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. P201

…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

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

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 principles P382

...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…

(First)... 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

(ror - this tradition is criticised by Fowler, Plato's Acadamy - regardless of history as truth, Fowler's mathematics on incommensurables has an independent validity of interest.)

"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."

(quoting 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

(speaking about John von Neumann)

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

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

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,James R.
The World of Mathematics Vol I

(Anthology)

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

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: 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:

and thus to embody in a basic formula the conjecture that the aesthetic measure is determined by the density of order relations in the aesthetic object.

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

(ror - p 2193 Further mathematical considerations lead to;)

***

(ror - There is more in the article but the following seems a good summary: citation lost - 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.)

 

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

“…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

Overbye, Dennis
Lonely Hearts of the Cosmos
Back Bay Books
Little, Brown and Company, 1991 Originally published by HarperCollins, 1991
ISBN 0-06-015964-2 (HC)
ISBN 0-316-64896-5 (Back Bay ed.)

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.

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.

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

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

(ror - NOTE: The current academic view is that Bach used "Well Tempered Scales" - different than today's Equal Temperament. The evidence seems convincing but regardless, Schroeder's mathematical assessment of Equal Temperament is independently valid.)

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:

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:

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:

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:

The solution of this approximate equation is:

One-third of the octave, or:

(r = 4, frequency ratio approximately equal to 1.260), for example is equivalent (modulo the octave) to k = 4 fifths (frequency ratio approximately equal to 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,

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,

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:

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.)

END

***

(ror) Continued fractions are discussed in two animations, Continued Fractions 2 and Continued Fractions 3 based on the above and the following web pages:

Knott, Rusin, Dunn, Wikipedia, Temple, Weisstein, Mathworld, Mit PDF, Waterloo

Some mathematicians using continued fractions come out with a different set of answers - this has to be reconciled sooner or later

Schroeder, 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 Schroeder, 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.

The idea of a natural frequency is absolutely fundamental to our understanding of musical instruments. P46

( cf w/ Galileo pendulum below)….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”)

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