On Mathematics and Music The Wave Structure of Matter (WSM) in Space

On Mathematics and Music

The Wave Structure of Matter (WSM) in Space

mathematics and musicEverything 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

Leonhard Euler

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 Pythagorean Musical Scale

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.

Music, Mathematics and Philosophy

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 Harmony of the Spheres

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.

The Mozart Effect

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."