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commute

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Journal Article
Journal of Music Theory (2010) 54 (2): 143–177.
Published: 01 October 2010
...Robert Peck In Generalized Musical Intervals and Transformations (1987), David Lewin describes the structure of commuting groups for transformation groups that have simply transitive actions. We extend Lewin's notion to transformation groups that have any type of action, including merely transitive...
Journal Article
Journal of Music Theory (2004) 48 (1): 99–141.
Published: 01 April 2004
... a simply-transitive-group system. Part five outlines relationships between atonal theory’s system of transpositions and inversions, neo-Riemannian triadic systems, and Lewin’s simply-transitive-group systems.4 The last part of the article considers “non-commutative” GISs. Two of Lewin’s innovative...
Journal Article
Journal of Music Theory (2002) 46 (1-2): 57–126.
Published: 01 October 2002
... to G≥ minor is (3, the same element of Γ that happens to represent D≥ minor (under the canonical identifica- tion of (0, +) with C major). The following theorem makes this dual na- ture of the set Γ more explicit. 1.3. Theorem. The set Γ forms a commutative group with multiplica- tion defined...
Journal Article
Journal of Music Theory (2005) 49 (1): 109–140.
Published: 01 April 2005
... of parsimonious UTTs will do likewise.12 Fig- ure 1a illustrates this non-commutative behavior.13 The figure depicts two cycles alternating parsimonious UTTs 4, 8 with 9, 3 (or L alter- nating with R). The inner cycle matches D≤ minor with its 0, 0 (or P) sibling D≤ major. In both cycles, clockwise...
Journal Article
Journal of Music Theory (2005) 49 (1): 1–43.
Published: 01 April 2005
... system is a commutative group (Lewin 1987, 49). This is a key feature of his critique of our thinking about transposition.10 Accordingly, as we pursue “transposition” as a suitable theoretical framework for understanding the depth invoked by contexts such as Fig- ure 5, we must extend...
Journal Article
Journal of Music Theory (2006) 50 (1): 111–127.
Published: 01 April 2006
... of the GIS model in each case, adjusting them so as to characterize the particularity of the given phenomenon as sharply as possible. This might involve revisiting questions of commutativity and noncommutativ- ity as well as redefining the interactions between S...
Journal Article
Journal of Music Theory (2004) 48 (1): 1–24.
Published: 01 April 2004
... scs). In the analy- ses, I use the variables more freely. 11. G1 is commutative, not cyclic, and isomorphic to the Klein 4-group. 12. G2 is commutative, cyclic, and isomorphic to the cyclic group C6. The group gen- erator is either T4L or T8L. 13. 048 is identical to Boros’s (1990, 37...
Journal Article
Journal of Music Theory (2007) 51 (2): 277–332.
Published: 01 October 2007
... of the arrow vectors. addition Since is arrow commutative, addition we can add the arrows in any order. For example, to find the Fourier transform of P = 0, 5, 15 is commutative, we can add the arrows in any order. For example, to {find the } with respect to a 6-cycle...
Journal Article
Journal of Music Theory (2011) 55 (1): 1–41.
Published: 01 April 2011
... to the beginning. The conclusion of Animation 1 makes explicit these two paths toward C≥/D≤. Because the UTTs in K(1, 11) are commutative operations (Lewin 1987, 206), and because of its replicative structure (Lind 2009), Figure 15 consti- tutes a “product...
Journal Article
Journal of Music Theory (2020) 64 (2): 241–281.
Published: 01 October 2020
... For example, E + 1 = B = 5 + 1 = 4; E + 2 = F = 5 + 2 = 3. All transpositions in this article are commutative. Table 1. The central portion of the line of fifths (Hook 2011: 85) Letter name Spelled pitch class Pitch class Generic pitch class Accidental index spc (spc) 12 (spc) 7 (spc...
Journal Article
Journal of Music Theory (2002) 46 (1-2): 153–205.
Published: 01 October 2002
..., in this case f preceding g, is important since the full group TTO24 is not commutative. 2.4 L-relation (L-class) In general, the J-relation is not symmetric. However, certain pairs may be related in both directions, by J-relations for suitable f’s. We will call these L-relations. However, it requires...
Journal Article
Journal of Music Theory (2015) 59 (2): 273–320.
Published: 01 October 2015
... groups are commutative, allow for 67  Gollin (2007, 159) makes a similar claim, remarking that paths formed by the combination (words) of generators, “Bartók exploits the ordered, pairwise distribution of pitch and yield the following relations: if f a = sb and pa = sc, then classes of the (3, 5, 3...
Journal Article
Journal of Music Theory (1999) 43 (1): 21–82.
Published: 01 April 1999
... produces a partition indicates—practically rather than logically— that p ∨ q is indeed a partition. Because the definitions and algorithms for partition meet and join are symmetric so far as p and q are concerned, partition meet and join will be commutative: that is, p ∧ q = q ∧ p, and p ∨ q = q ∨ p...