Math
September 22, 2007
Note: if you are an actual mathematician, you might want to skip right down to the numbers in the Shroeder article and the links that follow. There is nothing very advanced here in any event. Most of the other passages refer to ideas about math and physics I never got in school (I was inattentive) and want to keep handy for perspective.
Bell, Eric Temple
The Magic of Numbers
Whittlesey House-
McGraw-Hill Book Company, Inc.
New York: London, 1946
Two
consequences of lasting significance from the positive achievements of
Pythagoras:
1. ...the belief that “number” may be so defined that at least
the physical universe can be consistently described in terms of “number.”
2.
…the common belief that conclusions reached by mathematical reasoning have a
greater certainty than those obtained by any other means.
Together they
still are complementary postulates of one as yet unverified hypothesis: a
rational account of (at least) the physical universe is possible which, when
finally given, will agree with sensory experience and empower human beings to
predict the course of nature. P201
Bronowski, J
The Ascent of
Man
Little, Brown and Company
Boston/Toronto
© 1973 by
J.Bronowski
ISBN0-316-10930-4
Physical phenomena consists always of the interaction of energy with matter. P224
Bronowski, Jacob
A Sense of the
Future
Edited by Piero E. Ariotti in collaboration with Rita
Bronowski
©1977 by The Massachusetts Institute of Technology
ISBN
0-262-02128-5
…truth is not the only aim of science,. We want more than mere truth: what we look for is interesting truth. P100
We want to feel that the world can be understood as a unity, and that the rational mind can find ways of looking at it that are simple, new, and powerful exactly because they unify it. P101
The facts are endless chaos; science is the activity of finding in them some order. And this order is not merely a shorthand for the facts; it is what gives them meaning, it is their meaning. Science is the human activity of finding order in nature by organizing the scattered meaningless facts under universal concepts. P255
Brown, Lloyd A.
The Story of
Maps
© 1949, 1977 by Florence Brown
Dover Edition is published by
special arrangement with Little, Brown and Company
34 Beacon St., Boston,
Mass 02106
Dover Publications, Inc., 180 Varick Street, New York, NY
10014
ISBN 0-486-23873-3
LCCN 79-52395
There is no such thing as an ideal, all purpose map or chart; every projection must sacrifice accuracy and tolerate distortion of one kind or another. p138
Cole, K.C.
Sympathetic
Vibrations
© 1985 by K.C. Cole
Permissions Department
William
Morrow and Company, Inc.
105 Madison Ave., New York, N.Y. 10016
LCCN
84-60547
ISBN 0-688-03968-5
Take a jump rope, or violin string, secured at both ends. If you pump energy into it and set it swinging, it can vibrate only in a certain number of ways, taking a few predetermined shapes…in its characteristic harmonics
Weisskopf, Victor – …what’s simple is what’s understood. What you don’t understand always seems complicated—no matter how simple it may seem to someone who does understand it. (The simplicity, in other words, comes from the clarity of understanding, from the ability to see through the distractions and focus in on the essential elements, to explain many unconnected things with one “simple” idea…) “What’s beautiful in science is the same thing that’s beautiful in Beethoven, there’s a fog of events and suddenly you see a connection. It expresses a complex of human concerns that go deeply to you, that connects things that were always in you that were never put together before.” P229,230
RESONANCE
The
key to resonance is pushing or pulling in time with the way (…something) wants
to go. P264
Resonance is the physics lesson all children learn when they
try to pump themselves on playground swings. The trick they soon learn is
timing. Pushing forward or leaning backward at the wrong place or time gets them
nowhere. P263,264
Courant, Richard and Robbins,
Herbert
What is Mathematics?
Oxford University Press
London,
New York, Toronto
© 1941 by Richard Courant
What matters and what corresponds to verifiable “fact” is structure and relationship…
While the Greeks chose point and line as the basis of their mathematics, it has become the modern guiding principle that all mathematical statements should be reducible ultimately to statements about the natural numbers, 1,2,3…P1
Rational
Numbers
…we need not only to count individual objects, but also to measure
quantities such as length, area, weight, and time. If we want to operate freely
with the measures of these quantities, which are capable of arbitrarily fine
subdivision, it is necessary to extend the realm of arithmetic beyond the
integers.
The first step is to reduce the problem of measuring to the problem of counting. First we select, quite arbitrarily, a unit of measurement…(then count them)…In general, the process of counting units will not come out even…we introduce new subunits…by dividing the original unit 1 into n equal parts (1/n); and if a given quantity consists of exactly m of these subunits, its measure is denoted by the symbol (m/n). This symbol is called a fraction or ratio. (Also written m:n) When comparing the magnitudes of two line segments, it may turn out that while no integral multiple of a equals b, we can divide a into say n equal segments, each of length a/n, such that some integral multiple m of the segment a/n is equal to b:
When this
equation holds, we say that the two segments are commensurable since they have a
common measure, the segment a/n which goes n times into a and m times into
b.
(the Greek Pythagorean school discovered …that) There exist incommensurable segments, or if we assume that to every segment corresponds a number giving its length in terms of the unit, irrational numbers. This revelation was a scientific event of the highest importance…certainly it has profoundly affected mathematics and philosophy from the time of the Greeks to the present day. P59
The concept of function enters whenever quantities are connected by a definite physical relationship.
The whole domain of periodic phenomena—the motion of the tides, the vibrations of a plucked string, the emission of light waves from an incandescent filament—is governed by the simple trigonometric functions sin x and cos x. P272
It was observed that stable equilibrium of a mechanical system is attained if the system is arranged in such a way that its “potential energy” is a minimum. P382
Not only the laws of equilibrium, but also those of motion are dominated by maximum and minimum principlesP382
Dunham, William
Journey Through
Genius
Penguin Books USA Inc.,
375 Hudson Street, New York, NY
10014
First published in the United States of America by John Wiley &
Sons, Inc., 1990
Published in Penguin Books 1991
© John Wiley & Sons,
Inc., 1990
ISBN 0-471-50030-5(hc)
ISBN 014 01.4739X(pbk.)
... tradition credits the Pythagorean Hippasus with discovering that the side of a square and its diagonal are not commensurable. That no matter how small one goes, there is no magnitude...dividing evenly into both the square’s side and its diagonal….it shattered those Pythagorean proofs that rested upon the supposed commensurability of all segments. Secondly, it had an unsettling impact upon the supremacy of whole numbers, for if not all quantities were commensurable, then whole numbers were somehow inadequate to represent the ratios of all geometric lengths. Consequently, the discovery firmly established the superiority of geometry over arithmetic in all subsequent Greek mathematics…Far better, thought the Greeks, to avoid the numerical approach altogether and concentrate on magnitudes simply as geometric entities…A final result of the discovery of irrationals was that the Pythagoreans, incensed at all the trouble Hippasus had caused, supposedly took him far out upon the Mediterranean and tossed him overboard to his death. If true, the story indicates the dangers inherent in freethinking…PP9, 10
Gleick, James
Chaos
© James
Gleick, 1987
Published by the Penguin Group
Penguin Books USA Inc.
375
Hudson Street
New York, NY 1014
ISBN 0 14 00.92501
The sciences do
not try to explain, they hardly even try to interpret, they mainly make models.
By a model is meant a mathematical construct, which with the addition of certain
verbal interpretations describes observed phenomena. The justification of such a
mathematical construct is solely and precisely that it is expected to
work.
John von Neumann P274
Above possibly from ‘Method in the Physical Sciences’ from Collected Works by John von Neumann, Vol. 6 – Pergamon Books Ltd, © 1962
Macrae,Norman
John von
Neumann
A Cornelia & Michael Bessie Book
Pantheon Books (Random
House, Inc.), New York,
1992
ISBN 0-679-41308-1
To some extent
Aristotle invented pure mathematics by seeing that (the primary question needs
to be not What do we know? but How do we know it? P114
And he was very
serious when he said several times why the mathematics inherited from the Greeks
had been crucial to the development of civilization. The greatest virtues were
that mathematics remained rigorously free from emotional content, free from
ethical content, and free from political content. It allowed people to rise to
the top by being reasoning scientists and scholars, instead of bullying priests.
P116
Motz, Lloyd and Weaver,
Jefferson Hane
The Story of Mathematics
Avon Books
A
division of
The Hearst Corporation
1350 Avenue of the Americas
© 1993
by Lloyd Motz and Jefferson Hane Weaver
Published by arrangement with Plenum
Publishing Corporation
LCCN: 93-26527
ISBN: 0-380-72458-8
for copyright
information
Plenum Publishing Corporation
233 Spring Street
New York,
NY 10013
By mathematical thinking we mean seeking and finding relationships among entities…P3
(Descartes’ coordinate system)... which has become perhaps the most important theoretical construct in the evolution of theoretical physics as well as mathematics…can be used to express graphically the relationship between any two quantities that are functionally related to each other. PP106, 107
…The actual functional relationship may have nothing at all to do with distances. The distances are then merely scale factors. P114
Newman,James R.
The World of Mathematics
Vol I
Jourdain,
Phillip E. B.
The Nature of Mathematics
Mathematical methods are contrived for the…”convenient handling of long and complicated chains of reasoning.”…though often suggested by natural events are purely logical.
…we should discover what is constant and what is variable in the processes of nature; that we should discover the same law in the molding of a tear and in the motions of the planets. This is the essence of nearly all science… P9
Whitehead, Alfred North – Mathematics as an Element in the History of Thought
The birth of
modern physics depended upon the application of the abstract idea of periodicity
to a variety of concrete instances.
Weyl, Hermann (1885-1955) – Symmetry
Editor’s
Commentary
Footnote 1 – Vitruvius defines: “Symmetry results from
proportion…Proportion is the commensuration of the various constituent parts
with the whole
In one dimensional time repetition at equal intervals is the musical principle of rhythm…Reflection, inversion in time, plays a far less important part in music than rhythm does. A melody changes its character to a considerable degree if played backward. All musicians agree that underlying the emotional element of music is a strong formal element…if so...we have probably not discovered the appropriate mathematical tools…PP702, 703
Kepler …We still share his belief in a mathematical harmony of he universe. It has withstood the test of ever widening experience. But we no longer seek this harmony in static forms like the regular solids, but in dynamic laws. P720
Newman
The World of Mathematics Vol III
Eddington, Sir Arthur Stanley – The Theory of Groups
That is the way with all models and pictures and familiar descriptions; they show the property that we are interested in, but they connect it with irrelevant properties which may be erroneous and for which at any rate we have no warrant. P1565
Mach, Ernst – The Economy of Science
1.It is the object of science to replace or save experiences, by the reproduction and anticipation of facts in thought…Science is communicated…in order that one man may profit by the experience of another…Language, the instrument of this communication, is itself an economical contrivance. Experiences are analyzed, or broken up, into simpler and more familiar experiences, and then symbolized at some sacrifice of precision. P1787
9. Although we
represent vibrations by the harmonic formula,… no one will fancy that vibrations
in themselves have anything to do with the circular functions. It has simply
been observed that the relations between the quantities investigated were
similar to certain relations obtaining between familiar mathematical functions,
and those more familiar ideas are employed as an easy means of supplanting
experience. Natural phenomena whose relations are not similar to those of
functions with which we are familiar, are at present very difficult to
reconstruct. P1794
Newman
The World of Mathematics, Vol IV
Poincaré, Henri – Mathematical Creation
To create consists
precisely in not making useless combinations…Invention is discernment,
choice…the mathematical facts worthy of being studied are those which, by their
analogy with other facts, are capable of leading us to the knowledge of a
mathematical law…They are those which reveal to us unsuspected kinship between
other facts long known, but wrongly believed to be strangers to one another.
P2043
Von Neumann,
John - The Mathematician
Re: the nature of intellectual effort in
mathematics
I think it is a
relatively good approximation to truth—which is much too complicated to allow
anything but an approximation—that mathematical ideas originate in empirics,
although the genealogy is sometimes long and obscure. But once they are so
conceived, the subject begins to live a peculiar life of its own and is better
compared to a creative one, governed by almost entirely aesthetical motivations,
than to anything else and, in particular, to an empirical science.
P2063
Birkhoff, George David – Mathematics of Aesthetics
Re: Nature of the Aesthetic Experience
The typical
aesthetic experience may be regarded as compounded of three successive phases:
1. a preliminary effort of attention…which increases in proportion to what
we shall call the complexity (C )of the object
2. the feeling of value or
aesthetic measure (M) which rewards this effort
3. a realization that the
object is characterized by a certain harmony, symmetry, or order (O), more or
less concealed, which seems necessary to the aesthetic effect.
Re: Mathematical Formulation of the Problem
This analysis of the aesthetic experience suggests that the aesthetic feelings arise primarily because of an unusual degree of harmonious inter-relation within the object. More definitely, if we regard M (aesthetic measure), O (order), and C (complexity) as measurable variables, we are led to write
The well-known aesthetic demand for ‘unity in variety’ is evidently closely connected with this formula. The definition of the beautiful as that which gives us the greatest number of ideas in the shortest space of time (formulated by Hemsterhuis in the eighteenth century) is of an analogous nature. P2186.
In order that the act of perception be successfully performed, there is also required the appropriate field of attention in consciousness. P2186
The actual types
of formal elements of order which will be met with are mainly such obvious
positive ones as repetition, similarity, contrast, equality, symmetry, balance,
and sequence, each of which takes many forms. P2190
On the other hand, ambiguity, undue repetition, and unnecessary imperfection are formal elements of order which are of strongly negative type. P2191
Final mathematical considerations lead to;
Jeans, Sir James –Mathematics of Music
…regularity is the essential of a musical sound-curve. Yet the regularity can be overdone, and absolute unending regularity produces mere unpleasing monotony…our aesthetic sense calls for a certain amount of regularity, rhythm and balance. Yet these qualities carried to excess produce monotony and boredom. P2281
The octave interval is fundamental in the music of all ages and of all countries…P2282
…in a vibration the restoring force is exactly proportional to the distance the particle has moved from its position of equilibrium..P2284
Overbye, Dennis
Lonely Hearts of the
Cosmos
Back Bay Books
Little, Brown and Company
Boston, New York,
London
© 1991 by Dennis Overbye
Originally published by HarperCollins Publishers, 1991
ISBN
0-06-015964-2 (HC)
ISBN 0-316-64896-5 (Back Bay ed.)
“…but one
can see lines in noise, the eye is such a good pattern maker.” Jim Peebles
P147
…the real discovery was always how you got the answer, not the answer itself. (Ascribed to an interview with JimPeebles) P436
Partch, Harry
Genesis of a Music, Second
Edition
Da Capo Press
New York 1974
A system of
music is an organization of relationships of pitches or tones, to one another,
and these relationships are inevitably the relationship of numbers. Tone is
number, and since a tone in music is always heard in relation to one or several
other tones—actually heard or implied— we have at least two numbers to deal
with: the number of the tone under consideration and the number of the tone
heard or implied in relation to the first tone. Hence, the ratio.
It is well
to plunge at once into ratio nomenclature and to disregard the more familiar
“A-B_C” terminology…The advantages of doing so, in opening new tonal vistas, ,
in getting to the analyzable root of music and the core of the universe of tone,
are inestimable. P76
The handling of small-number ratios, representing the intervals to which the ear is most responsive, involves nothing more than simple multiplication and division of improper fractions. Only when the expedient of temperament is introduced do the computations become at all complicated, when logarithms are employed to produce deliberately chosen irrational percentages of the factor of 2. P78
Pierce, John Robinson
The Science of
Musical Sound
© 1983 by Scientific American
Scientific American
Books, Inc.
Distributed by W. H. Freeman and Company
41 Madison Avenue,
New York, New York 1010
The ultimate
test is always how things sound…a good musician is always right about sound,
though the details of what he says may be wrong. Pxi
Mersenne found the correct (frequency ratios) by counting the number of vibrations per second of long strings, including a hemp cord 90 ft long…and a brass wire 138 feet long… P22
It is in the spirit of science to try to get behind the regularities of complex phenomena and find their explanation in simple terms. P23
For periodic musical sounds, the pitch is tied firmly to their periodicity, the frequency of the first harmonic partial. P37
However, trying
to represent actual sounds as sums of true sine waves, which persist from the
infinite past to the infinite future, is a mathematical artifice… A sum of
harmonically related sine waves doesn’t correctly represent (a musical sound),
because the sound starts, persists a while, and then dies away.
In practice,
we use the ideas of sine waves and their frequencies and amplitudes to
characterize musical sounds, and other sounds as well. The measurements we
really make are those suitable for our purposes and are as accurate as they need
be. (…the artificial and actual sounds will sound just the same, we will hear
them as being the same.) P45
The relative
strengths of the various harmonics depend on whether we pluck or strike, and on
where along its length we pluck or strike it. We can hear the difference in the
sound quality or timbre. P46
...if we change sine waves too fast, we will
hear a click or the twang of a plucked string. P56
(It is the)
experiences of consonance and dissonance that underlie the evolution of the
musical theory of harmony. P74
I believe, however, that to look at
experiences of consonance and dissonance as arising from rules is to look at
things the wrong way round. Rather, I believe that the rules and customs are
based on experiences of consonance and dissonance that are inherent in normal
hearing. Of course, musical training and sophistication will change the
subjective experience. The trained ear hears much in harmony that escapes the
musically untrained. Sometimes the trained ear hears things that aren’t there…
P75
Helmholtz tried
to explain consonance and harmony entirely in terms of beats…the work of Plomp
(1976) and others has shown that this is too simple a view. Slow beats do not
give a sense of dissonance, but merely a tremolo, a rising and falling of
amplitude. The range of frequencies in which we hear beats or roughness is
called the critical bandwidth.
When frequency components are separated by
more than a critical bandwidth we can hear them separately…But frequency
components that lie within a critical bandwidth interact, and give us sensations
of beats, roughness, or noise. P76
It may be that the major triad and its fundamental bass, which was first recognized by Rameau, is a sort of scaffolding on which very elaborate structures of harmony can be built. The major triads on the tonic, the dominant, and the subdominant (C, G, and F, for example) contain all the notes of the scale…Sounded in proximity, these three chords indicate the key unambiguously. P93
Appendix D Mathematics and Waves
…the propagation of waves along a string or through the air (is) a traveling disturbance. Such a disturbance involves continual changes in momentum (mass times velocity) caused by a force. The force may be associated with the bending of a stretched string or the compression of air. Continual changes in the force occur because the stretched string is bent as the wave travels along, or because the air is compressed when the velocity associated with the wave is lower ahead than it is behind.
One result of
such mathematics is the fact that waves behave in a very simple fashion only for
small amplitudes; that is, when the stretched string along which a wave travels
isn’t bent too sharply or when a sound wave traveling through the air raises or
lowers the pressure only by a small fraction. The behavior of such small
amplitude waves is linear. In essence, this means that when two waves are
present in the same medium (string, air) they don’t interact with one another.
Each goes its own way as if the other weren’t present. The total motion
(displacement, velocity, or pressure) is simply the sum of the motions
associated with the two (or more) waves.
Pierce, J.R.
Symbols, Signals and
Noise
The Nature and Process of Communication
© 1961 by John R.
Pierce
Harper Modern Science Series
Edited by James R
Newman
The
connection between numbers and collections of objects seems so natural to us
that we may overlook the fact that arithmetic is itself a mathematical theory
which can be applied to nature only to the degree that the properties of numbers
correspond to properties of the physical world. P7
Sampling theorem: This theorem states that a continuous signal can be represented completely by and reconstructed perfectly from a set of measurements or samples of its amplitude which are made at equally spaced times. The intervals between such samples must be equal to or less than one-half of the period of the highest frequency present in the signal. P66
Shlain, Leonard M.
Art and
Physics
© 1991 by Leonard M. Shlain
Permissions Department
William
Morrow and Company, Inc.
1350 Avenue of the Americas,
NY, NY 10019
ISBN
0-688-09752-9
Although logic itself is timeless, the process of logic depends heavily upon time. Logic proceeds one step after another. P32
(in the
Renaissance) When logic merged with experimental data, the scientific method was
born. P56
Pythagoras demonstrated that intervals had a mathematical,
which meant a rational foundation. P273
(Printing Press – Musical Notation)…Once music could be seen…it could be analyzed…Much like the anatomists who were their contemporaries, fifteenth-century composers began to dissect harmony in an attempt to learn the nature of its underlying structure. They teased apart its components and carried out experiments until they perfected polyphony. P275
Coincident with the invention of the printing press, emphasis on analysis informed all disciplines. P277
Shroeder,
Manfred
Fractals, Chaos and Power Laws
1991
W.H. Freeman and
Company, New York
Used with permission of W.H. Freeman and Company, New York, from Shroeder, Manfred, Fractals, Chaos and Power Laws, copyright 1991. Any reproduction or further use of this article requires the permission of W.H. Freeman and Company.
More Self-Similarity in Music: The Tempered Scales of Bach (PP99-101)
The Ancient
Greeks, with their abundance of string instruments, discovered that dividing a
string into two equal parts resulted in a pleasant musical interval, now called
the octave. The corresponding physical frequency ratio is 2:1.
“Chopping off”
one-third of a string produced another pleasant musical interval, the perfect
fifth, with a frequency ratio of 3:2.
The Pythagoreans asked themselves
whether an integral number of octaves could be constructed from the fifth alone
by repeated application of the simple frequency ratio 3/2. In mathematical
notation, they asked for a solution of the equation
in positive
integers n and m. But the fundamental theorem of number theory tells us that no
positive power of 3 can equal a power of 2, that is, that the equation
3n = 2k has no integer solutions for n > 0.
However,
the Greeks were not discouraged and by trial and error, found an excellent
approximate solution:
which is based
on the near equality of 3 1/19 and 21/12 .
A systematic
way of finding such near-coincidences is to write the ratio of the logarithms of
the two integer bases (2 and 3) as a continued fraction:
=
[1,1,1,2,2,…]
where the bracket notation is a convenient way to write the continued fraction
Continued fractions generally yield good rational approximations to irrational numbers; for example,
.
This excellent approximation to pi using not very large integers was known already to the ancient Chinese. Breaking the foregoing continued fraction off after the fifth term (as shown) yields the excellent approximating fraction for the musical fifth:
from which follows
Another important fact here is that the exponents 12 and 7 are coprime, so that repeated application of the perfect fifth modulo the octave (the “circle of fifths”) will not be close to a previously generated frequency until the twelfth step. These 12 different frequencies within an octave are all approximate powers of the basic frequency ratio
the semitone. Thus, there is always some value of k for which
r = 1,2,3…
The solution of this approximate equation is
One-third of the
octave, or
(r = 4, frequency ratio
1.260), for example is equivalent (modulo
the octave) to k = 4 fifths (frequency ratio
1.266).
The third part of the octave is also close to the pure major third (frequency ratio 5/4). This is the lucky result of another, independent, number-theoretic near-coincidence,
relating the next prime number above 3, namely 5, with the smallest prime, 2.
The fifth itself is approximated by seven semitone intervals with an accuracy of 0.1 percent:
The resulting
shortfall from the exact value 1.5 is called the Pythagorean Comma. It is
interesting to note that not only do seven semitones make one fifth, but modulo
the octave, seven fifths make one semitone. This coincidence results from still
another number-theoretic fluke, namely that 7 is its own inverse modulo 12, to
wit, 7 * 7 = 49
1 mod 12.
To ensure that
fixed note instruments, such as the piano, can be played in many different
musical keys. The frequencies of the different keys should be selected from the
same basic set of frequencies. This led to the development of Bach’s tempered
scales, based on the semitone with a frequency ratio of
. A musical instrument tuned according to
the tempered scale thus has frequencies approaching the following multiples of
the lowest note:
1,
,
,
,
,
…
up to some highest note.
Thus we see that
the frequencies of a well-tempered instrument form a self-similar sequence, with
the similarity factor
.
If all these notes were sounded simultaneously, the instrument would produce an
acoustic output (to put it mildly) that approximates a self-similar Weierstrass
function. (Actual tuning of pianos differs from exact self-similarity, the
tuning being somewhat “stretched” to minimize the beating of overtones, which
are not precisely harmonic, as a result of the finite bending stiffness of the
strings.)
Continued fractions are discussed in two animations, Equal Temperament Ratios 1, and 2 based on the above and the following web pages:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html%20
http://www.math.niu.edu/~rusin/papers/uses-math/music/
http://en.wikipedia.org/wiki/Continued_fraction
http://www.math.temple.edu/~yury/calendar/node2.html
http://www.ericweisstein.com/encyclopedias/music/CommaofPythagoras.html
http://www.mathworld.wolfram.com/ContinuedFraction.html
http://www.math.mit.edu/phase2/UJM/vol1/COLLIN~1.PDF
http://www.math.uwaterloo.ca/JIS/oldindex.html
Birkhoff’s
Aesthetic Theory
George David Birkhoff (1884-1944) Am. Math
Birkhoff’s
theory, in a nutshell, says that for a work of art to be pleasing and
interesting it should neither be too regular and predictable nor pack too many
surprises. P109
Taylor, Charles
The Science and
Technology of Tones and Tunes
IOP Publishing Ltd. 1992
Institute of Physics Publishing
Techno House
Redcliffe
Way
Bristol BSI 6NX, UK
US Editorial Office: Institute of Physics Publishing
The
Public Ledger Building, Suite 1035
Independence Square
Philadelphia, PA
19106
The idea of a
natural frequency is absolutely fundamental to our understanding of musical
instruments. P46
( child on a swing cf w/ Galileo pendulum)….a child on a
swing behaves like a pendulum and has a natural period. To keep the swing going,
a push must occur at the right moment…just at the turning point…we now have a
way of feeding energy in to keep an oscillation going…it is not necessary to
give a push at the right moment in every cycle…the pendulum can be kept going if
a short push occurs at twice, three times, or any other multiple of the natural
frequency…P46,47 (this is an aspect of “resonance”)
Alexander Wood’s (1879-1950)
The Physics
of Music
Revised by J.M. Bowsher
Seventh Edition,
1944,1975
Chapman and Hall, Ltd.
11 New Fetter Lane
LondonEC4P 4EE
©
1975 by Chapman and Hall Ltd.
A Halstead Press Book
John Wiley & Sons,
Inc., New York
ISBN 0 412 13250 8 (cased edition)
ISBN 0 412 21140 8
(Science Paperback Edition)
The principle underlying the phenomenon of resonance was clearly demonstrated by Galileo (1564-1642). It. Astron, & phys.
‘ Even as a boy, I observed that one man alone, by giving impulses at the right instant, was able to ring a bell so large that when four, or even six, men seized the rope and tried to stop it they were lifted from the ground, all of them together being unable to counterbalance the momentum which a single man, by properly-timed pulls had given it’. P25, 26