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Tonal Pitch Space

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Tonal Pitch Space Mapping Tone Helixes to Cylindrical Lattices using Chiral Angles Hanlin Hu, Brett Park, David Gerhard – University of Regina, Canada Euler’s Tonnetz (1739) Riemann’s Triangular lattice (1914) Three types of cylindrical hexagonal tubes • Euler and Riemann tried to map the harmonic arrangement of tones on laces. • According to the dual of Riemann’s triangular lace the hexagonal isomorphic layouts are generated to consistently present iden;cal musical constructs such !!" n, m as chords regardless the beginning pitch. • A chiral tube , is defined by a chiral vector , indicang the orientaon of ( ) !!" !!" Ch!!" !!" !!" • By curling a planar the hexagonal lace on the tube: , where and are two Ch = n•a1+ m•a2 a1 a2 hexagonal lattice which in basis vectors separated by 60°. specific direc;on, along the " % 1 3m edges of hexagons, the • The laces grouped by dis;nguishing chiral angle , as the angle θ = tan− $ ' m + 2n resul;ng sheet becomes a between chiral vector and the Zigzag direc;on. # & cylinder. • The vector that in isotone • Laces: “Armchair”, ; “Zigzag”, ; Other chiral, θ = 30° θ = 0° 0° <θ < 30° axis direc;on (a ray from Three types of hexagon lattice cuttings one note of par;cular pitch passes through all same notes) is the chiral vector, which looks like spiral on the cylinder. • Aim to map tone helixes we present cylindrical hexagonal laces which have been extensively studied in context of carbon nanotube. Hexagonal Lattice Cutting to implement Shepard’s Helical Model Chew’s Spiral Array Model (2000) Cutting required for Chew’s Model • According to Drobisch (1855)’s helix, Shepard introduced an equal-spaced helical model to arrange chromac pitches over a regular, symmetrical, transformaon-invariant surface and noted that this model is isomorphic. • Considering in this model the pitch increases by semitones around the spiral, comple;ng one turn of the spiral once per octave, the chiral angle can be " % 1 3•12 calculated as θ = tan− $ ' = 23.2° # 12 + 2 & • The adjacent hexagons in one direc;on corresponding to semitones shown in • Chew explored an abstract spiral model for mapping Tonnetz-based representaons the hexagonal lace cung. to the helix, providing an iden;cal distance between each perfect fi[h and another Shepard’s Helical Model(1982) • Shepard‘s model allows a differen;al two between major / minor third by fixing angle with 90°. stretching or shirking of ver;cal extent • Since the chiral vector for this arrangement of the notes is not circumferen;al to the of an octave of the helix relave to its resul;ng tube, chew’s model cannot be implemented by adjacent hexagons. diameter. So, we can accomplished this • If we make a modificaon by rotang and mirroring the model so that major thirds by allowing duplicates of the cung, are along the spiral, and perfect fi[hs are in the ver;cal direc;on, it works properly. resul;ng in a larger-diameter tube. • It may also possible to implement Chew’s original pitch helix by allowing addi;onal notes to appear between each note on the helix, and removing or ignoring the Resulting Chiral Tube (Shepard’s) interspaced notes. Resulting Chiral Tube (Modified Chew’s) Cutting to implement modified Chew’s Spiral Array Model [1] L. Euler, Tentamen novae theoriae musicae ex certissismis [5] C. L. Krumhansl, “Perceptual Structures for Tonal Music,” Music harmoniae principiis dilucide expositae. Saint Petersburg Academy, 1739. Perception, vol. 1, no. 1, pp. 28–62, Fall 1983. [2] B. Park and D. Gerhard, “Discrete Isomorphic Complete- ness and a [6] E. Chew, Mathematical and Computational Modeling of Tonality, Theory Unified Isomorphic Layout Format,” in Proceedings of the Sound and and Applications, 1st ed., Springer. International Series in Operations Music Computing Conference, Stockholm, Sweden, 2013. Research and Management Science. Springer US, 2014, vol. 204. [3] F. Lerdahl, Tonal pitch space. Oxford Univ. Press, 2001. [7] P. von Janko, “Neuerung an der unter patentirten Kalviatur,” 1885. [4] R. Shepard,“ Geometrical approximations to the structure [8] L.-C. Qin, “Determination of the chiral indices (n,m) of carbon nanotubes by of musical pitch,” vol. 89(4), 7 1982. electron diffraction,” in Phys. Chem. Chem. Phys., 2007, vol. 9, pp. 31–48. .
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