SYMMETRY FESTIVAL 2013

Delft, The Netherlands

2-

7 August, 2013

Anatomy of consonance

 

Nasretdinov Alexey

P

hysicist

musician 

(b. 

Sverdlovsk

Russia

, 19

67

).

Address

Beriozovoy roschi pr. 12-573, Moscow, Russia

.  

E-mail: 

anasretdinov

@

yandex

.

ru

 

.

Fields of interest

physics, mathematics, music, biology, neuroscience, history, philosophy

.

Awards

Moscow Institute of Physics and Technology, student, 

graduate, 

lecturer

Publications

 

and/or

 

Exhibitions

:  

Nasretdinov Alexey. 

Physics and anatomy of music

”, 

Boslen, Moscow, Russia, 2012

Nasretdinov Alexey. 

Musical mathematics of anc

ient people. Search for harmony”, Boslen, Moscow, Russia, 2012

Abstract

S

ince ancient times 

t

he question of harmony in music and in many 

natural

scientific

 

disciplines 

concerned

 the

 human

kind

. U

nderstanding what 

the

 consonance (euphonic sound - the central concept of musical harmony)

 is

, came from the 

m

athematics and 

p

hilosophy (Pythagoras), cosmology and religion (ancient Chinese), anthropocentrism (ancient Indians). Now 

the 

common approach based on knowledge of biology, psychoacoustics, neuroscience

 and

 information theory.

 

In this paper, 

we will consider 

the basic theor

ies

 of the height of a single sound

 and

 consonant sound musical intervals

 

determining

: the Pythagorean theory, the 

common

 bass 

theory

 of 

Jean-Philippe Rameau

Hermann von Helmholtz

 theory, 

N.

 

Garbuzov’s 

band theory of pitch hearing, psychoacoustic theory, the theory, based on the characteristics of human short-term memory (Stefanovich M

.

). 

Also the

 little-known experimental data

 are shown here

.

Keywords

Music, harmony, consonance, anatomy

1. BACKGROUND

ALL ANCIENT MUSICAL SYSTEM

S

 (PYTHAGORAS, CHINA, INDIA) 

ARE

ORGANIZED SO

,

 THAT THE 

SCALE HAS MAXIMUM

 NUMBER OF PURE CONSONANCES, BASED ON THE FIRST HARMONICS 

OF 

FREELY VIBRATING STRING.

Figure 1

Harmonics 

of 

freely vibrating string

THE QUESTION OF CONSONSNCE 

ARAISED AND 

BEC

O

ME AN IMPORTANT ISSUE AGAIN AFTER THE DEVELOPMENT OF MISUCAL IN

S

TRUMENTS WITH FIXED SETTINGS 

AND/OR COMPLEX TIMBRES

,

 PLAY IN ORCHESTRAS 

AS WELL AS 

A

 WIDER RANGE OF VARIANTS OF TEMPERED

SCALES DEVELOPMENT.

1.1 BASIC THEORY OF MUSICAL CONSONANCE

 

1.1. 1 HEIGHT OF A SINGLE TONE DETERMINATION

BEFORE TALKING ABOUT THE CONSONANCE (EUPHONIOUS SOUNDING PAIRS OF SOUNDS), I MUST SAY A FEW WORDS ABOUT HOW 

THE 

ONE DETERMINES THE HEIGHT OF A SINGLE SOUND.

RANGE OF PERCEIVED MUSICAL SOUNDS 

IS FREQUENCIES 

FROM 30 HZ TO 5000 HZ

DECOMPOSITION OF SOUND IN 

FOURIER SE

RIES 

IN THE 

HUMAN

INNER EAR. T

HE CONVERSION 

IN

TO DIGITAL FORMAT.

THEOR

IES

 AND PROBLEMS OF PERCEPTION OF SOUNDS: THE LACK OF SOUND-SENSITIVE CELLS

 IN THE EAR 

PERCEIVING

FREQUENCY

 

BELOW 100 HZ

, THE FORMATION OF SUBJECTIVE HARMONICS AND DIFFERENCE TONES, 

SPLITTING TONE

, THE 

MISSING FUNDAMENTAL PHENOMENON

, THE THEORY OF 

PLACE

PERCEPTION OF PITCH, TIME THEORY, THE MODERN THEORY.

1.1. 2 BASIC THEORIES OF CONSONANCE

 

1.

PYTHAGOREAN THEORY. 

(VI BC) 

THE FAMOUS FOUR NUMBERS: 6-8-9-12. OCTAVE, FIFTH, QUART. THE ARITHMETIC MEAN AND THE HARMONIC MEAN.

2.

JEAN PHILIPPE RAMEAU (

XVIII

). 

SEARCH

 THE VERY LEAST

MULTIPLIER OF

 SEVERAL FREQUENCY SOUNDS.

3.

HELMHOLTZ (

XIX

). THE THEORY OF RESONANCES AND BEATS BETWEEN THE FUNDAMENTAL FREQUENCY AND/OR HARMONICS AND OVERTONES.

4. GARBUZOV 

N. 

THEORY (

XX

). 

I

NTERVALS 

PURITY 

AS

FREQUENCY RELATIONS, WHICH

 HIT

 IN THE

 CERTAIN FREQUENCY BAND. INTERVALS WITHIN THIS ZONE 

ARE

CONSIDERED TO BE PURE, BUT ONE FEELS THE CHANGE OF INTONATION.

5. PSYCHOACOUSTIC THEORY. CONSONANCES CONSIDERED FROM THE POINT OF VIEW OF THE FREQUENCY DIFFERENCE BETWEEN HARMONICS AND OVERTONES IN SOME PART OF THE CRITICAL STRIP OF THE ORGAN OF CORTI SNAILS.

6. THEORY STEFANOVICH M

.

 

(XXI) 

"DELIGHTS" CONSONANCES INTERPRETED IN TERMS OF THE STORED INFORMATION ABOUT THE DIFFERENCE BETWEEN SUBJECTIVE FREQUENCY HARMONICS SOUND MUSICAL INTERVALS

 

MINIMIZING

.

7. EXPERIMENTS SHOWING CRUCIAL TIMBRE PERCEPTION OF CONSONANCE (TENOR-BASS, VIOLIN-TRUMPET, PIANO-VIOLIN, ETC.)

 

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