Matrices and Vectors

	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	T	U	V	W	X	Y	Z	AA	AB	AC	AD	AE	AF	AG	AH	AI	AN	AO	AP	AQ	AR	AS	AT	AU	AV	AW	AX	AY
1
2	MATRICES and VECTORS									[Click on the "Toolkit" tab below for some basic tools to explore sets, transformations, and complements.]
3
4	Matrices are arrays of numbers. Two kinds of matrices -- addition matrices and inversion matrices -- help us to display an abundance of information about the relationships between pitch-classes (or beat-classes) in the analysis and composition of post-tonal (and post-metric, and/or "phase-shift"-oriented) music. This worksheet focuses on how useful vectors can be drawn from these matrices. This resource doesn't stand alone: consider this a supplement to lectures and in-class demonstrations.															THE BASIC IDEA:
5
6																ADDITION MATRICES: find common tones under inversions (at varied indices/axes/transpositions)
7																INVERSION MATRICES: find common tones under transposition
8
9
10																Why would I want to do this?
11		12-tone inversion matrix:															1. In pc-set-oriented ("free atonal") composition -> matrix vectors allow you to plan "overlapping" pc-sets so that you can saturate a piece with a particular chord type. Lots of overlap helps slow the "harmonic movement" through the aggregate, less overlap speeds it up. Yay harmonic unity! You can also do this intuitively, but you're likely to miss golden opportunities.															7
12
13			11	1	5	3	9	8	6	10	7	4	0	2
14
15			0	2	6	4	10	9	7	11	8	5	1	3			2. In 12-tone serialism -> matrix vectors allow you to build a row with combinatorial properties, by using the vector position marked "0" (a transformation with zero common tones). Combinatoriality is the property by which one part of the row is essentially a transformation (usually a transposed inversion) of the other. That means you get the Schoenbergian poetic value of reaching from H ("hexachord") to H-c ("hexachord complement") in two different ways...(a) by horizontal or "melodic" motion through the aggregate, and (b) by vertical transformations (transposition/inversion). Think of (a) as a evolving/developing tune, and think of (b) as the "vertical" relationship between two keys or two versions of a melody. Combinatoriality lets Schoenberg, Webern, and Berg get a special kick out of unity and contrast going simultaneously.															Enter another 12-tone row here to create a matrix automatically.
16			10	0	4	2	8	7	5	9	6	3	11	1																		10	1	5	3	11	8	9	6	7	4	0	2
17			6	8	0	10	4	3	1	5	2	11	7	9
18			8	10	2	0	6	5	3	7	4	1	9	11																		0	3	7	5	1	10	11	8	9	6	2	4
19			2	4	8	6	0	11	9	1	10	7	3	5																		9	0	4	2	10	7	8	5	6	3	11	1
20			3	5	9	7	1	0	10	2	11	8	4	6																		5	8	0	10	6	3	4	1	2	11	7	9
21			5	7	11	9	3	2	0	4	1	10	6	8																		7	10	2	0	8	5	6	3	4	1	9	11
22			1	3	7	5	11	10	8	0	9	6	2	4																		11	2	6	4	0	9	10	7	8	5	1	3
23			4	6	10	8	2	1	11	3	0	9	5	7																		2	5	9	7	3	0	1	10	11	8	4	6
24			7	9	1	11	5	4	2	6	3	0	8	10			3. In beat-class sets -> matrix vectors allow you to superimpose rhythms in interesting ways. Beat-classes (bcs) are specific positions in the "meter" of a piece, and beat-class sets (bc-sets) are like particular characters of rhythm that can be transformed. Milton Babbitt and Donald Martino have used beat-class sets in the same way Schoenberg used pc-sets. Steve Reich and other "minimalist" composers have devised tesselations — superimpositions of repeated rhythms that neither occupy common beat-classes (i.e. they don't 'touch'), nor leave any beat-classes unoccupied: thus while the rhythm by itself works as a motivic riff, two or more combined phase-shifts (Tn) of the rhythm form the "aggregate" of beat-classes...together they completely fill the metric space exactly once.															1	4	8	6	2	11	0	9	10	7	3	5
25		(P11)	11	1	5	3	9	8	6	10	7	4	0	2																		4	7	11	9	5	2	3	0	1	10	6	8
26			9	11	3	1	7	6	4	8	5	2	10	0																		3	6	10	8	4	1	2	11	0	9	5	7
27						(RI4)																										6	9	1	11	7	4	5	2	3	0	8	10
28						(RI4)																										10	1	5	3	11	8	9	6	7	4	0	2
29																																8	11	3	1	9	6	7	4	5	2	10	0
30
31	1	Enter a hexachord in the yellow squares below:
32																HOW TO MAKE & USE ADDITION MATRICES
33											(Inversion T			4	)	1		Find the TnI set-class (TnI prime form) of a set.
34
35	PH:		3	4	2	8	11	2		1	0	2	8	5	2	2		Make a matrix using the prime form on both the top and left edges.
36	ascending order:		2	2	3	4	8	11		0	1	2	2	5	8	3		Count occurences of each value and write them in a vector.
37
38	2	Enter a normal-form set in the yellow squares below:
39	normal form:		9	10	11	2	3	4		7	6	5	2	1	0	4		Find the "normal inversion" of your original set: start from its first member and trace the intervals in the opposite direction. (In other words, invert it around the "axis" of its first member.)
40	Tn set class:		0	1	2	5	6	7		0	1	2	2	5	8
41	TnI set class:		(choose the most compressed of the above Tn set classes)
42			3
43																5		Careful in this step!: (I always mess this one up.) The vector refers to transpositions of the "normal inversion" you created in step 4. (It also refers to the normal inversion of any other transposition of the original set.) But you are measuring common tones between that transposition and the original set, not the normal inversion.
44	HOW TO MAKE & USE INVERSION MATRICES
45	1		Find the TnI set-class (TnI prime form) of a set.
46	2		Make a matrix using the prime form on top, and inversion (12-x) on the left edge.
47	3		Count occurences of each value in the matrix, and write them in a vector.
48	4		The number in each position of the vector represents the number of common tones you will find between an original version of the set and a transposition by the value associated with that position.													ADDITION MATRICES							Note the symmetry!
49																	TnI of H-->									TnI of HI-->
50																	0	1	4	5	11	2				0	2	4	6	8	9
51																	1	2	4	6	8	10				2	4	6	8	10	11
52																	4	4	6	8	10	0				4	6	8	10	0	1
53																	5	6	8	10	0	2				6	8	10	0	2	3
54																	11	8	10	0	2	4				8	10	0	2	4	5
55																	2	10	0	2	4	6				9	11	1	3	5	6
56		INVERSION MATRIX															No elevens in this matrix ^									No sevens in this matrix ^
57			Note the diagonal zeros here, and in the 12-tone matrix example above.														(Thus T7I yields the complement)									(Thus T11I yields the complement)
58
59						0	1	3	5	7	9					ADDITION VECTOR
60						11	0	2	4	6	8							Tells you how many common tones available under each inversion.
61						9	10	0	2	4	6							Count the occurences of each number in the matrix:
62						7	8	10	0	2	4							T:
63						5	6	8	10	0	2					for H:		1	2	3	4	5	6	7	8	9	10	11
64						3	4	6	8	10	0							2	3	2	4	2	4	2	4	2	5	0
65
66																Where H=(89e135), transpositions of the "Normal Inversion" (e13578) are:
67		INVERSION VECTOR									(Also called a "transposition vector".)						11	0	1	2	3	4	5	6	7	8	9	10
68				Tells you how many common tones available under each transposition.													1	2	3	4	5	6	7	8	9	10	11	0
69				Count the occurences of each number in the matrix:													3	4	5	6	7	8	9	10	11	0	1	2
70				T:													5	6	7	8	9	10	11	0	1	2	3	4
71				1	2	3	4	5	6	7	8	9	10	11			7	8	9	10	11	0	1	2	3	4	5	6
72				1	4	2	4	2	4	2	4	2	4	1			8	9	10	11	0	1	2	3	4	5	6	7
73
74		Now check your work, for example…																T:
75		...where H is the hexichord in bold on the left, transpositions are as follows:														for HI:		1	2	3	4	5	6	7	8	9	10	11
76		0	1	2	3	4	5	6	7	8	9	10	11					2	4	2	4	2	5	0	5	2	4	2
77		1	2	3	4	5	6	7	8	9	10	11	0
78		3	4	5	6	7	8	9	10	11	0	1	2			Where HI=(e13578), tranpositions of the "Normal Inversion" (23579e) are:
79		5	6	7	8	9	10	11	0	1	2	3	4				2	3	4	5	6	7	8	9	10	11	0	1
80		7	8	9	10	11	0	1	2	3	4	5	6				3	4	5	6	7	8	9	10	11	0	1	2
81		9	10	11	0	1	2	3	4	5	6	7	8				5	6	7	8	9	10	11	0	1	2	3	4
82																	7	8	9	10	11	0	1	2	3	4	5	6
83		(Note that vector predict the number of common tones between H and any transposition.)															9	10	11	0	1	2	3	4	5	6	7	8
84																	11	0	1	2	3	4	5	6	7	8	9	10
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100