Harmonics Music Pythagorus Universe

8/14/2019 Harmonics Music Pythagorus Universe

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Harmonics, Music, Pythagoras and the Universe

The following is text has been produced with permission of the participants fromconversations that took place in the group Alexandria City in August 1996. Thediscussion group is based on the city of Alexandria and so is a centre for anything

relating to philosophy, history, music, mathematics or cosmology with particularreference to Greek knowledge. The main participants in this particular discussion werethe listmistress of Alexandria, Cynndara Morgan, Andrew Green, Joseph Milne, and me,Ray Tomes. Several others also participate.

This first part contains my presentation and the following 3 pages have the discussion.

Ray Tomes wrote:

Harmonics, Pythagoras, Music and the UniversePart 1: Musical Background

After researching what notes sounded pleasant together Pythagoras worked out thefrequency ratios (or string length ratios with equal tension) and found that they had aparticular mathematical relationship.

The octave was found to be a 1:2 ratio and what we today call a fifth to be a 2:3 ratio.Pythagoras concluded that all the notes could be produced by these two ratios as(3/2)*(3/2)*(1/2) gave 9/8 which is a second and so on.

The problem was that after applying these ratios repeatedly he was able to move throughthe whole scale and end up back where he started... except that it missed by a bit, calledthe Pythagorean comma. After twelve movements by a fifth (and adjusting down anoctave as required) he got back to the same note but it had a frequency of 3^12 / 2^19[Note ^ means to the power of] which is 1.36% higher in frequency than it should be.

Although Pythagoras did a wonderful job he did get it slightly wrong. The correctsolution was worked out by Galilei (the father of the famous Galileo Galilei) whoconcluded that the best frequencies were in the proportions

do re mi fa so la ti do1 9/8 5/4 4/3 3/2 5/3 15/8 2

Which may be represented as whole number proportions as24 27 30 32 36 40 45 48

These proportions are called the Just Intonation music scale and are the most pleasingproportions for note frequencies for any one key. The differences from Pythagoras aresmall, so that mi is 5/4 (=1.250) rather than 81/64 (=1.266).


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It is interesting to look at the ratios between the notes. do-mi-so are 24-30-36 which cancancel to 4:5:6. This same proportion links the notes fa-la-do which are 32-40-48cancelling to 4:5:6. Again, so-ti-re (re from the next octave) gives 36-45-54 whichcancels to 4:5:6 again. So every note is linked to "do" by three major chords which haveratios of 4:5:6.

However when music contains modulations, that is, changes of key, then some of thenotes need to change frequency. As many instruments cannot do this it was necessary tomake a compromise. Many systems were developed for this compromise and it is calledtemperament.

Instruments such as pianos, guitars and trumpets have fixed frequencies while violins andthe human voice can vary to any note required.

An example of a chord which requires a change is re-la which have 27-40 above. Thisneeds to change to the ratio 2:3 so either the 27 must become 26+(2/3) or the 40 must

become 40+(1/2). Human voices and string quartets do this adjustment automaticallybecause they listen for the harmony. Guitars and pianos just cannot do it hence thecompromise.

Bach popularised a system called "equitempered" which is used almost exclusively today.It is a compromise between all keys and uses a common ratio between every semitone of2^(1/12). This gives frequencies of:

equitempered 1.000 1.122 1.260 1.335 1.498 1.682 1.888 2.000

just int. 1.000 1.125 1.250 1.333 1.500 1.667 1.875 2.000

which are nearly right as you can see. Bach popularised this tuning by some very cleverpieces such as the well-tempered clavier and so on. As pointed out to me by a friend, this

piece is full of musical puns. In fact many times the puns have three possible meanings.My friend was reduced to rolling about the floor laughing when he attempted to playguitar chords along with a piano playing this piece.

Pythagoras and his followers and later Kepler were to consider that these musicalrelations or harmonies had wider application in the universe. This idea was almostforgotten or dismissed for many centuries. However I will hope to show you that there ismuch evidence that the universe is completely organised on a system of mathematicalharmony and that it shows up in every branch of scientific study.

Part 2: Cycles Background

Back in 1977 I was using computers to try and predict various economic variables forcorporations in New Zealand. In the course of doing this I found that many aspects of theeconomy showed quite clear cycles. After designing a method to search out the mostconsistent cycles they turned out to be ones with periods of 4.45, 5.9, 7.15 and ~9 years.These worked well for making forecasts.


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After a while I noticed that the periods that I was using were all very near exact fractionsof 35.6 years. Also, other cycles existed at other fractions of this period such as ~12 yearsand a fraction under 4 years. The literature showed that there were other shorter cyclesknown as well as longer ones. I acquired some weekly data to look for shorter cycles andfound that there were similar patterns at shorter periods and that often they had

proportions of 2 and 3 in them.

Then it struck me. These fractions of 35.6 years were in fact frequencies of 4:5:6:8 whichis exactly a major chord in music. Also, the shorter cycles turned out to be exactly in theproportions of the just intonation musical scale plus a couple of back notes (E flat and Bflat if we are in the key of C).

35.6/8=4.45 35.6/6=5.93 35.6/5=7.12 35.6/8=8.9 years

I realised that the Kondratieff cycle of about 54 years also fitted in that 2*54 is very nearto 3*35.6.

There was of course the question "Why 35.6 years?" and the answer almost surely hadsomething to do with causes from beyond the earth. For Jupiter's orbital period is 11.86years which is very close to 35.6/3 and the node of the moons orbit takes 8.85 years totravel once around the earth. There are other astronomical periods which fit also.

This was very weird and for some time I didn't tell anyone because I was sure they wouldthink I was weird. However, I heard about a place called the Foundation for the Study ofCycles in the late 1980s and visited there in 1989.

Edward Dewey had formed the Foundation in about 1940 and had unfortunately died

before I got there. He had left behind an enormous legacy of research into cycles. In oneof his articles I was to find the following diagram. Dewey found many relationships withproportions 2 and 3 in cycle periods starting from a period of 17.75 years, in an enormousvariety of different time series. His table of periods in years is:-

142.0 213.9 319.5 479.3

-----

71.0 106.5 159.8

-----

35.5 53.3 x2 x3

---- ---- \ /

17.75 \ /

-----

5.92 8.88---- ---- / \

1.97 2.96 4.44 / \

---- ---- ---- /2 /3

0.66 0.99 1.48 2.22

---- ---- ----

0.22 0.33 0.49 0.74 1.11

---- ---- ---- ----

Underlined figures are commonly occurring cycles.


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Interestingly Dewey, using data from different countries, different time periods anddifferent fields of study had arrived at a table which included a very good match to myfigures. There was 35.5 years looking at me along with 4.44, 5.92 and 8.88 years.Although this table didn't show 7.12 years, his catalogue of reported cycles showed aclear concentration of reports at this figure.

The above table shows several of the periods, such as 142, 53.3 and 17.75, 5.93 years,similar to those found by Chizhevski in the cycles of war, namely 143, 53, 17.7, 6.0years. However it doesn't show the 11 and 22 year cycles and some others. To find theseit is necessary to introduce a ratio of 5 just as was done by Galilei to Pythagoras' musicscale. For 22.2 years is 5 times 4.44 and 11.1 is 5 times 2.22 years. When the aboveperiods are multiplied by 5 they also produce many other commonly reported cycles suchas 178 years which is found in the alignment of the outer planets, in solar activity and inclimatic variations.

It is worth mentioning that these cycles have been found in every aspect affecting life on

earth. Wars, economic fluctuations, births and deaths, climate, geophysics, animalpopulations, social variables, stock and commodity prices. We literally live inside a giantmusical instrument which is playing notes, chords and scales in such slow motion thatonly the Gods could hear it.

Dewey wrote a very touching piece late in his life where he likened himself to TychoBrahe who gathered and catalogued the information about the planetary motions. He saidthat he had so wanted to solve the riddle but was then very old and knew that he wasleaving it for some later Kepler to explain.

In my next post I will stake my claim to being the Kepler or the Newton of Cycles and

you can be the judge.

TURN! TURN! TURN!

Words: Book of Ecclesiastes

Adaptation and Music: Peter Seeger

To everything (Turn, Turn, Turn)

There is a season (Turn, Turn, Turn)

And a time for every purpose under heaven.

A time to be born, a time to die;

A time to plant, a time to reap;

A time to kill, a time to heal;

A time to laugh, a time to weep.

A time to build up, a time to break down;

A time to dance, a time to mourn;

A time to cast away stones,

A time to gather stones together.

A time of love, a time of hate;

A time of war, a time of peace;


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A time you may embrace,

A time to refrain from embracing.

A time to gain, a time to lose;

A time to rend, a time to sew;

A time to love, a time to hate;

A time for peace, I swear it's not too late.

Part 3: The Harmonics Theory

After finding that Dewey had observed very similar cycle periods and also the samemusical/harmonic relationships between the cycle periods to those that I had found Iknew that it was real effect and not some delusion on my part. I found several othersources for similar observations and in all cases they fitted the same periods. Thequestion to be answered was why?

The musical relationships are characterised by the quality that there are as many small

number ratios between the frequencies of notes as possible. This indicated that the causeof it all was related to the formation of harmonics. The word "harmonics" has a slightlynarrower meaning in physics than in music and means "frequencies which are a multipleof some fundamental frequency".

It is well known in mathematics/physics that a non-linear system will develop harmonics.Non-linear simply means "not exactly proportional". For example gravity is non-linearbecause it is not proportional to distance. In the real world almost everything is non-linear.

To begin with I assumed that some long period cycle existed in something and then

looked at what would happen as that something affected other things. The universe is fullof ways for things to affect each other and I was not concerned with the details, just thebroad idea. I quickly proved that an initial long cycle could only ever produce othercycles which were harmonics, that is, had multiples of the original frequency or fractionsof its period. That was fine, but it could produce any harmonic, not just the observedfavoured ones of 2, 3, 4, 6, 8, 12 etc.

Perhaps I should say something about the use of the word "frequency" for cycles. Usuallywe use "period" for long cycles (as in period of 5 years) and frequency for short ones (asin frequency 440 cycles per second) but it is equally valid to say a frequency of 0.2cycles/year or a period of (1/440) second. Frequency is used here because it makes the

maths much simpler.

Consider an initial frequency 1 in such a system. It will generate harmonics offrequencies 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, etc.

Now consider each of these frequencies in turn. They will each create harmonics ofthemselves which will be frequencies of:

1 --> 1 2 3 4 5 6 7 8 9 10 11 12 13 ...


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2 --> 2 4 6 8 10 12 ...

3 --> 3 6 9 12 ...

4 --> 4 8 12 ...

5 --> 5 10 ...

6 --> 6 12 ...

7 --> 7 ...

8 --> 8 ...

9 --> 9

10 --> 10

11 --> 11

12 --> 12

13 --> 13

etc

Now what is immediately obvious here is that some frequencies are produced in manymore ways than others; 4, 6, 8, and especially 12 are produced often while 11 and 13aren't.

The number of ways each number can be factorised is a measure of how much power wecan expect to find in that harmonic (after allowing for the general drop-off in power for

higher level harmonics). It turns out that when the spectrum of this function is examined(AT ALL SCALES) it produces strong frequencies which have relationships exactly inthe proportions of major chords in music, and moderately strong frequencies in exactlythe proportion of the musical scale (the old just intonation scale, not the modernequitempered scale). An example is the range of harmonics from 48 to 96 shown belowwith relative power after allowing for the drop off with higher harmonic number.

I I I

I I I I relative

I I I I I I I I power

I I I I I I I I I I I I I I

I I I I I I II I I I I II I II I I I I I

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII48 60 72 96


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single initial frequency will generate harmonics AND EACH OF THESE WILL DO THESAME. Please excuse the caps, but that is the important bit.

What then is the longest cycle? I already knew that there were some very long cycles like2300 and 4600 years in both climate and astronomy, but also the Milankovitch cycles of

100,000, 40,000 and 25,000 years which relate to the earth's orbit and axis and alsodetermine ice ages. But not so long ago someone reported a 27,000,000 year cycle in theextinction of species and geologists find even longer cycles.

This all seemed to be leading towards a conclusion which I initially joked about and thenfinally embraced; the fundamental cycle was the cycle of the universe!

I had a false start in trying to calculate the very large harmonics and at one time had to goback 2 years in my research and do it again. However what came out of that is therealisation that there is an especially important harmonic which is 34,560. This number34,560 is 2*2*2*2*2*2*2*2*3*3*3*5 and you can see why Pythagoras and Dewey

found lots of 2s and 3s but only Galilei found the 5.

The harmonics theory predicts that, compared to the entire observable universe taken asthe fundamental oscillation, the 34560th harmonic will be an especially important one. Italso predicts that at further ratios of 34560 in size there will be important oscillations andsizes.

To understand how harmonics divide space as well as time, consider a stringedinstrument. It can oscillate at a fundamental frequency which has just one wave in thestring. It can also oscillate at the 2nd harmonic. In that case both the length of the stringand the time of the oscillation are divided by 2. Likewise, if we could get the 34560

harmonic going in the string it would divide both the length and oscillation period by34560.

ratios

V V V V V V V V V V


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picture of the universe is very similar. The standing waves are electromagnetic waves(which means radio waves, light and x-rays etc).

Things are a bit oversimplified above. In fact there are many other moderately strongwaves predicted but the above ones are the super strong ones. The other waves turn out to

explain galaxy clusters and other things. In each scale there are multiple strong wavesand for the distances between the stars for example they are 4.45, 5.93, 8.9 and 11.86light years. These are the same periods that were found by Dewey and I in cycles onearth. They are indeed "influenced by the stars" but not in the way that astrologersnormally mean.

I reached this stage in about 1993. Since then I have found that the detailed predictions ofthe Harmonics theory are confirmed by observations in cosmology, geology, atomicphysics, economics, climate, biology and human affairs.

The universe is a musical instrument and everything in it is vibrating in tune with the

larger things that contain it. I believe that there are no other laws in the universe than this.All the other laws of physics appear to be the result of the wave structure that leads to theHarmonic law.

Part 4: Predictions and Verifications of the Harmonics Theory

This is the last article in this series except to answer any questions arising. Anyone whofinds it interesting can find a lot more material under the Harmonics theory. This part willjust briefly describe some of the detailed findings and give references for more.

Previously I mentioned that the harmonics calculated in the 48 to 96 range exactly fittedthe just intonation musical scale and that the strongest of these; 48, 60, 72, 96; are amajor chord (ratios 4:5:6:8). Further examples of major chords happen at other places inthe harmonic structure.

There are also minor chords found. These happen in the transition zones between theplaces where the major chords are found. It is this transition which I believe gives theminor its quality.This graphic shows the harmonics from 20 to 360 and shows some ofthe strong harmonics 240, 288, 360, 480 which makes a minor chord (ratios 10:12:15:20).

If the electromagnetic zone around the earth vibrates, it does so with a frequency of 7.5

Hz because the speed of light is 300,000 km/s and the circumference of the earth is40,000 km. Therefore the predicted strong harmonics of this vibration should havefrequencies of 7.5 Hz times the various harmonics numbers. Interestingly, the frequenciesresulting exactly match those used in Indian music.

Harmonic h 48 54 60 64 72 80 90 96

Frequency h*7.5 Hz 360 405 450 480 540 600 675 720

Indian note pa dha ni sa ri ma ga pa

Western scale F G A Bb C D E F
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The modern standardised scale has A=440 Hz and the others adjusted according to theequitempered scale which does not quite fit this table. However the trend has been for Ato increase with time and it had got to 450 Hz before the standard was set. Based onIndian music, the earth's natural resonance, a study of the rhythm speed for greatcomposers and on other evidence, I believe that 450 Hz is the true and correct A. It is in

harmony with the earth. For indian scales relationships seegraphic.

Redshifts are what astronomers use to tell how far away galaxies are and are believed tobe based on the velocities of galaxies relative to us and caused by the big bang. I don'tbelieve in the big bang or that redshifts are due to velocity. The harmonics theory predictsthat the redshifts of the galaxies should favour the following values which are in km/s:

144 72 36 18 9

48 24 12 6.0 3.0

(16) 8.0 4.0 2.0

(2.67)

The prediction of galaxy redshift distributions is shown in a graphic.

Two years ago I put a message in the sci.astro usenet group which predicted that galaxiesshould come at these favoured redshifts. I knew that the 72 km/s value had been observedbut not any of the others. Those observations were not taken seriously by mostastronomers because they could not reconcile them with their beliefs in the big bangtheory. As a result of that post, someone directed me to the work of W G Tifft who hadobserved the following redshift quanta (or tendencies for redshifts to come in multiples):

144, 72, 36, 24, 18, 16, 9.0, 8.0, 6.0, 3.0, 2.67 km/s

You can see that they match almost perfectly. I hadn't included the 16 and 2.67 km/s

values in my original list because they were slightly weaker values but Tifft had foundthem anyway.

If Tifft's observations were really the results of "noise" in the data as most astronomersbelieve, or if my theory was not correct about the universe, then the chance of such agood match between the numbers would be 1 in about 1,000,000,000,000,000,000. Inother words, most astronomers believe in something that is incredibly unlikely.

Similar calculations show that the stars should favour certain distances. The followinghistogram is based on the distances between all pairs among the nearby stars. Each "*" isone star distance.

Number Of Star Pairs At Distance *

*

*

* *

** * *

** * ****

* * * ** ** * * ****

* * * * * * *** **** * *** ******

* * * **** * ***** **** ************** ***** ******
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0 1 2 3 4 5 6 7 8 9 10 11 12

A A A A A A

4.45 5.93 7.12 8.9 9.6 11.86

Distance between star pairs in light years --->

Also shown (by A's) the expected universal harmonics, and the

common cycles periods from Dewey's catalogue.

It is quite clear that stars do favour the distances predicted by the harmonics theory andthat these distances in light years exactly match the period of common cycles on earth asreported by Dewey.

Likewise the distances of the planets favour multiples of two distances, near 10 and 0.35astronomical units (1 a.u. = earth-sun distance). The accuracy of this agreement is shownin a graphic. These two distances correspond to waves that oscillate in 160 minutes and5.8 minutes. The sun has a strong oscillation at 160 minutes and a set around 5 to 6minutes. This shows that such waves exist in the solar system even though we cannotdirectly detect them.

The Harmonics theory also works at the atomic and sub-atomic scales. In 1994 at alecture at Princeton I predicted that there should be a particle with a mass 68 times anelectron or 1/27 of a proton. In 1995 just such a particle was discovered and it wasunexpected and unpredicted by any other theory.

Last year while travelling by plane I noticed some very regular cloud formations, likeploughed fields. As near as I can estimate the distances between the rows were 1/34560of the earth's circumference, or about 1.16 km. More research is needed into regularstructures on the surface of the earth and other bodies.

It seems appropriate to close on a note related to the ancient Greeks. As mentioned in aprevious post in reply to Mary Lynn Richardson, it seems that neolithic people andancient greeks used a system of measurements which had many ratios of 2 and 3 and alsoof 12. Their units include feet, yards, chains and such and the entire pattern of these isextremely similar to the pattern of wave sizes predicted by the harmonics theory. Afterfinding some rocks near my home that have exactly these dimensions (yards, cubits,spans, feet etc) I am now convinced that these ancient units were based on naturallyoccurring dimensions which reflect the electromagnetic wave sizes in the universe.

Harmonics Discussion

After part 1, Andrew Green wrote:
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I'm going to speak only of the classical Greek seven-stringed lyre - not of monochords, ormathematics. And I'm going to start with a few general assumptions.

1) You can play music on the lyre, i.e. the strings should each have a different pitch!

2) You must be able to tune the instrument without reference to an external source. Thatmeans that each string MUST be capable of sounding an harmonic which can be foundon at least one other string - to allow for cross-tuning.

3) I've limited this to the fifth harmonic, otherwise it gets too difficult to play - and alsoalmost impossible to hear.

Having said that, one can treat the lyre as a 7 by 5 matrix which has (I think) a uniquesolution imposed by my second assumption.

string: 1st 2nd 3rd 4th 5th 6th 7th

----------------------------------------------------------------------

open 1 9/8 5/4 4/3 3/2 5/3 15/82nd harmonic 2 9/4 5/2 8/3 3 10/3 15/4

3rd harmonic 3 27/8 15/4 4 9/2 5 45/8

4th harmonic 4 9/2 5 16/3 6 20/3 15/2

5th harmonic 5 45/8 25/4 20/3 15/2 25/3 75/8

The fact that the instrument ends up with the open strings tuned nicely to a diatonic scaleis a consequence only of the solution to the matrix.

There is a single discord. The ratios 27:8 and 10:3 have almost the same numerical value.Furthermore, it becomes apparent that the octave is divided into twelve, more or lessequally spaced parts - although only eleven of those notes are available on the oneinstrument. Interestingly, adding more strings (using similar principles) increases thenumber of discords but does not produce the elusive twelfth tone.

............

The point of this is that I can't imagine how else Pythagoras can have solved the problem.We don't know what he did - or even his original solution. What we do know is that theexperiment involving the "blacksmith's shop" does not work - Galilei, again, seems tohave been the first to have published a refutation of that. What the lyre tuning adds to ourknowledge is the values for the ratios of the chromatic scale.

There has been an unfinished correspondence with Ben, and I was going to show how thelyre tuning may be derived from first principles. But that died out while we argued overwhether we should discuss music at all. Maybe the time has arrived.

Ray Tomes wrote:


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The difference between the ratios 27:8 and 10:3 is a ratio of 27*3:10*8 or 81:80 and thisis the same difference as between Pythagoras' 81/64 and Galilei's 5/4. This 81/80difference and another one of 64/63 are two common discrepancies which occur as wemove around the keys. Certain notes need to change by these ratios. Sometimes 36/35(=81/80*64/63) can also occur.

Interestingly, in Indian music they do in fact have other notes located at these places.This is most easily described by a graphic on my WWW pages. See Indian Music ratios.

In that graphic the darker shaded area shows the 7 notes of the scale. When a modulationoccurs the small shaped area moves one position. If it moves one position to the left, thenwe see that the note "dha" will change form 405 Hz to 400 Hz which is a ratio of 81:80.

Mary Lynn Richardson wrote:

Ray, didn't Pythagoras himself say that "a stone is frozen music"? I don't know thecontext of the quotation, having only this, from George Leonard's _Silent Pulse_. Do youknow where it comes from?

Ray Tomes wrote:

Mary, I am quite ignorant about what Pythagoras said, but it certainly wouldn't surpriseme.

Last year I studied the rock formations on the coast about 30 km from my home. The

rocks look like someone has laid them and ever so neatly fitted together square andrectangular tiles. After measuring quite a few, because they seemed to be of certainconsistent sizes, I found that the common sizes were in proportions of 1:2 and 2:3 witheach other.

Then it struck me that I was measuring in metric units and that if I converted to the oldbritish units something wonderful happened. The rock sizes became 36", 18", 9" and 24",12" and 6". Of course there are names for these units. A yard is 36", a cubit is 18" and aspan is 9" while a foot is 12". [Note: " means inches] In other words the english units arein fact natural sizes.

The megalithic people used various measurements which it seems are related to thenatural sizes of rock formations. This extends also to a common use of measurementswhich relate to the modern chain which is 22 yards or 66 feet. Many megalithic sites aremultiples of 33 feet. I found that books about greek temples had pictures which showedthe structures were based on units relating to this also.

Other people have reported a neolithic yard which is about 2.75 modern feet. This valueis in fact 33 feet divided by 12 and so also connects to chains.
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A photo of the rock formations showing the highly regular structure.

So if Pythagoras said that rocks are frozen music then he is absolutely right. All thoseproportions of 2 and 3 are there just the same. There is another possible interpretation ofrocks being frozen music, and that is the atomic theory. I will return to that another time.

Bernard X. Bovasso wrote:

Rocks or stones as frozen music may derive from the habit of Greek mathematiciansusing movable pebbles to calculate (*caculus* means pebble or stone, as in *calx,* stoneused in gambling). In the Pythagorean tradition the *tetraktys* of the dekad wasdemonstrated with such pebbles as unit markers. Dried beans were also used as calc tocalculate but as the Pythagoreans knew, if defrosted by digestion could lead to noisyflatus down below. Hence their taboo. Since the mathematicians were not prone to eatpebbles, calculation was safe. One the other hand, *psyphoi* (pebbles), *psychros*

(frozen) and *psyche* as soul may tell us something about the music of the soul which is*psychros* until thawed in death and psyche grows wings and becomes pneumatic. Howelse to reach the heavenly spheres (and music of, thereby) and break the (reincarnational)wheel of births to attain ontological permanence in the cosmos which the Pythagoreansheld preferable to endless tranmigrational becoming.

In other words, they were impatient to end the train of serialized karma. Eating thetabooed beans could only propitiate this rather than allow ontological permanence and theeternity of Being. These, of course, are also Orphic notions that are handed down inChristian theology and which we unfortunately have no way of unravelling from what weknow of the Caberoi of Thrace and what would be consistent to such notions, a concept

of monotheism (e.g., the monotheistic Thracian Zalmoxis reported by Herodotus). Since Ientertain a notion of proto-Hebraic people originating in Northern Europe I cannot helpnotice the consistency of inferred correlation between "Pythagoreans," "Orphics," andJews. Since we are ending the third millennial binarium (6000 years) this may havesomething to do with the event of the Holocaust, Germans, Jews and the legend of a losttribe.

But I cannot speculate further on this, not at least, until I get another endopsychic hit.And for the while I shall refrain from beans which have a habit of always(metempsychotically) winding up in my big toe. In any case, it is known that hoppingaround on one foot is one way of waiting for a hit.

After part 4 Joseph Milne wrote:

This is fascinating stuff. There are several on this list learned in this subject - Andy, forexample. I am interested that you have referred to the Renaissance tuning, which Zarlinoadopted and which was employed by the Renaissance architects. I was very interested inthis some years ago, but particularly in the different modes and their qualities and reputed
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effects. I would be interested if you could tell us something about how these variousmodes (ancient Greek or the later Church modes) might relate to your investigations. Youhave mentioned in this message a distinction between the major and minor chords, so Iwondered if the various modes might open up avenues of investigation for you too.

Ray Tomes wrote:

Thanks Joseph, here are a few more thoughts on tuning systems. While I am aware thatthere were special church tuning systems and a little about the historic context, there areprobably some present who know the history much better, so I will concentrate on thetheory. I am not sure how this will go in ascii, but I will try.

Let us go back to Pythagoras to begin. Using only proportions of 2 and 3 we can get thefollowing frequency relationships:

F 5+ 11- 21+ 43- 85+ 171- (+/- mean by 1/3)

C 2 4 8 16 32 64 128 256 512

G 6 12 24 48 96 192 384 768

D 18 36 72 144 288 576

A 54 108 216 432 864

E 162 324 648 all in Hz

B 486

This starts from C as being a power of 2 (for simplicity) and each perfect fifth is a ratio of3/2. Therefore a second becomes 9/8 and a sixth 27/16. We know that this breaks down ifwe try to go all the way round the 12 keys and return as Pythagoras showed.

In fact it has already broken down by the time we get from C to E and B. The funny thingis that C-G-D are correctly related, G-D-A are correctly related, D-A-E are correctlyrelated and A-E-B are correctly related and yet C is incorrectly related to E and B. Howdoes that happen?

Well as Galilei observed, C and E want to be in the proportion 4:5 (which is 64:80) andwe have 64:81 so it just misses. The thing is that the right note depends on the key we arein. If we are in C then Galilei is right, but if we are in D then Pythagoras is.

Basically the best rules can be summed up by saying that if we take a section of musicand take the key that it is in (which may be a modulation from another key) then the notesin that section will generally want to have small whole number ratios to the key note.

The more we modulate music the more the problems will be with notes changingfrequencies on us. This is no problem for voices of unfretted stringed instruments wherethey can accommodate naturally without thinking about it. It is only a problem fororgans, pianos and such fixed note instruments.


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The more keys that we travel around the worse this problem gets. But even if we stay inone key it won't go away. In the key of C it turns out that D want's to be different thingsdepending on what other notes are present at the same time.

So at one time the church organs allowed for this problem. As far as I know they had a

few notes that could vary and they were all black notes (please correct me if anyoneknows better). It was easy to say that D# was different to Eb, and have the black keydivided into two parts. It was not too difficult to learn to play such a keyboard.

To sidetrack briefly here, some years ago I invented a system which I call AJI forAutomatic Just Intonation. I realised that an electronic keyboard doesn't have to makecompromises when selecting frequencies as there is no reason that a single key cannotproduce different frequencies depending on the circumstances. The circumstances includethe other notes played at the same time and the other notes played recently. So if we havebeen playing in C and play D with a G it should be 288 Hz to give a 3:4 ratio with G at384 Hz while if we play D with an F then it should be 284.4 Hz to give a 5:6 ratio with F

at 341.3 Hz.

I took out a provisional patent on this idea and tried to interest some keyboardmanufacturers. However if I wanted to keep up with it full patents would have cost inexcess of $50,000 and so I decided to just make it public so that anyone can use it forfree. It would not be difficult to implement this on a computer with a MIDI interface.

So what do I mean by small whole number ratios. I mean ratios made from numbers like1, 2, 3, 4, 5, 6, 8, 9 and maybe 7, 10, 12, 14, 15 and 16. I have found one piece, "Concertoin C Major" by Mozart, that has two wonderfully bitter-sweet chords that I think areintended to have ratios of 11 and 13. Such values do not fit the equitempered (or any

other) scale and so it would be fascinating to know what they should sound like or if thatwas indeed Mozart's intention. In other words I am guessing that Mozart really intended anote that is not normally considered to be part of any scale. It would be interesting tohave a string quartet record this and see what frequency ratios they actually played.

The equitempered scale is designed around the ratio 2 because each semitone is a ratio of1 to the twelfth root of 2. As it turns out this accommodates ratios of 3 almost perfectly,which is of course why it was chosen. It also can deal with a ratio of 5 roughly, althoughthe chords that want ratios of 5 are not called "perfect" like the ones that have ratios ofonly 2 and 3. A ratio of 7 is a bad fit however and 11 and 13 even worse. They simplyfall down the cracks in the keyboard.

Joseph Milne wrote:

Would it be possible to give us the exact chord sequence of this Mozart passage? I have asuspicion that Mozart knew exactly what he was doing with sounds and that if he doessomething harmonically unusual he has good reason. In some of his Masonic music thereare very strange and potent harmonies. If you could send me privately a midi file of the


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chord sequence I could reproduce it on my computer sound card - if that is not too muchtrouble to you. Or an image file of the page in the score.

I notice you mention the number 7 above and am curious where this might occur in aharmony.

One final question. Have you given any consideration to rhythmic ratios and whatsignificance they may hold, since music is also proportion in time?

Again, many thanks for presenting this rich material.

Ray Tomes wrote:

I don't have a midi file for this (but you might find it on one of the classical midi sites). Iwill scan the music and send as a GIF file to you by private email. I will let you find the

relevant chords yourself first and see if you get the same conclusion. Application of thejust intonation scale will likely get a ratio of 45/4 rather than my 11 and something elsefor the 13. I don't know whether you can produce the sound with a ratio of 11 on yourMIDI instrument. Can you?

As I mentioned, equitempered scales don't produce 7 at all accurately, but I believe thatthe meaning of a dominant 7th chord such as G-B-D-F-G is the ratios 4:5:6:7:8 but thecorrect frequency is between F and Fb.

In answer to the question "Have you given any consideration to rhythmic ratios and whatsignificance they may hold, since music is also proportion in time?"

Ah yes, another whole rich field of study. It occurred to me that rhythm can beconsidered as slow vibrations and could be in tune with the key. For example, if a pieceof music is in A then as A=440 Hz it is also 220, 110, 55, 27.5, 13.75, 6.88, 3.44, 1.72Hz. But 1.72 Hz is 1.72/sec*60sec/min = 103 beats/minute. Therefore if a piece is playedat 103 bpm then its rhythm is in A.

My guess was that good music should have a match between the music key and therhythm with possibly one ratio of 3 among the ratios of 2 used to get down from the notefrequencies to the rhythm. It is only possible to compare when the tempo is accuratelystated, although I suppose that it would be possible to time recordings. In fact the great

composers such as Beethoven and Mozart did do this right most of the time.

Not all composers get it right, but some modern composers such as Billy Joel do also. Heis an especially interesting case because many pieces have accurately stated tempo and Ifound that he played them about 2 to 3% faster than would be expected. But adding thispercentage to the standard A of 440 gives 450 Hz. I regard this as another good piece ofevidence that the true and correct A is 450 Hz. This was in fact the extra evidence that Imentioned earlier.


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I explained the above by email to a friend who is into drumming with others and wasinvestigating putting different ratio rhythms together. He sent me an email about a weeklater saying "Check out this web page; someone else has been thinking your thoughts!http://arts.usf.edu/music/wtm/art-sj.html". [link no longer working]

This is a description of rhythms as chords reduced in frequency by octaves until we hearthe oscillations as rhythm. The writer describes how many of the common rhythms are infact simple chords played in slow motion.

The harmonics theory does also produce rhythm due to different cycles coming togetherat different points. For example the strong beats are the ones with periods of 2, 3, 4, 6, 8,12 say

2 = * * * * * * * * * * * * *

3 = * * * * * * * * *

4 = * * * * * * *

6 = * * * * *

8 = * * * *12= * * *

adding these up gives

6011203031105011303021106 or something like

# . * * # * * . #

Including more harmonics, such as 5, 10, 20, 30, 60, 24 and so on leads to a moredetailed structure. I have calculated the sum of thousands of harmonics in the correctproportions according to the Harmonics theory and produced a graph of the result. Itought to represent the rhythm of a complete cycle of the universe :-)

Another way of getting a similar result is to look at the interaction of harmonically

related periods and there are many different rhythms produced such as boom-boom-titty-boom and so on. The factorisation of numbers can give this also. If you set out thenumbers from 1 to 24 (or 48 or 60 or 144) and then make a bigger mark beside all theones that have many factors then you get interesting rhythms. They do approximatelyrepeat after 12 or 24 etc steps but not exactly. Just like real rhythms. e.g.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

. . . + . * . # + * . # . * * # . # . # * * . #

Note that these numbers represent which cycles come together. At 12 you get 1, 2, 3, 4, 6and 12 all in step while at 8 there are 1, 2, 4, and 8 while at 24 all of these come together.[You can probably guess that the symbols . + * # represent degree of loudness of therhythm] So the above example looked at as 4=crochet gives tick-boom-boom-boomptyboom-boom and so on.

The example of Mozart's use of a ratio of 11 is included in the description of my AJI(Automatic Just Intonation) invention.

Joseph Milne wrote:


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Thanks again for your illuminating message. Unfortunately I can't reproduce MIDI filesin anything but tempered tuning, but can get round this with 'real' instruments! All I haveon my computer is a score writing programme which can play the score with syntheticnoises, but I can play MIDI files separately. I have a rule when composing never to usesounds but write straight onto paper by ear. The score programme is just a good way of

printing the music.

I'm afraid you are going to get me into all this again after I have left it for some time withso many other things to do! A few years ago some friends used to meet to play stringquartets (mostly Mozart) and we experimented a lot with the just tuning. We found that ifwe tuned the open strings to just tuning in C major (for works in C major) the harmonywas very clear. This meant that the ratio between D and A was 40/27 and that the DFAchord was 'dissonant' but actually quite interesting and we allowed it as a 'natural tuning'.In part the effect was to make the tonic a much more powerful sense of 'home'. Also theinstruments resonated better being tuned in this way. Modulating much beyond therelative major or minor led to some odd sounds though!

Ray Tomes wrote:

This is all very interesting to me as I am more of a theoretician than a practical musician(translated, I am a lousy pianist) and so practical experiments provide guidance on thetheory side. I am interested to hear more if it is not too much like repeats for the othershere.

I have sent by private email 2 GIFs of the Mozart piece and 1 GIF of the rhythm of theuniverse as I compute it. Note that the correct tempo is to take 14.1 billion years for the

full piece! Actually that depends on the Hubble constant and it might only be only 9.4billion years. Needless to say, when some of this big universal drum rolls come aroundthe universe is a wild and wooly place to be.

Thanks for lots of interesting questions, comments and thoughts.

Andrew Green wrote:

Most of my instruments, including Yamaha, let you tune to any scale - but it's a laboriousprocess. Automatic control of pitch is (in my experience and opinion) a blind alley.

Rather, concentrate on Vincenzo Galilei (the father of Galileo). Changing "key" hasnothing to do with Pythagoras - changing "mode" is everything. Vincenzo Galilei notedthat "modern" (to him) music just didn't work. That is - no effect.

Correcting pitch has little effect, changing key has little more. For the big effect classical"Western" composers will shift from major to minor (or back) - and they ended up with amost valuable, but quite different kind of music.


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I'm really using this as no more than a placeholder. I've read what Ray has written, andalso his web pages, and also many of the leads from there. A good deal of this is newmaterial, and I can't give a considered response immediately. But it is most interesting.

Ray asked for historical information, and I would recommend, again, Joscelyn Godwin's

"Harmonies of Heaven and Earth", as a starting point. Then Carl E. Seashore's"Psychology of Music" (Dover) which includes analysis of the pitches musicians sing andplay - relative to those indicated in the written music - together with opinions as to whatsounds best. One of the best books looking backward at the legacy of Pythagoras is S.K.Heninger's "Touches of Sweet Harmony - Pythagorean Cosmology and RenaissancePoetics" (The Huntington Library, San Marino), and if you want stuff about how peopleperceive pitch, try Carol L. Krumhansl's "Cognitive Foundations of Musical Pitch"(OUP).

For a thorough consideration of all the ancient texts on Greek music, there is M.L. West's"Ancient Greek Music" (Oxford), and if you want translations of all the ancient texts they

are to be found in Sir John Hawkin's "General History of Music" (1776, although therewas a reprint in 1875).

And, finally, Mark Lindley's "Lutes, Viols & Temperaments" (Cambridge) - whichcomes with a cassette to illustrate what the different tunings sound like.

Most of what Ray has written about would not have been known to these writers - butmight not have come as a surprise to some of them. How far must one go before onebegins to draw conclusions? - and realizes what Joseph had quoted from Shakespeare:

>Such harmony is in immortal souls,

>But whilst this muddy vesture of decay>Doth grossly close it in, we cannot hear it:

Joseph also wrote:>One final question. Have you given any consideration to rhythmic ratios and>what significance they may hold, since music is also proportion in time?

And on that subject I noticed a most valuable lead from Ray's web pages. Ray will replyno doubt.

Joseph also expressed interest in MIDI files to illustrate musical points. This is just to say

that I have about 250 MIDI files of the Renaissance lute repertoire, and these could bemailed privately to anyone interested. I also have lute samples if you have a sampler(Maui, S1000, or .wav files), and would like something more realistic than the "classicalguitar" you get on normal computer sound cards.

Ray Tomes wrote:


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Andrew thanks for all the references. I can see that I am due to make a trip into theuniversity music library. Like Joseph, this isn't what I was intending to be doing, but it isa fascinating subject.

I am still hoping to get some similar feedback on the cosmological and physical aspects

of my posts as well as the musical.

Having typed that word "aspect" it also brought to mind the other subject where thatword has such importance. Harmonics is also related to astrology and I have found thatmy theory does produce the various aspects with exactly the degrees of importanceattributed to them by many astrologers. Of course most of astrology as practiced is wrongbut there is a core of truth as discovered by Gauquelin. Is this a fit subject for this list?

The reference for the article on rhythmic ratios that Andrew mentions is byStephen Jay.[link no longer working].

>Automatic control of pitch is (in my experience and opinion) a blind alley.

I would agree that any fixed tuning is a blind alley as there are always some conditionsthat stuff it up (as the 40/27 mentioned by Joseph). However some Yamaha instrumentsallow individual notes to be reset on the fly. Now it would be laborious to have to do thiseven in a MIDI file without a computer doing all the grunt for you. But with a somewhatclever program it could recognise that 40/27 and convert to 3/2 by retuning one of thetwo notes just before playing it. There is a bit of thought required to decide which one toretune but it isn't too bad.

I have a document that I wrote on this but unfortunately I did it on an Amiga (since given

away) and so only have the printout now. I will scan it and OCR it and set up some extraWWW pages.

Let me pose a question to you. Given a skilled string quartet do you agree that for anypiece there is a correct and true harmony for them to use for each and every chord? If so,then there must be an algorithm for finding that correct harmony. It is of course possiblethat computers are too dumb for this but never underestimate what a cunning programmercan achieve by asking the experts hundreds of questions.

Ray Tomes wrote:

Today I went to the University Music Library to try and find the books that Andrewrecommended. On Sun, 18 Aug 96 Andrew Green wrote:

>I would recommend, again, Joscelyn Godwin's "Harmonies of Heaven and>Earth", as a starting point.

Unfortunately this one was out. Will try again another day.
http://arts.usf.edu/music/wtm/art-sj.htmlhttp://arts.usf.edu/music/wtm/art-sj.htmlhttp://arts.usf.edu/music/wtm/art-sj.html


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>Then Carl E. Seashore's "Psychology of Music" (Dover) which includes>analysis of the pitches musicians sing and play - relative to those>indicated in the written music - together with opinions as to what>sounds best.

Unfortunately not available, but there were 3 other books of the same. The best of thesewas edited by Diana Deutsh and had some very interesting stuff.

A graph showed the psychological judgement of consonance/dissonance for intervals upto an octave. The interval was continuously varied in the experiment, not just bysemitones. The ratios that have definite peaks of consonance are 1, 6/5, 5/4, 4/3, 3/2, 5/3and 2 though there are almost peaks at two other places. The text says that chords withlow ratios are consonant and that ratios of 7 (e.g. 7/4 and 7/5) are on the borderlinebetween consonance and dissonance.

Singers apparently don't follow the equitempered scale, but neither do they follow just

intonation exactly.

One odd thing that was there was that people with the ability to judge absolute pitch oftenfound that around 50 years of age everything suddenly started to sound about 1 to 2semitones sharp. Presumably there internal clock suddenly slows down by 5 to 10%.Weird. I guess this interested me because I am 49, but I can't judge pitch anyway so whatthat proves is beyond me. To quote Homer, "Duh!"

>For a thorough consideration of all the ancient texts on Greek music,>there is M.L. West's "Ancient Greek Music" (Oxford),

This was the only one that I actually found. The part about auloi tuning was interesting asthe intervals appear to be somewhere between just tuning and an equal 7 interval scale.

Ray Tomes wrote:

Joseph wrote: (in reply to David)>... I take your point about changing modes.

I didn't get this. Could someone explain please.

>I have only ever heard one little piece by Galilei, but that was quite special.

I have never heard such a piece. Where would I find one?

Joseph Milne wrote:


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compose music that moves through several modes but all keeping the identical tuning foreach note. The possibilities here are immense. I have composed a lot of music this waywithout using a single accidental. Once your mind gets into this tuning, accidentalsbecome 'foreign' sounds. Their use can only be to change the pitch of the modes - aswhen classical composers modulate from one major key to another by introducing a sharp

or flat. Some composers say that each major key has a different quality. This is not a'modal' fact, but is connected (in my view) with some sense of absolute pitch. How that isknown is rather mysterious - it may be connected with your idea of the harmony of theuniverse.

As to the Galilie piece of music, I will have to ask a friend about that, who has it on anold record. It is a little piece for harp. I will get back to you on this. I hope this goes someway to answering your questions - and that it has not been too boring for others on thislist. I am afraid that to us votaries of the Muses this is the most fascinating and importantthing in the whole world!

Andrew Green wrote:

Regarding changing modes, I think Joseph explained admirably. I will probably addsomething though.

Vincenzo Galilei was associated with Mei and the Camerata in Florence, engaged in theresurrection of ancient Greek music. He published a number of books (available in a fewlibraries, and in facsimile), and engaged in arguments over the emerging iniquity of whatwould become "equal temperament". A couple of bits, in translation, follow:

Music was numbered by the ancients among the arts that are called liberal, that is, worthyof a free man, and among the Greeks its masters and discoverers, like those of almost allthe other sciences, were always in great esteem. And by the best legislators it wasdecreed that it must be taught, not only as a lifelong delight but as useful to virtue, tothose who were born to acquire perfection and human happiness, which is the object ofthe state.

But in the course of time the Greeks lost the art of music and the other sciences as well,along with their dominion. The Romans had a knowledge of music, obtaining it from theGreeks, but they practiced chiefly that part appropriate to the theatres where tragedy andcomedy were performed, without much prizing the part which is concerned with

speculation; and being continually engaged in wars, they paid little attention even to theformer part and thus easily forgot it. Later, after Italy had for a long period suffered greatbarbarian invasions, the light of every science was extinguished, and as if all men hadbeen overcome by a heavy lethargy of ignorance, they lived without any desire forlearning and took as little notice of music as of the Western Indies.........

For all the height of excellence of the practical music of the moderns, there is not heardor seen today the slightest sign of its accomplishing what ancient music accomplished,


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nor do we read that it accomplished it fifty or a hundred years ago when it was not socommon and familiar to men.

Thus neither its novelty nor its excellence has ever had the power, with our modernmusicians, of producing any of the virtuous, infinitely beneficial, and comforting effects

that ancient music produced. From this it is a necessary conclusion that either music orhuman nature has changed from its original state.

Vincenzo Galilei: Dialogo della musica antica e della moderna, 1581, from Strunk.

He had two sons. One, Michaelangelo, was a lutenist, and a number of his works areavailable. The other was Galileo (and the rest is history!).

Vincenzo should not be underestimated. Without him there may have been no Opera!

His published lute music is available, but I'm afraid I don't have much in my collection.

(Ray, if you email me I can arrange to let you have material, and at least one recording).

Joseph Milne wrote:

I wonder if you have considered the Vedic description of the four Yugas or ages of theworld? They are reckoned in divine years and are multiplied by 360 to obtain the sum of'human' years.

"Each cosmic age is preceded by a "dawn" (sandhya) and is followed by a "dusk" ofequal length (sandhyansha). each of these two periods constitutes one tenth of the

respective yuga. The four yugas are (1) Krita- or Satya-Yuga (1,728,000 human years);(2) Treta-Yuga (1,296,000 years); (3) Dvapara-Yuga (864,000 years) (4) Kali-Yuga(432,000 years). The total is 4,320.000 human years (12.000 divine years) equals onemaha-yuga, or 'great age.' Two thousand maha-yugas equal one day and one night in thelife of Brahma." (Rider Encyclopedia of Eastern Philosophy and Religion)

Ray Tomes wrote:

The Vedic cycles periods are very interesting in relationship to the periods that I calculateby the Harmonics theory. Although the actual periods do not match, the pattern of ratios

of 2 and 3 are similar and many of the numbers are my very strong harmonics with twozeros added. For example, 4320, 8640, 12960, 17280 are all strong harmonics, being my34560 number divided by 8, 4, 8/3 and 2 respectively. Generally the Vedic periods havemore ratios of 10 (or 5) than I believe is correct.

About 18 months ago I had some email correspondence with a Hare Krishna chap whowas able to answer some of my questions about the subject. My conclusion, as withseveral other aspects of ancient knowledge, is that originally the basic knowledge appears


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to be very similar to the Harmonics theory but that with the passage of time some of themeaning has been lost. I suspect that the use of 100 year multiples is a misinterpretationalong the way.

This has made me think about the old testament numbers. I have been told that the old

testament contains lots of astronomical cycle knowledge but in coded form. For examplesthe planets are referred to as animals. I believe that Daniel is the main book concerned.Someone who I find generally trustworthy said that there is mention of a 2300 year cyclethere which is fascinating, because there is a 2300 year cycle in climate and inastronomy, but the astronomical one depends on knowing about Uranus and Neptune! Ittakes a large amount of recorded history to discover a 2300 year cycle or some goodinspiration. Does anyone know more about biblical codes and such like?

In the last 20 years there have been 2 or 3 different ancient (pre 1500 BC I think) textsfrom africa and the middle east decoded by someone who realised that they were in factastronomical calculations. It seems that the method used is more or less equivalent to

recognising that the planets move in ellipses and the method allows quite accuratepredictions.

To return to Vedic knowledge and the musical relationships mentioned earlier. As well asthe octaves (ratios of 2) they had up to eight ratios of 3, three ratios of 5 and one ratio of7. In the harmonics theory, beginning from the fundamental, after about 19 ratios of 2there should be eight of 3, three of 5 and one of 7 (although the next ratio of 7 would benearly due). In other words, Indian music has the ratios in it to cover a range offrequencies of more than a million million times. As human hearing cover a range ofabout 1000 times (20 to 20,000 Hz) this seems excessive to say the least.

Cynndara Morgan wrote:

I haven't been able to contribute anything to your discussion of tuning because myunderstanding is so limited compared to yours, but you have been exploring a number ofideas that have just started working on me in the last couple of years. For one thing, Rayand Andy have finally explained why it is that when I am tuning my harp by ear, thedamned chords just don't match up -- if I get C, G, and Dm chords perfect off of eachother, then the Am and F chords won't fall in, and vice-versa. It's so nice to know that thereason I can't do something is that it really can't be done, not just a tin ear.

But as a magician, I have other questions, and one I have been working on is listening tothe effect of rhythms, modes, and pitches through my to analyse the effects of music.This is incredibly complex and effects are far from predictable, but I seem to sense a fewthings that flew in the face of my conventional music "training", such as it is.

First, of course, with a tempered scale, key changes aren't supposed to affect the music. Itshould be possible to transpose and play a piece in any key, to fit the instrument (or moreoften, the voice). But "pitch" has a distinct effect on the emotional/physical perception of


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I'm afraid that I'm wandering all over the field, here, with no data to back me up at all.But I have been interested in the light which your knowledge throws across these murkytheories, and would be interested if you have any connections or (sigh) corrections tomake with them.

Curt Lang wrote:

I would like to hear more of your subjective thoughts on the physiological and emotionaleffects of music. They may not be easy to defend but they _are_ interesting. Have youobserved other keys and time signatures that touch other chakras?

Ray Tomes wrote:

This is another whole fascinating area. It is, as you say, incredibly complex but there are

snippets of information about the effects on the body.

Several years ago I saw a paper on some experiments where people were sat on a chairattached to a machine which could vibrate the chair and person at various frequencies (Isuppose that sound could be used directly instead) and they explored the effect of all thelow frequency ranges (about 1 to 100 Hz). It listed the frequencies and which parts of thebody had what sorts of feelings in them. Many or the responses are emotional as well asphysical. Unfortunately, later when I went back to make a copy of the paper it was nolonger there.

I suspect that the effects are due to two main factors. One is the size and shape of the

various organs and bones in our bodies. This determines the distance that sound musttravel in a bone for example and back again and so it will have a series of naturalvibration modes just like a string. The other is similar, being natural resonances, but notnecessarily sound. Nerve impulses have a very definite speed and so are capable ofhaving resonances also. No doubt there are others.

It follows that because different people have different dimensions and possibly differentnerve characteristics that there may be some variations between people in the responsesthat they get.

Another related research is the subject of ELF waves. ELF stands for Extra Low

Frequency and means around 1 to 30 Hz radio waves. As mentioned previously, the earthsupports a natural 7.5 Hz standing wave and harmonics of this. Because of the variationsin the size of the zone that separates the earth's e/m field from the solar wind (the earthhas a sort of bag that sometimes has a cometary type tail) there are variations in thenatural frequencies.

I believe that human brain waves are in the ~7.5 Hz range because of the effect of thesewaves around the earth. Experiments have shown that exposing people to slower or faster


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e/m fields (say 5 or 10 Hz) does either slow down or speed up their reaction times. Inother words our brain clocks itself to the signal.

Exposure to slower or faster ELF waves also produces unpleasant or exciting feelingsrespectively. I believe that changes in the ELF waves present around the earth are

possible the causes of such mass human actions as stock market panics and the like.Maybe this is the mechanism for starting wars also, as it is known that solar activity doesaffect the ELF waves.

This is confirmed by another research (in Budapest) which found that when 3 Hz ELFwaves are strong around the earth the number of accidents that people have isconsiderably increased. My interpretation is that the old clock is running too slow and webang into things before we have time to think about changing direction.

With this background I think that it is possible to at least partly understand your otherobservations.

>First, of course, with a tempered scale, key changes aren't supposed to>affect the music. It should be possible to transpose and play a piece in>any key, to fit the instrument (or more often, the voice). But "pitch">has a distinct effect on the emotional/physical perception of music.>So transposing affect how the piece is heard.

This is a generally accepted concept. Each key is supposed to have a particular set ofemotional responses associated with it. I can't find my book which lists the traditionalassociations at present.

To link this to what I said above, it is necessary to recognise that for any given key,especially when chords are played, there is a fundamental frequency which all others area multiple of. This fundamental is NOT the key note itself, but the 4th note in the scale.Let me explain.

If we are in the key of C then the relative frequencies of the notes are

C D E F G A B C

24 27 30 32 36 40 45 48

And so the note that has a frequency of 1 (and therefore divides into all the others) is Fbut 5 octaves below. In fact we have to consider the set of notes in each individual chord(or if just a melody is played we can consider a stretch of melody) and we can work out

the "fundamental frequency" associated. If we play C4 G4 E5 G5 then the fundamentalfrequency is C3 because when C3 is taken as 1 the other notes are 2:3:5:6 in this case. Inmost music the fundamental note moves around very little.

What is the meaning of this fundamental note? It is the time interval over which thesound repeats. So, even though we play C4 G4 E5 G5, the sound is repeating over a timeinterval that is characteristic of C3 which wasn't even played. In terms of body responses,the C3 resonance will be important. Most of the body responses are in the low range and


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even below the threshold of hearing. The fundamental frequency is usually below 100Hz. For example in C, using the notes in the octave from middle C will give afundamental of 262 Hz / 24 = 10.9 Hz.

So, returning to you observations, the key is very largely determining this "fundamental

frequency" and so establishing the body responses. This idea of a fundamental frequencyis my own term, but it is probably known to other people also (though I haven't seenreferences to it anywhere). It is important in my automatic just intonation invention as allthe other frequencies are then played as exact multiples of this notes frequency.

>... This seems to be related to perception in>the "chakras", but now I hesitate, because some performers and pieces>*don't* seem to be rhythmically impacting one of the seven classic chakras,>but instead positions between them.

I would very much like to hear from you (and others) what keys or pieces of music affect

each chakra. It would lead to some interesting study.

> Nevertheless the perception has been confirmed by>several of my colleagues. Next, rhythm seems to develop a harmonic with>pitch, so that according to tempo and rhythm, a pitch that normally impacts>in one zone may be shifted either slightly or even widely, something that>again I wonder if it has a harmonic basis.

Yes, this came up briefly before. Rhythm has a rate such as 103 beats per minute which is1.72 beats per second. Although we cannot hear this frequency, we can work out its keyby doubling it repeatedly to get say 440 Hz (1.72, 3.44, 6.88, 13.76, 27.5, 55, 110, 220,

440) which is the key of A. Therefore if a piece is played in A at 103 beats per minutethen the rhythm is in tune with the key. The great composers get this right most of thetime (I checked them out :-)

Therefore if you switch this piece from A to C to accommodate someone's singing rangethen you are either out of tune or need to sing it 19% slower (heaven forbid!) to have it bein tune.

> For instance, say, most pieces played on piano seem to>affect the third chakra area, and increase the ego-sense/self-esteem. This>relationship holds up and down the scale and across keys, relating to the

>"voice" of the instrument itself.

Ah yes, this is neat! Every different instrument has its own echo cavity. The overtones ofall notes are much stronger when they fit evenly into the echo cavity of the instrumentand weaker when they fit something and a half times as they then cancel themselves out.I found this out by digitally recording my mouth organ, guitar and piano on a computerand analysing the overtones. The cavity is either the length of the instrument (for mouthorgan) or the depth (for the others).


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At a public presentation, the people who had developed the machine said that the headend of the body was more "yang," and the perineum more "yin", and thus the head got theelectricity and the perineum the magnetism. I thought this was a bit reminiscent of 19thcentury European occultism ("electricity" and "magnetism" were seen as polarities -- anotion that did not occur as far as I know in traditional China, or in Europe prior to the

19th, or maybe late 18th, century -- though I'd have to check Benz and a few other placesto make sure), so I asked the rationale.

They supported my suspicions by saying that electricity was yang and magnetism was yin(no pedigree for this notion unfortunately was provided) -- and added that they had triedreversing the two and found that putting the magnetic field near the head could produceshort-term memory loss, so they decided it was a bad idea.

I had this filed away in the cross-cultural curiosa gallery of my memory theater untilrecently, when I came across work involving the use of highly focused magnetic fields asa preferred replacement for electro-convulsive therapy. It is possible to be much more

precise in affecting only small parts of the brain, it is possible to reduce the incidence ofactual seizures -- and the procedure can cause short- term memory disruption.

Curt Lang wrote:

I have been reading your very lucid essays on harmony and cycles and feel that I canfollow part of what you say but I suffer from not being able to hear the notes andharmonies you describe and enumerate. Have you ever considered writing an essay onyour observations and theories in the form of an audio file that would contain your voice-- narrating the ideas -- and an audible version of the notes and chords you describe? I

don't know enough about digital audio and midi to suggest how, and I can't imagine howbig an audience there might be, but I would find it interesting.

Ray Tomes wrote:

I did consider making a tape some years ago when I was working on the automatic justintonation idea. Unfortunately the programming effort was substantial to do it withcomputer wave sounds and it needs good equipment to do it with a keyboard. I have anold Kawai but it cannot vary its frequency on request. The people at Yamaha wereinterested in the idea but their new model with the necessary features was about $7000 at

the time. They said they would maybe get me a second hand model for about $1500 but itnever eventuated. I did some work on the Amiga that I had at that time but the quality ofsound was not good enough to truly show the effects and programming in BASIC made ita bit slow.

If any of the people in the group with good yamaha MIDI systems (that can tuneindividual notes on the fly) are interested enough in this to do the work then I would be


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very happy to supply the details of the automatic just intonation idea and assist in thelogic.

Actually I have a son who is a sound engineer but he keeps pretty busy doing jobs that hegets paid for (unlike his old man). He is interested in the idea so one day maybe....

I hope to put all the details together on my WWW pages so maybe some clever personwill come along and put it in to practice.

Andrew Green wrote:

I must be crazy - because I know what I'm letting myself in for - but why not? Nopromises on time scales though, I'm already programming against the clock.

I have an AKAI S1000, and a MAUI, both of which allow pitch bend up to half a

semitone. Judicious use of that parameter will allow us to change the pitch of each note"on the fly", and save the more laborious procedure of having to change the tuning of theinstrument.

I also have software which can be used more or less directly to include your algorithm,and it already bases its decisions on the harmony etc. of the previous 4 bars - which mustbe about what you need.

The bad news (if it is bad) is that my database of music only includes music for theRenaissance lute.

Let me look at your algorithm, BASIC will do, and I'll see if there's any hope.

Ray Tomes wrote:

Following Andy's kind (or foolish) offer to have a go at producing music based on myAJI (automatic just intonation) invention I have got together my old documents(produced on an Amiga with a dot matrix printer), scanned them, OCRed them and thennearly retyped them to fix the errors. There are also some associated graphics, somewhich I have redone.

I will put a copy on my WWW pages.

Curt, you will have to share the blame with me if Andy goes mad :-}

Harmonics & Beats are better with Just Intonation.


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where there are black notes and the "b" (flat) ones are correct in my view as to what theflat frequencies should be. The "#" (sharp) ones are unusual ratios but then again there isno real meaning to these notes in the key.

They do have the unusual ratios including 11, 13 and 19. Some of these are produced by

the cases like fa-ti (careful how you pronounce that one :-) which is a dreadful chordanyway, and others by the cases like re-la which is our old friend D-A when in C, that is27-40, which wants to be 2-3.

I haven't yet worked out whether the 11 and 13 ratios are consistent with the cases whereI thought Mozart intended these ratios in his "Concerto in C Major".

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