $0.38$ $[0.12,0.62]$ ; Bach $\to$ Mendelssohn $-0.64$ $[-0.86,-0.36]$
Paired fugue $-$ prelude, $p$	Bach 1.0; Mendelssohn 0.94; Shostakovich 0.002
Interaction trend, $p$	0.003
$\rho$ with fugue/prelude length ratio	$0.14$ ( $p=0.30$ )

Table S12: Prelude & Fugue tests at

m=4

. Fugue trend: one-sided Jonckheere–Terpstra, Bach

<

Mendelssohn

<

Shostakovich, Holm over

m=4

–

6

.

\delta

: Cliff’s delta with

90\%

stratified-bootstrap interval. Paired: exact Wilcoxon signed-rank on fugue

-

prelude

\Delta\eta

.

Figure S6:

\eta

(top) and

\Delta\eta

(bottom) against melody length, Prelude & Fugue set,

m=4

–

6

;

\rho

: Spearman correlation with

\log L

.

Figure S7:

\eta

of preludes and fugues by composer,

m=3

–

6

; titles give the Jonckheere–Terpstra fugue trend (Holm over

m=4

–

6

).

Figure S8: Paired prelude and fugue of each number,

\eta

(top) and

\Delta\eta

(bottom),

m=4

; thick line: medians;

p

: exact Wilcoxon signed-rank test on

\Delta\eta

.

	$m=3$		$m=4$		$m=5$		$m=6$
class	u	real	u	real	u	real	u	real
asymmetric	58.3	41.3–47.7	63.9	42.2–47.3	93.2	74.4–82.4	94.9	70.7–76.2
retrograde	20.8	33.3–35.4	13.9	24.0–27.2	3.4	13.0–17.3	2.2	11.6–14.0
retro-inversion	16.7	16.2–16.8	12.5	12.0–13.1	3.4	4.1–5.5	2.1	5.2–5.8
inversion	0	0	4.6	2.0–2.6	0	0	0.5	0.3–0.4
translation	0	0	0	0	0	0	0.1	0.7–1.1

Table S13: Share of windows per stabilizer class (%): uniform alphabet (u, all

12^{m}

windows) and the range of the four period means in the corpus (real).

	shuffle		IAAFT		Markov-1
Class	$m=4$	$m=6$	$m=4$	$m=6$	$m=4$	$m=6$
retrograde	1.20	2.36	1.07	1.74	0.99	1.16
RI	1.10	2.25	1.21	2.11	1.10	1.53
inversion	0.59	0.77	0.67	0.77	0.58	0.86
multiple	2.12	8.18	1.65	4.44	1.12	1.80

Table S14: Observed over expected share of windows per symmetry class (pieces weighted equally), against 199 surrogates per piece.

Null	$m=3$	$m=4$	$m=5$	$m=6$
shuffle	-0.06 (15/29)	+0.10 (34/9)	+0.22 (49/3)	+0.56 (80/1)
IAAFT	-0.09 (12/36)	+0.07 (23/9)	+0.10 (33/11)	+0.45 (70/2)
Markov-1	-0.01 (0/0)	-0.06 (4/20)	+0.07 (20/2)	+0.19 (39/3)

Table S15:

H_{\text{stab}}

of real lines against 199 surrogates per piece: median of

H_{\text{stab}}-\overline{H}_{\text{stab,null}}

in bits, with the percentage of pieces significantly above / below the null after Benjamini–Hochberg control in parentheses.

$m$	transposition	inversion	retrograde	retrograde-inversion
3	1.10 [1.06, 1.14] (54)	0.90 [0.88, 0.92] (41)	1.00 [0.98, 1.02] (54)	1.21 [1.18, 1.23] (79)
4	1.56 [1.47, 1.66] (75)	0.84 [0.80, 0.88] (36)	0.97 [0.94, 1.01] (46)	1.33 [1.28, 1.39] (71)
5	2.00 [1.86, 2.15] (76)	0.75 [0.70, 0.80] (30)	1.06 [1.01, 1.11] (46)	1.38 [1.31, 1.46] (66)
6	3.17 [2.92, 3.44] (73)	0.57 [0.51, 0.64] (22)	0.84 [0.77, 0.92] (30)	1.17 [1.06, 1.29] (48)

Table S16: Relation types against the within-orbit shuffle, pairs of asymmetric windows: corpus

O/E

with composer-bootstrap

90\%

interval and, in parentheses, the percentage of pieces above the null.

Figure S9: Relation types relative to the within-orbit shuffle (filled,

90\%

composer-bootstrap interval) and to the shuffle, IAAFT, and Markov-1 surrogates (open),

m=3

–

6

, pairs of asymmetric windows.

Figure S10: Transposition intervals relative to the within-orbit shuffle, keyboard corpus and Prelude & Fugue fugues by composer,

m=3

–

6

.

Figure S11: Composer means of each relation type relative to the within-orbit shuffle,

m=4

, shrunk toward the period mean (filled,

\pm 1

s.d.) with the unshrunk mean (open) and the period mean (dotted). Composers are ordered by transposition in every panel.

Figure S12: Rate of each relation type per pair relative to the within-orbit shuffle, by the distance

\Delta

between the starts of the two windows, grouped in doubling ranges

[m,2m),[2m,4m),\dots,\geq 32m

(every integer

\Delta\geq m

counted). Every rate exceeds one because rotated relations are rarer than under the null, so types are compared within a distance range.

Figure S13: Share of relations that include a cyclic shift, per piece by period (boxes), with the median under the within-orbit shuffle (cross) and the uniform expectation

(m-1)/m

(dashed).

Figure S14: Retrograde-inversion relative to the within-orbit shuffle over all windows (open) and over asymmetric windows only (filled), by period. Over all windows, transpositions of constant-step runs are also counted as retrograde-inversions, which dilutes the comparison.

Figure S15: Repetition relative to uniform forms and to the three surrogates,

m=3

–

6

, with

90\%

composer-bootstrap intervals. The within-orbit shuffle keeps repetition by construction.

Figure S16: Entropy of the operations relating windows,

m=4

, pairs of asymmetric windows. (a) Type entropy and interval entropy within each type, observed and under the within-orbit shuffle (median and interquartile range over pieces). (b) Deficit of the transposition intervals, shuffle minus observed

H(k\mid\text{T})

, by period (boxes) with composer medians (dots). (c) The same deficit in the Prelude & Fugue set, fugues (squares) and preludes (circles) by composer, with medians as bars.

Figure S17: Share of segments with each exact symmetry class, per census corpus.

Figure S18: Enrichment

O_{L}/E_{L}

under the note-shuffle (filled) and interval-shuffle (open) nulls, per exact length

L

(log scale; dashed line: no enrichment).

Symmetric Structure and Motivic Transformation in Melodies, Phrases, and Themes

Abstract

Figures & tables

Symmetry Discovery

Geometric Representations for Transformed Pattern Matching in Music

Probabilistic Multilabel Graphical Modelling of Motif Transformations in Symbolic Music

vega-mir: An information-theoretic Python toolkit for symbolic music, with applications to harmonic graphs and rubato spectra

Abstract

Symmetry Discovery

Explore similar work

Geometric Representations for Transformed Pattern Matching in Music

Probabilistic Multilabel Graphical Modelling of Motif Transformations in Symbolic Music

vega-mir: An information-theoretic Python toolkit for symbolic music, with applications to harmonic graphs and rubato spectra