Interesting Items

Generally, things for which I have not yet found a home.

July 2, 2011 (Format)
October 7 , 2010

Marin Mersenne

He seems to have been at the center of everything that was going on in the early scientific revolution and was himself the author of an influential music treatise.

For now:http://galileo.rice.edu/Catalog/NewFiles/mersenne.html
Mersenne (pronounced Mayr-senn). 1588 - 1648

He insisted on careful specifications of experimental procedures, repetition of experiments, and publication of numerical results of actual measurements distinct from those from theory...

From 1623 Mersenne began to make the careful selection of savants who met at his convent in Paris or corresponded with him from all over Europe and as far afield as Tunisia, Syria, and Constantinople. His regular visitors or correspondents came to include Peiresc, Gassendi, Descartes, the Roman musicologist Giovanni Battista Doni, Roberval, Beeckman, J.B. van Helmont, Fermat, Hobbes, and the Pascals. This correspondence is being published. It was in Mersenne's quarters that in 1647 the young Blaise Pascal met Descartes. His role as secretary of the public of scientific letters, with a strong point of view of his own, became institutionalized in the Academia Parisiensis, which he organized in 1635.

***

http://www.mathpages.com/home/kmath578/kmath578.htm

Pitch and Color Recognition
Roughly speaking, the coiled cochlea of the human ear has a varying elasticity along its length, so it can be regarded as a series of oscillators of different resonant frequencies, and these perform a fairly detailed spectral analysis of incoming sound waves, transmitting to the brain something (like) a 3000 point spectral profile

..., the frequencies of the cochlea are determined by the fluid pressure in the inner ear, and many other factors that could be sensitive to temperature, humidity, barometric pressure, and so on.

...(a)... sense of the “cycle” of audible tones is based on the harmonic relations modulo the octave. We associate each tone with its “equivalent” in other octaves. Since the range of audible frequencies covers ten octaves, each tone has ten audible “equivalents”. Placing the frequencies on a logarithmic basis, each octave is subdivided into the twelve tones of our traditional musical scale (so the frequency of each semi-tone differs from that of its neighbors by the factor 2^1/12), and then we place all the tones into equivalence classes modulo twelve (i.e., modulo one octave).

(ror - 2^1/12 = two to the one twelfth power)

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Euler's book on music and math 'Tentamen novae theoriae musicae' published in 1739.

"... it had no great success as it contained too much geometry for musicians, and too much music for geometers."

Another version

"... for musicians too advanced in its mathematics and for mathematicians too musical."

http://www-groups.dcs.st-and.ac.uk/~history/Extras/Archibald_music_3.html
http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html

***

Logarithms:

http://links.jstor.org/sici?sici=0049-4925(197701)8%3A1%3C22%3AABHOL%3E2.0.CO%3B2-J

The concept of a logarithm made its first appearance in ancient Babylon where baked clay tablets have been found which contain tables of successive powers of whole numbers. In some of these records the question is asked: "To what power must a certain number be raised in order to yield a given number?"

Archimedes made an observation that is the basis of our modern logarithms. (He defined the "order" of a number to be equivalent of the exponent where the base is 100,000,000.) Archimedes then observed that the addition of orders corresponds to finding their product, a result which we know as the first law of exponents.

***

Chinese-Indian Music

(Still looking this one over.)

The following site takes a mystical approach and will need a closer look to see if belongs here. Some of the following ideas are of interest in any event: http://home22.inet.tele.dk/hightower/scales.htm

Chinese music
Music was the cornerstone in the Chinese civilization, which was the longest living culture in history. It was considered to embody within its tones, elements of the celestial order. The audible sound, including music, was but one form of manifestation of a much more fundamental form of Super physical Sound. The fundamental Primal Sound was synonymous to that which the Hindus call OM. The Chinese believed that this Primal Sound, Kung or "Huang Chung" (directly translated "yellow bell") was, though inaudible, present everywhere as a Divine Vibration. ...Further more and very important, the whole spiritual being of the musician himself was crucial.

Indian music
That applies also to Indian music...The ancient Indians had a less formalized approach to their music than the Chinese. Generally speaking they emphasized the personal inner contemplation more than the outward organized rituals...There is no sound without a meaning, so the Indians consider the emotions that different intervals evoke as exact as sound ratios.

December 4, 2006

From Ochoa and Corey,The Timeline Book of Science, The Stonesong Press, Inc, 1995, ISBN 0-345-38265-X

Weber's Law, Weber-Fechner Law Ernest Heinrich Weber (1795-1878)
Gustave Fechner...the just-noticeable increment in stimulus intensity is a constant fraction of the intensity already present.

p113 Stanley Smith Stevens(1915)...the Power Law of Psychophysics.
His developements in auditory scaling methods determne that physical continua usually conform to a psychophysical power law rather than Gustav Fechner's logarithmic law. The power function will prove controversial in psychphysics for more than thirty years. p247

(ror - Current status unknown to me, I have seen controversy on the internet but nothing definitive yet - I like the music produced by the logarithmic scale on my guitar, a lot of people have liked such music for some time now - but not everybody does. Some of the above may be a reason but the fashion and prejudices of the times and locations may be more important, there are other logical structures involved; human beings can be very fickle and get bored easily.)

November 16, 2006

Math In Music (excerpts - I would consider each of the following statements to be of equal value in an unresolvable, possibly unavoidable, discussion)

http://www.newmusicbox.org/chatter/chatter.nmbx?id=4425

"...In the composer's quest for formal unity, or whatever reason that calculator is turned on, absolutely nothing profound is instilled in the music...

...really it's about proportion, symmetry, balance, weight - all aesthetic concepts, not (merely) mathematical ones...

...What good is music if we truly understand it?..."

"Operating "without a system" is simply operating in ignorance of the system one is using - that's the nature of, well, everything..."

***

A cite from Reginald Bain: http://www.music.sc.edu/fs/bain/atmi02/index.html

"The structure of recognizable diatonic tunings is basically an array of intricate interconnections...which are the very foundation of what is perceived as tonal harmonic motion, are shaped by the short-term span of human memory, the tolerance range of the human ear, and the peculiar manner in which intervals are perceived."

Easley Blackwood
The Structure of Recognizable Diatonic Tunings

***

http://www.earlyromanticguitar.com/

"Wound strings were also not adopted initially because the frets were made of gut and tied on to the fingerboard, and metal strings tended to cut the gut frets. Presumably, metal frets were introduced to allow the use of metal-wound strings. Guitar publications in 1730, 1762, and later cite the benefits of metal-wound versus plain gut bass strings. Tyler & Sparks state: "They were expensive items (two sets could cost as much as a new guitar)... [but] from 1785 onwards, references to metal-wound bass strings can be found in [various inventory records of the period] ..."

My thinking here is that after the introduction of embedded metal frets to accomodate metal-wound strings, it was no longer possible to make fret adjustments and equal temperament became the only option on the instrument.

However...

http://wideopendoors.net/middleages_original/LifeTimes/Guitars.html

The BAROQUE GUITAR apparently came on the scene in the very early Seventeenth Century. ... the frets are permanent, whether wood, ivory, or metal.

(ror - There is also some reference (here or elsewhere) that regardless of metal or gut strings, these instruments had embedded frets on the soundboard. Whether sufficient evidence exists to determine original tuning on them is at present unknown to me. Very few have survived without modification. From mid-18th century there doesn't seem to be much in the way of meantone fretting)

***

http://en.wikipedia.org/wiki/Guitar_tuning
Standard tuning has evolved to provide a good compromise between simple fingering for many chords and the ability to play common scales with minimal left hand movement.

...It also yields a symmetry and intelligibility to fingering patterns.

***

http://www.cranfordpub.com/otis/craft_science_art.htm
Reflections-Otis A. Tomas
And it is still up to the artist to decide the ends toward which he strives. The complex unique and individual qualities that we find fascinating and beautiful as we play, will probably remain forever beyond the measurements of the physicist.

***

Music is the pleasure the human mind experiences from counting without being aware that it is counting.

Gottfried Leibniz

April 5. 2005

http://www.speech.kth.se/music/acviguit4/part4.pdf at page 25
Acoustics For Violin and Guitar Makers
Erik Jansson

The important conclusion from the presented material is that the played scales do not follow simple mathematical formulas - one plays neither in the Pythagorean scale, nor the pure scale nor the equal tempered scale. It seems rather that the played tone frequency is a part of the musical speech and one chooses the frequency approximately as the mathematically defined scales but with minor deviations for the effects one wants to stress. When playing several instruments together the possibilities of selection for the single musician is strictly limited to avoid unwanted sound effects.

*****

Citation lost:

Regarding Equal Temperament:...the intervals are multiples, not sums; so the scale is logarithmic, not linear.

The intervals are slightly mistuned (for example, the ratio of the fifth is 1.498 instead of 1½, and the third is 1.259 instead of 1¼), but the great advantage is that all keys are equally usable.

True Equal Temperament became possible only in the early 20th Century with electronic frequency generators.

*****

http://www.campanellaacoustics.com/faq.htm#basic_sound

Movements of the ear drum as small as the diameter of a hydrogen atom can be audible!

http://www.cs.fredonia.edu/~wilson/307/307S04.Syl.htm
From a college course syllabus on math and music - now offline by Dr. Julia Wilson. The quotation would make an interesting introduction to the proposed animation on "alternate tunings".

".... One of the most striking connections between math and music is the observation that if one string on an instrument is twice as long as another, then the longer one will produce a tone that is an octave lower than the shorter one. Thus the ratio of 2:1 is associated with the octave. Other harmonious intervals are also associated with ratios of small whole numbers. This observation leads to mathematical methods for defining musical scales, such as the Pythagorean scale. Over the last two thousand years, many scales have been proposed, each with its own influence on compositional style and music theory. In addition, many mathematicians, scientists, and musicians have devoted considerable effort to trying to explain this association of number with interval and to justifying their favorite scales..."

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