Alex

Is Beauty Quantifiable ?

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There was a great TV special on The Learning Channel (here) that talked about common occurance of a number called the golden section, or the golden ratio in things that are generally accepted are visually appealing. Their examples range from faces, to architecture, to credit cards, to shells. There are also many other occurances and methods that this number pops up in.

I was interested on what peoples opinions were on the idea of esthetics being quantifiable. Generally esthetics is often ascribed personal opinion to a large extent, but if it is in fact quantifiable, personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder.

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There was a great TV special on The Learning Channel (here) that talked about common occurance of a number called the golden section, or the golden ratio in things that are generally accepted are visually appealing.  Their examples range from faces, to architecture, to credit cards, to shells.  There are also many other occurances and methods that this number pops up in. 

I was interested on what peoples opinions were on the idea of esthetics being quantifiable.  Generally esthetics is often ascribed personal opinion to a large extent, but if it is in fact quantifiable, personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder.

I can't speak to the golden ratio, but as an additional data point on this I recall watching a show a number of years ago that discussed beauty, and one of the most interesting examples was that it appears that symmetry plays a large role in people's opinion of the attractiveness of the human face. One startling example they gave was a face that, while not ugly, would not be considered beautiful either. They took a photo of the face, cropped out half of it, mirrored the half they did keep, and pasted it back together. The "new" face was immediately much more attractive.

If there exists any science or mathematics on symmetry I'm unaware of it, but it seems at first glance that there could be something quantifiable there. Just something else to consider on this topic.

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I was interested on what peoples opinions were on the idea of esthetics being quantifiable.  Generally esthetics is often ascribed personal opinion to a large extent, but if it is in fact quantifiable, personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder.

To be sure I understand where you are coming from, can you give a definition of what you mean by "quantifiable?"

I would say that beauty is most definitely quantifiable, but the quantities designated are non-numerical, ordinal quantities. I would also say that the particular form of measurement involved in determining the quantity-of-beauty in an object is a personal one.

I can definitely say that the paintings found at the Quent Cordair gallery are more beautiful than any scrawl I have seen by Jean-Michel Basquiat. This comparison is based on the ordinal, non-numerical quantity I assign to the beauty in each.

The nature of such measurements are that they can vary from person to person. I know many people who find the work of Jackson Pollack or Basquiat beautiful, while I vehemently disagree. The personal variance of such measurements does not imply that they are non-objective. Such measurements are entirely objective if they are based on an assessment of the actual object one is measuring. An example of a non-objective measurement in this regard would be for me to say, "Basquiat was a famous, immensly popular artist. His art sold for millions of dollars and was praised by critics everywhere. Because Jean-Michel Basquiat is celebrated, his art is beautiful."

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There was a great TV special on The Learning Channel (here) that talked about common occurance of a number called the golden section, or the golden ratio in things that are generally accepted are visually appealing.  Their examples range from faces, to architecture, to credit cards, to shells.  There are also many other occurances and methods that this number pops up in.

The golden ratio is a truly fascinating relationship/number, one which bears a remarkable relationship to the Fibonnaci numbers, seen throughout structural forms in nature. The golden ratio is seen in Pythagoras, is a problem found in Euclid, and it was used by the Greeks in architecture. Kepler coined it "the divine proportion" and da Vinci illustrated a book on it, De Divina Proportione. There is a marvelous little popular book on it named, aptly enough, The Divine Proportion (H.E. Huntley, Dover Publications, Inc., 1970.)

If you like the mathematical aspects of this, and also enjoy mathematical relationships in nature, a really great book is D'Arcy Wentworth Thompson's On Growth and Form, Dover Publications, 1992.

I was interested on what peoples opinions were on the idea of esthetics being quantifiable.  Generally esthetics is often ascribed personal opinion to a large extent, but if it is in fact quantifiable, personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder.

You seem to be asking from the perspective of mathematical relationships, and in that sense I do not think that mathematics has any key to unlock the door to esthetics. There are aspects of design, in terms of shapes that are pleasing, that have been studied, and there are several theories that incorporate mathematics such as the golden ratio. But this is design, more related to perception than related to the conceptual arts that we call the study of esthetics.

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If there exists any science or mathematics on symmetry I'm unaware of it, but it seems at first glance that there could be something quantifiable there. Just something else to consider on this topic.

If you have some interest in this, there is a marvelous old popular book by Jay Hambidge that introduces this subject (The Elements of Dynamic Symmetry, Dover Publications, Inc., 1926/1967). This book was first brought to my attention by a magnificent young Objectivist artist, Ralph Hertle, who, sadly, gave up painting twenty-five years ago. His work, Until Now, is on my office wall (Ayn Rand had the original painting), and his Dynamis is on my living room wall, flanked by a Capuletti and a Maxfield Parrish.

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If you have some interest in this, there is a marvelous old popular book by Jay Hambidge that introduces this subject (The Elements of Dynamic Symmetry, Dover Publications, Inc., 1926/1967).

Excellent - I'll check it out. Thanks!

Or maybe you could just upload your mental bibliography, if not all the rest of your knowledge, directly into my brain. Save time...

B)

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Or maybe you could just upload your mental bibliography, if not all the rest of your knowledge, directly into my brain. Save time...

B)

Or, you could just purchase the soon to be released sixty DVD set, ten DVDs for each decade. :)

But, you know, this transfer of knowledge directly to the brain has always bothered me. The science fiction movies and stories often have the encylcopedia being transferred to a brain through some sort of head contraption, and then the person becomes a genius. But, why is this so? Everything from A to Z is received, but what of the connections and relationships among the data? Of what use is an alphabetic storage and recall? It all remains unintegrated, unprocessed. When we learn things, what is so important is how we relate our new knowledge to what we already know, and how we store those connections in our mind. Else the subconscious would just randomly spew out whatever the content of A thru Z.

Someone needs to work this procedure out! :D

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... Everything from A to Z is received, but what of the connections and relationships among the data? Of what use is an alphabetic storage and recall? It all remains unintegrated, unprocessed. When we learn things, what is so important is how we relate our new knowledge to what we already know, and how we store those connections in our mind.

Quite so.

I'd like to strongly recommend the Hugo Award winning "Lensman" series, by E.E. (Doc) Smith, first called to my attention by the late Jim Davidson, who loved it. The basic pattern of the series is escalation, in a battle between two diametrically opposed cultures, on a scale that gradually becomes wider than the galaxy itself. Its chief value for me (and, I think, for Jim) is that the major escalation is psycho-epistemological.

This is done in science-fiction terms, which do not exactly correspond to human psychology as known. However, most of the skills that the few, most advanced Lensman gain (as champions of Civilization) have parallels with actual psycho-epistemological skills.

And in the Lensman series, actual transfers of content from one mind to another do take place ("wide open 2-ways"), but they stagger the recipient's mind and have to be followed by weeks of slow, diligent sorting-out and establishing connections.

The series consists of:

Triplanetary

First Lensman

those first 2 titles can be skipped; the heart of the story is the next 4 titles:

Galactic Patrol

Gray Lensman

Second Stage Lensmen

Children of the Lens

There is a final title, Masters of the Vortex, that takes place in the same fictional universe, but does not deal with the central conflict of the series.

Highly recommended. I own a number of pulp fiction magazines from the 1930s and 40s that these novels originally appeared in, with great cover art.

P.S. After you've finished the series, for an amusing surprise look up what E.E. Smith got His Ph.D. in. No, it wasn't engineering, as you'd expect!

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The series consists of:

Triplanetary

First Lensman

those first 2 titles can be skipped; the heart of the story is the next 4 titles: ...

You say can be skipped. Do you recommend doing so? If I skip these two, is there anything lost in terms of continuity or integration?

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You say can be skipped. Do you recommend doing so? If I skip these two, is there anything lost in terms of continuity or integration?

Chapter 1 of Triplanetary begins 2 billion years ago, setting up the background, concerning the two hostile planets of Arisia and Eddore; so I would read that much, at least. The rest of the book describes how the Arisians and Eddorians manipulated Atlantis, Rome, and World Wars 1, 2, and 3 behind the scenes, each strugglng to nourish or destroy civilization; and the discovery of the inertialess space drive by Bergenholm. It does involve some questionable premises, about the selective breeding of humans until mankind reaches a high enough level to be trusted with Lenses.

First Lensman, not published till the 1950s, was an afterthought of the author's, almost entirely concerned with how earth was consolidated under a world government, under the Galactic Patrol. I would skip it entirely.

Galactic Patrol and the 3 following books are the heart of the story. Here's where the escalation of psycho-epistemologies takes center stage. Chapter 1 does sum up the requisite background info: mankind knows nothing about Arisia, except that the Arisians give mankind the mysterious Lenses ...

Happy reading!

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Here's a website with scans of those beautiful old pulps from the 1930s and 40s, in which the Lensman novels first appeared:

http://www20.brinkster.com/pariahpress/lensman/main.htm

The bibliographical data on this site confirms that Triplanetary was originally several unconnected stories, which E.E. Smith later attempted to tie in with the Lensman series, for book publication. And that First Lensman was an afterthought, not part of the series as originally conceived.

In case you're wondering what Z9M9Z means: that was the flagship of the Galactic Patrol, which saw action in the battles of Jalte, Jarnevon, the defense of Tellus, the invasion of the Second Galaxy, the battle of Klovia, the defense of Arisia, and the battle of Ploor.

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[...]

I was interested on what peoples opinions were on the idea of esthetics being quantifiable. Generally esthetics is often ascribed personal opinion to a large extent, but if it is in fact quantifiable, personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder.

The topic of personal opinion and its role in the appreciation or valuation of works of art needs to be discussed objectively. Far too many arguments by Objectivists, I have found, rely upon individual rights at their conclusion. I wasn't able to present a concept of objective beauty, or as an artist, say that one work of mine was more sophisticated and more beautiful than the previous work. The 'rights' proponents say that 'Its my right to have the preferences that I want, and that there are no other types of opinions that can supercede my preferences in type or importance.' They have not provided an opinion using esthetics principles, say regarding order, morality, or beauty, and instead they have opted for a political base for their opinions. These ownership experts have changed their ethical principles from those regarding the objective principles of esthetics to those of Positivism, a subjective philosophy that claims that the speaker is right because he says so.

Objectivism has said a lot regarding morality in art, although the topic name of, morality, has not been used so much as, individualism, for example. Objectivism has said little regarding the structures of works of art, for example, the inventive timed melodic structures of music, the systems of motion in dance, or the dramatic ordered arrangements in architecture and design. In my theory of art and design, these structures would be dealt with in the Formal Arts [as opposed to the representative arts, the Conceptual Arts].

Objectivism also cannot deal with the topic of beauty as a universal matter for all esthetics until it deals with the Formal Arts. That is, the broad principles of order and beauty cannot be seen to be universal until the Formal and Conceptual Arts have been defined. Beauty, however, until then, may be dealt with in particular, and numerous specific examples should be evaluated by everyone.

Objectivism is miles away from dealing with the matters, of proportion, dynamic proportion, Dynamic Symmetry, order, dramatic relationships, or beauty, for example, and it won't be able to create a coherent general theory until these core matters of the Formal Arts have been identified and written.

These basic formal concepts need to be established in a Helmholtz - like fashion prior to dealing with music. The architecture of music needs to be expressed in detail prior to establishing relationships to the other Formal Arts. Only then may the major innovations of esthetics take place, for example, the discovery of a system of music that is based upon Dynamic Symmetry proportional systems. That would result in a type of music that would be the equivalent of the Dynamic Symmetry - based architecture of the Parthenon or of Art Deco in architecture and design. At this point in time one could not create a general theory, only a selected number of examples that would have some of the appropriate attributes.

I have been researching the subject of dynamic proportional harmony systems, Dynamic Symmetry, the principles of geometry and architecture of the Ancient Greeks, Art Deco, order, and geometric drama for a book. I am somewhat of an expert in the Dynamic Symmetry system of geometry. I have even re-written the musical keyboard for the piano to correct specific proportional harmonic errors.

Objectivists who wish to discuss these matters may contact me at, ralph.hertle@verizon.net.

Whether beauty is quantifiable, may not be a matter of number. Nonetheless, it may be a matter of measurement. Surely. conceptual identification and objectification of the causes of works of art, and also, the means of evaluating works of art, are possible. My opinion at this time is that these matter must wait for the fundamentals to have been first identified.

Inventor

.

.

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Whether beauty is quantifiable, may not be a matter of number. Nonetheless, it may be a matter of measurement.

Could you expand a bit on this, Inventor? How can something be measured without a number?

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I have even re-written the musical keyboard for the piano to correct specific proportional harmonic errors.

Oh, I'm quite curious about this. How did you go about it? What did you choose as an alternative scale?

The even toned keyboard was gradually chosen over the natural scale keyboard because music - common practice harmony, classical, "Formal Arts" music - was entering and departing too many keys, and being composed in too many keys, to have a single harmonic scale on every keyboard. If your keyboard is in C, it becomes tricky to play something in D major or perform a work which visits the relative minor/major key. I played a fair bit of natural horn (this: http://www.corno.de/schmid/deu-eng/lorenzhornk.jpg) and this was an acute problem - the instrument cannot depart from its tube's set tonality (so e.g. in symphonic works you change tubes in rests; this gets quite ridiculous in the late Classical/early Romantic works as the natural horn was overlapping the first valved instruments, the valves being a way to change tubes quickly).

So in a way I already held the even toned keyboard as being, whilst perhaps not the "ideal solution", the best possible compromise. Indeed it is considered the accepted form in natural horn playing to bend the notes back to an even scale.

The only thing I can think of would be for you to set the home key of the natural harmonics keyboard DURING a performance, setting it to whatever key the piece is visiting. That should be quite interesting to hear.

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Whether beauty is quantifiable, may not be a matter of number. Nonetheless, it may be a matter of measurement.

Could you expand a bit on this, Inventor? How can something be measured without a number?

I should have added that I was using the concept, "measurement", in the same sense that the Ancient Greeks used the idea. When their concept for "scientific concept" got translated by the later Platonists and religious writers from Ancient Greek to English they wrote, "magnitude". Today, astronomers still use the English term. "magnitude" to mean the amount on a scale of brightness for stars. Today, historians of mathematics say that Pythagoras said, "The universe is made of numbers," howevers, Pythagoras really meant that the universe is governed by, or measured by, scientistific measurements, identified principles, or concepts." He meant, and this has its parallel in Aristotle's usage, that when one says one measures something one may mean describing the amount of something or identifying some property of something.

Aristotle said, for example, that, "one cannot measure the universe." That was an error of translation. He meant that, because the universe was a plurality of things, and not one thing, oned could not identify any property of the universe. He meant that one can only identify the properties of individual existents, and of course, selected properties one step removed that are relationships of things, for example, the sizes or lengths of things.

I am too lazy at the minute to find the references in the writings of Pythagoras, Aristotle, and Euclid that use the Ancient Greek word for, "magnitude".

Recall, for example, that when asked, "if love may be measured," Ayn Rand said, "And how." We may concur that means identification and amount, yes; and particular numbers, e.g., 4398560391, no.

To summarize. Some attribute of a work of art may be measured in the sense of identifying a property of the work, e.g., is the work beautiful? That's what I meant.

Inventor

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I have even re-written the musical keyboard for the piano to correct specific proportional harmonic errors.

Oh, I'm quite curious about this. How did you go about it? What did you choose as an alternative scale?

The even toned keyboard was gradually chosen over the natural scale keyboard because music - common practice harmony, classical, "Formal Arts" music - was entering and departing too many keys, and being composed in too many keys, to have a single harmonic scale on every keyboard. If your keyboard is in C, it becomes tricky to play something in D major or perform a work which visits the relative minor/major key. I played a fair bit of natural horn (this: http://www.corno.de/schmid/deu-eng/lorenzhornk.jpg) and this was an acute problem - the instrument cannot depart from its tube's set tonality (so e.g. in symphonic works you change tubes in rests; this gets quite ridiculous in the late Classical/early Romantic works as the natural horn was overlapping the first valved instruments, the valves being a way to change tubes quickly).

So in a way I already held the even toned keyboard as being, whilst perhaps not the "ideal solution", the best possible compromise. Indeed it is considered the accepted form in natural horn playing to bend the notes back to an even scale.

The only thing I can think of would be for you to set the home key of the natural harmonics keyboard DURING a performance, setting it to whatever key the piece is visiting. That should be quite interesting to hear.

The revised piano keyboard may be a good solution to fix the seeming unique tonality of the piano's musical scale. I'm not a musician, composer, or technologist of music, however, my limited study gave me just enough knowledge that, after working with the keys, I could make a possible discovery.

I won't spell out all the pitches for all the keys, and I have yet to do that. I want to find a musician who can set up an MS EXCEL workbook file with cell formulas that will generate all the numerical pitches in the correct ratios and the new sequence. Actually, the ratios will remain unchanged, and the names of the keys will be changed in two instances. Only one new pitch is added to fill a gap in the sequence of ratios, and the remainder of the notes in sequence will stay unchanged.

Here are some of the characteristics of the revised keyboard.

1. The new octave is made of thirteen keys from C to C', for example. That seems odd, however, it works harmonically quite well.

2. The piano keyboard has the same number of octaves.

3. At the left end of the octave trade the positions of one black key with one white key.

4. At the left end of the octave add one black key.

5. All the positions between white keys will then have been filled.

6. All music that has been written for the conventional tone scale and piano keyboard will be able to be played unchanged. The keyboardist or pianist will have to reckon with the new key arrangement.

7. All other types of instruments will be able to be played without re-learning their respective means of entering or selecting notes.

8. Stringed instruments that are played with a bow or pick may play the conventional instrument unchanged, and they may find the location for the added note.

9. Horns and possibly all wind instruments will be able to play all music written for the conventional twelve - tone scale as before, however, the fixed physical structures of the instruments may not permit playing the single new note. I don't know if a trombone with its variable pitch can find and play the single added note.

The method of playing pairs of harmonizing notes will demonstrate and validate the new keyboard arrangement.

Currently stringed instruments that play chords with the left hand can either play the added note or find a chord that can render a similar sound. A piano cannot do that, and a new physical key must be added. For reasons of continuity in playing pairs of notes on the keyboard the two existing black and white keys are changed in positions.

The way to generate the new scale is to begin by playing C and C' above, simultaneously. The two notes harmonize. Next select the inside two notes, C+1 and C'-1. Do this for every inside pair of notes in the octave, including both black and white notes, e.g., C+2 and C'-2, and C+3 and C'-3, and so on. Find that almost all the notes harmonize.

When you find notes that harmonize, you will find that some harmonizing notes are on black keys and some are on white keys. Change the positions of the black and white keys, and you will find that the those positions reconcile the arrangement of the keys.

One one will be found to be missing. Add that note to complete the reconciliation.

A fully harmonized and completed keyboard will be the result.

All existing music may be played from existing written arrangements without change.

Newly written music may include the missing/new note, and new graphic notation may be created. The labeling of the two repositioned notes on the piano keyboard will be changed in the written notation.

The result is that the new musical scales played on the piano seem more evenly spaced and more measured. To me, the sound of the scale is that a greater sense of gradualness is evident. Because of the change of the pitches the piano melodies no longer have the quality of discontinuity, and they seem more melodious. I believe that composers may benefit from the new scheme.

Inventor

.

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The topic of personal opinion and its role in the appreciation or valuation of works of art needs to be discussed objectively. Far too many arguments by Objectivists, I have found, rely upon individual rights at their conclusion.

In all the years I have been around Objectivists, including Objectivist artists, art historians, and art critics, I can't say I've ever heard anyone base their esthetic opinions on individual rights. Most esthetic analysis by Ayn Rand and those who agree with her involve the concept of sense of life instead.

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The topic of personal opinion and its role in the appreciation or valuation of works of art needs to be discussed objectively. Far too many arguments by Objectivists, I have found, rely upon individual rights at their conclusion.

In all the years I have been around Objectivists, including Objectivist artists, art historians, and art critics, I can't say I've ever heard anyone base their esthetic opinions on individual rights. Most esthetic analysis by Ayn Rand and those who agree with her involve the concept of sense of life instead.

Hi Betsy,

Numerous times I have tried to explain to individuals just why a certain piece was better than others. They may have felt cornered when I tried to explain, and when they wanted to hold to their preferences. The attitude that I found was that there is no objective criteria for better art or for beauty, and the beholders would say that they like a certain piece because they like it. I would say, what if the other work is better and give specific reasons. They would say, well, I don't care, and its my right to have any opinion that I want.

I will agree with you that the sense of life property of a work is highly important, and possibly the most important, and I would further qualify the discussion to say that a person's rights to their opinion are not a valid exuse for one or the other esthetic opinion or preference. The discussion has probably been closed to me long before I would hear the argument from individual rights regarding art.

I wanted the person to learn more, and to see and enjoy more. A hopeless type of dicussion is that a work is better than another because it is sunny, has golden mean proportions, is hard edged, is representational, has beautiful color patterns or harmonies, and so on.

At a certain point a person's preferences are indeed important.

The worst thing that I found is to try to alter an artist's way of seeing reality, or to condemn them for this and such reasons. For example, that their work is bad, or evil, because they made blurry or imprecise imaginary illusions, and, for example, that they included representational things in their music, or that they made paintings that had no representational portrayals. Their artistic task may have been some entirely different quest, even one that Objectivism has had little to speak about, and that is beauty, for example. To me it is more important to get the artist to prosper in the terms of his of her quest. We need what the artist has to say now and for the future. Maybe the artist's rights to be free to create are highly important. They have to get their own educations too.

On balance, now that I think about it, and weigh some of the examples that come to mind, that the sense of life property of a work may be a primary. Yes, that is a concept that Ayn Rand discovered and explained. It is important for happiness, too. If so, then should not all moral and stylistic characteristics of a work serve that point?

That's a ramble.

Inventor

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The topic of personal opinion and its role in the appreciation or valuation of works of art needs to be discussed objectively. Far too many arguments by Objectivists, I have found, rely upon individual rights at their conclusion.

In all the years I have been around Objectivists, including Objectivist artists, art historians, and art critics, I can't say I've ever heard anyone base their esthetic opinions on individual rights. Most esthetic analysis by Ayn Rand and those who agree with her involve the concept of sense of life instead.

Hi Betsy,

Numerous times I have tried to explain to individuals just why a certain piece was better than others. They may have felt cornered when I tried to explain, and when they wanted to hold to their preferences. The attitude that I found was that there is no objective criteria for better art or for beauty, and the beholders would say that they like a certain piece because they like it. I would say, what if the other work is better and give specific reasons. They would say, well, I don't care, and its my right to have any opinion that I want.

I've have never heard anyone base their aesthetic opinions on individual rights either. Whether or not they felt "cornered" by you, their reliance on individual rights to not have to argue with you or satisfy you does not mean that for their aesthetics they "rely upon individual rights at their conclusion" and does not mean that they had not identified any objective criteria categorizing what they do and don't like. Silence or a refusal to be pressured does not mean no good reasons for rejection or a subjectivist position such as basing one's principles of aesthetics or anything else on "individual rights".

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I have even re-written the musical keyboard for the piano to correct specific proportional harmonic errors.

What did you choose as an alternative scale?

The revised piano keyboard may be a good solution to fix the seeming unique tonality of the piano's musical scale. I'm not a musician, composer, or technologist of music, however, my limited study gave me just enough knowledge that, after working with the keys, I could make a possible discovery.

I won't spell out all the pitches for all the keys, and I have yet to do that. I want to find a musician who can set up an MS EXCEL workbook file with cell formulas that will generate all the numerical pitches in the correct ratios and the new sequence. Actually, the ratios will remain unchanged, and the names of the keys will be changed in two instances. Only one new pitch is added to fill a gap in the sequence of ratios, and the remainder of the notes in sequence will stay unchanged.

If there is in fact some principle behind all this it would be a trivial matter to generate the values of the frequencies, and it doesn't require Excel to do it. The frequencies can even easily be inserted into any standard audio dsp editing program on your pc to generate the corresponding pure sinusoidal tones to show what they sound like in sequence or together in chords.

But the description you have given is incoherent. Do you change the pitches or not? Pitches are determined by the ratios of the frequencies, which ratios you say don't change, in which case there is nothing to calculate. And there is no "gap" in which to put a supposedly missing note. The frequency of each half tone in the modern equal tempered scale is 5.95% higher than the previous note, uniformly for every one of them -- because an octave jump doubles the frequency and 1.05946^12=2. Because there are only twelve half-tones in an octave, for the eight notes in the diatonic major scale each note is separated from the next lowest note by two half tones except for two of them -- the 3rd and the 6th -- which is why the number of sharps or flats designating a half tone change must change when the key is modulated and the scale starts in a different position of the 12 half-tone sequence. All the notes are accounted for and any musical piece can be played in any key. There are no missing notes. You can play in any key off written music with the proper transposition by taking into account the new key signature. You can't do this only for instruments that can't replicate the full twelve tones in an octave -- like a bugle, English post horn, or natural horn mentioned by rtg24.

The pitch ratio convention of the equal tempered scaled is very close to the older just scale and the results are indistinguishable for almost anyone's pitch discrimination ability. The difference would be swamped by inserting a 13th tone, which would make a drastic change far in excess of that.

This is all very basic information that you can find in elementary books like Ian Johnston's Measured Tones: the Interplay of Physics and Music, 1989, or the more detailed Benade, Fundamentals of Musical Acoustics, 1976.

Here are some of the characteristics of the revised keyboard.

1. The new octave is made of thirteen keys from C to C', for example. That seems odd, however, it works harmonically quite well.

Not in any usual meaning of the term harmonically it doesn't. Neither this characteristic nor most of the subsequent items in your list of assertions have any meaning without your explaining what you are doing so we can see what you are talking about.

2. ...

9. Horns and possibly all wind instruments will be able to play all music written for the conventional twelve - tone scale as before, however, the fixed physical structures of the instruments may not permit playing the single new note. I don't know if a trombone with its variable pitch can find and play the single added note.

If you don't understand enough of how something as simple as a discrete versus continuously varying slide works to change the resonant frequencies of an instrument you should not be speculating about these things. Pressing a valve on a valved instrument adds a length of tubing causing the fundamental resonant frequency and its overtones to drop by very close to a half tone, with slight variations possible depending on the base pitch. A continuous trombone slide changes the resonant frequency distributed over seven positions (plus overtones), just as 3-valved instruments have seven unique valve combinations, except that a trombone slide can be in any position in between (within the resolution of the resonance peaks). That is how a trombone can 'smear' when the slide is moved more slowly. Many valved instruments also have krooks and triggers to impose slight variations in the valve tubing lengths invoked in real time coordinated with pressing the valves in order to compensate for slight deviations prominent for certain notes in the lower register (and which is why a french horn and a tuba have a fourth valve to obtain the proper half steps on the proper pitch).

One one will be found to be missing. Add that note to complete the reconciliation.

A fully harmonized and completed keyboard will be the result.

All existing music may be played from existing written arrangements without change.

Newly written music may include the missing/new note, and new graphic notation may be created. The labeling of the two repositioned notes on the piano keyboard will be changed in the written notation.

The result is that the new musical scales played on the piano seem more evenly spaced and more measured. To me, the sound of the scale is that a greater sense of gradualness is evident. Because of the change of the pitches the piano melodies no longer have the quality of discontinuity, and they seem more melodious. I believe that composers may benefit from the new scheme.

None of this talk about "reconciliation", "fully harmonized and completed keyboard", "scales played ... seem more evenly spaced and more measured", "a greater sense of gradualness", "quality of discontinuity", etc. has any objective meaning without defining your terms, explaining what you think you are doing, and showing how it is different from the current distribution of frequencies playable in all Western scales used today.

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[...]

I don't see why you are so negative. You apperar to want to diminish the discovery. If I have indeed made the discovery I claim to have made, why wouldn't that be a prospect that would enable a more refined music?

I presented a recipe for a result. Not a string of numbers mapped to pitch names for keys. A recipe that may be tried and demonstrated in order to discover for one's self the particulars of the new scale.

Why isn't that good?

I need to re-try the recipe and to validate once again the results. I hope that I haven't made an error.

I will, however, check out the books and computers. I checked with an electronic keyboard manufacturer, and their frequency generators to physical key locations are fixed by the hardware PC board. The computer SW approach may be a good idea.

If I find the time, I may be able to answer some more of your questions.

Thank you for your response.

Inventor

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Because there are only twelve half-tones in an octave, for the eight notes in the diatonic major scale each note is separated from the next lowest note by two half tones except for two of them -- the 3rd and the 6th -- which is why the number of sharps or flats designating a half tone change must change when the key is modulated and the scale starts in a different position of the 12 half-tone sequence.

Sorry, I counted these incorrectly. In the major scale the intervals are all full steps except for the half steps between the 3rd and 4th and between the 7th and 8th (not 6th and 7th). For example in C major the half steps occur between E and F and between B and C.

Also, to be more specific, the 12 note half-tone sequence (the chromatic scale) refers to 12 steps, not 12 notes; if you count all the notes in the chromatic scale for an octave there are 13 notes, again with the name of the top note repeating the name at the bottom. For example, starting at C the chromatic scale for one octave runs as C, C#, D, Eb, E, F, F#, G, Ab, A, Bb, B, C. Those are all a half step apart, with the frequency of each 5.95% percent higher than the previous note. There are no gaps or missing notes.

The comparison of the C major scale and the chromatic scale, showing where the half steps are, is:

C  C# D  Eb E  F  F# G  Ab A  Bb B  C 
C D E F G A B C

To change to any other key you slide the major scale to the starting point you want and adjust it to keep the two half steps in the right place. For example, the F major scale is:

C  C# D  Eb E  F  F# G  Ab A  Bb B  C  C# D  Eb E  F 
F G A Bb C D E F

Because there are identical half steps in the chromatic scale, with ratios of the frequencies all the same, you can always do this for any key and any scale. Scales other than the major scale simply use a different distribution of notes in sequence of increasing (or decreasing) frequency. The same holds for chords, in which different distributions of notes are played simultaneously for different "harmonies". Any chord and any scale can be transposed to any key because the frequency ratios between notes are uniform and there are no missing notes. To obtain more harmonies than are available under this system would require using different frequencies than are used in the modern Western scale, which would normally be perceived as dissonances.

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[...]

I don't see why you are so negative. You apperar to want to diminish the discovery. If I have indeed made the discovery I claim to have made, why wouldn't that be a prospect that would enable a more refined music?

Facts and established theory are not negative. You have not shown that you have made any discovery and have not described your claim in terms that anyone can understand what it is supposed to be. I don't agree that inserting an additional frequency, whose nature you have not defined, means a more "refined" music.

I presented a recipe for a result. Not a string of numbers mapped to pitch names for keys. A recipe that may be tried and demonstrated in order to discover for one's self the particulars of the new scale.

Why isn't that good?

Why is it good? Frequencies of the pitches define the notes. You cannot ignore that.Your claimed recipe is not a well-defined algorithm that tells us what you are doing. You speak of a need to calculate new frequencies for the notes, then contradict it by saying that they stay the same. You claim to have introduced an additional note without saying what it is. You say that this leads to a new quantity of 13 keys from C to C' in the 12 tone chromatic scale, but there are already 13 keys, with the top note repeating the name of the bottom note; there are 12 steps between the notes.

I need to re-try the recipe and to validate once again the results. I hope that I haven't made an error.

You didn't say what you did to validate it the first time. If you are introducing the insertion of a new note how did you hear it on a piano? Or have you not inserted a new note at all, only renaming the keys and shifting them across the entire keyboard? It is not at all clear what you are doing.

I will, however, check out the books and computers. I checked with an electronic keyboard manufacturer, and their frequency generators to physical key locations are fixed by the hardware PC board. The computer SW approach may be a good idea.

Electronic keyboards follow a standard called MIDI (Musical Instrument Digital Interface). There are such devices that are more flexible (in the name of "special effects") but it is easier and cheaper to use software to synthesize wave forms.

A well known and highly recommended free audio program (which I haven't used) is Audacity.

Two other very good ones I have used are Sound Forge, which was bought and further developed by Sony, and Cool Edit, which was bought and further developed by Adobe as Audition. They are much more expensive now, costing many hundred of dollars, but there are older editions of Cool Edit still available for free at http:/oldversion.com. They also have many versions of Audacity.

For explanations of audio editing software try Derry, PC Audio Editing from Broadcasting to Home CD; 2000, Riley, Audio Editing with Cool Edit, 2002; Garrigus, Sound Forge Power, 2002; or newer books you may find on amazon. These books on audio editing in software do not include the basic theory of musical scales and musical instruments I cited earlier.

If I find the time, I may be able to answer some more of your questions.

Thank you for your response.

Inventor

I think that your time would be better spent for now on first understanding the elementary theory behind scales and chords, formulating in such terms more objectively what you are trying to do, and learning and experimenting with basic dsp software for audio editing. Then you can report back to us.

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There was a great TV special on The Learning Channel (here) that talked about common occurance of a number called the golden section, or the golden ratio in things that are generally accepted are visually appealing. Their examples range from faces, to architecture, to credit cards, to shells. There are also many other occurances and methods that this number pops up in.

I was interested on what peoples opinions were on the idea of esthetics being quantifiable. Generally esthetics is often ascribed personal opinion to a large extent, but if it is in fact quantifiable, personal opinion drops as a criterion to judge beauty and beauty is no longer in the eye of the beholder.

The Golden Ratio is the positive solution to the quadratic equation arising from the ratio 1:x :: x:(x-1) and is (-1 + sqrt(5))/2. This ratio pops up in various length ratios in the human skeleton (and some other mammalian skeletons too). I have a hunch why this keeps happening. It so happens that the characteristic equation of the Fibonnaci Sequence is non other than the above quadratic x^2 -x +1 = 0. The Fibonnaci Is ubiquitous in physical processes.

See the article in Wikipedia on the Fibbonacci Numbers.

http://en.wikipedia.org/wiki/Fibonnaci_Seq...mbers_in_nature

Bob Kolker

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