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 idencal 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 direcon, along the " % 1 3m edges of hexagons, the • The laces grouped by disnguishing chiral angle , as the angle θ = tan− $ ' m + 2n resulng sheet becomes a between chiral vector and the Zigzag direcon. # & cylinder. • The vector that in isotone • Laces: “Armchair”, ; “Zigzag”, ; Other chiral, θ = 30° θ = 0° 0° <θ < 30° axis direcon (a ray from Three types of hexagon lattice cuttings one note of parcular 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, compleng 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 direcon 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 idencal distance between each perfect fih and another Shepard’s Helical Model(1982) • Shepard‘s model allows a differenal two between major / minor third by fixing angle with 90°. stretching or shirking of vercal extent • Since the chiral vector for this arrangement of the notes is not circumferenal to the of an octave of the helix relave to its resulng 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 fihs are in the vercal direcon, it works properly. resulng in a larger-diameter tube. • It may also possible to implement Chew’s original pitch helix by allowing addional 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

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