reference book on mathematics for engineers and students of technical colleges. Bronstein and

I. N. Bronstein and K. A. Semendyaev’s handbook on mathematics for engineers and students of higher educational institutions has firmly gained popularity not only in our country, but also abroad. The eleventh edition was published in 1967. Further edition of the reference book was suspended, as it no longer met modern requirements.

Decimal logarithms.
Explanations for tables of logarithms and antilogarithms. Table 1.1.1.7 is used to find the decimal logarithms of numbers. First, for a given number, the characteristic ei about the logarithm is found, and then the mantissa from the table. For three-digit numbers, the mantissa is located at the intersection of the line at the beginning of which (column N) are the first two digits of the given number, and the column corresponding to the third digit of our number. If the given number has more than three significant digits, linear interpolation must be applied. In this case, the interpolation correction is found only on the fourth significant digit of the number; it makes sense to make a correction for the fifth digit only when the first significant digit of the given number is 1 or 2.

To find a number by its decimal logarithm, use table 1.1.1.8 (table of antilogarithms) *). The argument in this table is the mantissa of the given logarithm. At the intersection of the row, which is determined by the first two digits of the mantissa (column m), and the column corresponding to the third digit of the mantissa, the digital composition of the desired number is found in the antilogarithm table. An interpolation correction must be applied to the fourth digit of the mantissa. The characteristic of the logarithm allows you to put a comma in the result.

Free download e-book in a convenient format, watch and read:
Download the book Handbook of Mathematics for engineers and students of higher educational institutions, Semendyaev K.A., Bronstein I.N., 1986 - fileskachat.com, fast and free download.

• Handbook of mathematics for engineers and students of higher educational institutions, Bronstein I.N., Semendyaev K.A., 1986
• Non-standard methods for solving equations and inequalities, Reference book, Olehnik S.N., Potapov M.K., Pasichenko P.I., 1991
• Mathematics, School reference book, grades 7-11, Definitions, formulas, schemes, theorems, algorithms, Chernyak A.A., Chernyak Zh.A., 2018

The following tutorials and books.

The previous, 12th edition (1980) came out with a radical revision made by a large team of authors from the GDR, edited by G. Grosche and W. Ziegler. Numerous corrections have been made to this edition. For students, engineers, scientists, teachers.

1.1.3.3. Table of indefinite integrals.

General instructions. 1. The constant of integration is omitted everywhere except when the integral can be represented in various forms with various arbitrary constants.

Editorial
1. TABLES AND GRAPHS
1.1. TABLES
1.1.1 Tables of elementary functions
1. Some common constants A1) 2. Squares, cubes, roots A2). 3. Powers of integers from 1 to 100 B9). 4. Reciprocals of C1). 5. Factorials and their reciprocals C2). 6 Some powers of numbers 2, 3 and 5 C3). 7. Decimal logarithms C3). 8. Antilogarithms C6) 9. Natural values ​​of trigonometric functions C8) 10. Exponential, hyperbolic and trigonometric functions (for x from 0 to 1.6) D6). 11. Exponential functions (for x from 1.6 to 10.0) D9). 12. Natural logarithms E1). 13. Circumference E3). 14. Area of ​​a circle E5). 15. Elements of a circle segment E7). 16. Converting a degree measure to a radian F1). 17. Proportional parts F1). 18. Table for quadratic interpolation F3)
1 1.2. Special Function Tables
1. Gamma function F4). 2 Bessel (cylindrical) functions F5). 3. Legendre polynomials (spherical functions) F7). 4. Elliptic integrals F7). 5 Poisson distribution F9). 6 Normal distribution G1). 7. X2-distribution G4). 8. /-student distribution G6). 9. z-distribution G7). 10. F-distribution (distribution v2) G8). 11. Critical numbers for the Wilcoxon test (84). 12. X-distribution of Kolmogorov-Smirnov (85).
1.1.3. Integrals and sums of series
1 Table of sums of some numerical series (86). 2. Table of expansion of elementary functions into power series (87). 3 Table of indefinite integrals (91). 4 Table of some definite integrals (PO).
1.2. GRAPHS OF ELEMENTARY FUNCTIONS
1.2.1 Algebraic functions FROM
1 Entire rational functions A13). 2. Fractional rational functions A14). 3. Irrational functions A16).
1.2.2. Transcendent Functions
1. Trigonometric and inverse trigonometric functions A17). 2. Exponential and logarithmic functions A19) 3. Hyperbolic functions A21).
1.3. KEY CURVES
1.3.1. Algebraic curves
1 3rd order curves A23). 2. 4th order curves A24).
1 3.2. Cycloids
1.3.3. Spirals
1.3.4. Chain line and tractrix
2. ELEMENTARY MATHEMATICS
2.1. ELEMENTARY APPROXIMATE CALCULATIONS
2.1.1. General information
1. Representation of numbers in positional number system A30). 2. Errors and rules for rounding numbers A31)
2.2. COMBINATORICS
2 2 1 Basic combinatorial functions 1 Factorial and gamma function A34) 2 Binomial coefficients A34). 3 Polynomial factor A35)
2 2 2. Binomial and polynomial formulas 1 Newton's binomial formula A35) 2 Polynomial formula A35)
2 2.3 Statement of problems of combinatorics
2 24 Substitutions
1. Substitutions A36). 2. The group of permutations to elements A36). 3. Fixed Point Substitutions A36). 4 Permutations with a given number of cycles A37) 5 Permutations with repetitions A37)
2 2 5. Placements 137 1 Placements A37) 2 Placements with repetitions A37). 2 2 6 Combinations 1 Combinations A38). 2 Combinations with repetitions A38).
2.3. FINITE SEQUENCES, SUMS, PRODUCTS, AVERAGES
2 3 1 Notation of sums and products
2 3.2 End sequences 1 Arithmetic progression A39) ^2 Geometric progression A39)
2 3 3 Some finite sums
2 3 4 Average values
2.4. ALGEBRA
2 4 1. General concepts 1 Algebraic expressions A40) 2 Meanings of algebraic expressions A40) 3 Polynomials A41) 4 Irrational expressions A41). 5 Inequalities A42) 6. Elements of group theory A43)
2 4.2 Algebraic equations 1 Equations A43) 2 Equivalent transformations A44) 3 Algebraic equations A45) 4. General theorems A48). 5 System of algebraic equations A50)
24 3 Transcendental equations
2.4 4 Linear algebra 1. Vector spaces A51) 2. Matrices and determinants A56). 3. Systems of linear equations A61) 4 Linear transformations A64). 5 Eigenvalues ​​and eigenvectors A66)
2.5. ELEMENTARY FUNCTIONS
2 5 1. Algebraic functions 1 Entire rational functions A69) 2 Fractional rational functions A70) 3 Irrational algebraic functions A74)
2 52 Transcendent functions 1. Trigonometric functions and their inverses A74). 2 Exponential and logarithmic functions A79). 3 Hyperbolic functions and their inverses A80).
2.6. GEOMETRY
2 6 1. Planimefia
26 2 Stereometry 1 Straight lines and planes in space A85) 2 Dihedral, polyhedral and solid angles A86) 3 Polyhedra A86) 4 Bodies formed by moving lines A88)
2.6.3. Rectilinear trigonometry 1. Solving triangles A90) 2. Application in elementary geodesy A91)
2 6 4. Spherical trigonometry
1. Geometry on the sphere A92). 2. Spherical triangle A92) 3 Solution of spherical triangles A92).
2.6.5. Coordinate systems
1. Coordinate systems on the plane A95). 2 Coordinate systems in space A97)
2.6.6. Analytic geometry
1. Analytic geometry in the plane A99) 2 Analytic geometry in space B04)
3. FUNDAMENTALS OF MATHEMATICAL ANALYSIS
3.1. DIFFERENTIAL AND INTEGRAL CALCULUS OF FUNCTIONS OF ONE AND SEVERAL VARIABLES
3.1.1. Real numbers
1. The system of axioms of real numbers B10) 2. Natural, integer and rational numbers B11) 3 The absolute value of a number B12). 4. Elementary inequalities B12)
3.1.2. Point sets in R"
3.1 3. Sequences
1. Number sequences B14) 2 Point sequences B15)
3.1.4. Real Variable Functions
1. Function of one real variable B16) 2 Functions of several variable variables B23).
3.1 5. Differentiation of functions of one real variable
1. Definition and geometric interpretation of the first derivative Examples B25) 2 Higher order wires B26).
3. Properties of differentiable functions B27) 4 Monotonicity and convexity of functions B28).
5. Extrema and inflection points B29) 6 Elementary study of the function B30).
3.1.6. Differentiation of functions of several variables. N 2M
1. Partial derivatives, geometric interpretation B30) 2. Total directional differential, gradient B31) 3. Theorems on differentiable functions of several variables B32)
4. Differentiable mapping of the space Rn into Rm, functional definitions i el u. implicit functions; existence theorems B33) 5 Change of variables in differential expressions B35). 6. Extrema of functions of several variables B36)
3.1 7. Integral calculus of functions of one variable
1. Definite integrals B38) 2 Properties of definite integrals B39) 3 Indefinite integrals B39). 4. Properties of indefinite integrals B41) 5 Integration of rational functions B42)
6. Integration of other classes of functions B44) 7 Improper integrals B47) 8 Geometric and physical applications of definite integrals. B51)
3.1.8. Curvilinear integrals
1. Curvilinear integrals of the 1st kind (integrals over the length of a curve) B53) 2 Realization and calculation of curvilinear integrals of the 1st kind B53) general view) B54) 4. Properties and calculation of curvilinear integrals of the 2nd kind B54).
5. Independence of the curvilinear integrals oi of the integration path B56) 6. Geometical and physical applications of the curvilinear integrals B57)
3.1.9. Integrals depending on a parameter
1. Definition of integral depending on parameter B57) 2 Properties of integrals depending on oi parameter B57). 3. Improper integrals depending on parameter B58) 4 Examples of integrals depending on parameter B60)
3.1.10. Double integrals 2b0
1. Definition of a double integral and elementary properties B60) 2 Calculation of double integrals B61).
3. Change of variables in double integrals B62) 4 Geometrical and physical applications of double integrals B63)
3.1.11. Triple Integrals
1. Definition of the triple integral and elementary properties B63) 2 Calculation of multiple hhicirals B64). 3. Change of Variables in Triple Integrals B65). 4 Geometrical and physical applications of triple integrals B65).
3.2. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
3.2.1. Calculus of variations
1. Statement of the problem, examples and basic concepts B87). 2. Euler-Lagrange theory B88). 3. The theory of Hamilton - Jacobi B94). 4. Inverse problem of the calculus of variations B95). 5. Numerical Methods B95).
3.2.2. Optimal Control
1. Basic concepts B98) 2. Pontryagin's maximum principle B98). 3. Discrete systems C03) 4. Numerical methods C04).
3.3. DIFFERENTIAL EQUATIONS
3.3.1. Ordinary differential equations
1 General concepts. Existence and uniqueness theorems C05) 2. First order differential equations C06). 3. Linear differential equations and linear systems C13). 4. General non-linear differential equations C25). 5. Stability C25) 6. Operator method for solving ordinary differential equations C26) 7. Boundary value problems and eigenvalue problems C27).
3.3.2. Partial Differential Equations
1. Basic concepts and special methods solutions C31) 2. Partial differential equations of the 1st order C33). 3. Partial differential equations of the 2nd order C39).
3.4. COMPLEX NUMBERS. FUNCTIONS OF A COMPLEX VARIABLE
3.4.1. General remarks
3.4 2. Complex numbers. Riemann sphere. Areas
1. Definition of complex numbers Field of complex numbers C57). 2. Conjugate complex numbers Modulus of a complex number C58). 3. Geometric interpretation of C58). 4. Trigonometric and exponential forms of complex numbers C58). 5 Degrees, roots C59). 6. Riemann sphere. Jordan curves. Regions C59).
3 4.3. Functions of a complex variable
3.4.4. The most important elementary functions
1. Rational functions C61) 2 Exponential and logarithmic functions C61) 3 Trigonometric and hyperbolic functions C64).
3.4.5. Analytic Functions i. Derivative C65) 2 Cauchy-Riemann differentiability conditions C65) 3 Analytic functions C65).
3.4.6. Curvilinear integrals in the complex domain
1. Integral of a function of a complex variable C66). 2. Independence of the path of integration C66).
3. Indefinite integrals C66) 4 Basic formula of integral calculus C66). 5. Cauchy integral formulas C66)
3.4.7. Expansion of analytic functions in a series
1. Sequences and series C67). 2 Functional rows. Power series C68). 3. Taylor series C69). 4 Laurent series C69). 5. Classification of singular points C69). 6. Behavior of analytic functions at infinity C70).
3.4.8. Deductions and their application
1. Residues C70). 2. Residue theorem C70). 3. Application to the calculation of definite integrals C71).
3 49 Analytic continuation 1 Principle of analytic continuation C71). 2 Symmetry principle (Schwarz) C71)
3 4.10 Inverse functions Riemann surfaces
1 Univalent functions, inverse functions C72) 2. Riemann surface of the function z = |/w C72). 3. Riemann surface of the function z - Ln w C73).
3 4 11 Conformal mappings
1 The concept of a conformal mapping C73) 2. Some simple conformal mappings C74).
4. ADDITIONAL CHAPTERS
4.1. SETS, RELATIONS, MAPPINGS
4 1 1 Basic concepts of mathematical logic
1 Algebra of logic (propositional algebra, propositional logic) C76) 2 Predicates C79)
4 1 2. Basic concepts of set theory
1. Sets, elements C80). 2 Subsets of C80)
4 1 3 Operations on sets
1 Union and intersection of sets C81). 2. Difference, symmetric difference, complement of sets C81) 3 Euler-Venn diagrams C81) 4. Cartesian product of sets C82) 5. Generalized union and intersection C82)
4.1.4 Relations and mappings
1. Relations C82) 2 Equivalence relation C83) 3 Order relation C83). 4. Mappings C84).
5. Sequences and families of sets C85) 6 Operations and algebras C85).
4.1 5 Cardinality of sets
1. Equivalence C86). 2 Countable and uncountable sets C86)
4.2. VECTOR CALCULUS
4 2 1 Vector algebra
1 Basic concepts C86). 2. Scalar multiplication and addition C86). 3. Multiplication of vectors C88).
4 Geometric Applications of Vector Algebra C89).
4 2 2. Vector analysis
1 Vector functions of scalar argument C90) 2. Fields (scalar and vector) C91). 3. Scalar field gradient C93). 4. Curvilinear integral and potential in a vector field C94). 5 Surface integrals in vector fields C95). 6. Divergence of a vector field C97). 7. Vector field curl C98).
8. Laplace operator and vector field gradient C99). 9. Calculation of complex expressions (Hamilton operator) C99). 10. Integral formulas D00) 11 Definition of a vector field by its sources and vortices D01) 12. Dyads (tensors of rank II) D02)
4.3. DIFFERENTIAL GEOMETRY
4 3.1 Flat curves
1 Ways to specify plane curves. Plane curve equation D05). 2 Local elements of a plane curve D06) 3 Points of a special type D07). 4 Asymptotes D09) 5 Evolute and involute D10). 6 Envelope of a family of curves D10).
4 3 2 Spatial curves
1 Ways of specifying curves in space D10). 2 Local elements of a curve in space D10)
3 Main theorem of the theory of curves D11).
4.3.3. surfaces
1. Methods for defining surfaces D12) 2 Tangent plane and normal to the surface D12).
3. Metric properties of surfaces D13). 4 Surface curvature properties D14). 5. Main theorem of the theory of surfaces D16). 6 Geodesic lines on the surface D17).
4.4. FOURIER SERIES, FOURIER INTEGRALS, AND THE LAPLACE TRANSFORM
4 4.1. Fourier series
1 General concepts D18). 2. Table of some Fourier expansions D19) 3 Numerical harmonic analysis D23).
4 4 2. Fourier integrals
1 General concepts D25). 2 Table of Fourier transforms D26).
4.4 3 Laplace transform
1 General concepts D37) 2 Application of the Laplace transform to the solution of ordinary differential equations with initial conditions D38) 3 Table of the inverse Laplace transform of fractional rational functions D38)
5. PROBABILITY THEORY AND MATHEMATICAL STATISTICS
5.1. PROBABILITY THEORY
5 1 1 Random events and their probabilities
1 Random events D41) 2 Axioms of the theory of probability D42). 3 The classic definition of faith! event probability D43) 4 Conditional probabilities D43) 5. Total probability Bayes formula D43)
5 1 2 Random variables
1 Discrete random variables D44) 2 Continuous random variables D45)
5 1 3 Moments of distribution
1 Discrete case D46) 2 Continuous case D47)
5 1 4 Jurassic random ages (multivariate random variables)
1 Discrete random vectors D48) 2 Continuous random vectors D49) 3 Boundary distributions D49) 4 Moments of a multidimensional random variable D49) 5. Conditional distributions D50)
6 Independentib random variables D50) 7 Regression dependence D50) 8 Functions oi of random variables D51)
5 1 5 Characteristic functions
1 Properties of characteristic functions D52). 2 Inversion formula and uniqueness theorem D52) 3 Limit theorem for characteristic functions D52) 4 Generating functions D53)
5 Characteristic functions of multidimensional random variables D53).
5 1 6 Limit theorems
1 Law of large numbers D53) 2 De Moivre-Laplace limit theorem D54) 3 Central limit theorem D54)
5.2. MATH STATISTICS
5 2 1 Samples
1 Histogram and empirical distribution function D55). 2 Sample function D56) 3 Some important distributions D57)
5 2 2 Parameter evaluation
1 Properties of point estimates D57) 2 Methods for obtaining estimates D58). 3 Confidence estimates D59)
5 2 3 Hypothesis testing (tests)
1 Statement of the problem D60) 2 General theory D60) 3 r-test D61) 4 /-test D61) 5 Wilcoxon test D61). 6 X-criterion D62) 7. Case of additional parameters D63) 8 Kolmogorov-Smirnov agreement criterion D63)
5 2 4 Correlation and regression
1 Estimation of correlation and pei ression characteristics by samples D64) 2 Checking innoiejbi р = 0
in the case of a normally distributed 1general population D64)
6. MATHEMATICAL PROGRAMMING
6.1. LINEAR PROGRAMMING,6 11 Statement of the problem of linear programming and the simplex method
1 General setting of giving, i eoms! logical interpretation and solution for sch with noisy variables D66)
2 Canonical view of the LLP, image of the vertex in the simplex table D68) 3 Simplex method with a given initial table D69) 4 Obtaining the initial vertex D71). 5 Degenerate case and its treatment using the simplex method D73) 6 Duality in linear programming D73).
7 Modified methods, additional change to task D75)
6.2. TRANSPORT CHALLENGE
6 2 1 Linear transport problem
62 2 Omitting the initial solution
62 3 Transport method
6.3. TYPICAL LINEAR PROGRAMMING APPLICATIONS
6.3.1 Capacity utilization
6.3.2. Mixture problem
6.3.3. Distribution, planning, comparison
6.3.4. Cutting, shift planning, coating
6.4. PARAMETRIC LINEAR PROGRAMMING
6.4 1 Problem statement
6 4.2. Solution Method for the Case of a One-Parameter Objective Function
6.5. INTEGER LINEAR PROGRAMMING
6 5 1. Statement of the problem, geometric interpretation
6.5.2. Gomory Section Method
1. Purely integer linear programming problems D87). 2. Mixed integer linear programming problems D88).
6.5.3 Branch method
6.5 4 Comparison of methods
7. ELEMENTS OF NUMERICAL METHODS AND THEIR APPLICATIONS
7.1. ELEMENTS OF NUMERICAL METHODS
7.1.1. Errors and their accounting
7.1.2. Computational methods
1. Solution linear systems equations D91). 2. Linear eigenvalue problems (D95).
3. Nonlinear Equations D96) 4. Systems of Nonlinear Equations D98) 5 Approximation D99) 6 Interpolation E02) 7 Approximate Calculation of Integrals E06) 8 Approximate Differentiation E10). 9 Differential Equations E10).
7 1.3 Implementation of the numerical model in electronic computers
I. Criteria for the choice of method E16). 2. Control methods E16). 3. Calculation of functions E17).
7.1 4 Nomography and slide rule
1 Relations between two variables - functional scales E18) 2. Slide rule E19). 3. Nomograms of points on straight lines and grid nomograms E19).
7.1 5 Handling empirical numerical material
1. Method of least squares E21). 2. Other alignment methods E22).
7.2. COMPUTER ENGINEERING
7.2.1. Electronic computers (computers)
1. Introductory remarks E23) 2. Representation of information and computer memory E23) 3 Exchange channels E24). 4 Program E24). 5. Programming E24). 6. Computer control E26). 7. Mathematical (software) E26). 8. Performing work on a computer E26)
7.2.2 Analog computers
1. The principle of the analog device computer science E27). 2 Computing elements of an analog computer E27). 3. Programming Principle for Solving Systems of Ordinary Differential Equations (E29). 4 Quality programming E30)
Bibliography
Subject index

S. N. Bronstein "Theremin and Electrola". Moscow, publishing house "NKPT", 1930

Restored from a printed book using OCR and manual proofreading.
The current OCR version is 3.0 from 11/10/2017.

In the electronic version, the spelling has been updated to modern, spelling errors have been corrected. Units of measurement are left unchanged.

The capacitances of capacitors are indicated in the CGS system - in centimeters ( cm), and not, as has become customary since the 1960s in the International System of Units (SI), in farads.

Please report any typos you see.

Back cover (advertisement of the book "Vacuum tube as a detector")

Title page

S. N. BRONSHTEIN

THERMENVOX AND ELECTROLA
(THEORY AND PRACTICE OF ELECTRIC MUSICAL INSTRUMENTS)

PUBLISHING HOUSE NKPT

MOSCOW 1930

Back of title page

"Mospoligraf",
13th type-zincography
"The Printer's Thought"
Moscow, Petrovka, 17
Mosoblit № 59328
Circulation 2500
Order No. 4074

PREFACE.

Interest in the theremin, the first musical instrument with cathode lamps, is extremely high. Its demonstrations in the USSR and abroad are accompanied by invariable success both among specialists in musicians and radio engineers, and among general public.

However, despite the fact that more than eight years have passed since its invention, the theremin has not been released for sale. Also not published by Ing. L. S. Theremin to this day the data of his construction, the principles of which are generally known.

Meanwhile, the need to popularize such an apparatus, which is a kind of renewal of modern musical instruments frozen in their forms, is undoubtedly overdue. This, on the one hand, will expand the scope of the hitherto known application of radio engineering; on the other hand, the birth of a new cadre of musicians - "theremin players" will benefit the instrument itself, which is still far from perfect.

The author, on the basis of the available separate fragmentary information in foreign literature, as well as on the basis of his own experiments, developed a detailed design of a musical apparatus of the theremin type, the manufacture of which is within the power of every more or less trained radio amateur.

At the same time, the final section of the book is devoted to a new instrument designed by the author - the "electrole". This device, which gives, in general, the same results as the theremin, but is built on completely different principles, is extremely simple, as a result of which it can contribute to the musical development of the amateur radio masses.

Chapters I-VI introduce the reader to the basic principles of the origin of sound and the operation of an electric musical instrument.

Moscow, August 1929

I. ELECTRICITY AND MUSIC.

Electric music - it sounds somewhat unusual for our ear. What is common, at first glance, between technology and art? Engineers, as is commonly thought, are not musical people. Even the very term "electric music" corresponds more to the idea of ​​some kind of mechanical automaton than to a genuine musical instrument.

Indeed, if we trace the history of the use of electricity in music, we will see that at first electricity played a purely applied role here - it, so to speak, "electrified" already known instruments, without introducing anything new into them.

An organ is given as an example in such cases. As you know, playing the organ to force air into the pipes requires the expenditure of a certain muscular strength. In small organs or harmoniums, this is done by pressing the pedals with the player's feet; in larger instruments, a special person stood on the bellows, and sometimes there were even several of them.

Electricity naturally replaced human work in this case with a small motor.

Further, in the same organ there is a rather complex mechanism that opens the corresponding pipe when a finger is pressed on one or another key. In the latest systems, this is done electrically, and the keyboard and pipe system can be located at a great distance from each other and even in different rooms.

Another example is the so-called "pianola" (mechanical piano). In the pianola, any piece of music is recorded by punching holes on a paper tape. This tape is passed at a known speed in front of a series of tubes into which air is supplied under pressure. Depending on the nature of the perforation of the tape, one or another tube sends air shocks to the system of cams located above the keyboard of a normal grand piano.

In the "pianola" the movement of the tape and the injection of air is carried out by foot pedals. In the improved Mignon piano, these functions are again performed by an electric motor.

Such examples can, of course, be given a large number and they will all be of the same order.

At the next level, already higher, is the telephone, intended, however, at first not to reproduce musical sounds, but to transmit human speech. Only later did the telephone mechanism become an indispensable part of radio music and electric music in the truest sense of the word.

Finally, we move on to the discovery of radio communication. However, even in the radio, with which we can listen to any piece of music, the human voice, concerts, operas, etc., electricity does not play a dominant role; a singing person or some musical instrument is still needed. The radio here performs the function of transmitting sound or receiving it, but it is not the source of sound.

We received a real electric musical instrument only with the appearance of the “theremin”, invented by the Leningrad engineer L. S. Termen.

This apparatus was shown at the beginning of 1921, still in a laboratory state, but even then it aroused great interest. Only in 1927 did Termen demonstrate a more or less finished device, made in several versions, on which the inventor performed relatively simple pieces of music. In the future, the "theremin" is shown first at the Frankfurt Music Exhibition, and then in a number of cities in Europe and America; "concerts" are accompanied by constant resounding success.

From the outside, the "theremin" does not at all resemble a musical instrument in our view. The newest model of it, in essence, is an ordinary multi-tube receiver, mounted in a box in the form of an inclined console, on which the notes lie. At the base are several control buttons and measuring instruments. A metal rod is placed on the right side, and a small metal arc on the left. The apparatus is connected to one or more loudspeakers. Under the table on which the theremin is located, there are accumulators for heating and anode, familiar to the eye of a radio amateur (Fig. 1).

Rice. one. L. S. Theremin playing the theremin.

The game is played as if on an air neck - by approaching the hands to the rod and the arc. When the hand approaches the rod, the height changes, to the arc - the strength of the sound. To give it a more lively color, tremolation is necessary, achieved by a slight oscillation of the right hand.

In another model, the sound intensity is adjusted by pressing the foot on the pedal, while the left hand rests on a special interrupter, which contributes to intermittent tones.

The combination of "theremin" with various kinds of amplifiers allows you to increase the transmission power to any limits.

The inventor not only "soloed" on his instrument to the accompaniment of the piano (violin and cello repertoire), but showed experiments in playing together with another performer on two devices, as well as with string instruments and a human voice.

A similar design was simultaneously constructed by the Leningrad engineer V. A. Gurov, who demonstrated it at the Nizhny Novgorod Fair in 1922. In this apparatus, the pitch was adjusted by moving the finger of the right hand along the usual wooden violin neck located on the table. With the left hand, the movement of the handle changed the strength of the sound. If we discard the usual interest of the general public in any entertaining innovation, the question naturally arises: is the theremin an amusing toy, or does it really hold great potential as an instrument of the music of the future.

It should be pointed out, of course, that in its modern execution this apparatus is still far from ideal: the nature of the sound, sometimes reminiscent of singing with a closed mouth for one vowel, sometimes a somewhat monotonous howl, from a musical point of view, still leaves much to be desired. A major drawback is the monophonic melody and the absence of chords. The game is also somewhat difficult, as it does not yet allow the performance of even relatively virtuoso pieces. True, here it should be recognized that neither a developed school, nor a technique for playing the instrument itself in the “youth” of the instrument itself is yet available.

However, if we discard all these features and shortcomings inherent in every still unimproved apparatus, then it should be recognized that the theremin should give a lot of new musical art, and it is equally interesting for both the technician and the musician. Its main advantage is the breadth of the range and the richness of the sound palette. From this small box, you can extract sounds as thin as the highest harmonics of a violin, and thick bass tones of a double bass. The nature of the sound, at the request of the player, resembles string instruments of various timbres and colors, and some wind instruments, and even a human voice. At the same time, these sounds are not like any of the existing ones, differing in some kind of extreme airiness and weightlessness. It is felt that there is nothing connected with matter in them; these are, indeed, the sounds of the ether.

Unlike instruments with fixed sounds (piano, organ, etc.), in which the so-called. "tempered system". "Theremin" makes it possible to expand our musical system, easily reproducing smaller intervals than those that are accepted among Western peoples. The need for such an expansion in modern musical circles is long overdue, so the appearance of the "theremin" in this regard turned out to be extremely useful.

Finally, there remains the relative ease of control and power of transmission - a slight movement of the hand in space gives all the necessary transitions and changes the strength of sound to a huge extent: such freedom to extract sound literally “out of thin air” contributes to the complete subordination of the instrument to the musician playing it. Next in line are polyphony and chords, a sharper and more colorful change in timbres and shades, greater sound saturation, the use of various kinds of resonator boxes, the development of the playing technique itself, the use of theremin ensembles with various sound characters in combination with other instruments and the human voice, and, finally, "radio orchestra", etc.

II. SOUND AND MUSICAL INSTRUMENTS.

In order to get the fullest possible idea of ​​the principles that underlie the construction of an electric musical instrument, it is necessary to get acquainted in general with the nature of the origin of sound. What is required to receive it? To do this, we need to bring some body (solid, liquid or gaseous) into rapid oscillatory motion, i.e., one in which we would have a periodic (through the same time intervals) change in the direction of motion. A good example is the oscillation of a pendulum. The time during which the pendulum, deviated, let's say, to the right, swings to the left and returns to its original position again, we call the period of oscillation. The number of these oscillation periods per second is the oscillation frequency.

A body to which a certain oscillatory movement is communicated, for example, a violin string or human vocal cords, in turn causes oscillatory movements of air in the form of air waves propagating in a circle. These waves run at a known speed, approximately 330 meters per second for air. Similar waves, in the form of diverging concentric circles, are formed in the water of a pond if a stone is thrown into it.

Upon reaching our ear, the waves vibrate the eardrum and create the physiological impression of sound.

The oscillation frequency, which we spoke about above, plays a very large role here; if the frequency is not high, then we will not hear anything; only when the frequency increases to at least 16 vibrations per second does our consciousness feel a very low musical sound.

As the frequency increases, the pitch rises; the opposite limit is (depending on the sensitivity of the ear) between 25.000-35.000 oscillations per second. With a further increase in frequency, we again cease to hear. In practice, in the music that we currently use, the oscillation frequency ranges from 26 to 4000.

Rice. 2. The vibration frequency of the individual tones of the piano keyboard.

On fig. 2, for clarity, a piano keyboard is shown, near the keys of which the frequencies corresponding to each note are placed. The range of different instruments and the human voice is not the same. So, for example, the voice volume of a bass singer lies between 85 and 341 frequencies, baritone - 96 and 384, tenor - 128 and 480, female soprano voice - 240 and 1152 (not counting the so-called "falsetto"). In the double bass, the lowest stringed instrument, we have a space between 40 and 240 frequencies, and in the violin from 192 to 3072. The bass trumpet gives the thickest note in wind instruments (42 vibrations per second), the highest is the piccolo flute (4608 vibrations) and so on. Thus, we see the greatest range in the piano or organ, but the "theremin" can give an even wider range.

Except heights musical tone, are still important for us strength and especially, timbre. Even sounds of the same height can differ from each other in hue, which is obtained due to the fact that the main tone of the sounding body is accompanied by a number of additional tones (the so-called overtones). Depending on the number and nature of these overtones, the quality of the sound also changes very diversely.

Thus, we see that in order to pronounce any sound, it is necessary to vibrate an elastic body. Depending on the mode of occurrence of these oscillations, we get different types musical instruments, divided into three main groups: wind, string and percussion.

In wind instruments, the sound is obtained from the vibration of the air column in the pipe when air enters it under pressure (lungs of a musician or bellows of an organ). pitch in this case depends on the length of the air column enclosed in the tube, and also on whether the tube is open at both ends or only at one end. This change is achieved by opening and closing the holes located along the tube (done directly with the fingers or with the help of special valves). This is the case in woodwinds (flute, oboe, cor anglais, bassoon, clarinet).

In brass instruments, the air column, for the most part, is not shortened, but lengthened due to the inclusion of additional tubes (horn, trumpet, cornet, trombone, tuba).

A complex wind instrument, which is a combination of a number of wind pipes into which air is blown by bellows, is called an organ.

In stringed instruments, sound is produced by vibrating the strings. Strings are also divided into two types: bowed and plucked. In the first, the string is brought into vibration by friction with a bow (violin, viola, cello, double bass). The sound can be received any duration and any force.

The pitch of the sound here depends on the length of the string (the shorter it is, the higher the frequency of vibration and, therefore, the tone is higher). The change in length is achieved by pressing one or another place of the string to the fingerboard.

In plucked types, the strings vibrate when struck with a hammer (piano) or touched with a finger (harp, guitar, balalaika, zither, etc.). The sound is called short and gradually fading.

Percussion is divided into noise (drum, tam-tam, castanets, tambourines, triangles, cymbals, etc.) and tuned (timpani, bells, xylophone, metallophone, cymbals, etc.). Sound is caused by the vibration of stretched skin, metal, wooden plates, etc.

III. ELECTRIC OSCILLATIONS AND THEIR ROLE IN RADIO ENGINEERING.

Sound, as we have seen, is the vibrations of the air felt by our ear. The basis of sound propagation is the wave-like movement of the air medium. Similar processes occur in electricity during the transfer of electrical energy. Here we are also dealing with a wave, only not with an air wave, but with an electromagnetic one, and this type of wave does not need an elastic medium familiar to us for its propagation, but moves in the so-called. world air; the latter fills all substances, all the space around us, including the airless one (recall that electromagnetic waves propagate even in a vacuum, their propagation speed is 300,000 kilometers per second).

The same definitions of the period and frequency of vibrations, which we have already met when considering the phenomena of sound propagation, apply to electromagnetic waves. However, the frequency with which radio engineering operates during transmission is much higher and ranges from several tens of thousands to several tens of millions per second (the so-called high frequency oscillations).

Electromagnetic waves, due to their speed and, in contrast to sound waves, their slight attenuation over distance, are, as is known, used in radio communications. The source of these waves is most often a cathode lamp, which is an indispensable generator of high frequency oscillations. Such a lamp, suitably connected to the oscillating circuit and the filament and anode batteries, excites undamped oscillations of a known frequency, which depends on the self-induction data and the capacitance in the circuit. The smaller the values ​​of these latter, the shorter the length of the waves excited by the lamp through the antenna device and, consequently, the greater the frequency. With an increase in capacitance and self-induction, reverse phenomenon.

In order to understand the phenomena associated with the construction of an electric instrument, let us briefly trace all the processes that take place in radio transmission and reception.

It must be pointed out that high frequency oscillations in the radiotelephone play an essentially secondary role. This frequency lies well above the limit that could be translated into the language of audio frequencies. Therefore, their direct use for sound reproduction is not possible, and they are only a kind of means for recording sounds. This becomes clear when considering Fig. 3, 4 and 5; the first of them shows graphically the high frequency current excited in the transmitter antenna. In the following figure, we see the current curve of some pure sound produced in front of a microphone. Sound vibrations are transformed after the microphone into low-frequency electrical vibrations; the latter are superimposed on high-frequency oscillations, the correspondingly changed amplitude of oscillations of which is shown in fig. 5. In this figure, we have obtained “recorded” or, as they say in radio engineering, “modulated” oscillations.

Rice. 3. High frequency oscillations.

Rice. four. Pure sound.

Rice. 5. Modulated high frequency oscillations.

Modulated oscillations propagate in all directions in the ether, are caught by a receiving antenna and excite fast alternating currents in the oscillatory circuit. It remains to transfer such high-frequency currents to a lower level, i.e., to convert them into sound ones. This is necessary because, as we pointed out above, a high frequency in our auditory organ will not give the impression of sound, and also because the metal membrane of the telephone cannot respond to such frequent vibrations.

For conversion, a detector is used, which is used of two types: 1) crystalline (imperfect contact of a metal tip with some crystals or a pair of crystals) and 2) the same cathode lamp, placed in special operating conditions. The detector is a kind of valve, allowing vibrations to pass only in one direction; it cuts them in half thanks to this, turning the alternating current into a constant pulsating one (see Fig. 6). Thus, rectified oscillations of already audio frequency come out of the detector, which can act on the membrane.

Rice. 6. detector action.

Rice. 7. Phone cut.

The telephone is a direct converter of electric current fluctuations into air. The cut phone is shown in fig. 7 and consists of a membrane and an electromagnet located in front of it. The diaphragm is therefore subjected to the constant attractive force of the steel magnet and the varying force of the iron core being magnetized by the coils. A rectified current from the detector is passed through the latter, due to which the membrane begins to attract and move away, i.e., to oscillate in time with changes in the current oscillations. The membrane, in turn, is an ordinary oscillating elastic body capable of exciting sound waves.

If you want to get loud sounds, you must first turn on a low-frequency amplifier after the detector, made up of the same universal cathode lamps. In the latter case, the range of sound vibrations will increase many times over, and the membrane, under their influence, will vibrate the nearest layers of air more intensively. An ordinary telephone is overloaded, which is why in the latter case special mechanisms are used with membranes or horns of a special design (loudspeakers).

All these elements: a cathode lamp in three roles - as a high-frequency generator, a low-frequency detector and amplifier, and a loudspeaker are the components of the "theremin".

IV. ELECTRIC OSCILLATIONS AS A SOURCE OF SOUNDS.

So, we have seen in previous chapters that sound and electricity are based on oscillations, and oscillations of an electric current, with the help of devices known to every radio amateur, can perform mechanical work and excite, although not directly, a sound wave.

In an ordinary musical instrument or a human voice apparatus, there must necessarily be some elastic body, which can be brought into a relatively fast oscillatory motion by mechanical action. Striking a string with a hammer, touching it with a bow, directing a jet of compressed air from our lungs to the metal reed of a wind instrument, we make these bodies vibrate at a certain frequency we need, which is already transmitted to the surrounding layers of air. In radio engineering, we also have an ideal constant exciter of oscillations, namely, a cathode lamp. The only trouble is that usually the frequency of these oscillations is too high; even if we could build such a perfect telephone mechanism and such an elastic membrane that could follow high frequency vibrations, we would still not hear anything with our imperfect ear.

Here, of course, it must be pointed out that it is possible to put the cathode lamp under such operating conditions under which the frequency generated by it would drop from its heights to the limits we need. The reader will find more detailed instructions about such devices below, in chapters VI and X-XII.

Let's return to the starting position, to the high-frequency generator, and try to translate its oscillations, so to speak, "transpose" them to a more acceptable range for the ear. It turns out that this is possible. The main method used in this case by Theremin and most radio engineers who construct devices similar to the theremin is not particularly new - it is the principle of detecting undamped oscillations using interference (oscillation addition) and the resulting beats.

Let's explain this phenomenon using an example from the field of acoustics: let's press two adjacent keys on the harmonium in the lower octave, for example, "si" and "do". The frequency of vibrations of the first note is 32 per second, the second 34. It seemed that we should have heard two sounds forming an interval of half a tone. In fact, in addition to this interval, we will hear an additional periodic amplification and weakening of the sound, felt in the form of some shocks. If we take a second interval, wider, for example, "si" and "re" (frequency 32 and 36), then these shocks will become more frequent. At the same time, we will notice that the frequency of these shocks exactly corresponds to the difference in the frequencies of the two fundamental tones we have caused: in the first case 2 and in the second 4. Therefore, the greater this difference, the more often the shocks follow each other and vice versa. With two notes coinciding in frequency, no shocks will follow.

These shocks are the beats we need. The latter arise from the interference of two sound waves, the frequency of which differs slightly from each other.

Let's go further - to high frequency oscillations. And here also we can use the same beats for our purpose. The simplest example from this area is given by amateur radio practice. Suppose that you are receiving a station on a well-known regenerative receiver operating at a certain wavelength or, in other words, with a certain oscillation frequency. If you tune the receiver exactly to this station and bring the grid and anode coils closer, i.e. increase the feedback, then at a certain position of these coils a high whistle will be heard in the phone. With further convergence of the coils or with a change in the capacitance of the variable capacitor in the tuning circuit, the pitch of this whistle will decrease until it disappears completely. As the feedback continues to increase, the whistle reappears at a low note, which will now begin to rise, reaching the highest note and finally disappear there.

This whistle, which is so disliked by the neighbors of a radio amateur making such experiments, was the result of the interference of two waves: one wave is sent by the transmitting radio station that you receive, and the other was the result of the fact that your regenerative receiver, with increased feedback, and in turn, became a miniature transmitter with a wavelength very close to that of the received station.

So, here we repeated the previous experiment with the addition of sound waves, but the whistle that we discovered is beats.

Let us assume that the transmitter at the station emits a wave with a frequency of 1,000,000 oscillations per second, which corresponds to a wavelength of 300 meters. Your transmitter-receiver "works" on a wave that differs by a very small fraction from the first, for example, with a frequency of 1.002.000 per second, i.e., somewhat shorter. Interfering, these oscillations will give beats, the frequency of which is equal to the difference in the oscillation frequencies of both transmitters, namely, 2000 oscillations per second.

This frequency, as we see, is already of the sound order, which, acting through the detector on the telephone, will cause the membrane of the latter to vibrate accordingly. Therefore, we will now hear a tone (whistling) of a certain height. At the same time, it should be noted that we felt the beats from the addition of sound waves not in the form of a musical note, but in the form of clicks, due to the fact that their frequency was below 16 per second.

By changing the loop setting or bringing the grid and anode coils closer together, we will thereby change the wavelength of the “local” transmitter. As the frequency difference decreases, the beat frequency will decrease and the pitch will therefore decrease. Having reached a certain limit, at which the wavelengths of both transmitters will be exactly equal, we will not hear anything, since the difference in frequencies will be equal to zero (the so-called "zero beats"). When this boundary is crossed to the other side, the beats reappear; their frequency will gradually increase and the pitch will rise again. When this difference crosses the "sound limit", i.e., there will be more than 25,000 vibrations per second, the sensation of sound will disappear, since the ear will not feel it.

Rice. eight. Interference of two waves.

Graphically, this phenomenon is shown in Fig. 8, where both upper bands show two oscillations with periods slightly different from each other, and the lower one is the result of interference (the sinusoidal line of decrease and increase of the 3rd type of oscillations - beats) is circled with a dotted line. When passed through the detector, these latter are rectified, as usual, turning into a current pulsating in time with beats in one direction, acting on the telephone membrane.

V. THEORETICAL PART OF THE THERMENVOX DEVICE.

Thus, the key to solving the problem set before us has been found. It is enough to construct two small transmitters, connect them to a detector and a telephone, and control the pitch of the beats by changing the tuning of one of the transmitters; in this way we can get the musical phrases of any pattern.

This method of changing the beat frequency by detuning the contours is not new and has already been used in radio engineering, at least for measuring extremely small changes in self-induction and capacitances (Widdington, Herweg, Pungs, Vvedensky, etc.). L. S. Termen had the good idea to use this method to create a new musical instrument, which he managed to do extremely beautifully and witty.

To complete our theoretical premises, let us dwell a little more on the transmitters themselves, or, as we will call them in what follows, generators. It would seem that there is no need for the constructive implementation of the "theremin" to pile up a large number of lamps and install independent generators. In fact, one could use a conventional regenerative receiver, which is an unusually simple source of whistles of various tonalities; It would be possible to “play” on such a receiver by changing the setting of the receiving circuit in one way or another. Of course, this idea is easy to use, one has only to disconnect the antenna and ground from the receiver terminals and remove the screen on the panel that interferes in this case. By tuning the receiver in unison with the incoming oscillations, we can easily get a certain range of tones by moving our hand closer and further away from the handle of a variable capacitor or by adjusting the vernier.

However, this turns out to be insufficient for producing a truly artistic impression. The “contamination” of the air by a large number of simultaneously operating telephone and, in particular, telegraph stations does not make it possible to single out pure notes of a certain height; in the absence of transmitting stations, the instrument would have to be silent. In addition, obtaining low tones would be possible with great difficulty.

For the last reason, going further, it is inconvenient to use only one generator instead of two, which theoretically would also seem possible (generator-receiver, that is, simply speaking, a regenerative receiver and an additional local oscillator, similar to a classical superheterodyne). This method, as practice has shown, somewhat worsens the results; the receipt of tones is unstable, and therefore it is necessary, despite the extra costs, to design two independent generators.

Essentially, individuals with a normal 0-V-1 or 0-V-2 tube receiver can build a "theremin" by placing two high frequency oscillators in front of the receiver.

How does the Theremin change the pitch? As we pointed out earlier, "playing" is carried out by approaching and removing the performer's hand from a small metal rod located on the right side of the apparatus. This method, of course, is much more convenient than turning the knob of a variable capacitor. With the Theremin method, the hand makes approximately the same movements as the hand of a violinist or cellist on the fretboard of an instrument, with the only difference being that it remains freer, and the sound is more easily amenable to the movements of the hand and even the body of the player.

This method of control is in full accordance with the phenomena that occur in every unshielded regenerative receiver (remember the historical "radio links"), in which it is extremely difficult to tune into distant stations, since the approach of the hand to the tuning controls is very intensively reflected in the behavior of the receiver. Here it is all the more easy because the change in the oscillation frequency required for the entire tonal range, and, consequently, the change in the capacitance of the circuit of one of the generators, should be completely insignificant.

The design of V. A. Gurov’s apparatus (see Ch. I), in which the pitch is controlled by moving the hand along the fingerboard, generally gives the same results: here, too, the hand approaches and moves away from the contour, with the only difference that it loses not in space, but rests on a wooden neck. With Theremin, in his original devices, the setting, and in some cases, was also achieved by moving the hand along the lid of the table on which the apparatus was located.

In addition to changing the pitch, to complete the musical impression and give expressiveness to the game, it is necessary to adjust the volume of the sound. Theremin in his newest model this is done by the action of the left hand on a special wire arc; This method, which reduces muscular effort to a minimum, is extremely rational due to the fact that it is free from any kind of mechanical influence on the sound source, allowing for very subtle nuances. Whether this is due to a change in the capacitance of the connection between the circuits or something else, it is difficult to say. One French radio engineering journal, interested in such a peculiar method of putting "soul" into performance, cites the following hypothesis: the frequency of one of the generators is kept strictly constant by means of a quartz crystal. The self-induction coil of this generator is divided in the middle into two parts; the end of one half of the coil and the beginning of the other are brought out and attached to one large coil of thick wire with a diameter of 20-25 cm. By approaching the hand to this coil, more or less strong attenuation is introduced into the circuit, which leads to a drop in the intensity of the oscillations; quartz, at the same time, does not allow the oscillator setting to change due to a change in capacitance (this explanation is hardly true.) For our part, we will indicate further more primitive methods that are used in our design to obtain the effect of amplifying and attenuating sound.

It remains to say a few words about timbre as well. When getting acquainted with acoustic vibrations, it has already become clear that it is extremely difficult to obtain an absolutely pure tone, free from any overtones. Both violin notes and the sounds of the human voice are essentially complex sounds in which a number of softer-sounding "overtones" are attached to the main loudest tone. The same is true for electrical vibrations. And here, additional “electrical overtones” are added to the main oscillation, which are oscillations with shorter periods, the so-called. "harmonics". (As an example, we can indicate the "harmonics" of some of our stations, which, in addition to the main wave, let's say, at 1000 meters, have weaker "accompaniments" on waves with a length of 500, 250, etc. meters).

By combining these "overtones" and changing the mode of the tubes accordingly, as well as using loudspeakers with different resonators, you can get sounds that differ greatly in timbre from each other.

VI. FOREIGN ELECTRIC MUSICAL INSTRUMENTS.

After Termen demonstrated his invention abroad, a number of similar musical instruments appeared there.

Some types are built, like Termen's, on the principle of using two high-frequency generators and the beat phenomenon. As the most interesting, one can point to the design of the professor of the Paris Academy of Music Maurice Martenot, who is not only a musician, but also a radio engineer. The scheme of his "spherophone" is shown in Fig. 9. G 1 and G 2 are two high-frequency generators already familiar to us, M is a detector and V- low frequency amplifier; R is a sound intensity regulator by means of a special kind of variable resistance, L 1 and L 2 - loudspeakers. The method of changing the pitch of the sound, i.e., playing, is peculiar, which differs sharply from the method used by Theremin.

Rice. 9. Diagram of the Martino apparatus.

The device, which in appearance resembles a conventional multi-tube superheterodyne receiver, is located on a small table; on the front side there is a drawing of a keyboard 1½ meters long. A thin thread passes over the keyboard, on which a red ball is fixed at 5 mm diameter. On the right side of the table comes a string stretched through the block; at the end of the cord there is a horn ring and a celluloid plate with several metal keys. On the left side, next to it, there is a second small table, on which lies a small box with six keys or buttons.

The method of playing is as follows: the horn ring is put on the index finger of the right hand; by pulling the cord out of the apparatus, the red ball is forced to move along the keyboard drawn in front of the apparatus. The pitch will correspond to the key of the keyboard before which the ball stops playing. The left hand rests on the second drawer with keys that serve to adjust the volume of the sound and change timbres. In the depths there are several loudspeakers of various designs (horn and non-horn), which together form various sound combinations.

The pitch control device, as is clear from the diagram, is a very small variable capacitor connected in parallel to the capacitor of one of the generators. It is formed from thin steel wire D passing over a metal plate R. On one side, this wire is connected to a coil spring. F, and on the other hand, with a cord ending in an insulated ring H. By pulling the wire out of this ring, we thereby change the capacitance of the additional capacitor formed by the wire D and a record R. When the ring is released, the wire is pulled back under the action of the spring. On the wire, a pointer is fixed in the form of a ball located in front of the keyboard. To.

The device is designed in such a way and the plate R bent in such a way that the size of the keyboard divisions is the same throughout.

The change in timbres is carried out, as mentioned above, by turning on various loudspeakers, and also simultaneously by changing the mode in which the amplifying lamps operate. Using different sections of the characteristics of the lamps and combining the resulting distortion and overtones, we obtain a variety of shades of transmission over a very wide range. This is achieved by changing the incandescence, the anode voltage and the additional voltage on the grids. If there is no distortion, then the tone is extremely clear, reminiscent of the human voice and woodwinds. When distortion is introduced, the sound begins to resemble the tone of stringed instruments, etc. These changes are achieved by turning on the four keys in different ways, on which there is left hand playing.

A non-inductive resistance device that regulates the strength of sound is kept secret by Professor Martenot. This resistance works, as eyewitnesses report, flawlessly, changing the strength of sound over very large limits.

To obtain trills and jerky notes, three metal plates are used, located at the horn ring worn on the right hand. These plates are connected by a flexible conductor to a wire D. Touching these plates with the fourth and fifth fingers of the right hand, we turn on small additional containers formed by the body of the player and the plates; thanks to this, it is possible to raise or lower a certain tone by ½ tone or a whole tone (finger pressure on one or two plates).

Before playing, the red ball is placed on the note "A" and the apparatus is tuned like a violin to the same piano note; the adjustment is made by turning the knob of the capacitor, placed on the front wall of the apparatus, and the filament rheostat.

In addition to such systems, there are others (M. Bertrand's dynaphon, Givelet's apparatus, etc.) built according to a slightly different principle, namely, by using generation at a low frequency (see Chapter X). There is only one generator here, which directly produces sound frequency oscillations, connected to an amplifier and loudspeakers. The pitch is adjusted by changing the circuit tuning of this oscillator as the capacitance changes. With such a system, a conventional keyboard with keys that directly turn on one or another capacitor can be supplied. You can also use a variable capacitor instead of the keyboard; turning its knob changes the capacitance and, consequently, the pitch. Under the pen pointer there is a round scale with divisions printed in the form of a miniature keyboard. The design of the capacitor is designed in such a way that the divisions of the keyboard would be the same throughout.

Since the change in the capacitance of such a capacitor can really only be within a maximum of one octave, the transition to other octaves is achieved by including additional auxiliary capacitors and other complex devices.

The timbre of the sound changes in these devices, approximately the same as in Martenot, by changing the number of overtones.

It should be pointed out that the method of playing and changing the strength of sound used by Theremin (removing and approaching the hand in space) is, however, the most witty from the point of view of technique and music.

VII. DEVICE OF HOME-MADE "TERMENVOX".

Having mastered the principles of the device of a radio musical instrument, you can proceed to its practical implementation. As for the technical side, this does not require any special devices and special knowledge - only the experience of an average radio amateur, experienced in assembling tube circuits and handling them, is sufficient. With the musical part, it will be much more difficult, but we will talk about this in more detail in the future.

Rice. ten. Schematic diagram of a homemade theremin.

Schematic diagram of the "theremin" of our design is shown in fig. 10. It has four lamps - two generators, one detector and one amplifying at a low frequency. This kit is quite enough for room performance. It's another matter if the question is about demonstrations in large rooms: here a more powerful amplifying part is needed, which is more convenient to separate from operational lamps.

Further, it must be pointed out that the third option is not excluded, which is beneficial to those radio amateurs who, due to their limited budgetary possibilities, do not want to build a special separate apparatus, but wish to use the receiving devices they already have for this, without prejudice to the reception itself. In the latter case, you can limit yourself to assembling one generator half.

In view of this, we have three types at our disposal, which we will describe in sequence. Let's start with the design built according to the scheme of Fig. 10, and we will analyze it in more detail, since it is, in essence, the main one.

The most important detail is the arrangement of generators. In order not to complicate matters, we will focus on the generator circuit, in which the oscillatory circuit is located in the grid circuit. This design, although not distinguished by any high qualities, is extremely simple and does not represent anything new in comparison with normal receiving circuits.

Of course, instead of such a scheme, one could put “push-pull push-pull, giving stronger and more stable oscillations, generators, due to which it is easier to achieve the same strength of sounds spaced along the sound ladder at a great distance from each other. In our opinion, for amateur radio use, installation should not be complicated, and besides, too powerful vibrations can "seriously" go beyond the room and create unwanted interference for neighbors. Therefore, the required sound power must be achieved by selecting appropriate low frequency amplifiers.

So, we will restrict ourselves to our primitive generator, which is essentially an ordinary feedback receiver with the only difference that the first one does not have a "grid-face" and a handset.

Next, we will analyze on which range it is more profitable to make the generators work, that is, which wavelength should be chosen. The resolution of this issue depends on the sound management system. Since in our case we are using very small changes in capacitance (due to the movement of the hand at a distance), the frequency of the oscillations must be relatively high and the wavelength of the emitted wave must be lower than the lengths of the powerful stations operating in the area. If this condition is not met, then we will often have cases of "climbing" of such waves directly into the detector circuit or, even worse, into the generator circuit. In the latter case, we will have a complex interference of oscillations not only from local generators, but also from incoming ones. As a result, instead of a harmonous sound scale, we will hear unexpected jumps and completely not included in the calculations of the performer of the sound.

For caution, of course, it would be necessary to apply complete shielding of the circuits from external influence, as is done, for example, in a superheterodyne, in order to protect the intermediate amplifier from receiving uninvited guests in the form of long-wave telegraph stations or their harmonics.

On the other hand, a very short waveform is inconvenient to control, since the manipulation of the hand will cause too strong effects in the tuning when operating at very high frequencies.

Therefore, keeping in mind that we need a chromatic scale with transitions from about 30 to 4000 vibrations, which corresponds to the piano keyboard, we can stop at the fundamental frequency, at least 1,000,000 vibrations per second; thus, the beat frequency in this figure is from 0.003% to 0.4%, which is freely obtained by moving the hand in a convenient area for playing.

Applied to this position, approximately select the value of both oscillatory circuits of the generators. Each of these circuits consists of a self-induction coil and a variable capacitor. In order to save money, you can limit yourself to placing such a capacitor in only one circuit, and leave the second circuit non-tuning by including a capacitor of constant capacitance selected once and for all. However, in order to expand the limits of experimentation and be able to obtain beats not only with fundamental oscillations, but also with harmonic ones, as well as to move within certain limits from one operating range to another, it is recommended to make both capacitors variable.

The question of harmonic beats plays an important role here. The fact is that in order to get bass, you need to adjust the tuning oscillator with respect to the stable one almost exactly in unison, with a difference of only a few tens or hundreds of vibrations per second. In practice, this turns out to be almost impossible, since, by gradually reducing the difference in frequencies, we reach a certain limit, after which the beats break off and no notes can be obtained. This is due to the fact that due to the direct interaction of both circuits with each other, the setting of one of the circuits, with a large convergence of frequencies, begins to act, in addition to the will of the player, on the second, i.e. their oscillation frequency is compared automatically.

To avoid such an undesirable phenomenon, one has to resort to somewhat artificial means and excite beats between the fundamental oscillations of the first generator and the nearest harmonic of the second. In this case, we tune one generator, for example, to a wave of 400 meters, and the second to almost 200 meters. Then, therefore, we can easily approach any, even the most insignificant, frequency difference and get all the necessary bass notes, without the interaction of circuits tuned, in reality, in completely different ways. Since our elementary transmitters are rich in harmonics, the beats will be almost as strong as if we interfered directly with strong fundamental vibrations.

Parts list.

• 50 m bell wire.
• Two variable capacitors ( From 1 at 500 cm and From 2 at 350 cm).
• Mica Fixed Capacitor From 3 (100-300 cm).
• Grid-Lick Resistance R 1 (1-2 megohms).
• filament rheostat R 2 in 10 ohm.
• 4 lamp panels.
• Low frequency transformer.
• Vernier handle.
• 3 phone jacks.
• 12 contact buttons
• Mounting wire.
• Wooden box.
• ½ m copper rod.
• 2 tuning knobs (small and large size).
• Sheet of cardboard.
• 4 lamps Micro.
• Dry battery or battery for heating (4-4.5 volts).
• anode battery.
• Switch.
• Small screws, wood screws, insulating rubber tube, piece of brass, etc.
• Loudspeaker.
• Cords for connecting batteries and loudspeaker.
• 2 plug-in feet.
• Rubber sponge for shock absorber.

Let us turn to the constructive implementation of generators; the main part here is the coils, which you need to make yourself as carefully as possible. As can be seen from the diagram, we have six coils, divided into two groups of three each. Coils L 1 and L 4 are mesh, coils and L 5 anode finally coil L 3 and L 6 are used for communication between the generators and the detector lamp. The connection between the coils in each system is made constant, although the possibility of changing their position relative to each other is desirable for experimentation.

To wind the coils, four cardboard cores should be made: two with an outer diameter of 100 mm and a length of 130 mm and two with an outer diameter of 85 mm and length 55 mm. The material is thin, dense, flexible cardboard, presspan, or another material suitable for this purpose.

The skeletons are made as follows: a wooden block or a bottle of the appropriate size is taken, four ribbons are cut out of cardboard: two in 130 mm wide and two 55 mm width. The length of these tapes is taken depending on the thickness of the cardboard so that the tape can be rolled up in two or three layers to obtain a sufficiently stable core. The edges of each tape are brought to nothing with a sharp knife so that when gluing there are no sharp protruding folds.

Lubricated on one side with syndeticone or carpentry glue, the tape is superimposed on the blank and tightly folded, after which the thuja is tied with twine so that the tape does not unravel. The skeleton should not stick to the blank, for which the latter is wrapped with a strip of paper before gluing.

The finished core must be covered with some kind of insulating substance, since hygroscopic cardboard easily absorbs moisture in damp air, which can cause large losses in the circuits. To avoid this, the cardboard is coated inside and out with asphalt or shellac varnish.

Winding is carried out with a bell wire or similar in double paper insulation (PBB) with a wire with a thickness of 0.8 without winding mm and with a winding, approximately 1.5 mm.

Let's start with the manufacture of mesh and anode coils, which are wound together on a common core of 130 mm length. To connect the coils to the rest of the parts, four small terminals are screwed in at the base of them, or, even cheaper, contact buttons. We drill holes in the appropriate place for the buttons at a distance of 2-3 cm from each other. To improve the insulation, these holes should be paraffinized or provided with small carbolite insulating washers now available for sale (instead of the latter, a celluloid or mica gasket can be made). Contacts are screwed heads inside; under the heads, from the inside, the beginning or end of the windings is brought, and metal washers are pre-laid on both sides. From the outside, the contacts are screwed tightly with nuts with metal washers. If the washers are not placed, then the insulating sleeves easily burst when screwing.

One pair of contacts is connected to the lower (grid) coil, and the second pair to the anode (upper); they are located at the level of one centimeter above the base of the coil.

Having fixed the beginning of the wire inside the coil at the corresponding contact, we will bring it out through a hole in the body of the coil at a height of 2 cm from the base. Let's make 25 turns and insert the wire inside through a new hole, fix it at the second contact and cut off the rest. The wire should be laid carefully, coil to coil, pulling it during the baby so that it does not come loose.

Retreating 15 mm from the side of the first winding, in the same way and in the same direction, we wind the anode coil, also in 25 turns, strengthening its ends at the second pair of contacts.

Communication coils L 3 and L 6 are wound individually in 15 turns on cores of 55 mm length from the same wire; their ends are connected to two contact buttons located at one of the sides of the coil opposite each other. Contacts are strengthened at a distance of 10 mm from the side; the beginning of the winding is placed at a distance of 20 mm From him.

Coils are the only homemade part, the rest are purchased ready-made.

Capacitors of variable capacity can be taken of any design; it is not required that they be quadratic or direct frequency, since this does not play a role in this case. It is only desirable that their initial capacity is not large. Capacitor From 1 taken with a capacity of 500-600 cm(Products of the Precision Mechanics or Electrosvyaz trusts, the head of Radio, the Metalist workshop, etc.). Capacitance of the second capacitor From 2 it is more convenient to take a smaller one, at 350-400 cm so that the first generator could, if desired, excite a wave larger than in the second (to obtain the proper harmonics). Cast capacitors are suitable for this purpose. "Radio". Both capacitors should be taken without pushers or additional plates, since the vernier fixtures are made independently. The exception is the new cast capacitor head. "Radio" with a serrated vernier that can be put in the first circuit to save on the purchase of an extra vernier handle.

We will talk about the arrangement of devices for fine tuning during assembly.

As a “grid-face”, you can take either a ready-made “grid-face” in a wooden frame (Precision Mechanics Trust), or make it up from separate ones - a resistance and a mica capacitor. The nature of the sound depends on the quality of the capacitor and leakage, so they must be sufficiently reliable and constant.

The filament rheostat is set common to all four lamps - 10 ohms. The latter is done to save money, since with the heterogeneity of our lamps it would be more rational to use separate rheostats, 25 ohms each. The most durable in operation are the products of Electrosvyaz.

Lamp panels must be of good quality with high insulation and no leaks: for mounting on a horizontal board, round sockets of the Elektrosvyaz Trust with terminals brought out on the sides are convenient. In order to avoid the appearance of howling notes at high amplification (microphone effect), it is necessary to use shock absorbers, since Micro tubes are very sensitive to all kinds of shaking. At present, even special low-capacity shock-absorbing panels of the Elektrosvyaz trust (on spiral springs) and on a sponge (of the Precision Mechanics trust) have gone on sale.

Such panels can also be constructed independently as follows: a piece of rubber sponge is taken (sold in Rezinotrest stores), from which mugs are made according to the size of the panel. When assembling, a piece of sponge is placed on the installation board and a lamp socket is placed on it, in which holes are pre-expanded for screwing the socket to the base. Through these holes, either thin studs with ends bent on one side or screws driven into the base are passed in such a way that the panel can move up and down (Fig. 11). Installation when using shock absorbers should be done with a flexible wire. With such a device, the panel, as it were, rests on springs (instead of studs, you can fix the panel across with two rubber bands).

Rice. eleven. Cushioned lamp panel.

In order not to bother with the device of such panels, with equal success it is possible to amortize the entire device directly by placing its bottom on four pieces of a flat sponge. These pieces hold up well when glued to the base with wood glue or, even better, rubber glue.

Let's move on to the low frequency transformer; the nature and beauty of the sound largely depends on the properties of the latter. Some types, due to improper amplification of sounds of different frequencies, will transmit low tones weaker than high ones. Therefore, one should stop at a transformer with a more or less even gain line. The best are the new armored transformers of the "Elektrosvyaz" trust, as well as the head of "Ukrainian radio" with a ratio of turns of 1: 4 or 1: 5.

It remains to make a box for our apparatus. In this regard, the radio amateur, of course, seems to have complete freedom, as long as the installation is expedient from the point of view of technology. You can build an apparatus like a receiver, or, on the contrary, hide, if possible, any reminder of radio engineering. In such a case, all parts should be mounted in a deep box, with a large inclined board, in the form of a console or music stand, on which notes would be placed. The lamps and all controls are hidden inside, so that the necessary adjustments would have to be thrown back the front cover.

Our design is made according to the first method in a conventional receiving box, according to the so-called. "American" type on three panels. All the lamps in it and other parts are located on the horizontal panel, and the control knobs are placed on the vertical one. The terminals are moved back to a special small socket.

The internal dimensions of the panels are as follows: horizontal - 210 × 350 mm, vertical - 160 × 350 mm power panel - 40 × 200 mm. Both vertical panels are sawn from even dry wood or plywood 8-10 mm thick. Since all critical parts of the installation are made on insulating gaskets or bushings, there is no need for waxing. In the absence of such bushings, the power panel should be cut out entirely from carbolite or ebonite (old gramophone records are suitable, which are easily cut with a jigsaw or a heated sharp knife). Finally, you can take a tree, and after drilling the necessary holes, it is impregnated for 10-15 minutes in molten, but not brought to a boil, chemically pure paraffin.

The horizontal base panel should be made of thicker wood so that it protrudes a couple of millimeters beyond the edges of the walls.

Rice. 12. Box.

Usually, with such a mounting system, the work panels, fastened with copper squares, are pushed into a special box open at the front. In this case, you can do it easier. Two side walls are attached to the panels assembled on screws, thanks to which the entire structure is given greater strength. The rear wall and top cover are hinged to facilitate installation and inspection. This eliminates the need for a special case. Details of the manufacture of the box are shown in Fig. 12; the finished box is stained and varnished.

It is desirable to shield the entire box so that the approach of the hand does not affect the setting.

Assembly of the theremin.

It remains to make the installation (see the wiring diagram in Fig. 13). Pre-place all the parts on a vertical panel. Capacitor on the left side From 1, from the right From 2, between them below the filament rheostat. From the outside of the panel, capacitor C 3 is supplied with a conventional large mastic pen with graduations. A device for fine tuning must be attached to the capacitor C 1, which facilitates the approach to the desired number of beats. For this purpose, a vernier handle mast is used. "Metalist", which reduces the speed of rotation of the axis by 10 times.

Rice. 13. Mounting diagram of the theremin.

In the absence of a handle, you can proceed as follows: an ordinary mastic handle of the largest possible diameter is mounted on the axis of the condenser. A hole is drilled under this limb. Into which the telephone jack is screwed. An ordinary carbolite plug leg is inserted into the socket. A small cone, cut out of drawing gum, is tightly fitted onto the latter. To protect the cone from slipping, the leg in the appropriate place should be sawn off, giving it a square shape, and smeared with thick glue. A washer is soldered onto the leg from the inside of the panel to prevent the vernier from falling out. The vernier should be placed in such a way that the rubber cone fits snugly against the limbus. For better adhesion, you can make a small notch on the edges of the limbus with a thin file (Fig. 14).

Rice. fourteen. Vernier.

However, such a vernier serves for a rough approach; to adjust the beat frequency before starting the game, it is necessary in parallel with the capacitor From 1 put a small capacitor with a capacity of 5-10 cm. Such an additional capacitance is formed from a small plate and fixed capacitor plates. From 1. Manufacturing details are clearly visible on the wiring diagram. The plate is oblong (width 1 cm, length 4-5 cm) is cut from aluminum or brass in 0.5-1.0 cm thick. A hole is made at one end of the plate, into which a metal axis with a screw thread at the end is inserted to secure the plate with a pair of nuts.

The axis is passed through the front panel (in the upper corner). For better contact, a telephone jack is inserted into the opening of the panel, through which the axle must pass with known friction. The socket is connected to the axis of the movable plates of the variable capacitor. From the outer side of the panel, a handle made of insulating material 5-10 is mounted on the axis cm length. To prevent the plate from dangling, a pair of wooden bushings are put on the axis on both sides. At the same time, it is necessary to ensure that the additional plate does not swing during rotation, as this will affect the tuning. Therefore, for greater stability, it is recommended to slightly lengthen the axis and make a second fulcrum at the free end, in the form of a small metal square, fixed next to the side wall.

Distance between additional plate and movable capacitor From 1 should be approximately one centimeter. The handle needs to be lengthened so that the beat frequency can be adjusted from a distance.

Rice. fifteen. Generator coils.

On the horizontal panel, standing in the rear extreme corners, the double coils of both generators are located. They are attached to it either with copper paws, or by means of round pieces of wood inserted into the inside of the coils (and cutouts are made in places in contact with the contact buttons).

Communication coils L 3 and L 6 inserted into generator coils. In order for the coils to hold tight enough, pieces of cork are driven between both cores. Both small coils should be approximately level with the anode coils of the generators (fig. 15 and 16).

Rice. 16. Section of the theremin.

The lamp panels are arranged symmetrically between the coils: a “grid-lick” is placed in the middle. To avoid leaks, the latter should be supported by weight; otherwise, an insulating gasket must be placed under it.

The low frequency transformer is mounted in front, next to the filament rheostat.

A pair of sockets for a loudspeaker (left) and current supply terminals (right) are screwed onto the power panel.

The "antenna" for adjusting the pitch is a flat copper rod ½ meter long and 5-6 mm thick. To connect to the generator circuit, the grid of the second lamp is connected by a wire to a terminal placed in front of the side wall at a height of 6-8 cm from the base. This terminal must be well insulated. One end of the rod is bent into a narrow ring, the plane of which is ground with a sharp file and attached to the terminal with a nut. In order for the antenna not to sway and thus not change the distance to the player’s hand, a piece of carbolite (for example, the body of a plug) is strengthened in the upper part of the wall, through which the rod is passed.

The antenna, of course, can be placed separately at a short distance from the box, fixing it in a porcelain socket from electric lighting and connecting the latter to the terminal with a thick insulated cord.

Installation is carried out with a copper, best of all, silver-plated wire (1.0-1.2 mm thick); in places of crossing, rubber tubes can be put on the wire.

The wiring diagram is designed in such a way that the conductors, with the exception of one connection, are led directly to the terminals and sockets (without soldering).

The turns in the anode and mesh coils must run in opposite directions. Therefore, when assembling, it is necessary to test various connection methods in order to achieve a position at which generation occurs most intensively. Also, the way the coils are turned on is not completely indifferent. L 3 and L 6 , and the method of connecting them with an end, which is also in practice.

The device, in order to avoid complicating the design, is made without full or partial shielding; the latter, of course, can be useful to reduce the interaction of contours. When shielding, all walls, the bottom and the cover should be pasted over with steel and a brass partition should be placed between the coils, connecting the shield to the “-4” terminal.

Let's move on to nutrition. Since the "theremin" has four lamps, a dry filament battery will quickly sink, which is why it is more profitable to put a 4 volt battery with a capacity of at least 20 ampere-hours. Dry batteries are taken on the anode. To excite generation, at least 80 volts should be given to the first two lamps, 45-80 volts to the detector lamp, and 80 volts to the amplifying lamp. To get bass notes, it is imperative to increase the anode voltage on the generators and low frequency to 125 volts. In the latter case, an additional voltage of 3-4 volts is supplied to the grid of the last lamp from the battery of a pocket electric torch.

It should be borne in mind that the following reasons affect the quality and nature of the sound: the magnitude of the anode voltage and incandescence and the size of the additional voltage on the grid. In general, by changing the mode of the lamps in any way, you can give the sound a different character. Since not all microtubes work in the same way, it is necessary, by trying different specimens, to select those that generate the most intensively. With the release by the Elektrosvyaz trust of an amateur powerful amplifying lamp, the volume of the transmission can be increased. In this case, the last cascade should be provided with a separate heating rheostat.

The device is assembled, you can start playing. However, some additional details are needed to create a greater artistic impression.

Since the transition from one pitch to another is achieved by moving the hand in front of the antenna, the playing takes on a somewhat creeping character (continuous "glissando"). For some musical phrases, this character is undoubtedly acceptable, but in most cases it is desirable to be able to obtain separate pure intervals, without going through the entire intermediate ladder of sounds.

The simplest way is to include a bell button in one of the wires going from the device to the loudspeaker. Playing in this way, it is necessary to press the button frequently during a fragmentary transition from one note to another, thus achieving the required sound duration.

Rice. 17. Breaker.

With a more or less fast pace, this method makes it difficult to perform, so Termen uses a more advanced type of "breaker" in one of his apparatus. For this purpose, two contacts are fixed on a wooden base at a distance of several centimeters from each other, connected by a wire led to a common terminal (Fig. 17). Above these contacts, an anchor made of a piece of brass is strengthened, having an axis in the center. The anchor is kept in balance by two springs placed on both sides of it. From the axis of the armature there is a conductor to the second terminal. This interrupter is included in the loudspeaker circuit like the bell button described above. The blow is made with two fingers of the left hand alternately on the right or left half of the anchor, due to which the loudspeaker circuit is closed each time.

With such a balanced arrangement, work is facilitated, since the interruption is obtained almost automatically and without any effort.

The proper distance between armature and contacts should be adjusted first. The outer surface of the anchor and the wooden base are pasted over with a piece of leather. In order not to tire the arm, a small pad is placed under the brush, or an appropriate curved shape is given to the base.

In fact, in order to first master such a relatively complex instrument as the theremin, this should be limited. Regulation in two directions (pitch and volume) presents a number of difficulties for a beginner, although, of course, the absence, for example, of the volume of sound gives the game a somewhat dispassionate character (cf. with an organ in which purely mechanical means are used to change the volume, such as opening and closing the lids of resonator boxes, changing from one pipe system to another, etc.).

To adjust the sound intensity, we use three methods, all related to a low-frequency amplifier. Experiments with the first three lamps showed that here we are dealing with an area that is too sensitive, in which any movement of the handle intended to change the force simultaneously affects the tuning, i.e., the pitch of the sound (unless, of course, there are no special devices used by Theremin).

On the contrary, amplifying lamps allow the use of lighter and more affordable means for an ordinary radio amateur.

Rice. eighteen. Capacitor in the circuit to adjust the volume of the sound.

The first method is to turn on a small variable capacitor in front of the grid of the amplifying lamp at 100-150 cm with a minimum initial capacity (Fig. 18). In practice, of course, it is inconvenient to use for this purpose a normal capacitor rotated by a handle, in view of which its design should be changed. It is possible, for example, to compose this capacitor or two round aluminum plates 10 cm across. One of them is fixed motionless on an insulated stand, and the second on a lever with a spring. When the lever is pressed, the plates approach each other (capacitance increases), when the pressure is released, the opposite occurs. It is also possible to hold a second plate, attached to an insulated handle and connected to the circuit with a flexible wire, directly in the left hand, etc.

To eliminate the noise that sometimes appears in this case, it is necessary to connect the grid with a glow using a resistance of 1-2 megohms.

You have to adjust the capacitance of such a capacitor with your left hand, which means that the device for making the sound jerky either disappears, or it has to be made by foot; in the latter case, its size increases in such a way that a balancing device with two pedals is obtained (the anchor is made in the form of a flat wooden lever of 20 - 25 cm length).

It is possible, of course, to combine both devices into one in such a way that the approach and removal of the capacitor plate would be done by pressing the brush, and jerkiness would be achieved with two fingers, but this will be somewhat difficult.

To switch on, two terminals are screwed into the front panel.

The connections are made short and untwisted, which creates additional capacitance.

With another method that gives nice results, a variable resistance is included in the loudspeaker circuit. The latter can be included either in one of the connecting wires (in this case, by reducing the resistance, we increase the sound intensity), or in parallel with the loudspeaker clamps (the reverse phenomenon is obtained). Its design may be different.

An exemplary device is made as follows: a strip of good thick paper with a width of 5 mm and a length of 30 mm. The strip is shaded with a pencil, after which a terminal is passed through one of its ends. For better contact between the terminal and the strip, a piece of steel is placed under the nut. A copper slider connected to the second terminal should walk along the strip. It is more convenient to adapt the resistance for the foot pedal in such a way that when the foot is pressed, the resistance decreases; when lifted, the slider should move away under the action of the spring.

We do not give here a detailed design, since it can be developed in various ways by each radio amateur, like the well-known variable megohms. It should only be borne in mind that the angle of movement of the slider should not exceed 30º, otherwise it will be difficult to work with the pedal. The value of resistance has to be selected in practice, shading the strip with different strengths or erasing the excess with an elastic band.

It is also possible to build this resistance according to the type of variable megohms of the Precision Mechanics Trust, in which the change in resistance is achieved by greater or lesser pressure on granular coal powder. The powder is in an insulated tube. A fixed copper bushing is inserted into one end, and a copper piston on a coil spring passes through the other. The composition of the powder must be chosen so that the resistance varies within wide limits. If pure charcoal powder (for example, used in elements) gives too little resistance, it can be mixed with a small amount of gypsum or the like (moreover, see Ch. XI).

Finally, there is also a third way, namely: changing the sound intensity by adjusting the degree of incandescence of the low-frequency amplifier lamps (although not within wide limits). The rheostat should also be made foot. This method can only be used with high-capacity filament batteries, in which the change in the incandescence of the amplifying lamps will not be reflected appropriately in the change in the mode of the generators, which affects the pitch.

It remains to say a few words about the speakers. The loudspeaker can be taken of any design, preferably the most sensitive ("Record"). In terms of beauty of transmission, the best results are obtained with horn systems, in which the sound acquires a warm character, reminiscent of the sound of a wind instrument. It is also good to combine horn and hornless loudspeakers, including them separately and together.

The nature of the sound can be changed within certain limits by shunting the loudspeaker clamps with various capacitors of constant capacitance ranging from 1000 to 15000, which softens the sharp tops and gives the sounds a somewhat muffled tone.

For this purpose, a box (the so-called "tone filter") is switched on in parallel with the loudspeaker. Under the panel of this box are five capacitors in 1000, 3000, 5000, 10000 and 15000 cm. A switch with six buttons is placed on the panel, connected to the ends of the corresponding capacitors; one button remains blank. The opposite ends of the capacitors are connected together. A pair of input and a pair of output terminals are screwed into the left and right sides of the panel. The connection diagram is shown in fig. 19. Having such an uncomplicated device, one can change, to a certain extent, the nature of musical phrases during a game in a purely mechanical way.

Rice. 19. Scheme of the "tone filter".

VIII. HOW TO PLAY THE THERMENVOX.

It is not easy to give a satisfactory answer to this question, since, as has already been pointed out, there are no schools, and even the players are numbered in units. You have to pave the way yourself.

Let's start with bringing the device to "combat readiness". Insert the lamps, attach both batteries and a loudspeaker. Let's put a capacitor From 2, to the maximum, and the capacitor From 1, to the middle position; turn on the heat. We try to slowly rotate the knob of the capacitor From 1.

If sound notes do not work, increase the intensity. With the correct assembly of the generators, beats should occur at a normal glow for microlamps of 3.6 volts. You need to manipulate the capacitor slowly so as not to slip past.

When the generation is detected, let's try to tune in to "zero beats". Let's assume that the apparatus sounds on a high note. Bringing the hand closer to the antenna, we make the tone go down, we reach a dip, after which the sound rises again. Now fine re-adjustment with an additional plate is necessary. Keeping a distance from the antenna, we carefully turn the knob of this plate, due to which the tuning of both generators will approach, the tone will begin to drop and reach the “dead center”, i.e. disappear. A slight movement of the knob will cause the tone to reappear.

When we have reached this position, the apparatus is brought into a state of unstable equilibrium; by bringing the hand closer to the antenna now, we will produce the deepest tone, and by further approaching the hand, an ascending chromatic sound scale is obtained (in the bass range, a step up will require more hand movement than in the upper register).

It turned out the desired air neck. Its length can be taken any, depending on the desire of the player, since the state of equilibrium has, figuratively speaking, a certain "length" depending on the adjustment with an additional plate: you can make the "theremin" sound already at a two-meter hand distance from the antenna, or reduce this distance to 30-40 centimeters.

Depending on whether the oscillation frequency of the first oscillator is less than or greater than the oscillation frequency of the second one, an ascending or descending scale can be called up. In practice, it is more convenient to use the first method, in which the highest note will be received at the shortest hand distance from the antenna. It is also more profitable not to increase the length of the neck too much, so that you do not have to make large movements with your hand (for example, no more than 30-40 centimeters).

The initial tuning should combine the different capacitor positions of both oscillators to produce the cleanest and loudest beats starting from the lowest bass note.

If we have an interrupter, then fine tuning to “zero beats” is not necessarily required, since in the latter case it does not bother the player if the transition point hits the neck itself (due to this, the working part of the neck can be made of insignificant length).

Further, it should be borne in mind that the sound will initially turn out to be somewhat lifeless, not much reminiscent of the sound of a musical instrument in general. To revive it, tremolation should be used (by analogy with a violin). This is achieved by a slight trembling of the hand. The correct jitter frequency is obtained after some practice. You should not get carried away with excessive trembling, since in this case the performance will begin to take on the character of a “howl”.

Rice. twenty. How to play the theremin.

What should be the “setting of the hand” in this case? It depends on the desire of the performer. You can hold your hand freely in space and play while standing. At the same time, the arm should be extended, fingers extended in the direction of the antenna.

On fig. 20 shows how to play a homemade theremin.

In another way, which is perhaps less tiring, the player sits with his arms bent and his elbow resting on the table. The fingers of the hand are bent (the thumb is pressed against the second) and the hand is directed towards the antenna with an edge. The size of the neck is taken small. The body of the player should be as far away from the apparatus as possible so that the movements of the body do not affect the setting.

Training should be done without devices for interrupting and changing the strength of the sound, since at first it will be difficult to coordinate the movement of both hands.

You don't need to know music to play, but you do need to have an ear. The playing process itself is complicated, since in this case we do not have a neck fixed once and for all, as in a conventional stringed instrument, but we play in the air. It is especially difficult if it is necessary to take tones that are far apart. It will, of course, be much easier for a player who plays the violin or cello, since he already has a feeling for the fretboard. All this, however, as with any instrument, is achieved with practice and skill.

To begin with, you should not take on the performance of musical things, but you need to master the instrument, that is, start with scales and arpeggios to the accompaniment of the piano. The difficulty for the beginner is to get pure tones of a certain pitch, since the slightest movement of the hand changes the tuning.

In general, it should be pointed out that it will not be difficult to assemble a theremin for a radio amateur; to achieve artistic performance is far from an easy task and requires thorough practice and the presence of musical abilities.

The choice of things should be approached with some caution. Best of all, the so-called. cantilena, but not phrases jumping over the entire sound range. Suitable melodic violin or cello repertoire or vocal works. To begin with, you should practice on things in which the piano accompaniment repeats the melody.

Sample repertoire:

1. Folk songs.
2. Arioso Canio from the opera Pagliacci by Leoncavallo.
3. Romance "Night" by Rubinstein.
4. Nocturne is his.
5. An old French song by Tchaikovsky.

In the future, you can also take special piano pieces, performing a melody.

After you have mastered the basic techniques of the game, you should move on to achieving expressive performance. In practice, the amplification and weakening of the sound is not melodic, but by maintaining a note of a certain height.

The interrupter is used during pauses, as well as if you want to get a jerky range of sounds.

Before starting to play, you should tune the instrument to one tone, determined once and for all, finding the already known position of the hand on the fretboard, otherwise it will be difficult to adjust each time.

IX. VARIANTS OF THE MAIN SCHEME OF THE THERMENVOX.

As we have already pointed out earlier, the construction presented by us can be performed in several versions. The simplest one is for people who have a conventional 0-V-1 tube receiver. In this case, you can restrict yourself to the device of only the generator part of the first two lamps. In the receiver, the oscillatory circuit (i.e., the coil and the variable capacitor) should be turned off. Connections are made with short wires. The wiring diagram remains the same, only the third and fourth lamps are thrown away with a "grid-face" and a low-frequency transformer.

In the second case, to obtain a more powerful transmission, the apparatus is made up of the first three tubes, removing the low-frequency amplifier. The latter is mounted in a separate box for two lamps or as a three-tube resistance amplifier. The latter is generally the best, as it produces less distortion.

Rice. 21. Lamp block.

As a low-frequency amplifier, we can recommend the two-tube amplifier of the trust "Electrosvyaz" UN - 2, which allows the transition from one to two lamps. To include a variable capacitor in it that regulates the sound intensity, you should use a special block for a lamp with two output terminals. The design of such a block is shown in Fig. 21. For this purpose, an insulated block with legs is removed from a burned-out cathode lamp; on the latter, the same lamp panel is fixed, which we use for installation. Fastening is carried out with a screw with a nut, passed through the centers of the block and panel. Panel terminals are connected by soldering insulated conductors to the corresponding legs. From the terminal and the legs of the grid, insulated flexible conductors are produced, attached to the terminals of the capacitor.

If desired, such a block can be placed on the first or second lamp of the amplifier.

Such an amplifying part can, of course, be assembled independently according to the scheme shown in Fig. 22. Low-frequency transformers are taken by the trust "Electrosvyaz" or "Ukrainradio" with a ratio of turns in the first transformer 1: 3 and in the second 1: 2. The rheostat is common to both lamps.

Rice. 22. Scheme of a separate amplifier n. hours for theremin.

The amplifier is mounted in any way (either with the lamps hidden inside, or taking them out). The loudspeaker can be plugged into the socket BUT(the first lamp works) or into the socket B(both lamps work). In the first case, if there are no separate rheostats, the non-working lamp is removed from the sockets. The grids of both lamps have leads for supplying additional voltage to them.

The primary windings of transformers can be shunted with various capacities, and the secondary winding of the second transformer with a resistance of 0.5-3 megohm. The combination of shunts changes the nature of the sound (to adjust during the game, put the corresponding sliders with buttons on the panel).

To get a more powerful amplification, you can build a "push-pull" amplifier or put the final amplification on powerful UT-1 lamps (with a corresponding increase in the anode voltage). In the latter case, the "Accord" should be taken as a loudspeaker, capable of filling a large audience.

Multi-tube low frequency amplifiers are often the source of very unpleasant overtones (low frequency generation, microphone effect, etc.). This is paralyzed by cushioning the panels or the box, putting heavy lead or wooden rings on the bulbs of the lamps and selecting the appropriate shunts.

The power terminals of the generator and amplifying parts are usually connected to each other and lead by a common cord to the batteries.

X. SOUND GENERATORS AT LOW FREQUENCY.

In addition to the methods of producing sounds using electrical vibrations, described in the previous chapters, there are some other possibilities that are of great interest to those who wish to experiment in this area.

One such method is low frequency generation. In a low-frequency amplifier, it often appears as a sharp, stable tone on a particular note, the pitch of which does not change depending on the tuning of the receiver circuit.

This generation can also be called artificially as follows: we take a high-frequency generator, turn off the tuning capacitor and replace the coils with others with a large number of turns. At known value coils, the oscillation frequency of the generator can be so reduced that these oscillations will affect our hearing directly, without any transposition. In practice, for this purpose, it is easy to use a conventional low-frequency transformer with a turns ratio of 1: 4 or 1: 5.

We remove the iron core from it. The primary winding is connected in place of the generator anode coil, and the secondary winding is connected in place of the grid coil. The direction of the turns, as usual, must go in opposite directions, otherwise generation will not occur. The glow and anode are normal.

On this principle, several types of radio musical devices were built abroad. One of the first is Garnsbeck's "radio piano" (1926 - America).

This apparatus has twenty-five keys connected to twenty-five separate low-frequency tube generators. Each of these generators is tuned once and for all to a certain note, and a chromatic scale of twenty-five semitones (that is, two octaves) is formed. In addition, each generator is connected in turn to a separate loudspeaker (practically, the design is made in the form of one large horn, equipped at the end with twenty-five powerful telephones). Thus, we have here an instrument similar to the piano, which can be played with both hands and take chords of any complexity. The tuning of each generator is carried out during the assembly of the instrument by introducing contours of iron wires of various thicknesses into the coils or by selecting constant containers. The keys are placed in the anode circuit and turn on the corresponding loudspeaker when pressed.

The designer of the "radio piano" is working on simplifying the instrument, in particular, on the use of one common loudspeaker, the circuit of which includes twenty-five coils connected inductively with all the generators (the device, however, has not yet been sufficiently stable, since the generators often begin to influence through coupling coils).

Such a device, even with one common loudspeaker, almost still seems too cumbersome, especially since a keyboard of eighty-eight keys is needed to play piano works. The combination of eighty-eight generators and the same number of loudspeakers on a common power supply in a modern technical design from the artistic and economic side can hardly be justified.

Another apparatus of the same kind (“radio trombone”), which is a trombone bell, at the end of which a telephone and a low-frequency generator are embedded, is essentially a toy, since its range is extremely insignificant.

French devices, as we have already indicated, are monophonic, since they have only one low-frequency generator. The adjustment in this case is carried out either by means of large capacitors of variable capacity, or by a system of selected constant capacities, switched on by means of keys (Givelet system).

Such designs, however, suffer from major drawbacks:

a) The range of the instrument is not large, since the sound reduction is achieved by the inclusion of progressively increasing capacitances, while with great importance capacitor in the circuit, the lamp loses its ability to generate. Usually the limit is 12 semitones (an octave).

b) During the game it is impossible to achieve "glissando" due to the fact that before pressing one key, you need to press the previous one (otherwise the capacitances will add up to get a lowered false sound). From the musical side, playing with jerky sounds is not very attractive.

c) In order to obtain a correctly tuned gamma, when assembling the instrument, an extremely painstaking adjustment of capacitances is required, or the presence of twelve variable capacitors. At the same time, a slight change in the incandescence of the generator lamp, the depletion of the anode battery, and, finally, a change in the lamp itself require a new restructuring or special and very complex devices.

In view of this, the French apparatus, as far as is known, did not find practical application.

The “electrol” apparatus designed by the author, free from the above disadvantages, is also a monophonic instrument built on the principle of using the phenomenon of low frequency generation. The range of the instrument is at least 5½-6 octaves, with a wide change in timbres and the nature of the sound.

Compared to the theremin, the electrola has the following qualities:

1. Extremely simple and cheap design and portable size.
2. Savings in the number of lamps and power supply (the sound strength of "electroly" on one lamp and "theremin" on four is the same).
3. Easy to handle and play that does not require much skill, except for the presence of some ear for music.
4. Lack of pre-tuning for "beats" and the constancy of the neck.
5. Absence of radiation on the air.

The sound, by its nature, reminiscent of "theremin", is distinguished by greater stability and density, free from "howling".

The "theremin" has its advantage - in terms of the way the sound is controlled by the movement of the hand in space (independence from the iron core, which has a known inertia).

XI. ELECTROLY DEVICE.

a) Simplified diagram.

The device can be made in two versions. According to the first one (the diagram is shown in Fig. 23), we have a single-tube generator, the sound power of which is still sufficient to fill a large room. In order not to complicate the device by winding coils, you can use windings from a conventional low-frequency transformer, from which the core has been removed.

Rice. 23. Schematic diagram of a single-tube electrolytic.

The pitch is adjusted, on the one hand, by pushing in and pulling out the iron core from the body of the coil (i.e., by changing the self-induction coefficient) and, on the other hand, by including high-capacity permanent capacitors in the circuit ( From 2 - From 4), changing registers, i.e., the frequency range (capacitor C, permanently connected).

By shunting the loudspeaker with containers From 5, From 6, From 7 and resistance R 2 You can change the tone of the sound. The nature of the sound is also regulated by changing the magnitude of the glow and the anode voltage and shunting the loudspeaker with an iron choke (not indicated in this diagram).

The circuit allows switching the anode coil in parallel with the loudspeaker clamps, which also dramatically changes the nature of the performance (with a normal leg regenerator 1-2 inserted into slots v-b, and with a modified scheme - into the sockets b-a).

Details. The main part of the "electroly" are self-induction coils L 1 and L 2 taken from a conventional low frequency transformer.

The secondary winding is connected to the grid circuit, and the primary winding is connected to the anode circuit. After a series of tests carried out on commercially available transformers, an armored transformer from the Radio factory was selected with a turns ratio of 1: 5 (primary winding 5000 and secondary 25.000 turns). Its advantage is its relatively large size, due to which the greatest effect (change in pitch) is achieved when the core is moved. With fewer turns in the secondary winding, the instrument will produce only very high whistling sounds.

The transformer is released from metal armor, for which the nuts of the four bolts fastening the core are unscrewed. The iron core is also removed. The core in this transformer is made up of iron frames with long branches inserted into the inside of the coil. In order to take them out, you have to bend the frames, after which they are easily pulled out alternately from both sides of the coil. This must be done very carefully so as not to damage the thin leads from the windings. To protect them from breakage, flexible conductors should be soldered at the ends and the junctions should be attached with sealing wax to the cardboard core of the coil, marking the corresponding conclusions of the primary and secondary windings.

Further, for the manufacture you need: a lamp panel of the Elektrosvyaz trust with contacts brought out to the outside, a filament rheostat R 1 in 25 ohms, five carbolite terminals, five telephone jacks, a plug, a slider with five contact buttons, some thin brass for the springs, four clamps for resistances, a resistance R 2 of 100,000 ohms and a set of fixed capacitors: From 1-350 cm, From 2-2500 cm, From 3-5000 cm, C 4 -10.000 cm, From 5-1000 cm, From 6-5000 cm and From 7-15.000 cm, Micro lamp; four volt incandescent battery, anode battery from 5 to 80 volts.

Rice. 24. Box wiring diagram.

Structural implementation. The apparatus is mounted in a small rectangular box measuring 170 × 110 × 90 mm. (Fig. 24 and 25). At the bottom of this box are placed; lamp panel (left) and transformer coils (near the right wall). An appropriately sized hole is made against the transformer (18 × 18 mm) to skip the core. The coil is reinforced with a small wooden plank (stop) screwed to the bottom of the box. A pair of screws is screwed into the side wall and prevents the transformer from moving sideways. For strength, you can fix it with a still dense cardboard tape that wraps around the body of the coil and is attached to the bottom of the box.

Rice. 25. Location of parts on a horizontal panel (top view).

Sockets are screwed into the front wall a, b, in and terminals G and d, and also made a hole for the output of the cord of the switching plug. The filament rheostat is fixed on the right, the loudspeaker nests are fixed in the left side post; in the rear wall - power terminals. A round hole is made in the lid for the lamp, which protrudes two to three centimeters outward.

Rice. 26. Horizontal panel wiring diagram (bottom view).

The box with the generator is placed on the second flat box with dimensions 330 × 170 × 33 mm so that in left part would have free space for placing the keys and the interrupter (see Fig. 26, which shows the bottom view of the box), the keys serve to turn on (separately or separately) capacitors From 2, From 3 and C4(capacitor C1 attached to the oscillatory circuit). The interrupter is needed in the same way as in the "theremin") to eliminate the not always desirable "glissando" and to obtain intermittent sounds and pauses.

On right there is a switch designed to change the timbres. It consists of a spring slider and five contact buttons. The first of them is idle, and the rest include capacitors in 1000, 5000 and 15000 in parallel with the loudspeaker clamps. cm or a resistance of 100,000 ohms.

Let us turn to the design of the keys and the interrupter. For simplicity, of course, it would be possible to put ordinary bell buttons instead of them, but this is both inconvenient and ugly. Therefore, it is best to make an independent design of the keys and the breaker.

Contact springs for keys are cut in the form of narrow strips of thin brass. To give the springs sufficient flexibility, they are stuffed for ten minutes with a wooden mallet. In total, three pairs of springs will be needed so that each key, when pressed, would be supported by a spring on a spring, and not on a solid contact; otherwise, during the game, an unpleasant knock will be heard and you will have to hit the keys hard, which quickly tires your hand. The same applies to the interrupter, the manufacture of which was discussed in the chapter on the "theremin".

Rice. 27. Breaker section.

Such a device has one drawback: when turned on and off, the Record-type loudspeaker clicks slightly. To avoid this, you can not interrupt the anode circuit, but short-circuit the grid coil of the generator. It is only necessary to change the design of the interrupter, since when pressed, in this case, not contact should occur, but separation. In view of this, it will be necessary to abandon the double-sided lever and confine ourselves to a button with a very light spring. The design of the button is shown in Fig. 27; here, as we can see, when the button is pressed, the spring moves away from the contact and thus turns on the generator.

Rice. 28. Key device.

Key manufacturing details are shown in fig. 28. Round heads from bell buttons are taken as keys. If the springs are mounted under the lid of the box, then holes are cut for the buttons; if the springs are placed at the top, as shown in the diagram, then a quadrangular strip of hard cardboard or thin plywood with corresponding holes for buttons is fixed above them on gaskets.

The buttons and the interrupter are arranged in such a way that the other hand with the first, fourth and fifth fingers can freely manipulate the keys, and the second and third - with the interrupter.

Capacitors are placed under the cover of a flat box. Outside there are spring clamps for resistance, which can be changed at will. In addition, there is also a second pair of clamps for an additional capacitor of the grid circuit ( e and and), if there is a need for it during the production of experiments and adjustment of the "electrodes".

Installation is done with a hard wire, preferably silver-plated. Capacitors are fixed under the panel with small copper screws, under which copper washers are placed. The panels on which the critical parts are mounted, after the necessary holes have been drilled, are recommended to be waxed. From the loudspeaker sockets, two flexible wires (for example, a cord from electric lighting) connected to the plug are led out through the front wall. Terminals G and d on the front wall are used for a possible conversion of the device into a keyboard (by attaching a system of permanent condensers of various capacities).

Figure 29. Iron core.

It remains to make the core, on which the range of the tool depends to a large extent. The length of the core is taken 100-120 mm with a tapering end (Fig. 29). The core should easily fit inside the transformer. The easiest way for this purpose is to use four iron crutches, folded in pairs with two bent ends up and two ends down. Crutches are tied with thin wire and covered with paper. Curved ends for convenience can be sealed in a wooden handle. Such a core works quite satisfactorily, although the connection between music and ... iron crutches is quite unexpected.

b) Concert "electrola".

The second type, more advanced, is adapted for "concert" performance (the circuit is shown in Fig. 30). Here, another lamp for a low-frequency amplifier is added, which significantly increases power, and a device for changing the sound intensity, which is, in essence, the soul of the instrument (expressiveness). This device is made in the form of a variable resistance, which is the most rational for this device. In a single-tube "electronic" such devices cannot be turned on, since any change in resistance sharply changes the magnitude of the anode voltage and, consequently, the pitch; this, of course And with a two-tube design, the anode circuits of both lamps are separated, and the resistance is included in the anode of the second lamp in front of the loudspeaker.

Rice. thirty. Diagram of a two-tube concert electroly.

The resistance should smoothly change within the range of approximately 25,000 to 3,000,000 ohms. It can be constructed in one of the ways indicated in chapter VIII. In addition to it, we point out one more method, which in this case gave very good results.

For this purpose, an ebonite tube with an internal diameter of 15 mm and 6 cm length. A wooden sleeve with a hole in the middle is tightly hammered into one end. A copper rod with a screw thread is passed through it; a round copper plate is soldered to the inner end of the rod exactly at 15 mm diameter, tightly included in the ebonite tube (see Fig. 31). From the outside, the rod is screwed with a nut; cloth or rubber pads are placed under the nut and under the plate.

Rice. 31. Variable resistance device.

On the opposite side, a wooden plug is inserted into the tube with a hole into which the telephone jack is screwed. A second movable copper rod with a soldered thickened tip is passed through it 8-9 mm diameter. Outside, a carbolite flat head is screwed onto the rod from the terminal; a spiral spring is put on the rod under the head.

Pure glycerin is poured into the tube up to half. Connections are made from the bottom nut and the movable rod. When you press the head, the resistance decreases. Glycerin should be changed from time to time, as it often decomposes under the influence of current.

The second change is introduced in the design of the generator coil. Its length is doubled - up to 100 mm, due to which one passage of the core produces a continuous scale of 30 semitones (2½ octaves), while in the previous apparatus - only 20 semitones. The inclusion of a system of permanent capacitors, the capacity of which is selected in practice (approximately 5.000, 12.000 and 30.000 cm), the tessitura moves down one octave each time, so that the overall range increases to 5½ - 6 octaves. This is quite sufficient, especially since any vocal work fits into 2½ octaves (covered even by one movement of the core).

The number of turns in this case is increased: in the anode up to 12,000 turns and in the grid up to 36,000 turns (ordinary enameled transformer wire with a thickness of not more than 0.08 mm). The grid winding is divided into two halves of 18,000 turns, which can be connected by means of a "jack" in parallel or in series, which also extends the range (optional).

A similar circuit can, if desired, be assembled from two factory transformers (armored) factory. "Radio" placed next to each other. The number of turns will have to be selected at approximately 10,000 in the anode and 40,000 in the grid windings (two transformers of 5000 - 20000 turns). Alteration of transformers is carried out in the same way as in the previous type. It is only necessary when connecting them to each other to ensure that the correct direction of the turns is observed (otherwise, in the same winding, the opposite direction of both halves of the windings may turn out). Usually, you have to try different connection options to find the one that gives you the maximum volume and range.

The low-frequency amplifier transformer must be of good quality, with a turns ratio of 1: 4 or 1: 5. Incandescent rheostats are installed at 25 ohms each, always separately for each lamp. It is useful to give an additional voltage of the order of 3-5 volts to the second lamp.

All parts are enclosed in a flat box (dimensions 25 × 15 × 2 cm), which is put on top of a semicircular cover with a height of 11-12 cm, in appearance resembling a case from a sewing machine.

Rice. 32. Location of parts on the base (top view).

Under the flat box panel, the entire installation is made and the filament rheostats, capacitors of the circuit and both shunts, as well as an iron choke (gives a sharp change in tone) are located. Shunts for changing timbres are placed at the primary winding of the transformer n. hours (capacitors in 1000 and 3000 cm) and in the anode circuit of the second lamp (capacitors in 1000, 5000 and 15000 cm and throttle). As the latter, a multi-ohm coil from a telephone with an iron core or its own magnet can be used.

Rice. 33. Mounting diagram of the base (bottom view).

Outside on the panel are placed: a generator coil, lamp panels (for indoor installation), a low-frequency transformer and the handles of both rheostats protruding outward (the incandescence of the lamps usually remains constant, and turning the current off and on is done by a separate switch or slider located in front on the side wall of the flat grounds).

During assembly, both side walls are attached to the base, connected at the top by a narrow crossbar. A cutout is made in the right wall to pass the core; the knobs of the timbre switches are placed on it. Under the cutout for the core, an oblong rubber wheel is fixed, in the form of a cylinder 2 cm, to facilitate the movement of the core.

The latter is assembled from thin iron plates insulated with varnish 15-16 mm wide and 15-16 cm length or wires enclosed in a cardboard case of appropriate thickness. The end is closed with a wooden handle (you can, of course, make the core solid from a square strip of iron). A breaker is placed on the handle, connected to the circuit by a flexible double cord. Interruption is thus carried out by pressing the thumb of the right hand supporting the core.

The left side wall is equipped with three keys (buttons) for turning on the circuit capacitors.

The volume control and "jack" are placed on the left side of the crossbar. The expressiveness of the performance is achieved by pressing the thumb of the left hand, and turning on the keys - by the second, third and fifth fingers.

Power terminals and two pairs of sockets for a loudspeaker (for 1 and 2 lamps) are screwed into the base wall from behind.

Rice. 34. Type of concert electro.

When the installation is completed, both halves of the semicircular cover are strengthened at the back and front. The front half is hinged so that you can change the lamps.

A metal handle for carrying the apparatus is attached to the crossbar.

The location of the parts on the horizontal sill and side walls and the installation of the base are shown in fig. 32-33, and appearance apparatus - in fig. 34.

XII. METHOD OF GAME ON ELECTRIC.

Ordinary Micro lamps are inserted into the device and power sources are connected. It should be pointed out that for playing under normal room conditions, 45 volts per anode is quite enough for a sensitive loudspeaker with a simultaneous slight decrease against the norm and the magnitude of the glow (per lamp). To increase the volume, the anode voltage rises, however, not higher than up to 80-90 volts, and the second lamp turns on.

Rice. Z5. How to play the electric.

It is much easier to play the electrole than the theremin. The tool is always ready for action; no painstaking tuning is required here, and there is also no very unstable air neck, which makes it very difficult to perform. A smooth change in pitch is achieved by moving the core: when the core is removed from the coil, the highest note is obtained, when pushed in, the lowest. The player's hand quickly gets used to finding the necessary positions of the core corresponding to certain sounds.

On fig. 35 shows how to play the electro.

A little practice is enough to master the technique of the game. In essence, it is more profitable to perform each piece of music with constant pressure on a specific key, since a sharp change in capacitance changes the timbres somewhat (high notes turn out to be of a sharper “lighter” character, while the lower ones sound somewhat thicker). It turns out the same phenomenon as in the harmonium, since the inclusion of capacitors in our case will correspond to some extent to the inclusion of registers that change the “color” of the sound.

It is difficult to accurately indicate the neck markings, since it depends on many factors: the quality and data of the transformer coils, the size of the core, the mode of the lamps, etc. It's all about a little practice and, of course, a musical ear.

Play best with piano accompaniment. As a repertoire, musical works of the "theremin" repertoire are most suitable.

By changing the registers, very great effects can be achieved, shading various phrases, which, of course, is possible only with a certain skill. You need to start with simple things with a lingering melody, for example, folk songs, etc., moving on to more complex works in the future.

When playing, the core should be slightly vibrated, as this gives the sound a more lively character. The interrupter serves, as mentioned above, for pauses and for accentuating and receiving intermittent notes. General change the timbre is achieved by turning on one or another shunting (loudspeaker and low-frequency transformer) capacitance or inductor (with a large capacitance, a soft, muffled tone is obtained).

The sound is varied. On a high stretch, without a shunt, he melts the saxophone like a NEP; on low notes, it represents a cross between a cello and a woodwind instrument. The device, by its musical properties, is suitable for typical ensembles (especially for jazz bands, etc., where variety and original sound are required), as well as for an orchestra.

An important role is played by the property of the loudspeaker, and the best results (in terms of quality and beauty of sound) are obtained with a horn loudspeaker.

The use of anode rectifiers worsens the sound, as the voltage in the electrical network constantly fluctuates and, in addition, an alternating current ripple leaks.

You should play sitting at a stable table, leaning your right elbows on the table top. It is convenient to hold the core with three fingers of the right hand.

"Electrola", in order to turn into an instrument that satisfies refined taste and increased musical requirements, of course, needs some constructive improvement, which can easily be done with the participation of collective amateur radio thought.

One of the most interesting tasks in this area is experimenting with obtaining complex harmonies. Whether this is possible, the future will show.

In the tempered tuning, the octave is artificially divided into twelve completely identical semitones, while in reality mathematically precise tuning gives an immeasurably greater number of intervals, the use of which, however, would greatly complicate the construction and playing of musical instruments.

According to available information, L. S. Termen, who is currently in America, is working on the organization of an orchestra consisting of several dozen devices.

Those interested in the theory of tube generators are referred to B. A. Vvedensky's book "Physical Phenomena in Cathode Lamps" (Chapter V).

The simplest microphone consists of a charcoal plate and powdered charcoal sprinkled behind it. Under the influence of air vibrations when talking or singing, the plate vibrates to the beat, due to which the resistance in the microphone circuit changes.

If a capacitor is taken "Mamza", you should put the vernier of the same factory, with a deceleration of 1: 24.

Patent awarded by the Committee for Inventions on July 29, 1929; application certificate No. 40042.

Cover back (advertisement of the book "High-Voltage Mercury Rectifiers")

I. N. BRONSHTEIN K. A. SEMENDYAEV
MATHEMATICS HANDBOOK FOR ENGINEERS AND STUDENTS
22.11B 88
UDC 51
Authors from the GDR who participated in the revision of the edition:
DIPL.-MATH. P. BECKMANN, DR. M. BELGER, DR. H. BENKER,
D.R. M. DEWEB, PROF. D.R. H. ERFURTH, DIPL.-MATH. H. GENTEMANN,
D.R. P. GOTHNER, DOZ. D.R. S. GOTTWALD, DOZ. D.R. G. GROSCHE,
DOZ. D.R. H. HILBIG, DOZ. D.R. R. HOFMANN, NPT H. KASTNER,
D.R. W. PURKERT, DR. J. VOM SCHEIDT, DIPL.-MATH. TH. VETTERMANN, D.R. v. WfjNSCH, PROF. D.R. E. ZEIDLER.
A Handbook of Mathematics for Engineers P university students.
Bronstein I. N., Semendyaev K. A.-M.: Science.
Main edition financial and mathematical literature, 1981.

Teubner Publishing House, GDR, 1979 ) Publishing house "Science",Main editionphysical and mathematical Literature, 1980

CONTENT
Editorial
1. TABLES AND GRAPHS
1.1. TABLES
1.1.1. Tables of elementary functions
1. Some common constants (12). 2. Squares, cubes, corn (12). 3. Degrees of integers from 1 to 100 (30). 4. Reciprocals (32). 5. Factorials and their reciprocals (34). 6. Some powers of numbers 2, 3 and 5 (35). 7. Decimal logarithms (36). 8. Antilogarithms (38) 9. Natural values ​​of trigonometric functions (40). 10. Exponential, hyperbolic and trigonometric functions (48). 11. Exponential functions (for x from 1.6 to 10.0) (51). 12. Natural logarithms (S3). 13. Circumference (56). 14. Area of ​​a circle (58). 15. Circle segment elements (60). 16. Converting a degree measure to a radian (64). 17. Proportional parts (65). 18. Table for quadratic interpolation (67).

1.1.2. Special Function Tables
1. Gamma function (68). 2. Bessel (cylindrical) functions (69). 3. Legendre polynomials (spherical functions) (71). 4. Elliptic integrals (72). 5. Poisson distribution (74). 6. Normal distribution (75). 7. chi distribution (78). 8. Student's r-distribution (80). 9. z-distribution (81). 10. F-distribution (distribution u3) (82). 11. Critical numbers for the Wilcoxon test (88). 12. Kolmogorov - Smirnov distribution (89).

1.1.3. Integrals and sums of series
1. Table of sums of some numerical series (90). 2. Table of expansion of some functions into power series (92). 3. Table of indefinite integrals (95). 4. Table of some definite integrals (122).

1.2. GRAPHS OF ELEMENTARY FUNCTIONS
1.2.1. Algebraic functions
1. Entire rational functions (126). 2. Fractional-rational functions (127). 3. Irrational functions (130).
1.2.2 Transcendent functions
1. Trigonometric and inverse trigonometric functions (131). 2. Exponential and logarithmic functions (133). 3. Hyperbolic functions (136).

1.3. KEY CURVES
1.3.1. Algebraic curves
1. Curves of the 3rd order (138). 2 Curves of the 4th order (139).
1.3.2. Cycloids
1.3.3. Spirals
1.3.4. Chain line and tractrix

2. ELEMENTARY MATHEMATICS 2.1. ELEMENTARY APPROXIMATE CALCULATIONS
2.1.1. General information
1. Representation of numbers in positional number system (147). 2. Errors and rules for rounding numbers (148).
2.1.2. Elementary Error Theory
1. Absolute and relative errors (149). 2. Approximate error limits for the function (149). 3. Approximate formulas (149).
2.1.3. Elementary Approximate Graphical Method
1. Finding the zeros of the function (150). 2. Graphical differentiation (150). 3. Graphical integration (151).

2.2. COMBINATORICS
2.2.1. Basic combinatorial functions
1. Factorial and gamma function (151). 2. Binomial coefficients (152). 3. Polynomial coefficient (153).
2.2.2. Binomial and polynomial formulas
1. Newton's binomial formula (153). 2. Polynomial formula (154).
2.2.3. Statement of problems of combinatorics
2.2.4. Permutations
1. Permutations (154). 2. The permutation group of k elements (155). 3. Permutations with a fixed point (156). 4. Permutations with a given number of cycles (156). 5. Permutations with repetitions (156).
2.2.5. Accommodations
1. Placements (157). 2. Placements with repetitions (157).
2.2.6. Combinations
1. Combinations (157). 2. Combinations with repetitions (158).

2.3. FINITE SEQUENCES, SUMS, PRODUCTS, AVERAGES
2.3.1. Notation of sums and products
2.3.2. End sequences
1. Arithmetic progression (159). 2. Geometric progression (159).
2.3.3. Some final sums
2.3.4. Averages

2.4. ALGEBRA
2.4.1. General concepts
1. Algebraic expressions (161). 2. Values ​​of algebraic expressions (161). 3. Polynomials (162). 4. Irrational expressions (163). 5. Inequalities (163). 6. Elements of group theory (165).
2.4.2. Algebraic equations
1. Equations (165). 2. Equivalent transformations (166). 3. Algebraic equations (167). 4. General theorems (171). 5. System of algebraic equations (173).
2.4.3. Transcendental Equations
2.4.4. Linear algebra
1. Vector spaces (175). 2. Matrices and determinants (182). 3. Systems of linear equations (189). 4. Linear transformations (192). 5. Eigenvalues ​​and eigenvectors (195).

2.5. ELEMENTARY FUNCTIONS
2.5.1. Algebraic functions
1. Entire rational functions (199). 2. Fractional-rational functions (201). 3. Irrational algebraic functions (205).
2.5.2. Transcendent Functions
1. Trigonometric functions and their inverses (206). 2. Exponential and logarithmic functions (212). 3. Hyperbolic functions and their inverses (213).

2.6. GEOMETRY
2.6.1. Planimetry
2.6.2. Stereometry
1. Straight lines and planes in space (220). 2. Dihedral, polyhedral and solid angles (220). 3. Polyhedra (221). 4. Bodies formed by moving lines (223).
2.6.3. Rectilinear Trigonometry
1. Solution of triangles (225). 2. Application in elementary geodesy (227).
2.6.4. Spherical trigonometry
1. Geometry on the sphere (228). 2. Spherical triangle (228). 3. Solution of spherical triangles (229).
2.6.5. Coordinate systems
1. Coordinate systems on the plane (232). 2. Coordinate systems in space (234).
2.6.6. Analytic geometry
1. Analytic geometry on the plane (237). 2. Analytic geometry in space (244).

3. FUNDAMENTALS OF MATHEMATICAL ANALYSIS
3.1. DIFFERENTIAL AND INTEGRAL CALCULUS OF FUNCTIONS OF ONE AND SEVERAL VARIABLES
3.1.1. Real numbers
1. System of axioms of real numbers (252). 2. Natural, integer and rational numbers (253). 3. The absolute value of the number (254). 4. Elementary inequalities (254).
3.1.2. Point sets in R"
3.1.3. Sequences
1. Numerical sequences (257). 2. Sequences of points (259).
3.1.4. Real Variable Functions
1. Function of one real variable (260). 2. Functions of several real variables (269).
3.1.5. Differentiation of functions of one real variable
1. Definition and geometric interpretation of the first derivative. Examples (272). 2. Derivatives of higher orders (273). 3. Properties of differentiable functions (275). 4. Monotonicity and convexity of functions (277). 5. Extreme points and inflection points (278). 6. Elementary investigation of a function (279).
3.1.6. Differentiation of functions of several variables
1. Partial derivatives, geometric interpretation (280). 2. Total differential, directional derivative, gradient (280). 3. Theorems on differentiable functions of several variables (282). 4. Differentiable mapping of the space R" into R"1; functional determinants; implicit functions; existence theorems for a solution (284). 5. Change of variables in differential expressions (286). 6. Extrema of functions of several variables (288).
3.1.7. Integral calculus of functions of one variable
1. Definite integrals (291). 2. Properties of definite integrals (292). 3. Indefinite integrals (293). 4. Properties of indefinite integrals (295). 5. Integration of rational functions (297). 6. Integration of other classes of functions (300). 7. Improper integrals (30S). 8. Geometric and physical applications of definite integrals (312).
3.1.8. Curvilinear integrals
1. Curvilinear integrals of the 1st kind (integrals over the length of a curve) (3I5). 2. Existence and calculation of curvilinear integrals of the first kind (315). 3. Curvilinear integrals of the second kind (projection integrals and general integrals) (316). 4. Properties and calculation of curvilinear integrals of the second kind (316). 5. Independence of curvilinear integrals from the path of integration (318). 6. Geometric and physical applications of curvilinear integrals (320).
3.1.9. Integrals depending on a parameter
1. Definition of an integral depending on the parameter (321). 2. Properties of integrals depending on a parameter (321). 3. Improper integrals depending on a parameter (322). 4. Examples of integrals depending on the parameter (324).
3.1.10. Double integrals
1. Definition of the double integral and elementary properties (326). 2. Calculation of double integrals (327). 3. Change of variables in double integrals (328). 4. Geometric and physical applications of double integrals (328).
3.1.11. Triple Integrals
I. Definition of the triple integral and the simplest properties (330). 2. Calculation of triple integrals (330). 3. Change of variables in triple integrals (331). 4. Geometric and physical applications of triple integrals (332).
3.1.12. Surface integrals
1. The area of ​​a smooth surface (333). 2. Surface integrals of the 1st and 2nd kind (334). 3. Geometric and physical applications of the surface integral (337).
3.1.13. Integral Formulas
1. Formula of Ostrogradsky - Gauss. Green's formula (336). 2. Green's formulas (339). 3. Formula. Stokes (339). 4. Improper curvilinear, double, surface and triple integrals (339). 5. Multidimensional integrals depending on a parameter (341).
3.1.14. Endless rows
1. Basic concepts (343). 2. Criteria for the convergence or divergence of series with non-negative terms (344). 3. Series with arbitrary members. Absolute convergence (347). 4. Functional sequences. Functional series (349). Power series (352). 6. Analytic functions. Taylor series. Expansion of elementary functions in a power series (357).
3.1.15. Endless works

3.2. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
3.1.1. Calculus of variations
1. Statement of the problem, examples and basic concepts (365). 2. Euler-Lagrange theory (366). 3. The theory of Hamilton - Jacobi (376). 4. Inverse problem of the calculus of variations (377). 5. Numerical methods (378).
3.22. Optimal Control
1. Basic concepts (381). 2. Pontryagin's maximum principle (383). 3. Discrete systems (390). 4. Numerical methods (391).

3.3. DIFFERENTIAL EQUATIONS
3.3.1. Ordinary differential equations
1. General concepts. Existence and uniqueness theorems (393). 2. Differential equations of the 1st order (395). 3. Linear differential equations and linear systems 404 4. General non-linear differential equations (420). 5. Stability 421 6. Operator method for solving ordinary differential equations (422). 7. Boundary value problems and eigenvalue problems (424).
3.3.2. Partial Differential Equations
1. Basic concepts and special methods of solution (428). 2. Equations in partial derivatives of the 1st order (431). 3. Equations in partial derivatives of the 2nd order (440).

3.4. COMPLEX NUMBERS. FUNCTIONS OF A COMPLEX VARIABLE
3.4.1. General remarks
3.4.2. Complex numbers. Riemann sphere. Areas
1. Definition of complex numbers. Field of complex numbers (466). 2. Conjugate complex numbers. Complex number modulus (467). 3. Geometric interpretation 468 4. Trigonometric and exponential forms of complex numbers (468). 5. Degrees, roots (469). 6. Riemann sphere. Jordan curves. Regions (470).
1.4.3. Functions of a complex variable
1.4.4. The most important elementary functions
1. Rational functions (473). 2. Exponential and logarithmic functions (474). 3. Trigonometric and hyperbolic functions 475
3.4.5. Analytic Functions
1. Derivative (476). 2. Cauchy-Riemann differentiability conditions (476). 3. Analytic functions 476
3.4.6. Curvilinear integrals in the complex domain
1. Integral of a function of a complex variable (477). 2. Independence from the path of integration (478). 3. Indefinite integrals (478). 4. Basic formula of the integral calculus (478). 5. Cauchy integral formulas 478
3.4.7. Expansion of analytic functions in a series
1. Sequences and series (479). 2. Functional rows. Power series (480). 3. Taylor series (481). 4. Laurent series (481). 5. Classification of singular points (482). 6. Behavior of analytic functions at infinity (482).
3.4.8. Deductions and their application
1. Deductions (483). 2. Residue theorem (483). 3. Application to the calculation of definite integrals (484).
3.4.9. Analytic continuation
1. The principle of analytic continuation (484). 2. Principle of symmetry (Schwartz) (485).
3.4.10. Inverse functions. Riemann surfaces
1. Univalent functions, inverse functions (485). 2. Riemann surface of a function (486). 3. Riemann surface of the function r=Lnw (486).
3.4.11. Conformal mapping
1. The concept of a conformal mapping (487). 2. Some simple conformal mappings (488).

4. ADDITIONAL CHAPTERS
4.1. SETS, RELATIONS, MAPPINGS
4.1.1. Basic concepts of mathematical logic
1. Algebra of logic (algebra of propositions, logic of propositions) (490). 2. Predicates (494).
4.1.2 Basic concepts of set theory
1. Sets, elements (496). 2. Subsets (496).
4.1.3. Operations on sets
1. Union and intersection of sets (496). 2. Difference, symmetric difference, complement of sets (496). 3. Euler - Venn diagrams (497). 4. Cartesian product of sets (497). 5. Generalized union and intersection 498
4.1.4. Relationships and mappings
1. Relations (498). 2. Equivalence relation (499). 3. Order relation (500). 4. Mappings (501). 5. Sequences and families of sets (502). 6. Operations and algebras 502
4.1.5. Power of sets
1. Equivalence (503). 2. Countable and uncountable sets 503

4.2. VECTOR CALCULUS 4.2.1. Vector algebra
1. Basic concepts (5.03). 2. Multiplication by a scalar and addition (504). 3. Multiplication of vectors (505). 4. Geometric applications of vector algebra (507).
4.2.2. Vector Analysis
1. Vector functions of a scalar argument (508). 2. Fields (scalar and vector) 510 3. Gradient of a scalar field 513 4. Curvilinear integral and potential in a vector field 515 5. Surface integrals in vector fields 6. Divergence of a vector field 519 7. Vector field rotor (520). 8. The Laplace Operator and the Gradient of a Vector Field (521) 9. Calculation of complex expressions (Hamilton operator) (522). 10. Integral formulas 523 11. Determination of a vector field from its sources and vortices 525 12. Dyads (tensors of rank II) (526).

4.3. DIFFERENTIAL GEOMETRY
4.3.1. Flat curves
1. Methods for setting plane curves. Plane curve equation (531). 2 Local elements of a plane curve (532). 3. Points of a special type (534). 4. Asymptotes (536). 5. Evolute and involute (537). 6. Envelope of a family of curves 538
4.3.2. Spatial curves
1. Ways of specifying curves in space (538). 2. Local elements of a curve in space 538 3. Main theorem of the theory of curves (540).
4.3.3. surfaces
1. Methods for defining surfaces (540). 2 Tangent plane and surface normal (541). 3. Metric properties of surfaces (543). 4. Surface curvature properties 545 5. Main theorem of the theory of surfaces (547). 6. Geodesic lines on the surface 548

4.4. FOURIER SERIES, FOURIER INTEGRALS, AND THE LAPLACE TRANSFORM
4.4.1. Fourier series
1. General concepts (549). 2. Table of some expansions in the Fourier series (551). 3. Numerical harmonic analysis 556
4.4.2. Fourier integrals
I. General concepts (559). 2. Tables of Fourier transforms (561).
4.4.3. Laplace transform
1. General concepts (571). 2. Application of the Laplace transform to the solution of ordinary differential equations with initial conditions (573). 3. Table of the inverse Laplace transform of fractional rational functions (574).

5. PROBABILITY THEORY AND MATHEMATICAL STATISTICS
5.1. PROBABILITY THEORY
5.1.1. Random events and their probabilities
1. Random events (577). 2. Axioms of the theory of probability (578). 3. The classical definition of the probability of an event (579). 4. Conditional probabilities 580 5. Full probability. Bayes formula (580).
5.1.2. random variables
I. Discrete Random Variables 581 2. Continuous random variables 583
5.1.3. Moments of distribution
I. Discrete case 585 2. Continuous case 587
5.1 4 Random vectors (multidimensional random variables)
1. Discrete random vectors 588 2. Continuous random vectors 588 3. Boundary distributions 589 4. Moments of a multidimensional random variable 589 5. Conditional distributions. 6. Independence of random variables 590 7. Regression dependence (591). 8. Functions of random variables 592
5.1.5. Characteristic functions
1. Properties of characteristic functions 593 2. The inversion formula and the uniqueness theorem (594). 3. Limit theorem for characteristic functions (594). 4. Generating functions 595 5. Characteristic functions of multidimensional random variables 595
5.1.6. Limit theorems
1. Laws of large numbers (595). 2. Limit theorem of De Moivre - Laplace (596). 3. Central limit theorem (597).

5.2. MATH STATISTICS
5.2.1. Samples
1. Histogram and empirical distribution function (598). 2. Sample functions (600). 3. Some important distributions (600).
5.2.2. Parameter Estimation
1. Properties of point estimates (601). 2. Methods for obtaining estimates (602). 3. Confidence estimates (604).
5.2.3. Hypothesis testing (tests)
1. Statement of the problem (606). 2. General theory 606 3. meriterium (607). 4. F-test (607), 5. Wilcoxon test (607). 6. Chi test (608). 7. The case of additional parameters (609). 8. Kolmogorov-Smirnov agreement criterion (610).
5.24. Correlation and regression
1. Evaluation of correlation and regression characteristics for samples (611). 2. Testing the hypothesis p = 0 in the case of a normally distributed general population (612). 3. General problem of regression (612).

6. MATHEMATICAL PROGRAMMING
6.1. LINEAR PROGRAMMING
1. General formulation of the problem, geometric interpretation and solution of problems with two variables (613). 2. Canonical view, image of the vertex in the simplex table (615). 3. Simplex method for given 7. Modified methods, additional changes to the problem (625).

6.2. TRANSPORT CHALLENGE
6.2.1. Linear transport problem
6.2.2. Finding the Initial Solution
6.23. transport method

6.3. TYPICAL LINEAR PROGRAMMING APPLICATIONS
6.3.3. Distribution, planning, comparison
6.3.4. Cutting, shift planning, coating

6.4. PARAMETRIC LINEAR PROGRAMMING
6.4.1. Formulation of the problem
6.4.2. Solution Method for the Case of a One-Parameter Objective Function

6.5. INTEGER LINEAR PROGRAMMING 6.5.1. Problem statement, geometric interpretation
6.5.2 Gomory cut method
6.5.3. Branch method
6.5.4. Comparison of methods

7. ELEMENTS OF NUMERICAL METHODS AND THEIR APPLICATIONS
7.1. ELEMENTS OF NUMERICAL METHODS
7.1.1. Errors and their accounting
7.1.2. Computational methods
1. Solution of linear systems of equations (649). 2. Linear eigenvalue problems 653 3. Nonlinear equations (655). 4. Systems of nonlinear equations 657 5. Approximation 659 6. Interpolation (663). 7. Approximate calculation of integrals (668). 8. Approximate differentiation 673 9. Differential Equations 674
7.1.3. Implementation of the Numerical Model in Electronic Computers
1. Criteria for choosing a method (681). 2. Management methods (682). 3. Calculation of functions (682).
7.1.4. Nomography and slide rule
1. Relations between two variables - functional scales (685). 2. Logarithmic (counting) ruler (686). 3. Nomograms of points on straight lines and grid nomograms (687).
7.1.5. Handling Empirical Numerical Material
1. Method of least squares (688). 2. Other methods of alignment (690).

7.2. COMPUTER ENGINEERING
7.2.1. Electronic computers (computers)
1. Introductory remarks (691). 2. Representation of information and computer memory (692). 3. Exchange channels (693). 4. Program (693). 5. Programming (694). 6. Computer control (695). 7. Mathematical (software) software (696). 8. Performing work on a computer (696).
7.2.2. Analog computers
1. The principle of the design of analog computing technology (697). 2. Computing elements of an analog computer (697). 3. Principle of programming in solving systems of ordinary differential equations (699). 4. Quality programming (700).

Literature
Universal designations
Subject index

EDITORIAL
Handbook of I. N. Bronstein and K. A. Semendyaev in mathematics for engineersand students of technical universities has firmly gained popularity not only in our country, butand abroad. The eleventh edition was published in 1967. Further publication of the reference book was suspended, as it no longer met modern requirements.The revision of the handbook was carried out at the initiative of the publishing house "Teubner», with the consent of the authors, a large team of specialists in the GDR (where previously referencedNick withstood 16 editions). A mutual decision was made to release this revisedtanny version co-published:in the GDR - the publishing house "Teubner" - in German;in the USSR - the main edition of the physical and mathematical literature of the publishing house"Science" - in Russian.As a result of the revision, the guide was not only enriched with new informationon those sections of mathematics that were presented earlier, but was supplementedand new sections: calculus of variations and optimal control, mathematical logic and set theory, computational mathematics and basicinformation on computing.At the same time, the general methodological style of the handbook was preserved, allowingand get factual help on finding formulas or tabular data, and familiarize yourself with the basic concepts (or restore them to memory); For a better understanding of the concepts, a large number of examples are given.In connection with such a thorough revision of the handbook, the entire text was rewrittentranslated from German.During the preparation of the Russian edition, some revision was made in order toto take into account, if possible, the requirements of the programs of domestic universities. This pererabotka is mainly associated with a change in the designations and terminology that we haveand in the GDR are not identical. Some sections for the Russian edition have been rewrittenagain - these are the first sections of the chapters on algebra, mathematical logic,set theory. The sections devoted to complex variables, the calculus of variations, and optimal control have undergone a less significant alteration.computational mathematics.To reduce the size of the handbook compared to originally plannedoption omitted some sections that are necessary for a narrower circle specialists. Some sections of the handbook were left without revision due tothe very short time allotted for the preparation of this publication. For example, in thisThe edition omits the section on tensor calculus. In this regard, section"Differential Geometry" should be rewritten in somewhat more detail andchange the presentation. The Computational Mathematics section says a lotabout computational methods and little is given to computational mathematics proper.In the section "Calculation of Variations and Optimal Control" there is not enough attentionniya is given to optimal control. However it takes a long time to complete this workand, most importantly, reader feedback. Therefore, the editorialwith a request to all who will use the guide to send their commentsand suggestions for improving the handbook so that they can be taken into account in furtherthe most work on it.Please send your proposals to the address: 117071, Moscow, Leninsky Prospect, 15, Main editorial office of physical and mathematical literature of the Nauka publishing house, editorialmathematical reference books.

Download the book Bronstein I. N., Semendyaev K. A. Handbook of mathematics. For engineers and university students. Publishing house "Science", Moscow, 1981