Learning Sonata Form Structure on Mozart’s String Quartets

Pierre Allegraud; Louis Bigo; Laurent Feisthauer; Mathieu Giraud; Richard Groult; Emmanuel Leguy; Florence Levé

doi:10.5334/tismir.27

Figures & Tables

*Andante con moto* of the String Quartet #16 in E♭ Major, K 428, 2nd movement. Encoded in Lilypond by Maurizio Tomasi for the Mutopia Project. This slow movement has a sonata form, as detailed in Section 2.1. Following notations of Hepokoski and Darcy (2006), the primary themes (P/P’) are followed by transitions (TR/TR’), ended with Medial Caesuras (MC/MC’) – they are here Half Cadences (HC) in the main tonality (I). In the exposition, the secondary theme (S) and the conclusion (C) are here in the tonality of the dominant (V, E♭ major). In the recapitulation, both S’ and C’ come back to the main tonality. In the exposition, the S theme ends with a perfect authentic cadence (PAC) named essential expositional closure (EEC), whereas, in the recapitulation, the S’ theme ends with an essential structural closure (ESC). Between the exposition and the recapitulation, the development (Dev) moves to other keys and is concluded by a retransition (RT) focusing on the dominant of the primary key.

Model topologies describing the most common sonata form structure at several resolutions. The set of states *Q_n* has n states. Q₃ and Q₇ model the basic sections of the sonata form. Q₁₄ (used by the model of Bigo et al. (2017)) and Q₁₈ further model Intro, TR, RT and Coda sections as well as transitional states between these sections, represented with squares: the medial cesuras MC and MC’, but also short transitions between the end of the closing zone and the complete end of the exposition (transition after the closing zone, TC), between the exposition and the development (d), between the development and the recapitulation (r), and between the recapitulation and the Coda (TC’). Initial and final states are circled twice.

Extract of the reference analysis for the second movement of the String Quartet #16 in E♭ major (K428.2), as viewed on http://www.dezrann.net/ (left) and represented as a `json` file (right). The Primary theme (P) ends with a half cadence in the primary key (I:HC). Here a Transition zone (TR) begins, which stops on different beats according to the references. The A analysis starts the secondary (S) theme after the HC in the primary key on measure 10, whereas the F analysis rather starts it on measure 14 (HC on the dominant key). Onsets in the `json` file are expressed in quarter notes.

Reference analysis A of 32 sonata-form and sonata-form-like movements in Mozart string quartets. The analyses are projected on the 18 states of Q₁₈. Vertical lines show cadences.

Medial Caesura in Allegro K80.2, measure 15. This half cadence (HC) has a very simple but very efficient tonic/dominant schema. It is reinforced by the sudden change of texture (`break`) between the unison in eighth notes and the *triple hammer blow* (`hammer`) that accentuates the dominant chord on D.

Table 1

Feature tallies for sections of the sonata form on the 32 movements of the corpus of Mozart string quartet movements in sonata form (see Section 4.1). The table shows, for each feature, the number of quarter notes where this feature occurs followed by its number of occurrences on quarter notes labeled as each of the sections in the reference annotation A, as well as, in gray, its expected number should the feature be random or uniformly distributed across the quarter notes. Bold, italic, and the ≫ and ≪ symbols indicate an estimation of the significance of their presence or absence compared to all the other sections as well as to adjacent sections. The total numbers of quarter notes can differ slightly from the sums of the different sections due to rounding of non-integer lengths on some sections (see Figure 3).

		Status
Features	quarters	Intro			P			TR			MC			S			C			TC			d			Dev			RT		…
`pat:P`	2448	35	20	≪	858*	193	≫	341*	235		15	13	≫	23*	250	≫	0*	200		0*	13		0*	12		10*	385		8*	50
`pat:S`	3008	0*	25		0*	237	≪	482*	289		36*	16		686*	308	≫	304*	246		6	16		0*	14		8*	473		0*	61
`ton:I`	4491	41	38		553*	354	≫	*311**	432		22	23		*229**	460		*222**	367		18	24	≫	0*	22	≪	*360**	707	≪	115	91
`ton:II`	510	0	4		18*	40	≪	74	49		12*	2		72	52		79*	41		1	2		14*	2	≫	107	80		2	10
`ton:III`	479	0	4		27	37		54	46		4	2		70	49		64*	39		8	2		5	2		125*	75		9	9
`ton:IV`	1734	31*	14		186*	136	≫	*104**	166		1	9		34*	177		36*	141	≪	21	9		4	8		*168**	273		25	35
`ton:V`	2514	0*	21		41*	198	≫	467*	242		36*	13		683*	257		468*	205	≫	1*	13		4	12		*327**	396		70	51
`ton:VI`	479	6	4	≫	12*	37	≫	54	46		2	2		98*	49		66*	39		0	2		2	2	≫	90	75	≫	0*	9
`ton:VII`	386	9	3		20	30		30	37		0	2		14*	39		23	31		4	2		0	1		94*	60		1	7
`ton:i`	892	3	7		84	70		52*	85		3	4		16*	91		23*	72	≪	12	4		22*	4		148	140	≪	42*	18
`ton:ii`	534	4	4		33	42	≫	10*	51		0	2		13*	54		17*	43		1	2		0	2		156*	84		8	10
`ton:iii`	356	6	3		5*	28	≪	70*	34		4	1		68*	36		46	29		0	1	≪	12*	1	≫	48	56		14	7
`ton:iv`	349	12*	2		6*	27		0*	33		0	1		16	35	≪	51*	28		0	1		8	1		114*	54		18	7
`ton:v`	460	0	3		22	36	≪	69	44		5	2		38	47		8*	37		3	2		1	2		162*	72	≫	2	9
`ton:vi`	1052	0	8		46*	83		73	101		2	5		112	107		51*	86	≪	14	5	≫	0	5	≪	368*	165	≫	0*	21
`ton:vii`	187	3	1		0*	14	≪	22	18		0	0		21	19		36*	15		0	1		4	0		14	29		0	3
`cad:PAC`	416	4	3		20	32		22	40	≪	9	2		48	42		72*	34		1	2		0	2		29*	65		4	8
`cad:rIAC`	142	2	1	≫	16	11		8	13		3	0		15	14		9	11		0	0		0	0		29	22		1	2
`harm:#`	144	2	1		13	11		18	13		6	0	≫	7	14		5	11		1	0		1	0		27	22		1	2
`harm:7`	1122	4	9		49*	88	≪	116	108		0	5		68*	115		86	91	≪	18*	6		3	5		271*	176		17	22
`ped`	971	10	8		116*	76	≫	76	93		0	5		42*	99		66	79		1	5		0	4		186	152		20	19
`rest`	331	6	2		35	26	≫	11*	31	≪	12*	1	≫	15	33	≪	38	27		4	1		2	1		39	52	≪	18	6
`seq`	1254	24	10		57*	99		61*	120		2	6		95	128		60*	102		3	6	≪	29*	6	≫	420*	197	≫	0*	25
`unison`	685	16	5		91*	54	≫	43	65		7	3		43	70		59	56	≪	24*	3		27*	3	≫	68*	107		12	13
`break`	482	1	4		24	38		50	46	≪	12*	2	≫	36	49		47	39		5	2		1	2		74	75		9	9
`hammer`	268	0	2		14	21		8*	25	≪	14*	1	≫	49*	27		20	21		0	1		0	1		52	42		7	5
Total	14318	122			1130			1378			76			1468			1171			78			71			2255			292		…

Table 1 (continued)

Feature tallies for sections of the sonata form on the 32 movements of the corpus of Mozart string quartet movements in sonata form.

		States
Features	quarters	…	r			P’			TR’			MC’			S’			C’			TC’			Coda
`pat:P`	2448		5	5	≪	770*	184	≫	317*	243		15	12	≫	20*	264	≫	0*	218		0*	17		32*	125
`pat:S`	3008		0	6		1*	227	≪	444*	299		34*	15		692*	324	≫	307	269	≫	0*	20		7*	153
`ton:I`	4491		20	10		582*	339	≫	524*	447		44*	23		640*	484		497*	401	≪	17	31	≪	295*	229
`ton:II`	510		0	1		16*	38		28	50		5	2		25*	54	≪	56	45		0	3		0*	26
`ton:III`	479		1	1		18	36		24*	47		0	2		16*	51	≪	46	42		6	3	≫	1*	24
`ton:IV`	1734		0	3		156	130		230*	172		9	9		269*	187		256*	155		16	12		188*	88
`ton:V`	2514		4	5		36*	189	≪	*132**	250		13	13		*112**	271		77*	224		10	17		34*	128
`ton:VI`	479		3	1		18	36		35	47		0	2		58	51		27	42		0	3		7*	24
`ton:VII`	386		0	0		17	29	≪	80*	38		3	2		35	41		27	34	≪	12*	2	≫	18	19
`ton:i`	892		5	2		115*	67		148*	88		10	4		109	96		46*	79	≪	21*	6	≫	32	45
`ton:ii`	534		0	1		44	40		64	53		2	2		81	57		36	47		8	3		58*	27
`ton:iii`	356		0	0		9*	26		18	35		0	1		26	38		28	31		2	2		0*	18
`ton:iv`	349		0	0		9*	26		9*	34		0	1		23	37	≪	78*	31		0	2		4	17
`ton:v`	460		0	1		24	34		43	45		0	2		33	49		27	41		2	3		21	23
`ton:vi`	1052		0	2		57	79		77	104		2	5		104	113		68	94		2	7		77	53
`ton:vii`	187		0	0		0*	14	≪	34	18		1	0		23	20		17	16		7	1		5	9
`cad:PAC`	416		0	0		26	31		25	41		8	2		46	44		73*	37		1	2		28	21
`cad:rIAC`	142		0	0		18	10		5	14		1	0		17	15		9	12		0	0		9	7
`harm:#`	144		0	0		14	10		14	14		4	0		16	15		7	12		2	1		6	7
`harm:7`	1122		1	2		53*	84		93	111		0	5		100	121	≪	196*	100		12	7		35	57
`ped`	971		1	2		131*	73	≫	67	96		0	5		47*	104	≪	84	86		2	6	≪	120*	49
`rest`	331		5	0		35	24	≫	14	32	≪	14*	1	≫	16	35		33	29		3	2		31	16
`seq`	1254		0	2		58*	94	≪	172*	125		3	6		118	135		120	112		0	8		32*	64
`unison`	685		8*	1		96*	51	≫	44	68		7	3		43*	73		49	61	≪	27*	4	≫	22	35
`break`	482		3	1		24	36		52	48	≪	14*	2	≫	37	52		52	43		5	3		36	24
`hammer`	268		1	0		14	20		8*	26	≪	12*	1	≫	46	28		16	23		0	1		10	13
Total	14318	…	32			1081			1427			74			1545			1280			99			732

Table 2

Classification results, with F₁-measures of the five studied HMMs as well as of baseline models on the 14318 quarter notes of the corpus against the reference A. The confusion matrix is detailed for $M_{18}$ : Each column denotes the quarter notes of a section in the reference analysis, and the rows show how these quarter notes are classified (after cross-validation (c-val.)) by $M_{18}$ . Underlined values are discussed in the text.

Q₁₈	Intro	P	TR	MC	S	C	TC	d	Dev	RT	r	P’	TR’	MC’	S’	C’	TC’	Coda
Intro	0	154	30	3	·	·	·	·	·	·	·	·	·	·	·	·	·	·
P	36	*751*	238	12	4	·	·	·	·	·	·	·	·	·	·	·	·	·
TR	1	86	*175*	10	121	47	·	28	35	·	·	16	32	4	29	·	·	·
MC	1	4	19	3	6	·	·	·	·	·	·	·	·	·	1	·	·	·
S	1	·	608	27	*588*	357	2	·	11	·	·	·	·	·	30	9	·	·
C	1	2	40	6	364	*355*	10	·	202	·	5	6	·	·	·	38	·	0
TC	·	23	68	·	5	1	0	21	114	·	·	12	·	·	·	·	·	·
d	3	·	29	·	5	2	6	6	101	·	·	9	·	·	·	·	·	·
Dev	49	85	134	11	268	353	60	16	*1320*	87	3	67	56	2	62	110	32	36
RT	30	24	20	·	30	12	·	·	393	*141*	14	57	51	3	12	·	·	5
r	·	·	·	·	·	·	·	·	20	25	2	49	2	·	·	·	·	·
P’	·	·	1	·	1	·	·	·	0	·	7	*713*	282	11	35	14
TR’	·	·	·	·	·	·	·	·	·	·	·	46	*161*	4	174	3	·	1
MC’	·	·	1	·	1	·	·	·	·	·	·	2	18	8	6	·	·	3
S’	·	·	14	3	73	14	·	·	·	·	·	7	549	20	*471*	393	·	16
C’	·	·	·	·	·	25	·	·	21	15	·	58	197	10	353	*213*	32	58
TC’	·	·	·	·	·	4	·	·	34	·	·	9	45	3	42	49	11	12
Coda	·	·	·	·	·	·	·	·	2	24	·	28	32	8	328	463	24	*587*
quarter notes	122	1130	1378	76	1468	1171	78	71	2255	292	32	1081	1427	74	1545	1280	99	732
F₁ ( $M_{18}$ , c-val.)	0.00	0.69	0.18	0.05	0.38	0.32	0.00	0.05	0.53	0.26	0.03	0.66	0.18	0.15	0.30	0.19	0.07	0.53
F₁ (equal)	0.00	0.56	0.14	0.04	0.29	0.24	0.00	0.00	0.30	0.12	0.20	0.42	0.02	0.00	0.19	0.15	0.00	0.26
F₁ (fixed)	0.02	0.15	0.18	0.01	0.19	0.15	0.01	0.01	0.27	0.04	0.00	0.14	0.18	0.01	0.19	0.16	0.01	0.10

Q₁₄	P	TR	MC	S	C	d	Dev	RT	r	P’	TR’	MC’	S’	C’
quarter notes	1130	1378	76	1468	1250	71	2255	292	32	1081	1427	74	1562	2095
F₁( $M_{14}$ , c-val.)	0.76	0.17	0.05	0.38	0.28	0.05	0.58	0.25	0.03	0.66	0.18	0.15	0.28	0.56
F₁ ( $M_{14}^{*}$ )	0.66	0.35	0.03	0.27	0.26	0.04	0.16	0.14	0.02	0.29	0.33	0.09	0.29	0.61
F₁ (equal)	0.40	0.05	0.00	0.20	0.04	0.00	0.16	0.08	0.11	0.23	0.00	0.00	0.12	0.31
F₁ (fixed)	0.15	0.18	0.01	0.19	0.16	0.01	0.27	0.04	0.00	0.14	0.18	0.01	0.20	0.26

Q7	P	S	C	Dev	P’	S’	C’
quarter notes	2582	1471	1321	2580	2580	1565	2095
F₁ ( $M_{7}$ , c-val.)	0.65	0.37	0.25	0.68	0.54	0.33	0.54
F₁ (equal)	0.50	0.36	0.23	0.44	0.39	0.18	0.37
F₁ (fixed)	0.31	0.19	0.17	0.31	0.31	0.20	0.26

Q3	Exp	Dev	Rec
quarter notes	5374	2580	6240
F₁ ( $M_{3}$ , c-val.)	0.76	0.57	0.85
F₁ (equal)	0.41	0.30	0.68
F₁ (fixed)	0.55	0.31	0.61

Table 3

Number of boundaries predicted exactly or within one measure (+), within between 2 and 3 measures (=), beyond 3 measures (–) or not predicted (!), compared to the reference analysis A. The bottom part of the table shows results obtained with a subset of features.

	main boundaries (total: 124)				all boundaries
	+	=	–	!	+	=	–	!
$M_{14}^{*}$	23	4	54	43	68	21	154	115
$M_{18}$	34	17	53	20	90	45	147	104
$M_{14}$	31	16	56	21	87	38	146	87
$M_{7}$	35	12	61	16	70	15	101	30
$M_{3}$	16	8	40	0	46	8	42	0
$M_{18}$ , no `pat:P/pat:S`	13	7	97	7	32	29	229	96
$M_{18}$ , no `ton:*`	3	11	100	10	32	31	236	87
$M_{18}$ , no `cad:*`	35	16	57	16	90	40	159	97
$M_{18}$ , only `ton:*`	3	8	104	9	24	27	247	88
$M_{18}$ , no break features	33	12	61	18	85	36	168	97

Detection precision (relative to the reference analysis A) of the five HMMs. Boundaries are predicted exactly or within 1 measure (green, + on Table 3), within between 2 and 3 measures (blue, =), more than 3 measures (red, –), or not predicted at all (gray, !). The lines at the left show the numbers of the spurious sections falsely predicted by the models.

Comparison between the reference analysis A (top) and the predicted analysis by $M_{18}$ (bottom) on six string quartet movements.

Learning Sonata Form Structure on Mozart’s String Quartets

Figures & Tables

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Table 1

Table 1 (continued)

Table 2

Table 3

Figure 6

Figure 7

Paradigm

My account