Everything
is determined by forces over which we have no control. It is determined
for the insect as well as for the star. Human beings, vegetables, or cosmic
dust - we all dance to a mysterious tune, intoned in the distance by an
invisible piper.
(Albert Einstein)
Music is the pleasure the human mind experiences from counting without being aware that it is counting. (Gottfried Leibniz)
The relationship between mathematics and music (vibrations / sound waves) is well known, and in hindsight it is obvious that mathematics, maths physics, music (sound waves) and musical instruments exist because matter is a wave structure of Space. This is why all matter vibrates and has a resonant frequency.
Below are some interesting articles and quotes that explain this relationship
between mathematics and music.
And for those of you who have children it is interesting to read about
the 'Mozart Effect', that listening to classical music improves both mathematical
and spatial reasoning skills.
The astronomer Galileo Galilei observed in 1623 that the entire universe "is written in the language of mathematics", and indeed it is remarkable the extent to which science and society are governed by mathematical ideas. It is perhaps even more surprising that music, with all its passion and emotion, is also based upon mathematical relationships. Such musical notions as octaves, chords, scales, and keys can all be demystified and understood logically using simple mathematics.
http://plus.maths.org/issue35/features/rosenthal/index.html
One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually integrate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.
In addition, Euler made important contributions in optics. He disagreed
with Newton's corpuscular theory of light in the Opticks, which was then
the prevailing theory. His 1740's papers on optics helped ensure that the
wave theory of light proposed by Christian Huygens would become the dominant
mode of thought, at least until the development of the quantum theory of
light.
(Wikipedia: Leonard Euler)
Note: Quantum Theory and the light 'photon particle' can now be explained with the Wave Structure of Light and Matter due to resonant coupling which is discrete, where E =nhf.
The ancient Greeks figured out that the integers correspond to musical notes. Any vibrating object makes overtones or harmonics, which are a series of notes that emerge from a single vibrating object. These notes form the harmonic series: 1/2, 1/3, 1/4, 1/5 etc. The fundamental musical concept is probably that of the octave. A musical note is a vibration of something, and if you double the number of vibrations, you get a note an octave higher; likewise if you halve the number of vibrations, it is an octave lower.
Two notes are called an interval; three or more notes is a chord. The octave is an interval common to all music in the world. Many people cannot even distinguish between notes an octave apart, and hear them as the same. In western music, they are given the same letter names. If you shorten a string exactly in half, it makes a note an octave higher; if you double its length, it makes a note an octave lower. You can think of the concept of octave and the number 2 as being very closely associated; in essence, the octave is a way to listen to the number 2.
If you shorten a string to 1/3 its length, a new note is produced, and the second most fundamental musical concept, that of a musical 5th emerges. We call it a 5th, because it is the 5th scale note of the Western do-re-mi scale, but it represents the integer 3. (Incidentally, the 5th is the only interval other than the octave that is common to all music in the world.) Strings of a violin are tuned a 5th apart. Men and women often sing a 5th apart, and most primitive harmony singing involves octaves and fifths.
If you build a musical system out of these integer notes, it is what is now called the Pythagorean scale, as used by the ancient Greeks. If you bore holes in a flute according to integer divisions, you will produce a musical scale. Oddly enough, if you try to build complex music from these notes, and play in other keys and using chords, dissonances show up, and some intervals and especially chords sound very out of tune. Our Western musical scale paralleled the evolution of the keyboard, and finally reached its modern form at the time of J. S. Bach, who was one of its champions.
After a few intermediate compromise temperings, as systems of tuning are called, the so called even-tempered or well-tempered system was developed. Even-tempering makes all the notes of the scale equally and slightly out of tune, and divides the error equally among the scale notes to allow complex chords and key changes and things typical of western music. Our ears actually prefer the Pythagorean intervals, and part of learning to be a musician is learning to accept the slightly sour tuning of well-tempered music. Tests that have been done on singers and players of instruments that can vary the pitch (such as violin and flute) show that the players and singers tend to sing the Pythagorean or sweeter notes whenever they can. More primitive ethnic music from around the world generally do not use the well-tempered scale, and musicians run into intonation problems trying to play even Blues and Celtic music on modern instruments.
The modern musical scale divides the octave into 12 equal steps, called half-tones. 12 is an important number on Western music, and it is oddly also an important number in our time-keeping and measurement systems. The frets of a guitar are actually placed according to the 12th root of 2, and 12 frets go halfway up the neck, to the octave, which is halfway between the ends of the strings. On fretted instruments we are playing irrational numbers! And any of you who have trouble tuning your guitars might get a clue as to why they are so hard to tune. Our ears don't like the irrational numbers, but we need them to make complex chordal music. The student of music must learn to accept the slight dissonances of the Western scale in order to tune the instrument and to play the music.
http://www.woodpecker.com/writing/essays/math+music.html
| Note | Ratio | Interval |
| 0 | 1:1 | unison |
| 1 | 135:128 | major chroma |
| 2 | 9:8 | major second |
| 3 | 6:5 | minor third |
| 4 | 5:4 | major third |
| 5 | 4:3 | perfect fourth |
| 6 | 45:32 | diatonic tritone |
| 7 | 3:2 | perfect fifth |
| 8 | 8:5 | minor sixth |
| 9 | 27:16 | Pythagorean major sixth |
| 10 | 9:5 | minor seventh |
| 11 | 15:8 | major seventh |
| 12 | 2:1 | octave |
Music is not considered one of the sciences today, but from the Middle Ages the study of music as a science (even if called a 'Liberal Art') was integral to the learned man's understanding of the world. Boethius helped establish it as one of the four disciplines of the Oxford quadrivium, in which music was studied together with arithmetic, geometry and astronomy. This was not, however, the kind of academic subject 'music' is today, but rather, was very much concerned with the old science of 'harmonics' - the study of the mathematical roots of harmony - in the context of Ptolemaic astronomy, which was itself a part of the quantitative harmony of the spheres 'tradition'. The universe (the motions of the planets and stars) was considered to be built on 'musical' harmonic principles - the same principles of harmony found in practical music.
The origin of this great 'tradition' is attributed to Pythagoras (c. 582 - 497 BC). One of its most important proponents was Plato, who was revered as a source of ancient wisdom, and whose Timeaus, which contains enigmatic references to the Pythagorean ideas, was known and studied before the renaissance. By the 17th century and the rise of the 'scientific age' music was still inseparable from science.
The idea of the 'harmony of the spheres' (harmonia mundi), or 'music of the spheres' (musica mundana), was largely received as the science of 'harmonics' - the study of the relationships between whole number 'harmonic ratios', musical intervals, and the orbital speeds and distances of the planets. The authority for this 'science' was referred back through its major proponents like Boethius or Ptolemy, to the 'ancient wisdom' of Plato or to its supposed originator, Pythagoras.
MUSIC POWER ENHANCES BRAIN FUNCTION
Music - either performing it or listening to it- has the power to enhance some kinds of higher brain function, a University of California research team has shown in new experiments with adults and preschool children. But it has to be the right kind of music.
"There is a causal link between music and spatial reasoning," co-author Frances Rauscher of the University of California at Irvine added in a telephone interview. "We now know it's true for the short term in adults, just from listening to music. It's true for eight months and probably longer in preschool children, by actually studying music. So there's no reason to expect it would not be true for older kids."
Rauscher and her colleagues at UC Irvine's Center for the Neurobiology of Learning and Memory attracted considerable attention last October with a report in the British journal Nature on what they call "the Mozart effect."
After listening for 10 minutes to a tape of Mozart's sonata for two pianos in D major, K. 488, college students in that earlier experiment scored approximately 9 points higher in IQ tests of abstract spatial reasoning than subjects exposed to 10 minutes of silence or a meditation tape.
Spatial reasoning tasks, which are generally processed by the brain's right hemisphere, involve the orientation of shapes in space. Such tasks are relevant to a wide range of endeavors, from higher mathematics and geometry to architecture, engineering, drawing and playing chess.
Interestingly, listening to other types of music did not enhance subjects' spatial test scores.
Neither Mozart nor the other music had any effect on subjects' ability to perform tests of short-term memory, which was consistent with the researchers' prediction about how the brain processes certain kinds of musical and spatial input.
The researchers believe that listening to Mozart's music, with its complex patterns of evolving musical themes, somehow primes some of the same neural circuits that the brain employs for complex visual-spatial tasks. They base their ideas on a "neural network" theory of music perception developed in 1990 by Gordon Shaw and Xiaodan Leng of UC Irvine and Eric Wright of the Irvine Conservatory of Music.
"In a nutshell, you have these neural pathways throughout your cortex," the higher brain centers involved in perception and thought, Rauscher explained. "The theory is when you experience something or learn something, these connections become stronger."
As provocative as the "Mozart effect" studies are, the researchers found that the effect is short-lived, 15 minutes at most. After that, Mozart listeners do no better on spatial tests than others.
To determine whether music can have more lasting benefits for spatial learning, the California researchers studied a group of 3-year-olds enrolled in a Los Angeles public preschool program. Of the 33 children, 22 received eight months of special music training -- daily group singing lessons, weekly private lessons on electronic keyboards and daily opportunity for keyboard practice and play.
When tested on a spatial reasoning task -- assembling pictures out of puzzle pieces -- "the children's scores dramatically improved after they received music lessons," the researchers reported. Among preschoolers without music training, spatial test scores remained unchanged over the eight-month experiment.
"We have shown that music education may be a valuable tool for the enhancement of preschool children's intellectual development," the researchers said. The group wants to show whether music training improves cognitive skills of school-age children, find out how long the effect lasts, and identify the mechanism behind it.
Others interested in the integration of music and other arts in school curricula were enthusiastic about the new studies.
"The main reason we teach music is because music itself is worthwhile," said Paul Lehman, dean of the University of Michigan school of music. "But at the same time music does a lot of other good things too, and especially in times when music is being cut back in school curricula."
(RICHARD A. KNOX, Boston Globe)
Philosophy / Metaphysics of Mathematics - I fully agree with you about the significance and educational value of methodology as well as history and philosophy of science. So many people today - and even professional scientists - seem to me like somebody who has seen thousands of trees but has never seen a forest. A knowledge of the historic and philosophical background gives that kind of independence from prejudices of his generation from which most scientists are suffering. This independence created by philosophical insight is - in my opinion - the mark of distinction between a mere artisan or specialist and a real seeker after truth. (Albert Einstein)
On the Mystical Foundations of Mathematics - Pythagoras, .. was intellectually one of the most important men that ever lived, both when he was wise, and when he was unwise. Mathematics, in the sense of demonstrative deductive argument, begins with him, and in him is intimately connected with a peculiar form of mysticism. The influence of mathematics on philosophy, partly owing to him, has, ever since his time, been both profound and unfortunate. (Bertrand Russell)
Mathematics and Music - The astronomer Galileo Galilei observed in 1623 that the entire universe "is written in the language of mathematics", and indeed it is remarkable the extent to which science and society are governed by mathematical ideas. It is perhaps even more surprising that music, with all its passion and emotion, is also based upon mathematical relationships. Such musical notions as octaves, chords, scales, and keys can all be demystified and understood logically using simple mathematics.
Famous Mathematics Quotes - As I work on these maths physics pages I collect mathematics / mathematical physics quotes that I think are important and add them to this page.
Covert Censorship by the Physics Preprint Archive - This page on censorship in Physics is here to show people that there are serious problems in getting new knowledge published in Journals (particularly if it contradicts current paradigm of 'particles' and 'fields', where it is just assumed that particle / wave duality is insolvable, waves are just 'probability waves', etc.). The particular example I have used relates to problems that Nobel Physics Laureate Brian Josephson had in getting articles published. But the problem is endemic and applies equally to philosophy.
Censorship at the Physics Forums http://www.physicsforums.com/ - This is a copy of a post that I replied to at http://www.physicsforums.com/. I don't think my reply was that bad given the criticism the Wave Structure of Matter received. However, when I went back to add a further comment, I found that I was BANNED for life from the physicsforums - the reason given was "crackpot" - and the post was deleted. I wrote to them asking for an explanation and have received no reply (funny about that!). And since then I have posted twice using new accounts. Each time my posts have been deleted and I have been banned. I wonder what people think of this form of censorship at the biggest physics forum on the internet? If you are like me and think this is wrong - please post at their forum and tell them so.
Mathematical
Physicist Dr Milo Wolff on the Wave Structure of Matter (WSM) in Space
I think all scientists would agree that to be certain
that the Spherical Wave Structure of Matter is true, we must demonstrate
that its Principles correctly deduce what we observe from observation and
experiment of the interconnected motion of matter in Space.
And here it is appropriate that I acknowledge the wonderful mind and pioneering
work of Mathematical Physicist Dr Milo Wolff on the Wave Structure of Matter
(1986 - 2007). His early work of applying the spherical scalar wave equations
to mathematical physics led him to deduce the fundamentals of both Einstein's
Special Relativity (mass increase) and Quantum Theory (de Broglie wavelength)
in one set of wave equations, thus both theoretically and metaphysically uniting
these two theories (metaphysics of Space and wave Motion - not Space and Time).
His work over the past 19 years has led to further deductions of the fundamental
laws of Nature from first principles (i.e. a
priori, rather than their current a posteriori empirical / inductive foundation which is always uncertain).
Wolff: Wave Structure of Matter - Summary of the Logic and Mathematics of the Wave Structure of Matter.
Wolff: Richard Feynman's QED - From Feynman's Advanced and Retarded Spherical Vector Electromagnetic Waves (Mathematical) to Spherical Scalar Quantum Waves (real waves).
Wolff: Binary Universe - The Binary Nature of Reality (Waves form from two balanced forces, Spherical In-Waves and Out-Waves). Relates this to digital computing.
Wolff: PhD Mathematical Physics / Biography - Milo Wolff - Astronomer, Electrical Engineer, retired Professor of Mathematical Physics, first formalised the Wave Structure of Matter in 1986.
Video Interviews of Dr Milo Wolff on Physics and the Wave Structure of Matter - The following interview between Philosopher Geoff Haselhurst and Physicist Milo Wolff on the Wave Structure of Matter was filmed in 2000 on the grounds of Berkeley University in California. Milo Wolff is a fine theoretical physicist and one day he will no doubt receive a Nobel Prize for his pioneering work on the Wave Structure of Matter. I think you will find these video interviews to be interesting (at times amusing) and very informative about the problems of modern physics and how they can be solved by the Wave Structure of Matter.
Physics Book: Milo Wolff: 'Exploring the Physics of the Unknown Universe' - Buy Milo Wolff's Physics Book on the Wave Structure of Matter from Amazon.
Milo Wolff's Physics Website on the Wave Structure of Matter - My name is Milo Wolff and I am a physicist, astronomer, and explorer of the wondrous cosmology of the universe. I have worked for MIT, Aerospace Corporation, the United Nations, as well as taught at various universities in the USA and Asia. Join the exciting and fascinating exploration of matter, the natural laws, and the universe. You may become a famous pioneer and, at the least, you will have fun! (Milo Wolff)
http://www-conf.slac.stanford.edu/Einstein/Talks/Wolff.pdf
- What is the Electron? Introduction:
In his later years, Einstein was asked his thoughts about the huge numbers
of short-lived heavy particles, kaons, pions, quarks, mesons, etc. found
using high-energy accelerators and enormous amounts of time and money. These
physicists thought they were finding important basic matter. They wanted
to know what Einstein thought of their work. Einstein was a careful thinker
and not given to theatrics so he was very serious when he replied, "I
would just like to know what an electron is."
This article will show, just as Einstein speculated, that the electron is indeed
the leading player in the universe and is intimately involved with light, matter,
the laws of Nature, and our lives. The path to the answer, is simple: Reject
the discrete material electron and replace it with a wave-structured electron
as proposed by Schroedinger and Clifford - using a scalar-wave equation. The
math and the proofs are straight-forward because scalar waves are the only
possible choice. Only two principles underlie all results. Despite this simplicity
the WSM explains the origin of the natural laws, new applications in micro-physics
and chips, understanding light energy exchange and lasers, plus the answers
to most current paradoxes of physics. It is breathtaking to find so many results
together – simply by answering Einstein’s question "What
is the Electron". (Milo Wolff, Geoff Haselhurst)
http://quantummatter.com/articles_html/body_point.html - The Wave Center forms the 'Point Particle' Effect. The dual particle/wave nature of the electron has long been a paradox in physics. It is now seen that the electron consists entirely of a structure of spherical waves whose behavior creates their particle-like appearance. The correctness of this structure is supported by the physical laws which originate from this wave structure, including quantum theory, special relativity, electric force, gravity, and magnetism. (Milo Wolff)
http://www.signaldisplay.com/Unified_Field_Theory.pdf
- The Derivation of a Unified Field Theory From a Model of Spherical
Quantum Waves.
Abstract: It is shown that if space is modeled as an elastic medium that propagates
spherical, scalar quantum-waves, then the ratio of the square of the wave velocity
to c2 reveals
the same results as the familiar time dilation formula that is produced from
the Schwarzchild G44 component. The Schwarzchild radius derived from the scalar-wave
model is shown to be equal to the radius of the observable universe, implying
that there are no gravitational singularities present within the observable
universe. The spherical wave model also produces a formula that calculates
the mass of a vector particle associated with each of the four forces and its
associated range. (Michael Harney)
http://redshift.vif.com/JournalFiles/V13NO2PDF/V13N2HAR.pdf - The Cosmological-Redshift Explained by the Intersection of Hubble Spheres (Observable Universe). The cosmological redshift is described by the intersection of two Hubble spheres, where a Hubble sphere is defined as a range over which spherical, quantum-waves interact, specifically Ru = 1.9 × 1026 m. (Michael Harney)
Deducing Heisenberg's Uncertainty Principle - Three short deductions of Heisenberg's Uncertainty Principle from Wave theorists, Mike Harney, Milo Wolff, and Chris Hawkings. The significant point is that by removing the 'particle' conception of matter and replacing this with the Spherical Standing Wave Structure of Matter we can deduce the Uncertainty Principle due to the spatially extended wave nature of matter. The confusion and paradox of Quantum Theory clearly results from the incorrect discrete 'particle' conception of matter. i.e. There is no discrete particle thus it is impossible to locate the exact position of something that does not exist (the continuous motion of a 'particle'!). In reality there is a discrete re-positioning of the wave center with each spherical in and out wave which gives the appearance of a continuously moving 'particle'.
http://www.ontostat.com/anglais/revisited_physics_gb.htm
- In this section on Physics, I would like to feed the debate on a certain
number of points, perhaps definitely established for physicists, but which
seem to me still fragile. Tackling with general questions, the first pages
(3.1 to 3.5) prepare the reader to this idea, matter is made of waves. From
the sixth (3.6) which present an undulatory model of particle, we really
enter into the subject.
The following pages display the basis of the Wave Structure of Matter (WSM)
Theory that I treat from the particle model I have conceived. Other authors
after Louis de Broglie, Milo Wolff, Gabriel LaFrenière,
Geoff Haselhurst, Chris Hawkings, Serge Cabala, have developed this theory,
broadly similar, but with differences in details. Nevertheless, the main difference
goes beyond this physical theory, it shows through the whole of ideas and facts
displayed on this site and is what leads the reader to this amazing finding:
a unique thing as simple as a wave, is likely to explain the entire Universe,
as well as the masse, the behaviour and the functioning of the objects it contains.
(Denys Lepinard)
Chris Hawkings - Could Matter and Matter Waves be Derivable? - 'The similarity in behaviour between matter and radiation suggests that matter may have an electromagnetic (wave) origin. It is shown that two light-speed waves with opposite directions of propagation yield a formalism which is identical to that describing the properties of matter. For spherical waves, particle localisation is also obtained and phase waves are generated, the properties of which match identically with matter wave properties.' Note: We do not agree with Chris in his use of vector electromagnetic waves (mathematical constructs which have no spherical solutions). We must instead work with scalar waves / real waves in Space.
Wave Structure of Matter Theorists Websites - Links to websites of people working on the Wave Structure of Matter. Physicist Dr Milo Wolff, Denys Lepinard, Mike Harney, Mike Weber, Ray Tomes (Harmonics), Serge Cabala, Gabriel LaFreniere.
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This is the profound new way of thinking that Einstein realised, that we exist as spatially extended structures of the universe - the discrete and separate body an illusion. This simply confirms the intuitions of the ancient philosophers and mystics.
Given the current censorship in physics / philosophy of science journals (based on the standard model of particle physics / big bang cosmology) the internet is the best hope for getting new knowledge known to the world. But that depends on you, the people who care about science and society, realise the importance of truth and reality.
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http://www.spaceandmotion.com/mathematical-physics/mathematics-music-waves-vibrating-space.htm
On Mathematics, Music and the Wave Structure of Matter (WSM) in Space