T&S Matrix transformations Matrix Equations for Twist & Shift Pitch Transformations on the Guitar Fretboard Neil J. Gunther Performance Dynamics Castro Valley, California 94552 March 25, 2024 © 2024 NJG March 30, 2024 1

T&S Matrix transformations Abstract We present the matrix representation for our Twist & Shift pitch transformations that define the underlying space group symmetry of the guitar fretboard. The basic matrix is the set of pitches belonging to all 6 strings within any 6-fret interval. A key point is that this pitch-matrix remains invariant under T&S transformations. This hidden symmetry determines such things as chord constellations, paths between enharmonic notes, and the relative position of individual notes within any constellation. T&S is intended to help guitarists understand how the 2D fretboard (matrix) processes music differently from single-note instruments (vector) like the piano and saxophone. © 2024 NJG March 30, 2024 2

T&S Matrix transformations Definitions Definition 1 (Fundamental Tuning). Strings 1, 2, . . . , 6 are tuned F4,C4,G3,D3,A2,E2, rather than standard tuning E4,B3,G3,D3,A2,E2. Definition 2 (Pitch Block). B is a symmetric matrix (Fig. 5) Bi,j , i, j = 1, 2, . . . , 6, of contiguous fundamental pitches between any 6 frets. B1 spans fret O (open) to fret V, across all 6 strings. Definition 3 (Virtual Block). B is a 6 × 6 matrix of complementary pitches that are unseen in B1 . They appear incrementally under the twisting action of Theorem 6. Definition 4 (Superblock). Superblock Q is a 12 × 6 matrix Q =   B B   (1) © 2024 NJG March 30, 2024 3

T&S Matrix transformations Twist and Shift Definition 5 (Shift). A discrete translation of the pitches in superblock Qk , by φ frets, is defined by Qk+1 = Qk + φ uuT , k = 1, 2, . . . (shifts) (2) where u is a unit column matrix and uT its tramspose. Definition 6 (Twist). The matrix twist operator, T, produces a cyclic permutation of the pitches in superblock Qk Qk+1 = Tk Qk , k = 1, 2, . . . (twists) (3) Remark 1. The number of discrete shifts along the fretboard and twists around the fretboard are both indexed by k because, as well shall see, the underlying group symmetry requires that they operate together. © 2024 NJG March 30, 2024 4

T&S Matrix transformations Theorem Theorem 1 (Pitch Invariant). Superblock Qk is invariant under k successive Twist and Shift (T&S) transformations, Qk + φ uuT − Tk Qk = 0 (4) Successive Qk pitch blocks execute a helical motion about a virtual axis lying parallel to the guitar truss rod (Fig. 1). Under (4), pitch block B1 retains the same pitches. However, as a result of the successive twisting motion (Fig. 3), B1 loses low strings as they seem to “disappear” off the low-pitch side of the fretboard, while new high-pitch strings seem to “appear” onto the high-pitch side of the fretboard, Remark 2. The helix “pitch” (not note frequency) or screw-thread distance is k = 12. After 12 T&S operations, the original block returns to the visible fretboard. In reality, an electric guitar only supports at most k = 5 blocks because of i) the logarithmically diminishing fret-spacing and, ii) the roughly two feet (60 cm) “scale length” for strings. © 2024 NJG March 30, 2024 5

T&S Matrix transformations Figure 1: Helical trajectory created by 12 successive T&S operations on block B1 (red). Similarly for the complementary block B1 (pale blue). These two complementary helices form the “DNA” of the guitar. © 2024 NJG March 30, 2024 6

T&S Matrix transformations Discussion The first block on the fretboard contains the following pitches B1 =              F4 Gb4 G4 Ab4 A4 Bb4 C4 Db4 D4 Eb4 E4 F4 G3 Ab3 A3 Bb3 B3 C4 D3 Eb3 E3 F3 Gb3 G3 A2 Bb2 B2 C3 Db3 D3 E2 F2 Gb2 G2 Ab2 A2              (5) Its complementary block contains pitches B1 =              B6 C7 Db7 D7 Eb7 E7 Gb6 G6 Ab6 A6 Bb6 B6 Db6 D6 Eb6 E6 F6 Gb6 Ab5 A5 Bb5 B5 C6 Db6 Eb5 E5 F5 Gb5 G5 Ab5 Bb4 B4 C5 Db5 D5 Eb5              (6) that are not visible collectively on the fretboard. © 2024 NJG March 30, 2024 7

T&S Matrix transformations Block B1 is the matrix comprising all 36 fundamental pitches (see Definition 1) across 6 strings in the first fret interval [O, V ]. Block Q1 is the superblock composed of B1 and B1 defined in (1). Remark 3. Translating a single pitch, e.g., middle C—denoted C4 in (5)—along the neck by 5 frets (perfect 4th interval), produces the pitch F4, which is higher than C4 due to shortening string-2. Similarly for any other pitch in the first column of block B1 . © 2024 NJG March 30, 2024 8

T&S Matrix transformations The Shift More generally, all the pitches in B1 can be translated collectively by applying (2) to produce the second block B2 on the fretboard. Bshift 2 =              Bb4 B4 C5 Db5 D5 Eb5 F4 Gb4 G4 Ab4 A4 Bb4 C4 Db4 D4 Eb4 E4 F4 G3 Ab3 A3 Bb3 B3 C4 D3 Eb3 E3 F3 Gb3 G3 A2 Bb2 B2 C3 Db3 D3              (7) Ignoring the redundancy between column 1 in B2 and column 6 in B1 (which gets removed by matrix concatenation), block B2 contains the correct pitches, across all 6 strings, in the fret interval [V, X] The same effect applies to the translation of virtual block B1 but, since those pitches also remain invisible on the fretboard, we don’t write them out explicitly. © 2024 NJG March 30, 2024 9

T&S Matrix transformations The Twist The same result can be attained by applying a Twist operation (3) to the superblock Q1 to produces Q2 after a single permutation. Qtwist 2 =                    · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Bb4 B4 C5 Db5 D5 Eb5 F4 Gb4 G4 Ab4 A4 Bb4 C4 Db4 D4 Eb4 E4 F4 G3 Ab3 A3 Bb3 B3 C4 D3 Eb3 E3 F3 Gb3 G3 A2 Bb2 B2 C3 Db3 D3                    (8) The action of T on Q1 is shown in (8). Clearly, the lower submatrix in (8) is identical to (7) but, the way it got there is very different. © 2024 NJG March 30, 2024 10

T&S Matrix transformations Qtwist 2 =                       E2 F2 Gb2 G2 Ab2 A2 · · · · · · · · · · · · · · · · · · Eb5 E5 F5 Gb5 G5 Ab5 Bb4 B4 C5 Db5 D5 Eb5 F4 Gb4 G4 Ab4 A4 Bb4 C4 Db4 D4 Eb4 E4 F4 G3 Ab3 A3 Bb3 B3 C4 D3 Eb3 E3 F3 Gb3 G3 A2 Bb2 B2 C3 Db3 D3                       (9) The twisting action makes it seem as though the original string-6 in block B1 , represented by row B6,j in superblock Q1 , has disappeared from the physical fretboard. In addition, a new string-1 (red) has appeared, seemingly out of nowhere. The action of T, however, gets applied to the entire superblock Q1 , not just the lower submatrix, Bi,j in Fig. 2. Because the permutation action is cyclic within the superblock: 1. the original string-6 in B1 has wrapped around to the top row © 2024 NJG March 30, 2024 11

T&S Matrix transformations (green) of the B1,j submatrix and becomes invisible. 2. the virtual string-6 in row B6,j has been displaced downward into row B1,j (red) and becomes visible as the new string-1. Figure 2: Movement of superblocks Q1 , Q2 , Q3 relative to physical fretboard. Cyclic wrap-around is not shown © 2024 NJG March 30, 2024 12

T&S Matrix transformations Proof Proof 1. From the foregoing discussion we see that starting with B2 in (7), a reverse Twist operation will reproduce B1 in (5). More generally, a positive Shift of any Bk increases the collective pitches by a perfect 4th in Bk+1 , while a negative Twist applied to Bk+1 decreases its collective pitches by a 4th and recovers Bk . Thus, any pitch block Bk is invariant under the combined action of the T&S operations. Each discrete T&S transformation preserves this invariant because it corresponds to moving by equal perfect fourth intervals both along the fretboard and across neighboring pairs of strings. This block-wise diagonal trajectory belongs to the 3D helical space group symmetry in Fig. 1. Remark 4. The “B” in B-string stands for: Breaks this symmetry. © 2024 NJG March 30, 2024 13

T&S Matrix transformations Application The T&S concept is more than a mathematical curiosity. It describes the inherent symmetry that determines how the guitar processes music very differently from single-note instruments, such as the piano or saxophone. In fact, this unique guitar symmetry can be made audible. Fig. 3 depicts the guitar fretboard from the viewpoint one has when playing the instrument. Each diagram shows the same 4-note chord shape located in successive pitch blocks. The top left diagram corresponds to physical manifestation of B1 . It involves 2 open strings (G and C) together with an E note on string-4. It therefore sounds as a C major chord. Recognizing this helical motion of pitches on the fretboard can be the starting point for a novel way to understand how notes are arranged on the fretboard and how to navigate them: something that every neophyte guitarist finds daunting because of its 2D matrix layout. © 2024 NJG March 30, 2024 14

T&S Matrix transformations Figure 3: T&S motion of invariant C chord across blocks B1 to B5 © 2024 NJG March 30, 2024 15

T&S Matrix transformations The top-right diagram shows the same chord in B2 . On the guitar chord shapes are “moveable” along the fretboard, i.e., the same chord shape produces increasing pitch constellations corresponding to the ordered pitches of a given scale. Conversely, the same chord shape in Fig. 3 produces exactly the same pitches. The chord shape is shifted up the neck by 5 frets and twisted downward by one string position. It moved up by a perfect 4th and down by a perfect 4th: so, the chord pitches in B2 are identical to B1 . Under T&S, transformations the same chord shape remains pitch invariant in a completely different position. This effect continues in the remaining diagrams as the chord moves diagonally between each block until it is reduced to a single note (C) on string-6 at fret 20, and finally disappears off the fretboard. This is the audible manifestation of the T&S invariant under the helical symmetry. The fretboard acts like a window that only allows us to see a partial 2D projection of the otherwise 3D helical motion. © 2024 NJG March 30, 2024 16