Literature

Augustine
City of God
abridged Version
Edited by Vernon J. Bourke
Image Books – A division of Doubleday & Company, Inc. Garden City, New York
Image Books Edition – 1958
© 1950,1952, 1954, 1958 by Fathers of the Church, Inc.
LCCN 58-5717
Chapter 21 P72

Citing from Cicero, De re publica 2.42

“Scipio, then had said: ‘In playing the lute, or the flute, or even in good vocal music, the different notes should be kept in harmony. If they are changed into discord, the trained ear cannot endure it. That agreeable harmony, however, is produced by the modulation of tones that are very dissimilar. In like manner, as in music, out of the highest and the lowest classes, and those that lie between, by a reasonable control, the state is fashioned into a concordant whole by the consent of very diverse elements. What musicians call harmony in music, in the State is known as concord, the closest and strongest bond of security in any commonwealth, and which can in no way exist without justice.’

Bronowski, J
The Ascent of Man
Little, Brown and Company
Boston/Toronto
© 1973 by J.Bronowski
ISBN0-316-10930-4

Art and science are both uniquely human actions…they derive from the same human faculty: the ability to visualize the future, to foresee what may happen and plan to anticipate it, and to represent it to ourselves in images that we project and move about inside our head …(or on a cave wall or a television screen) P56

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

So the later conflict in quantum physics between the behavior of matter as a wave and as a particle is a conflict between analogies, between poetic metaphors; and each metaphor enriches our understanding of the world without completing it. P13

Creation is the finding of order in what was disorderly. P16

(Citing Coleridge)…beauty is “unity in variety.” Now this to me is the creative process. Nature is chaos. It is full of infinite variety…there comes a moment when many different aspects suddenly crystallize in a single unity…you have found the path which organizes the material…That is the moment of creation. P17

… art in general…is composed essentially of self-reference, and takes its life from the duel tension between watching our own minds from the inside and watching someone else’s from the outside

. And this is one of the classical paradoxes in the theory of knowledge, how and when we know that others do indeed feel as we do. P69
Citing Popper, Karl (1902-1994)

Campbell, Joseph
The Power of Myth
With Bill Moyers, Betty Sue Flowers Editor
Doubleday, a division of Bantam Doubleday Dell Publishing Group, Inc.
666 Fifth Avenue, New York. New York 10103
ISBN 0-385-24773-7
ISBN 0-385-24774-5 pbk
© 1988 by Apostrophe S Productions, Inc., and Alfred van der Marc Editions

To be in accord with the grand symphony that this world is, to put the harmony of our own body in accord with that harmony. P55

James Joyce’s epiphanies – (his) formula for the aesthetic experience is that it does not move you to want to possess the object. Nor does the aesthetic experience move you to criticize...The aesthetic experience is a simple beholding of the object. Joyce says you put a frame around it and see it first as one thing, and that, in seeing it as one thing, you then become aware of the relationship of part to part, each part to the whole, and the whole to each of its parts. This is the essential aesthetic factor—rhythm, the harmonious rhythm of relationships. And when a fortunate rhythm has been struck by the artist, you experience radiance. You are held in aesthetic arrest. That is the epiphany. P220

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

“…almost anything that has to do with waves has these very specific properties reminiscent of quantum states. Not only violin strings, but air inside flutes and organ pipes vibrates in fundamental frequencies or exact multiples thereof—each time taking a quantum leap to the next harmonic. P117,118

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

Physical laws are nothing but statements concerning the way in which certain quantities depend on others when some of these are permitted to vary. Thus the pitch of the note emitted by a plucked string depends on the length, weight, and tension of the string….The task of the physicist is to determine the exact or approximate nature of this functional dependence. P275

Crick, Francis
The Astonishing Hypothesis
Touchstone
Rockefeller Center
1230 Avenue of the Americas
New York NY 10020
1994
ISBN 0-684-80158-2

Gestalt- an organized whole in which each individual part affects every other, the whole being more than the sum of its parts. In other words, your brain must actively build up these “wholes” by finding which combination of the parts seems the most likely to correspond to the relevant aspects of the object in the real world, basing its estimates on your previous experience and on the experience of your distant ancestors which is embedded in your genes. P36,37

Gestalt Laws of perception. P37,39
· We tend to group things together that are close to one another
· We tend to group things together that have some obvious property in common
· Continuation
· Closure
· Simplest, most regular, symmetric interpretation- prefer a sensible interpretation

(MW- Gestalt (Ger shape, form) – structure, configuration or pattern of physical, biological, or psychological phenomena so integrated as to constitute a functional unit with properties not derivable by summation of its parts.

Consciousness (MW)1. Perceiving, apprehending or noticing with a degree of controlled thought or observation 2. the quality or state of being aware esp. of something within oneself.)

3 basic ideas P15
1. Not all the operations of the brain correspond to consciousness
2. It involves some form of memory, probably a very short term one
3. Closely associated with attention
A formal definition is likely to be misleading or overly restrictive or both P20

Seeing 3 general statements P26
1. You are easily deceived
2. Information provided by your eyes can be ambiguous
3. Seeing is a constructive process

(the brain)…must find the best interpretation of the visual signals given its past experience…a many leveled interpretation,…usually in terms of objects and events and their meaning to us.P33

Attention – withdrawal from some things in order to deal effectively with others. P59

All forms of attention are likely to have both reflex and willed components
Attention filters out unattended events.P60

Attention can be finely focused in space or, alternately, made more diffuse.
Alters the way you see. P61

What is especially unclear is whether. In focused awareness, we are conscious of only one object at a time, or whether our brains can deal with several objects simultaneously. P209

What may prove difficult or impossible to establish are the details of the subjective nature of consciousness, since this may depend upon the exact symbolism employed by each conscious organism. P252

…another strikingly human characteristic, one that is seldom mentioned: our almost limitless capacity for self-deception. The very nature of our brains—evolved to guess the most plausible interpretations of the limited evidence available—makes it almost inevitable that without the discipline of scientific research, we shall often jump to wrong conclusions, especially about rather abstract matters. P262

Joyce, James
Ulysses
© 1914, 1918 by Margaret Caroline Anderson. © 1934 by the Modern Library, Inc. © 1942, 1946 by Nora Joseph Joyce - Random House, NY - 768 pages

-It is, Bloom said.
Numbers it is. All music when you come to think. Two multiplied by two divided by half is twice one. Vibrations: chords those are One plus two plus six is seven. Do anything you like with figures juggling. Always find out this equals to that, symmetry under a cemetery wall. He doesn’t see my mourning. Callous: all for his own gut. Musemathematics. And you think you’re listening to the ethereal. PP273, 274

Levenson, Thomas
Measure for Measure
© 1994
Touchstone
Rockefeller Center
1230 Avenue of the Americas
NY, NY 0020
Simon & Schuster, Inc.
ISBN 0-671-78730-6
0-684-80434-4 (Pbk)

Yo Yo Ma’s description of the Bach cello suites:

For Ma, the suites (and music in general) “are about the primary things we have to deal with, what we know and the terror and mystery of what we don’t: the terror and mystery of death.”…Expressing what he finds in the score Bach left him requires a balancing act. On the one hand, Ma pays extremely close attention to the elaborate structure of the pieces: “ You have to have an organized mind,” he says. “ If you go for beautiful sound and gorgeous melodies but ignore the rhythmic and harmonic pillars; if you are not constantly listening to the bass line; if you are not aware of the dance beats; if you don’t recognize the multiple patterns that sometimes go with and sometimes interfere with strong beats— then you can’t extract the joy from the music. You don’t get the feeling of large structure; you can’t do it, it just doesn’t happen.” But on the other hand, too precise a reading and the joy gets lost again. “Perfection means you hit every note, no scratches, playing totally reliably. But that squeezes the life out; it destroys a piece of music. The element of risk isn’t there. In performance you shape, mold, and sculpt. You follow whatever you can sense of the motion, the wave, the physicality of a smooth stream of notes. You have to build the piece with your fingers, mind, and instrument. It has to breathe. Perfection to me is static.” P231

Lin Yutang
The Wisdom of China and India
Random House NY 1942

Chuangtse

(from LV. The Virtues of the Child
To know harmony is to be in accord with the eternal. P612

Montaigne, Michel de
Selected Essays
Translated by Donald M. Frame;
Classics Club © by Walter J. Black, Inc. Roslyn, NY 1943

(Speaking of: Cato the Younger (95-46 B.C.) “…he who has touched one chord of him has touched all; he is a harmony of perfectly concordant sounds, which cannot conflict.” - Of the Inconsistency of Our Actions P120

Yet we must tune the string to all sorts of notes; and the sharpest is the one that least often comes into play. Of Presumption
P135

Newman, James R.
The World of Mathematics Vol I
Simon and Shuster, Inc.
Rockefeller Center
630 Fifth Avenue
New York 29, N.Y.
(C) 1956 by James R. Newman

Robert Recorde (1510-1558)
The Ground of Artes (1540)
…Musick hath not only great help of Arithmetic, but is made, and hath his perfectnesse of it: for all Musick standeth by number and proportion…P214

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

Harper & Brothers, New York

(The composer will not produce a simple and slow succession of random notes)… any more than a writer produces a random sequence of letters. Rather, the composer will make up his composition of larger units, which are already familiar in some degree to listeners through the training they have received in listening to other compositions. These units will be ordered so that, to a degree a listener expects what comes next and isn’t continually thrown off the track. Perhaps the composer will surprise the listener a bit from time to time, but he won’t try to do this continually. To a degree, too, the composer will introduce entirely new material sparingly. He will familiarize the listener with this new material and then repeat the material in somewhat altered forms. P252

To use the analogy of language, the composer will write in a language which the listener knows. P252

Successful art requires the appreciation of an audience as well as the talent of the artist. Audiences are influenced by other things besides the object of art before them. If a person sets his mind against anything, it will leave him cold. P266

…to be appreciated, …art must be in a language familiar to the audience.We can be surprised only by contrast with that which is familiar, not by chaos. P267

Plato (428-348 B.C.)
Apology,Crito,Phaedo
Translated by B. Jowett Edited, with introduction by Louis Ropes Loomis
Publishe for the Classics Club by
Walter J. Black, Inc. Roslyn, NY 1942

Phaedo PP83-153/ P121(the scene is Socrates’ final hours before execution – Editor’s comments- The personality of Socrates and the thought of Plato together make the Phaedo what it is, an unforgettable picture of one of the noblest scenes in European literature.)

“In this respect,” replied Simmias, “Suppose a person to use the same argument about harmony and the lyre—might he not say that harmony is a thing invisible, incorporeal, perfect, divine, existing in the lyre which is harmonized, but that the lyre and the strings are matter and material, composite, earthy, and akin to mortality? And when someone breaks the lyre, or cuts and rends the strings, then he who takes this view would argue as you do, and on the same analogy, that the harmony survives and has not perished—you cannot imagine, he would say, that the lyre without the strings, and the broken strings themselves which are mortal remain, and yet that the harmony which is of heavenly and immortal nature and kindred, has perished—perished before the mortal. The harmony must still be somewhere, and the wood and strings will decay before anything can happen to that…”P121

Phaedo
(Socrates…)”And does not the nature of every harmony depend upon the manner in which the elements are harmonized?”… “ I mean to say that a harmony admits of degrees, and is more of a harmony, and more completely a harmony, when more truly and fully harmonized, to any extent which is possible; and less of a harmony, and less completely a harmony, when less truly and fully harmonized.” P129

‘There is a perfection which all knowledge ought to reach, and which our pupils ought also to attain, and not to fall short of, as I was saying that they did in astronomy. For in the science of harmony, as you probably know, the same thing happens. The teachers of harmony compare the sounds and consonances which are heard only, and their labor, like that of the astronomers, is in vain.”
“Yes, by heaven!” he said; “and ‘tis as good as a play to hear them talking about their condensed notes, as they call them; they put their ears close alongside of the strings like persons catching a sound from their neighbor’s wall—
one set of them declaring that they distinguish an intermediate note and have found the least interval which should be the unit of measurement; the others insisting that the two sounds have passed into the same—either party setting their ears before their understanding.”
“You mean,” I said, “those gentlemen who tease and torture the strings and rack them on the pegs of the instrument; I might carry on the metaphor and speak after their manner of the blows which the plectrum gives, and make accusations against the strings, both of backwardness and forwardness to sound; but this would be tedious and therefore I will only say that these are not the men, and that I am referring to the Pythagoreans, of whom I was just now proposing to inquire about harmony. For they too are in error, like the astronomers; they investigate the numbers of harmonies which are heard, but they never attain to problems—that is they never reach the natural harmonies of number, or reflect why some numbers are harmonious and others not.” PP417,418

Plutarch
Selected Lives and Essays
Translated from the Greek by Louis Ropes Loomis
Introduction by Edith Hamilton
Published for Classics Club by Walter J. Black, Inc, Roslyn, NY 1951

Lycurgus (li-‘ker-ges) (ca 9th to 7th century B.C.) ( Lawgiver of Sparta)
14. But in his opinion the most important and noblest task of a lawgiver was education…Vol. 1, P13

(The Pythagoreans taught that the ultimate reality was a Sacred One or Monad.” Plutarch – Numa Pompilius I, 11, P44)

Pericles (ca 495-429 B.C.)
8. Pericles was anxious to acquire a style of speaking which, like a musical instrument, would harmonize with his manner of life and lofty spirit, and he made use of Anaxagoras, as another string, mingling with his own rhetoric touches of Anaxagoras’ natural science. Vol.1,P136

Coriolanus (first half of the fifth century B.C.)
1. The greatest benefit indeed which men gain from their friendship with the Muses is that softening of the nature which comes from their teaching and training, as we learn from them to prefer moderation and reject extremes. Vol.1, P235

Essay- Superstition
Music, says Plato, the creator of harmony and order, was given by the gods to man not in order to amuse him and tickle his ears but to disentangle gently, bring around and restore to its proper place again whatever in his body was maladjusted and disturbing to the courses and harmonies of his soul, and which often for lack of art and grace became unruly and mischievously lawless. Vol. 2, P370

Rigden, John S.
Physics and the Sound of Music 2nd ed.
1977, 1985
John Wiley & Sons

Strings, air columns, and membranes can vibrate with many frequencies simultaneously. P118

In order to determine which frequencies are present in the resulting vibration, we must determine which harmonics, when superimposed one upon the other, will add together to give a shape like the initial shape of the string. P118

Note: The initial phase of the third harmonic is negative while that of the fifth harmonic is positive; see Figure 9.2. This alternation in the initial phase continues. PP121

Two Conclusions:
1. …when a string is plucked, all harmonics having nodes at the pluck-point will be absent in the resulting vibration.
2. …the closer (the pluck is) to the end of the string, the richer will be the harmonic content of the resulting vibration. P122

Shakespeare (1564-1619)
The Complete Works
Edited By G.B. Harrison
Harcourt, Brace& World, Inc. - New York, Chicago, San Francisco, Atlanta
© 1968
ISBN 0-15-5 80530-4

Hamlet III, ii, 6 (P908)
For in the very torrent, tempest, and, as I may say, whirlwind of passion, you must acquire and beget a temperance that may give it smoothness

Love’s Labor Lost IV, iii, 344 (P416)
And when love speaks, the voice of all the gods makes heaven drowsy with the harmony.

The Taming of the Shrew, I, I, 36 (P337)
Music and poesy use to quicken you,
The mathematics and the metaphysics,
Fall to them as your stomach serves you.
No profit grows where is no pleasure ta’en.
In brief, sir, study what you most affect.

Taming of the Shrew III, I, 73 (P348)
‘Gamut’ I am the ground of all accord.

Troilus and Cressida I, iii, 85-111
The heavens themselves, the planets, and this center,
Observe degree, priority, and place,…
Take but degree away, untune that string,
And hark, what discord follows! Each thing meets
In mere oppugnancy.

Richard II, II, i, 5 (P442)
Oh, but they say the tongues of dying men
Enforce attention like deep harmony.

The Merchant of Venice V, i, 83 (P610)
The man that hath no music in himself,
Nor is moved with concord of sweet sounds,
Is fit for treasons, stratagems, and spoils.
The motions of his spirit are dull as night,

Sonnet 8 (P1596)
Music to hear, why hear’st thou music sadly?
Sweets with sweets war not, joy delights in joy.
Why lovest thou that which thou recivest not gladly,
Or else receivest with pleasure thine annoy?
If the true concord of well-tuned sounds,
By unions married, do offend thine ear,
They do but sweetly chide thee, who confounds
In singleness the parts that thou shouldst bear.
Mark how one string, sweet husband to another,
Strikes each in each by mutual ordering,
Resembling sire and child and happy mother,
Who, all in one, one pleasing note do sing.
Whose speechless song, being many, seeming one,
Sings this to thee: “Thou single wilt prove none."

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

One of the several striking features that separates young children’s thought processes from those of adults is “magical” thinking. Children blur the border between thinking and doing, between the inner space of imagination and the outer space of objectivity…wishing it, in effect—will affect the outcome of actual events…The linear alphabet and its equally linear comrades-in-arms, the numerals, are loosed like soldiers to destroy the child’s belief in discontinuous space and mythical time…As language, math, and logic take hold, they drive magic out of the child’s being…PP139-141

Every child is born with a desire to re-create the world in his or her own terms. This powerful motivation for producing art has always been a means of imposing order on the disjointed pieces of the child’s emerging worldview…Art translates curiosity and wonder into mastery over the environment. P141

…Newton himself retained a childlike curiosity and outlook, and saw himself as a youth engaged in play:

I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than the ordinary, while the great ocean of truth lay all undiscovered before me.
Quoted from Jacob Bronowski, The Ascent of Man (Boston: Little Brown, 1973), p236 (P142 in Shlain’s book)

Frierich Nietzche (1844-1900)
"Without music, life would be a mistake" P271

(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

Sudo, Philip Toshio
Zen Guitar
Fireside (Simon & Shuster)
1997

Zen = "Way" P14

…learning to play the guitar is inseparable from learning to harmonize body, mind, and spirit. To truly play from your soul, you must have all aspects of yourself working together as one.( cf ch’iyun-pg 31)” 15

To be obsessed with the destination is to remove the focus from where you are…put everything into …right now. 18

…carrying a empty cup… set aside all preconceptions and open your mind to learning as though for the first time 26

“The only opponent is within.” 36

practice…shugyo…mastering one’s deeds…There will never come a point where practicing ends… 39

Rhythm (hyoshi)…you cannot feel rhythm with your mind; you must feel it in your body…. All things have an underlying pulse, and their source is the same…. There is nothing deeper than this… 48

Each level of achievement brings a new set of problems. This should be understood beforehand. 56

Study hard, but stay relaxed. You never want the feeling of pressing; instead, learn the feeling of power held in reserve 82

…all you need to know is one thing: where your passion lies…103

Every note you play, everything you do, it’s all one take. 152

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

I have a suspicion, built up from experience in many different areas, that the ear-brain system seems to prefer something that is not quite perfect. Perfection is too bland and we need a little non-uniformity to add a little bite. P110 (seq)

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 source of a musical note is always some “system” in vibration. P1

…a law which is more or less true for all sensations- pressure, sight, &etc.—that the intensity of the sensation is proportional not to the stimulus but to the logarithm of the stimulus. This means that every time the stimulus is multiplied by the same factor the sensation goes up the same number of steps. Thus if the stimulus changes successively from 10 to 100, 100 to 1000, and 1,000 to10,000,&c., the sensation goes up by three equal steps. We get equal increments of loudness, not by adding equal increments of intensity but by multiplying intensity always by the same factor. P36

A note of any definite pitch being chosen as a starting point, a number of other notes can be picked out by the ear as in some way simply related to it. This kind of relationship was first studied by Pythagoras (Ca580-500 B.C.) (Gk philos. and math.) P50

Pythagoras found that whenever the string was divided by the movable bridge, so as to give two notes in some recognizable relationship, the ratio of the lengths of the segments was always the ratio of two small whole numbers. P50, 51

And if we wish to add two intervals, we have to multiply the corresponding ratios. Thus if we wish to add a fifth and a fourth, we see that the frequency of the upper note of the fifth bears to the frequency of the lower note the ratio 3/2. Now, the upper note of the fifth is the lower note of the fourth which is to be added, and the frequency ratio for the fourth is 4/3. The frequency of the upper note of the fourth must therefore bear to the frequency of the lower note of the fifth, the ratio 4/3 x 3/2 = 2. It is therefore the octave—a result which we should not have anticipated if we had tried to add 1.50 and1.33. P52

We have the same difficulty in subtracting one interval from another. We have to divide the corresponding ratios. P53

…sources of sound—e.g., stretched strings, air columns, &c.—have usually numerous possible modes of vibration. They can vibrate in a larger or smaller number of parts, and each mode of vibration has a different frequency. The greater the number of parts, the greater the frequency and the higher the pitch of the note. P61

Aristotle (384-322 B.C.) Gk philos., Book XIX, Problem 8 ‘Why does the low note contain the sound of the high note?’ And again in Problem 13, ‘Why is it that in the octave, the concord of the upper note exists in the lower, but not vice versa?’

Mersenne, Marin (1588-1648) Fr. cleric, math., a Minorite friar, wrote the first extended treatise on sound and music, Harmonie Universelle, Paris, 1636. In it he refers to these problems of Aristotle, and in a section headed ‘To determine why a vibrating string gives several sounds simultaneously’ (p.208) he says: ‘But it must be remarked that Aristotle did not know that the struck string gives at least five different sounds simultaneously, of which the first is the natural sound of the string, serving as fundamental to the others and to which alone attention is paid in singing; all the more because the others are so weak that only the best ears can hear them easily. It is necessary to choose a deep silence in order to hear them, although this will no longer become necessary when the ear has become accustomed. As for myself, I have no difficulty; I have no doubt that anyone can hear them who gives the necessary attention. Now these sounds follow the ratio of the numbers 1, 2, 3, 4, 5 because four sounds are heard different from the fundamental, of which the first is the octave above, the second is the twelfth, the third is the fifteenth, and the fourth, the major seventeenth.’ That Mersenne associated these partial tones with quality is clear from a query in the same section whether ‘the sound of each string is more harmonious and agreeable as it causes to be heard a greater number of different sounds simultaneously’. This is the first clear association of quality with partial tones. P65

Chapter 7 Vibrations of Strings
Probably the most important class of musical instruments is that based on the vibrating string, plucked, struck, or bowed. P90

There is reason to believe that it has its origin in the hunting bow. When the arrow was released the musical note of the bowstring must have been fairly obvious, and there is evidence that the bow was used as a musical instrument as well as a weapon. P90

Quality of Notes from Stretched Strings
There are three ways in which the quality of the note given by a stretched string may be varied:
1. By variation of the point of attack,
2. By variation of the method of attack,
3. By variation of the vibrating system to which the string is coupled. P92

The physical explanation of dissonance was first suggested by Sauveur (1653-1716), and afterwards developed by Helmholtz. P158

Chapter 11 – Scales and Temperament

Scales
In the process of the development of music, the first step was to select from the infinite variety of notes available the limited series to be used. The series of notes so selected is called a scale. P171

The smallest interval in our own scale is a semitone. Between the two notes forming the interval lie at least twenty-five other distinguishable notes, none of which is used. P171

It is due to a necessity in the nature of music itself—the necessity that the movement in pitch which constitutes melody should proceed by definite and appreciable steps, and not by indeterminate gliding, and that successive notes should stand to one another in a simple and easily apprehended relationship Only in this way can music make its appeal to the mind and to the emotions. P171

The origin of our own major and minor scales can be traced with fair certainty to the music of the Greeks…The records of musical culture show that there was considerable development as early as 1200 B.C. The instrument most closely associated with the development of the scale was the tetrachord, an instrument of four strings as the name suggests. P173

(The basic historical idea here is that (in theory) two tetrachords together would amount to eight strings with eight different notes and seven intervals (ratio of one frequency to another) between them, the highest note would be an octave (double the frequency, a ratio of 2:1) above the lowest note. All the intervals between would total that one octave. By changing the starting note from one string to the next a different order of intervals could be obtained that would help keep the music interesting by providing some variety. The order of intervals is called a mode. .P174)

The reaction of harmony on the scale was twofold.
1. In the first place, it fixed the pitch of the notes with greater precision. An inaccurate interval taken in melody may pass unnoticed, whereas the same inaccuracy in harmony would turn a smooth concord into a harsh discord.
2. Then again it puts a premium on the scale offering the largest choice of concordant intervals. P177

With the rise of harmony the principle acquired an altogether new importance. All other notes in the scale were related to the tonic, and in this way a new principle of unity was established. A few consecutive chords are sufficient to establish the key. The whole mass of tones and the consecutive harmonies must stand in a close and clearly perceptible relationship to the tonic, while the composition must be developed from it and finally return to it. P.178
Once this principle had become established the musical variety made possible by a large number of modes ceased to be important, and variety was sought rather in the use of the same mode at a different level of pitch—i.e., by modulation into a new key. Thus the only two modes to survive…have been the major and minor modes, which revealed themselves as more effective for harmonic development than any of the others. P178

In the case of the octave the upper note contains every even partial of the lower, and the possession of these partials in common may well account for the simple relationship which the ear appreciates when the two notes are sounded successively. A similar relationship holds for the fifth and the fourth, (see p160 et. Seq.) P 181

Pythagoras proceeded to derive a chromatic scale based entirely on octaves and fifths—the two simplest and most obvious intervals…Pythagoras worked out the ratio of lengths of strings for all the notes of a complete chromatic scale. P182.

Arithmetical simplicity in the ratios of the frequencies carries with it the possession of low-order partials in common, and therefore indicates a relationship which the ear recognizes and appreciates.

Most important of all, notes which stand to one another in a simple frequency ratio form a concord when sounded together. The rise of harmony has therefore enhanced the demand for pairs of notes satisfying this condition, and all the intervals already named form concords. Because it is built up in this way, the scale we have just considered is usually called the natural or true scale. P186

The Need for Temperament. — Music which remains always in one key gives a sense of monotony which is quite intolerable, and this is avoided by frequent ‘modulation’ or change of key, followed sooner or later by a return to the key of the composition. P188.

The process of compromise by which true intonation is sacrificed to the exigencies of the mechanics of keyed instruments in order that freedom to modulate may be secured is called temperament. P190

Musemath - Home