Wild Surmise

_____________________________________________________________Number 29 January 1999

An Almost Anonymous Informal Note

Sub Human Oinks and Grunts

Recently, coming through the great international airport near Paris, we had just unloaded our bags from the push cart and checked them through. I saw a place nearby to park the cart, and there were also two soldiers nearby with light machine guns. I had come to expect to see men with machine guns in Europe in places where American security forces would have carried pistols; recently more Europeans are carrying pistols. I think they are getting to be better shots or something. Anyway, I saw the chance to examine their weapons more closely if I took the cart over to the place it seemed to belong, so thither I went.

As I put the cart up, the two men started toward me. They passed no visible signal, moving if they were a single organism. I could see their brutally efficient weapons quite nicely now, thank you. Bolting did not seem like a promising option, so I simply turned back the way I had come just as the two came within grabbing distance. They separated slightly, one passing on one side of me and the other on the other side. They then closed rank and continued sauntering on their way.

At that moment, one of them hummed three notes. The little tune was the one we usually spell, "La dee dah." The communication was quite subtle. At the simplest level it meant, "We are completely relaxed and at ease; everything is just fine." Below that was the meaning, "Scared you, didn't we." And below that was, "We saw you pretending to have a chore that would let you sneak over and peek at our guns. You didn’t fool us a second. You are welcome to look at them closer. But we thought we'd tease you just a little, no offence intended." And yet below that, "You're American, aren't you?" I think that's not bad for three notes.

Perhaps I have not been alert, but it is the only time I have ever known a European to use the very characteristic American trick of talking with a tune.

Consider the notion that these are not words. You cannot really spell them, although we have traditional spellings. It is the tune that counts, not the sounds themselves. He hummed the tune. It would be spelled,
"Mmm m mmmm." Than means exactly the same as, "La dee dah." But the letters are different. With real words, the letters that is to say the sounds, chosen do matter. "Men" and "hen" mean entirely different things. "Mat" and "hat" are different. It makes a difference whether you start the word with your mouth open or your mouth closed. But for the tune, it does not matter.

The la dee dah tune of course carries a double meaning anyway. There is always an element of irony, a sense of mocking or gloating to it. It is an assertion of innocence while at the same time boasting of anything but innocence.

Another thing to notice is that the tune does not translate into the ordinary well-tempered chromatic scale that we use for our regular music. On the keyboard, the notes would be approximately, "E, F, D" with three counts on the first note, one on the second and four on the third. But that is only very approximate. The notes are not really pure tones. There is a little quaver on the first. The second rises slightly during its brief duration and the third trails off downward.

The well-tempered scale is an invention that was made popular by the great composer J. S. Bach. Here is my understanding of what they did.

Let us say that the note A above middle is the sound made by a string vibrating at 440 cycles per second. Now to my crude ear, throughout my life music, nominally the same music, has been played faster and faster at a higher and higher pitch with less and less expression. But no matter. Let us take as an assumption that the 440 cycle A is blazoned in letters of fire on some remote mountain, has always been there and always will be.

Now by venerable convention a string vibrating at twice as fast - 880 cycles per second - is "the same note an octave higher," and you will find a note on the keyboard that is approximately so. Be warned, though, that most notes in the piano are represented by more than one string, and those strings are tuned slightly differently. Being tuned differently, they do not vibrate exactly at the same time and rate, not exactly in step. If they did so vibrate, they would transmit their energy efficiently to the sounding board and the volume, the loudness of the note, would be a lot higher at first but decrease very rapidly. Also, some piano tuners will add excitement to their work by deliberately making higher octaves slightly higher than they ought to be. The description of how to tune a piano occupies an entire volume. For now, assume that A is 440, or double that or double that or, it is 220 or half that or half that.

There are twelve notes in the well-tempered scale. We will call them: A, A#, B, C, C#, D, D#, E, F, F#, G, G#. Don't worry. For now those are just some names; most people will put the sharp in front of the note as a courtesy to the performer. Now here is the rule. Each note has a frequency equal to the note just to its left times 1.059463 and a fraction. So if our first A has a frequency of 440, then the other notes have frequencies of:
 

Notice that we are a little off. The A we calculate from the G# is about nine vibrations per ten thousand seconds from what it ought to be. That's about one vibration per twenty minutes. Working out that fraction would take care of the difference, at least in principle.

Now music, Oriental as well as Western, is based on the convention of the "cycle of fifths." "Fifth" is just a name, like "hundred year old eggs." A fifth is an interval of seven "half tones," a half tone being like the interval between A and A#. The cycle of fifths goes: A, E, B, F#, C#, G#, D#, A#, F, C, G, D, A. Now the rule for the frequency of a fifth is: the higher (frequency) note is 1.5 the lower frequency. And if you go through the cycle you get back to A. Really?

Let's try: A has a frequency of 440 cycles per second. E, which is one "fifth" or seven "half tones" (or semi tones) above A. So E must be turned at a frequency of 660.

Before working out the rest of the notes, I hear the mutter, "Who cares?" Indeed there is something arbitrary about taste in music. The ancient Greeks distinguished between Apollonian music and Dionysian music. The names are from Greek gods and carry overtones of prefixes for "apart from" and "going through." Apollonian music is measured, stable and characteristically employs stringed instruments. Dionysian is more frenetic, insistent and early was characterized by the use of reed instruments.

A note is a sound with one lowest frequency vibration called the tone and a number of higher frequency vibrations called overtones. Dionysian music has more overtones. Tastes seem to run to more and more overtones, so that I eagerly await the release of "Concerto for Automobile Accident and Fingernails on Blackboard." However blackboards have changed, too, and I may be disappointed.

If you pluck a string on an acoustical guitar, it will vibrate at a rate determined by its length, mass per unit length and the tension on it. Keeping tension and mass per unit length constant, the rate of vibration is inversely proportional to the length; if you double the length then the frequency will fall by one half and the note will be an octave lower. Reduce the length by half and the frequency will double so you hear a note an octave higher.

Changing the length of the string changes the tone. But there are also overtones. To hear the first overtone, put the tip of your finger against the exact center of the string. The tone will die out at once; the center of the string must be moving to produce the tone. But you will hear a tone an octave higher. Each half of the string now vibrates, and of course the length is one half so the frequency is doubled. The two curves in the two halves of the string make an S shape to the work together.

Now pluck the string again and touch it exactly one third from either end. The string now vibrates in three segments so it is vibrating three times as fast as the original tone. This is the second overtone. But to keep things in the same octave, we divide the frequency by two and get "the same tone an octave lower." Well, the segment vibrating at the frequency of a fifth above the tone must itself have a first and a second overtone, the second being a fifth above a fifth above the tone. Continuing in this manner (and then lowering the tone into the proper octave be dividing by the proper number of 2's, we get the cycle of fifths.
 

Or, arranged on the scale:
 

Notice that the numbers are off. The notes as described by the cycle of fifths are different from the notes described by the "well tempered" scale in which each semi tone has a frequency of 1.059463 times the note next it. Neither comes out exact at the octave.

Well what would come out exact? The original tone is taken as a given. The first overtone is exactly an octave higher. The second overtone, defined by the string vibrating in three segments, we will take to be the fifth exactly (once we have divided by two to bring the frequency back to the same octave). The third overtone is the string vibrating in four segments with a frequency of four times the original tone. Divide by two twice to get back to the original octave, and we have the original tone.

The fourth overtone is the string vibrating in five segments, which is five times the frequency of the original tone. Divide by two until we are in the original octave and we have a frequency 5/4'ths of the original. If we started with a tone of 440, the fourth overtone has a frequency of 550. That is rather close to the frequency of 554.3651 called for as the note C# on the well-tempered scale. Similarly the second overtone: 660 is quite close to 659.2547 on the well tempered scale for the note E. And now if we play A (which is the tone as well as the first and third overtones), C# (which is the second overtone) and E (which is the fourth overtone) that is to say if we play the tone and first four overtones A, C#, E we get a very pleasing chord. Well I say we are on to something. I say the notes of the scale are approximations of the natural frequencies of a vibrating string. That is why they sound somehow "right." And if that is true, there is no limit to the notes we can legitimately add to our scale. The well-tempered scale and the cycle of fifths are only approximations based on theories that limit the number of notes we consider.

Well now what are the overtones of the note A? For each tone and overtone, how many segments vibrate, at what frequency does each vibrate, and what would be the frequency of each after being brought back to the original octave by dividing by the required number of 2's. We will calculate up to 32 vibrating segments, but there is no reason to stop there.
 

You can tell at a glance that all overtones below the 15^th are repeated between 15 and 30, so we will take our "natural" overtones as 15 to 30 and compare them with the well-tempered scale.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

We do not, unless we are being deliberately unkind, use all the notes in the well-tempered scale at once. In the key of A we use A, B, C#, D, E, F#, G#. Rounding off (my ear can not detect one cycle per second unless there is a note to compare it with) we get:
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Note Frequency Nearest harmonic difference

A 440 440 0

B 494 495 1

C# 554 550 4

D 587 578 9

E 659 660 1

F# 740 742 2

G# 831 825 6

So except for that D, the errors are less than 1 percent and a better approximation than the cycle of fifths. I submit for your consideration that the well tempered scale is an approximation of the cycle of fifths and the cycle of fifths is itself an approximate way of selecting some of the natural frequencies of the vibrating string.

What of those other frequencies? Let's see.

440 = A

468

495 = B

522

550 = C#

578 = D

605

632

660 = E

688

715

742 = F#

770

798

825 = G#

852

Obviously we have omitted a lot of tones that legitimately belong in the key of A. Notice that the first six in our list of overtones account for four of the notes in the scale, while the next ten account for only three. The lower end of the octave more nearly approximates the natural overtones.

Perhaps it is the missing overtones that have prompted us to use the minor scales. Minor scales that are related to the key of A are the harmonic and melodic minors F#. They include the notes F and F and D# respectively. F has a frequency of 698 on the well-tempered scale, which approximates 688 on the natural scale. D# is 622 on the well-tempered scale or close to the natural harmonic 632. Our list of overtones now looks like this:

440 = A

468

495 = B

522

550 = C#

578 = D

605

632 = D# difference is 10

660 = E

688 = F difference is 10

715

742 = F#

770

798

825 = G#

852

The two notes we added to get the minor scales are a minimally worse approximation of natural overtones than D. They are too far from natural overtones to be approximations. This approach does not seem to work.

Well what about the A minor scales? They include the notes A, B, C, D, E, F, F#, G, G#. C, F, and G are the notes that were not in the major scale. We have already noted that F is a poor approximation of an overtone of A. C has a well-tempered frequency of 523. That is gratifyingly close to the natural frequency of 522. G has a frequency of 784, which is right between two overtones.
 
 
 
 
 
 
 
 

440 = A

468

495 = B

522 = C

550 = C#

578 = D (sort of)

605

632

660 = E

688 = F (sort of)

715

742 = F#

770 = G (sort of)

798

825 = G#

852

Missing from the well-tempered scale is now A#, with a frequency of 466, which is close to 468, and D# with a frequency of 622, which is nearest to 632 but not very close.

440 = A

468 =A#

495 = B

522 = C

550 = C#

578 = D (sort of)

605

632 =D# (sort of)

660 = E

688 = F (sort of)

715

742 = F#

770 = G (sort of)

798

825 = G#

852

So at the end of the day, it is really delightful how well the well-tempered scale does approximate the natural harmonics with eight pretty good matches and four bad one that are mostly excluded anyway. On top of that, the composer can instantly change keys just by choosing different notes, and it is just as easy for the performer. It is small wonder that the well-tempered scale dominates western music.

Yet it also seems clear that the well-tempered scale is not arbitrary, but is an approximation of natural tone relationships that the ear can recognize and be comfortable with.

And there are missing frequencies. There are four notes in this set of overtones that are simply not represented on your keyboard, each of them more legitimately in the scale of A than is D, not to mention the three other bad matches. Counting the bad matches, that is eight missing tones. This is a lot, since there are only seven tones, six good ones, in the major scale. More notes are missing or misrepresented than are present. In other words, the natural harmonics should give us a scale of 16 notes. The well-tempered scale gives us six good notes and one poor one. There obviously is a lot of music missing.

Of course given modern computing techniques, changing keys becomes effortless. Adding new frequencies could be done easily. A far more rational, far more harmonically consistent and melodically sweet music could be made. But tastes run the other direction.

The point is that there are missing notes. And I suspect (although my ear is not good enough to be sure) that the tune "La dee dah" exploits those missing notes.

There are other tunes.

Julius Caesar, not one of the must cuddly of humans, was writing about a tribe that he was going to fight. He said they would discuss things and after each speech they would vote. Those in favor signified by cheering and clashing their armor while those opposed would moan is if in pain. The same tradition is still in evidence among small groups of young men.

I think two of the tunes we use may date back to those times; on is usually spelled "uh huh" meaning yes and the other "uh uh" meaning no. Again, the actual sound and spelling mean nothing. It is the tune. The uh huh tune is a pair of notes, the second about a semitone above the first and sounded with a little be more release of air. The uh uh tune is a pair of notes, the second about a semitone lower and no difference in the amount of air released. It is often spelled ah, and serves as a warning, but it still means no.

Another tune, generally spelled huh, is a single rising note. It means, "I don't understand that, I am surprised by that, I did not expect that."

Also the same spelling may be sounded as a level note in order to mean, "I am more surprised than pleased."

With less release of air, the same tune is spelled "uh." It means, "I find this disgusting, painful or a matter that requires effort."

Much the same tune, spelled "aw," means, "This is painful for one of us, and the other should sympathize." The pitch starts high, drops slightly and then goes back up.

Another tune starts lowish, goes up three semitones and then down one. It is spelled, "I dunno." It means, "I neither know nor am responsible for knowing."

Then there is what I take to be something called the "cat call." If I am correct, it is spelled nyah, nyah, nyah. It is six notes sounded to an eight count, one count to each of the first four and two to each of the last two. The notes are about: G, G, E, A, G, E with to my ear the A and last G being a little sharp. The meaning is, "You ought to be ashamed of yourself, and your embarrassment gives me delight."

Laughing and crying are done to the same tune. The tune is a series of notes at about the same pitch sounded at regular and rather quick intervals with an occasional note sustained up to four beats. The difference is whether the corners of the mouth are pulled up or down. You can do it with your fingers. Next time you are laughing or crying, try pulling up and then down on the corners of your mouth with your fingers, and see if it works. Try not to let anybody see you at it.

Since the position of the mouth, not the tune, carries the sense of laughter and crying, they are actually closer to being words than the other tunes.

A long sustained tone, spelled hum, means, "I am thinking about this." Sustained longer and spelled hmmm it means, "And having considered the question, I am happy, comfortable and affectionate."

There is a tune that starts with two counts on the same note, goes up a fifth for two counts and then down a minor third (three semi tones) for four counts. Sort of C, G E. This tune does not fit in the well-tempered scale. In fact, it is the greeting horn-tune in the Kirk Douglas movie The Vikings. It can be used for "be careful" or "I warned you." It carries also an overtone of, "I know what I am talking about and am giving you valuable information, but I am not sure you are showing proper respect." It can also be used as a greeting, as in the movie, or farewell or even "I love you," but still with a premonitory overtone. It is very similar to the tune for, "I don't know."

There is another tune, this one I am sure I have heard outside the United States, that starts high, may or may slide up a semi-tone and then slides down about two semi-tones without a specific count. It means approximately, "There is a ghost near here; in fact that might be it now." I have never heard it used seriously.

There are other classes of sound. There are sounds that look and sound like words, like the Swedish "ja, so," but don't really mean much of anything. Some sounds imitate animals, like the growl of a dog or the hiss of a snake to indicate displeasure. And many languages have characteristic sounds. I understand a quick grunt in Japanese means, "I am paying attention," and air sucked in through the teeth means, "You are asking a lot." But in these cases, it is the sound rather than the tune that carries the meaning.

Of course language has a music of its own. I once overheard a conversation between a man and a woman. The circumstances were such that it was proper for him to introduce himself. He spoke with the tone and cadence of a cross cut saw going through a sheet of plywood.

"Vair har yoo vrom?"

She replied in the same tone, "High ham vrom dGermany."

He then changed languages and said, "We could always speak German" in German. The striking thing was the change in the music of the language. Although he was now in a different language, his tone - that is his tune - was exactly what I might have used saying it in English.

If you have a vocabulary of a few thousand words, you can probably hold down a minimum wage job. And if you have a vocabulary of a quarter of a million words, either money is not the primary purpose of your work or you are probably being paid well indeed. So a word is probably worth a dollar a year to you, and the effort to learn a word is probably worth ten dollars. It is worth ten dollars to you whether in the end you sell it for money or use it for some noble end.

The down side of this is that, like the value of securities and even cash, the value of your vocabulary can decrease because of the actions of other people. For instance, there seems to be movement the change the term BC, historically meaning "before Christ" but actually simply meaning counting backwards from the year one on the calendar, to change the term to BCE meaning "before common era." There must be 200 million Americans who know what BC means. Dropping the term then will cost them two billion dollars; it's sort of like the government printing money at a rate that causes the cash in your pocket to lose value.

Another caution is that you can learn a lot of new words by studying a foreign language. I think that is a good thing to do to a certain extent, but studies have shown that it results in a cost in your own language. Bilingual children on average are not as good in either of their languages as their monolingual age mates. And just about anyone who speaks seven languages well will have an accent in every one of them.

That, of course, is no reason not to learn and grow.

Learn many words.

But in the end, few words will be as useful to you as a grunt.
 
 

Booty
 
 
 
 
 
 

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Ó Copyright December 1998, Wild Surmise

Ed

Mild Surprise

The Druid gestured we should gather on the hilltop and he pointed to the plain beneath. "Three thousand years ago, there was a lake there. The Celts who lived there built their houses over the water on pilings. There was a walkway from the shore to the village, but the walkway was slightly below the surface of the water. It did not go straight, so you had to know the way or go with great care to keep from falling in."

Communal living, major capital investment and an interest in mazes sounded more megalithic than Iron Age. And I thought the Celts arrived in Britain about twenty five hundred years ago.

He continued. "Tor hill. When the Celts looked up at it, they believed that inside was the palace of Gwyn ap Nudd, king of the fairies. Each night, just at midnight, he and his ghostly knights would burst out of the hill and scour the countryside for mortal souls to steal." Now that sounded Celtic.

That evening, everyone seemed to turn in early. Perhaps they were eager to lie asleep dreaming of demon steeds, huge eyes an opalescent blue-green in the dark and orange phosphorescence flaming from the mouths, dreaming of fairy knights outlined by moonlight, of the king clad in magic and armed with a soul hook.

I was not ready to go to bed.

My last boon companion pushed back his cider and left me with only the night and the directions from the Druid how to reach this side of the Tor. There did not seem to be any harm in just starting out. I mean it was just to see if the directions really worked … for the first few steps of course. I would not get into anything spooky.

The resolve to avoid anything spooky did not last. The first direction was to walk out the front of the hotel. That meant crossing the length of the hotel. The hotel was very, old and the floor had been laid out, not with any architectural prejudice as to where it ought to be, but simply in conformity with the hill on which the building grew. The hillside had been paved with the stones I now trod. It was a path of unthinkable antiquity.

Outside one turned left and walked to the top of the town. The street was dark and empty although on all sides there were sounds of merry making and carryings on in what must have been many pubs I never noticed

The air was cool, and the road went up hill at a nice slope. The road soon ended at what was arguably the edge of town. The directions were to turn right and walk to the Chalice Well. The high road was not perhaps so dark as the town but more deserted. Traffic was light and fast and had a rather indifferent feel to it after the cozy mirth of the young people I had been hearing.

The Chalice Garden, special we were told for women, was closed for the night. Water from the spring was ducted through a pipe through the wall and fell a couple feet into a drain. You could drink your fill or fill your vessel easily. I stood there measuring my next move. "Then you go up the hill," were the instructions.

The way was easy to guess. At the corner was a copse of densely packed trees, and the opening into them led into a tunnel of utter blackness. Out of the tunnel came the sound of running feet and of male voices shouting encouragement to each other. With a cheer two stout lads burst from the dark into the relative clarity of the gloomed street. Neither glanced my direction, but a mumble passed between them and they agreed, in young male talk, on two facts. They hadn't been scared, and they hadn't been seen.

I went to the opening. It was very dark. I had not brought a flashlight. Still, if two teenagers had come out at a gallop without falling, I supposed I could make my way through walking heel and toe. Any time orientation is an issue, it is best to look backwards frequently, to see what it will look like on the return trip, or - if one is returning - to be sure the way back looks like it did when it was the way forward. So I am fairly confident that the light from the street vanished at just about the time the first time light could be seen coming down from sky in the direction of the hill.

I emerged on the gentle grassy flank of the Tor.

It was another world. The street lights and traffic sounds were gone. The world was lit in shades of silver. The wind was raw enough so I happily tightened my shoulders under the coarse wool of my tweed jacket. And someone must have been singing at the top of the hill, because you could hear voices in a high happy wail coming from that direction.

The path was steeper than the street had been, but it switched back and forth in a way that kept the slope from ever being a problem. The greater work just balanced the greater cold. At last the path rounded a bend, and above a shoulder of the hill you could see the tower that stands on top of the Tor. It is a ruined tower, once part of a church. And I suppose the place has been a place of interest for a long time.

There were people up there. You could see light against the side of the tower. They were shining a flashlight on the stone wall, probably to make shadow pictures. That would be fun, but they had better be careful. The light from the flashlight was orange; the batteries were almost gone.

I continued to climb deliberately, looking out toward the Severn over fields and meadows where much of history and much of legend had happened. Out there was the Chalice Hill, and past that Wearyall. The path straightened and began to climb right along the backbone of the Tor toward the tower. It was farther than I had thought. I doubt that I heard voices from the tower before. It was too far away, particularly against the wind. Probably it was just the wind.

Half way along the ridge, you can see they have the flashlight on again. But the color is different. No, it is not a flashlight. There is a big hole in both sides of the ruined tower. That is the sky showing through. There is not enough light to see detail, but you can make out the silhouette of anything outlined against the sky.

For instance, I am about forty yards from the tower when a figure leans out around one corner and looks toward me. I make it to be a male, medium height, lean, teen age by his quickness and lightly dressed in spite of the cold. He looks my way a moment, and then his shadow vanishes into the shadow of the tower.

No one greets me when I reach to top. Curious, I circle the brow but there is no shadow descending in any direction. The stones of the old church could be hiding someone, but I doubt it. It would take the utmost care to be that quiet; and one would be cold very quickly holding so still on such a night.

I gazed into the night until my temperature fell to the lower edge of my comfort range. I started down.

I checked backwards as well as forward until the tower vanished for the last time behind the shoulder where it had first appeared. At last I could see the path clear all the way to the tunnel through the trees. I looked back.

The light was over the path about twenty yards behind me. It was the same orange I had seen against the tower, like a flashlight on its absolute last legs. As I watched, it was totally motionless. But if it had been following me down from the tower, it must have been moving at great speed. None the less, I am sure I reached the street before it could have overtaken me.

The Chalice Well whispered encouragement. I thrust in my hand to get a drink. I noticed my watch. 11:59:59.

I made delay to see if teenagers would emerge from the tunnel. They did not. Perhaps my enigmatic lone form had spooked them. People do get superstitious, you know. I am sure they have a wonderful story to tell of their adventures that night on the Tor.

I only hope they do not have to tell it before the court of Gwyn ap Nudd.

M