IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2019

Attractor Editing for Music Expression in Sophia Sound---Including Mapping Modulation Method in Cellular Flow --Mamoru Tanaka1, Masaki Bandai 1 and Yoshifumi Nishio2
Sophia University1

Tokushima University2

The expression technology including AI with machine learning has been attractive inorder to realize better timbre expression of the three elements of sound. A basic
technique is pitch shift processing, if PCM technique recording sound of expensive
live musical instruments is not used. The pitch shift processing is a process of shifting
the frequency set of the periodic function F(x) to the frequency direction in accordancewith the change of each pitch frequency fn of the note sequence {Nn }. As a direct method,
there is a frequency domain analysis in which a frequency set of the function F(x) is got
by Fourier transform, the frequency set is shifted, and an inverse Fourier transform isperformed to obtain a time domain function. There is also a time domain analysis inwhich the main period T of the function F(x) is changed so as to match the period
corresponding to each pitch frequency fn of the note sequence {Nn }. Also, the FM soundsource technique uses a method in which the pitch frequency fn is equivalent to thecarrier frequency f for the modulated frequency g with factor m as 𝑠𝑖𝑛(2𝜋𝑓𝑡 + 𝑚 ∗𝑠𝑖𝑛(2𝜋𝑔𝑡)).
In general, the function F(x) has a problem that the period T is not expressed explicitly.
We introduce the mapping modulation method as a new time domain analysis method.
The mapping modulation method is generally used as a nonlinear mapping function
𝑦𝑛 = 𝐹(𝐺(𝑡𝑛 )).
Here, for example, let{tn | n = 0,1,2, ..., tmax-1}
be a time sequence of one note, then, the nonlinear mapping method is expressed in a
discrete form as𝑟𝑎𝑡𝑒

xn = G(tn) = 2𝜋𝑓 sin-1(2πftn),
andyn = F(xn) in F(tn)≥0,otherwise, yn = F(tn) in F(tn)<0,- 21 -

in the region xn≥ 0, or

yn = F(-xn) in F(tn)≤0, otherwise, yn = F(tn) in F(tn)>0,
in the region xn<0.
The algorithm includes a delay element and a modulation factor (relative intensity ofthe function F with respect to the function G). Also the algorithm has a network
structure including a Fourier series expansion. Nonlinear function F is an arbitrary
function, especially a periodic function. The important thing is that the nonlinear
function F can be generated by called Cellular Flow (SPICE-like SDP circuit simulator)
which includes two-variable automatic differential expressions (calculation graphs asbasic technology of Google tensor flow), RLC elements, negative resistance, gyrators, etc.
The gyrator neuron is expressed by using activation function f(x) asg11(ii1 ) =f [(g221g12 v1 −v2g22 ) (v3 −v4)].

Each conductance gij for forward propagation can be determined in back propagationin the cellular flow learning process same as calculation graphs of the tensor flow. Error
propagation between teacher and output signals (v1 − v2 ) can be done by changing fromvoltage to current. The neuron element is included in the Jacobian matrix. TheFunction F is a waveform in the time domain generated by the circuit analysis whichuses implicit numerical integration method. In order to reduce the number of times foranalysis, it is important to edit the structure of the attractor which is composed of theleft and right sound waveforms. The sequence of output values {yn } is a time sequencewhich has frequency set shifted accordance with the change of each pitch frequency fn of
the note sequence {Nn } over the entire piano keyboard. In general, the MIDI sequenceobtained by automatic music transcription is incomplete, and the changes in loudness
for the expression are lost. In addition, the standard MIDI sound source (GM) has aproblem that it is insufficient for music expression because of its low quality. Therefore,
there is a need to convert MIDI sequences to Wav sequences to enrich the music
expression. By using Sophia Sound, we have tried the mapping modulation method tothe MIDI note sequence {Nn} for real Oscar Peterson's jazz music. As a result, a newmusical instrument sound was obtained with a rich sense of rhythm, which has a
mixture of piano, bass and drum sounds, (https://www.etlab.jp).
It will be expected that new instrument sounds with expressions will be generatedusing the resonance phenomenon of nonlinear elements and large-scale LC circuits.

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