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Braid Group
Canonical Transformation
Congruence Subgroup
Free Group
Group Action
Linear Transformation
Modular Group
Rewrite Rule
Non Commutative
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Musical intervals and special linear transformations
Musical intervals and special linear transformations,10.1080/17459730701375026,Journal of Mathematics and Music,Thomas Noll
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Musical intervals and special linear transformations
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Citations: 8
)
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Thomas Noll
This paper presents a transformational approach to musical intervals with particular focus on their constitutive role for wellformed scales. These scales have the property that their binary stepinterval pattern is maximally even. Transposition classes of wellformed scales are therefore characterized by two step intervals and their characteristic binary pattern, or, more abstractly, by four numbers: two step intervals and two associated multiplicities. The proposed transformational approach therefore studies group actions (1) on interval pairs, (2) on multiplicity pairs, such that the two intervals with their associated multiplicities form two pairs of canonically conjugated variables, and (3) on binary cycle words. The group is the same in all three cases: the
modular group
Γ=SL(2, ℤ). For any free commutative interval group G we have a faithful action of Γ on G×G through transvections. This action has a refinement in terms of an action of the
braid group
B3 on the product F×F of any noncommutative
free group
F with itself. The noncommutative interval group considers intervals as pathways rather than sums. The action of the
modular group
on ℝ through canonical transformations is given in terms of a representation of this group through symplectic 4×4matrices. This left action can be comfortably rewritten in terms of a right action on 2×2matrices. The submonoid SL(2, ℕ) exemplifies the Stern–Brocot tree and provides a link to the classical theory of wellformed scales. We recapitulate how the processes of approximating a scale generator g through its semiconvergents and of generating smaller and smaller step interval sizes are transformationally interconnected. The action of SL(2, ℤ) on cycle words with directed letter is based on parallel rewriting rules. The maximally even patterns form exactly one orbit of this group action: the orbit generated by the oneletterword . Looking to the future, the author indicates how the extension of this theory can be musically explored by the technology of Sethares spectra and how new questions arise from an inspection of intrinsic properties of the
modular group
Γ.
Journal:
Journal of Mathematics and Music  J MATH MUSIC
, vol. 1, no. 2, pp. 121137, 2007
DOI:
10.1080/17459730701375026
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Citation Context
(5)
...In
16
, the two dimensional special linear transformations are used to explore the wellformed scales which are shown to be related to secondary convergents...
Hermann Heßling
.
On the Euler scale and the μEuclidean integer relation algorithm
...The present article implements an announcement, which I made in an earlier transformational account to scale theory [
7
, footnote on p...
Thomas Noll
.
Ionian theorem
...Thomas Noll
16
linked symplectic transformations of integer pairs with continued fractions and developed in
17
a transformational model for
scale states
based on the Stern–Brocot tree (see also
12
13
)...
Franck Jedrzejewski
.
Generalized diatonic scales
...I thank Franck Jedrzejewski for an initial reference in word theory and Val´erie Berth´e for helpful comments on [
34
]...
Thomas Noll
.
STURMIAN SEQUENCES AND MORPHISMS A MUSICTHEORETICAL APPLICATION
...T. Noll introduced this term in [
10
] and established a recursive construction of these words, which represent the cyclic stepinterval patterns of maximally even sets...
...that transforms paths in the Christoffel tree into matrices of the monoid generated byL andR, which is isomorphic toSL(2;N); (See the proof given by T. Noll in [
10
])...
...Observe that the applicationSL(2;N) ! Q+ given by a c b d = a+d c+d (called mediant ratio by T. Noll in [
10
]) transforms each matrix Aw into juj jvj ;which is the slope of the dual word w . Denoting by A w the matrix d c b a (flipped the main diagonal), then (A w)=...
...3. The transformational theory for wellformed scales as proposed in [
10
] is mainly covered by the theory of Sturmian morphisms...
...It is therefore challenging to review the musictheoretical interpretations in [
10
] within the full algebraic context of Sturmian morphisms...
Manuel Domínguez
,
et al.
WF Scales, ME Sets, and Christoffel Words
References
(20)
The Grouptheoretic Description of 12Fold and Microtonal Pitch Systems
(
Citations: 76
)
G. J. Balzano
Published in 1980.
Representations of Space and Conceptions of Distance in Transformational Music Theories
(
Citations: 7
)
Edward Gollin
Published in 2000.
Introduction to Elliptic Curves and Modular Forms
(
Citations: 343
)
N. Koblitz
Published in 1984.
Gruppen und Kategorien in der Musik
(
Citations: 19
)
Guerino Mazzola
Published in 1985.
Geometrie der T¨on
(
Citations: 12
)
Guerino Mazzola
Published in 1990.
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Citations
(8)
On the Euler scale and the μEuclidean integer relation algorithm
Hermann Heßling
Journal:
Journal of Mathematics and Music  J MATH MUSIC
, vol. 5, no. 3, pp. 195215, 2011
Ionian theorem
(
Citations: 4
)
Thomas Noll
Journal:
Journal of Mathematics and Music  J MATH MUSIC
, vol. 3, no. 3, pp. 137151, 2009
Generalized diatonic scales
(
Citations: 2
)
Franck Jedrzejewski
Journal:
Journal of Mathematics and Music  J MATH MUSIC
, vol. 2, no. 1, pp. 2136, 2008
STURMIAN SEQUENCES AND MORPHISMS A MUSICTHEORETICAL APPLICATION
(
Citations: 2
)
Thomas Noll
Published in 2008.
Modes, the Height WidthDuality, and Divider Incidence  Draft
(
Citations: 2
)
David Clampitt
,
Thomas Noll