34.4: Twelve-Tone Matrix
- Page ID
- 117608
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)One tool analysts create to analyze a twelve-tone composition is a twelve-tone matrix, which shows all 48 row forms in a 12-by-12 grid. Below is a matrix for the row we’ve been dealing with in this chapter.
| I0↓ | I1↓ | I2↓ | I7↓ | I5↓ | I2↓ | I4↓ | I3↓ | I10↓ | I9↓ | I11↓ | I8↓ | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| P0→ | C | D♭ | G♭ | G | F | D | E | E♭ | B♭ | A | B | G♯ | ←R0 |
| P11→ | B | C | F | G♭ | E | D♭ | E♭ | D | A | A♭ | B♭ | G | ←R11 |
| P6→ | G♭ | G | C | D♭ | B | A♭ | B♭ | A | E | E♭ | F | D | ←R6 |
| P5→ | F | G♭ | B | C | B♭ | G | A | A♭ | E♭ | D | E | C♯ | ←R5 |
| P7→ | G | A♭ | D♭ | D | C | A | B | B♭ | F | E | G♭ | E♭ | ←R7 |
| P10→ | B♭ | B | E♭ | F | E♭ | C | D | D♭ | A♭ | G | A | F♯ | ←R10 |
| P8→ | A♭ | A | D | E♭ | D♭ | B♭ | C | B | F♯ | F | G | E | ←R8 |
| P9→ | A | B♭ | E♭ | E | D | B | C♯ | C | G | G♭ | A♭ | F | ←R9 |
| P2→ | D | E♭ | A♭ | A | G | E | F♯ | F | C | B | D♭ | B♭ | ←R2 |
| P3→ | E♭ | E | A | B♭ | A♭ | F | G | G♭ | D♭ | C | D | B | ←R3 |
| P1→ | D♭ | D | G | A♭ | G♭ | E♭ | F | E | B | B♭ | C | A | ←R1 |
| P4→ | E | F | B♭ | B | A | G♭ | A♭ | G | D | D♭ | E♭ | C | ←R4 |
| ↑RI0 | ↑RI1 | ↑RI2 | ↑RI7 | ↑RI5 | ↑RI2 | ↑RI4 | ↑RI3 | ↑RI10 | ↑RI9 | ↑RI11 | ↑RI8 |
To construct a matrix, write the prime form from left to right in the top row, then write the inverted form from top to bottom in the left column.
| I0↓ | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| P0→ | C | D♭ | G♭ | G | F | D | E | E♭ | B♭ | A | B | G♯ | ←R0 |
| B | |||||||||||||
| G♭ | |||||||||||||
| F | |||||||||||||
| G | |||||||||||||
| B♭ | |||||||||||||
| A♭ | |||||||||||||
| A | |||||||||||||
| D | |||||||||||||
| E♭ | |||||||||||||
| P1→ | D♭ | ||||||||||||
| E | |||||||||||||
| ↑RI0 |
From there, you can write the transpositions of the prime form, given the starting notes in the left column. One would continue with each transposition of the prime form until the matrix is complete.