A mathematical model for optimal tuning systems

Applied Soft Computing, 2014

Metaheuristic methods have been demonstrated to be efficient tools to solve hard optimization problems. Most metaheuristics define a set of parameters that must be tuned. A good setup of that parameter values can lead to take advantage of the metaheuristic capabilities to solve the problem at hand. Tuning strategies are step by step methods based on multiple runs of the metaheuristic algorithm. In this study we compare four automated tuning methods: F-Race, Revac, ParamILS and SPO. We evaluate the performance of each method using a standard genetic algorithm for continuous function optimization. We discuss about the requirements of each method, the resources used and quality of solutions found in different scenarios. Finally we establish some guidelines that can help to choose the more appropriate tuning procedure.

IFAC-PapersOnLine, 2017

Given a static plant described by a differentiable input-output function, which is completely unknown, but whose Jacobian takes values in a known polytope in the matrix space, we consider the problem of tuning the output (i.e., driving the output to a desired value), by suitably choosing the input. To this aim, we assume to have at our disposal a discrete sequence of trials only, as it happens, for instance, when we iteratively run a software, with new input data at each iteration, in order to achieve the desired output value. In this paper we prove that, if the polytope is robustly non-singular (or has full row rank, in the general non-square case), then a suitable discrete-time tuning law drives the output to the desired point. The computation of the tuning law is based on a convex-optimisation problem to be solved on-line. An application example is proposed to show the effectiveness of the approach.

Proceedings of the SMC Conferences, 2019

The tuning of a piano is a complicated and time-consuming process, which is usually left for a professional tuner. To make the process faster and non-dependent on the skills of a professional tuner, a semi-automatic piano tuning system is developed. The aim of the system is to help a nonprofessional person to tune a grand piano with the help of a computer and a motorized tuning machine. The system composes of an aluminum frame, a stepper motor, an Arduino processor, a microphone, and a laptop computer. The stepper motor changes the tuning of the piano strings by turning the pins connected to them whereas the aluminum frame holds the motor in place. The Arduino controls the motor. The microphone and the computer are used as a part of a closed loop control system, which is used to tune the strings automatically. The control system tunes the strings by minimizing the difference between the current and optimal fundamental frequency. The current fundamental frequency is obtained with an inharmonicity coefficient estimation algorithm, and the optimal fundamental frequency is calculated with a novel tuning process, called the Connected Reference Interval (CRI) tuning. With the CRI process, a tuning close to that of a professional tuner is achieved with a deviation of 2.5 cents (RMS) between the keys A 0 and G 5 and 8.1 cents (RMS) between G# 5 and C 8 , where the tuner's results are not very consistent.

We present some mathematical properties which determine tuning methods. We introduce the concept of fuzzy tuning systems and we analyze four of the systems coexisting within the current orchestras: Pythagorean, Just Intonation, H older's and Equal Temperament systems. We show that the theoretical and practical tuning methods are the same. We introduce the idea of compatibility between tuning systems and we give some suucient conditions to determine an appropriate number of notes into which the octave must be divided.

The Journal of the Acoustical Society of America

Chairman's Introduction-9:00 Invited Papers 9:05 R1. Spanning the two cultures 1550-1750: Scientists and humanists seeking a quantitative account of contemporary musical scales.

The larger research program, of which the present report is a part, involves approaching the analysis of world music from the vantage point of perception. With regard to tuning and scales, there have been three main approaches: via abstract numbers, via empirical measurements, and via perceptual responses. Since the pioneering work of Alexander Ellis in the 1880s, comparative musicologists, ethnomusicologists, acousticians, and music psychologists have measured fundamental frequencies and other partials in sonic spectra to characterize tunings empirically. Since at least as early as ca. 425 BCE, music theorists have employed abstract numbers to prescribe and describe tunings and scales. And since at least as early as ca. 1850 BCE, tunings have been prescribed and described in terms of perceptual responses. Whereas each of these three main approaches can be helpful in making sense of music, they are quite distinct. Nonetheless, these three distinct approaches have often been conflated and an aim of the present study is to disentangle them with a view to analyzing music perceptually. The present report illustrates this approach by focusing on equipentatonic tunings in general and Central Javanese sléndro in particular.

INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY, 2013

This paper, explains about the background study of the coupled tank and to model such tanks using Simulink blocks. It must explains, the coupled tanks are used to select the best tuning strategy for PID controller based on its performance and stability, and then the best tuning controller is obtained after comparing various tuning strategies like Ciancone, Cohen Coon & Ziegler- Nicholas tuning methods based on their performance in controlling the couple tanks. The couple tank is then designed on Simulink as well and three different tuning methods for PI & PID controller calculations are implemented. The controller which gives best performance corresponding their tuning parameters which is obtained from various tuning method, and then selected.

2020 IEEE Congress on Evolutionary Computation (CEC), 2020

Most of the algorithms designed for problem solving have many parameters which values determine their performance. Tuning methods or calibrators are algorithms whose goal is to automate the process of selecting the parameter values of heuristic based algorithms to efficiently solve complex search problems. However, many algorithms are still tuned by-hand either because of the execution time required or the number of scenarios to define before a calibrator is executed. In this work, we propose a practical tuning method that uses a local search procedure that allows obtaining good calibrations in a reduced amount of time, compared to other well-known calibrators. Our tuner has an opposite-inspired learning component used to focus on the most promising areas of the parameter values search space and gathers useful parameter information that is provided to the user. We compare our proposal with two well-known tuners to calibrate two classical optimization problems. We also evaluate the relevance of the opposite-inspired learning component during the search process. A convergence and statistical analysis are presented to confirm that our approach is a good option especially when the user does not have enough time for tuning.

2012

Mathematics, an enigma in numbers and calculations, often accompanied by feelings of rejection and disinterest while Music, a flow with emotions, feelings and life. Motivation for investigating the connections between these two apparent opposites’ poles is attempted in this paper. A correlation between Mathematics and Music is shown.. Music theorists sometimes use mathematics to understand music. Mathematics is "the basis of sound" and sounds itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical". In today’s technology, without mathematics it is difficult to imagine anything feasible. In this paper we have discussed the relation between music and mathematics. How piano keys are interrelated with mathematics, frequencies are correlated and discussed. With the aid of mathematical tools, regression, geometric progression, tuning frequency can be calculated and further re...

IEE Proceedings D Control Theory and Applications, 1991

The accuracy of the Ziegler-Nichols tuning formula is reviewed in the context of PID and PI autotuning. For PID autotuning, it will be shown that, for excessive overshoot in the setpoint response, set-point weighting can reduce the overshoot to specified values, and the original Ziegler-Nichols tuning formula can be retained. It will also be shown that set-point weighting is superior to the conventional solution of reducing large overshoot by gain detuning or set-point filtering. However, for excessive set-point undershoot, the tuning formula will have to be modified. For PI autotuning, it will be shown that the Ziegler-Nichols tuning formula is inadequate and has to be completely revised. The application of set-point weighting and the modification of the tuning formula can be based simply on the knowledge of the normalised gain or normalised deadtime of the process. These heuristic refinements, when incorporated, will give appreciable improvement in the performance of autotuners.