More on the Mathematics of Musical Scales

1974 ◽  
Vol 67 (1) ◽  
pp. 75-84
Author(s):  
Truman Botts
Keyword(s):  
Just Intonation ◽  
Equal Temperament ◽  
Musical Tone ◽  
Stringed Instrument ◽  
Harmonic Components ◽  
Basic Ideas ◽  
Compound Interest ◽  
Sine And Cosine Functions ◽  
Cosine Functions ◽  
Musical Scales

AN INTERESTING article by Paul S. Malcom entitled “The Mathematics of Musical Scales” appeared in the November 1972 issue of the Mathematics Teacher. The present article may be viewed as a sequel to that one, although it may also be read independently. We propose to explore in a bit more detail the just-intonation scale natural to stringed instrument players and singers and the relation to it of the equal-temperament scale used in tuning pianos. We shall make constant use of fructions, and along the way we shall encounter an analogue of the idea of compound interest and a simple exercise in exponents and logarithms. At the end, we shall comment very briefly on an advanced branch of mathematics, called harmonic analysis and involving trigonometric sine and cosine functions, some basic ideas of which are illustrated by the analysis of a musical tone into its harmonic components or overtones.

Some Mellin-type Definite Integrals Involving Sine and Cosine Functions

2019 ◽  
Vol 10 (1) ◽  
pp. 222-237
Author(s):  
M. I. Qureshi ◽  
Kaleem A. Quraishi ◽  
Dilshad Ahamad
Keyword(s):  
Definite Integrals ◽  
Sine And Cosine Functions ◽  
Cosine Functions

SINS Installation Error Calibration Based on Multi-Position Combinations

2011 ◽  
Vol 383-390 ◽  
pp. 4213-4220
Author(s):  
Zhen Huan Wang ◽  
Xi Jun Chen ◽  
Qing Shuang Zeng
Keyword(s):  
Theoretical Analysis ◽  
Least Squares ◽  
Rotation Rate ◽  
Scale Factor ◽  
New Method ◽  
Earth’S Rotation ◽  
Earth's Rotation ◽  
Error Calibration ◽  
Sine And Cosine Functions ◽  
Cosine Functions

A new method is proposed to calibrate the installation errors of SINS. According to the method, the installation errors of the gyro and accelerometer can be calibrated simultaneously, which not depend on latitude, gravity, scale factor and earth's rotation rate. By the multi-position combinations, the installation errors of the gyro and accelerometer are modulated into the sine and cosine functions, which can be identified respectively based on the least squares. In order to verify the correctness of the theoretical analysis, the SINS is experimented by a three-axis turntable, and the installation errors of the gyro and accelerometer are identified respectively according to the proposed method. After the compensation of the installation error, the accuracy of the SINS is improved significantly.

A Computer-Based Training Program for Developing Harmonic Intonation Discrimination Skill

1992 ◽  
Vol 40 (2) ◽  
pp. 139-152 ◽  
Author(s):  
Bruce F. Dalby
Keyword(s):  
Training Program ◽  
Matched Control ◽  
Control Group ◽  
Music Majors ◽  
Just Intonation ◽  
Computer Based Training ◽  
Equal Temperament ◽  
Computer Based ◽  
Experimental Subjects ◽  
Experimental Group

The purpose of this study was to determine the effectiveness of a computer-based training program for improving students' ability to make judgments of harmonic intonation. Twenty members of two undergraduate conducting classes participated in the Harmonic Intonation Training Program (HITP). An equivalent matched control group was selected from 156 other undergraduate music majors who had also taken the investigator-developed Harmonic Intonation Discrimination Test (HIDT). The HITP consisted of a body of drill-and-prac-tice exercises using intervals, triads, and brief three- and four-part musical passages. The exercises were played in both equal temperament and just intonation by a 16-voice digital synthesizer. After a 9-week treatment period, a two-way ANOVA on posttest HIDT scores revealed a difference (p= .005) in favor of the experimental group. Results of a questionnaire administered after the training to the experimental subjects indicated that attitudes toward the training program were mostly positive.

Controllability of stochastic nonlinear oscillating delay systems driven by the Rosenblatt distribution

2020 ◽  
pp. 1-23 ◽  
Author(s):  
T. Sathiyaraj ◽  
JinRong Wang ◽  
D. O'Regan
Keyword(s):  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Second Order ◽  
Delay Systems ◽  
Stochastic Delay ◽  
Finite Dimensional ◽  
Sine And Cosine Functions ◽  
Cosine Functions ◽  
Theoretical Results

Abstract In this paper, we study the controllability of second-order nonlinear stochastic delay systems driven by the Rosenblatt distributions in finite dimensional spaces. A set of sufficient conditions are established for controllability of nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices. Furthermore, controllability results for second-order stochastic delay systems driven by Rosenblatt distributions via the representation of solution by delayed sine and cosine functions are presented. Finally, our theoretical results are illustrated through numerical simulation.

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