[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 correct solution was worked out by Galilei (the father of the famous Galileo Galilei) who concluded that the best frequencies were in the proportions
do re mi fa so la ti do
1 9/8 5/4 4/3 3/2 5/3 15/8 2
Which may be represented as whole number proportions as
24 27 30 32 36 40 45 48
These proportions are called the Just Intonation music scale and are the most pleasing proportions for note frequencies for any one key. The differences from Pythagoras are small, so that mi is 5/4 (=1.250) rather than 81/64 (=1.266).
It is interesting to look at the ratios between the notes. do-mi-so are 24-30-36 which can cancel to 4:5:6. This same proportion links the notes fa-la-do which are 32-40-48 cancelling to 4:5:6. Again, so-ti-re (re from the next octave) gives 36-45-54 which cancels to 4:5:6 again. So every note is linked to “do” by three major chords which have ratios of 4:5:6.
However when music contains modulations, that is, changes of key, then some of the notes need to change frequency. As many instruments cannot do this it was necessary to make a compromise. Many systems were developed for this compromise and it is called temperament.
Instruments such as pianos, guitars and trumpets have fixed frequencies while violins and the 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. This needs 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 automatically because they listen for the harmony. Guitars and pianos just cannot do it hence the compromise.
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 of 2^(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 clever pieces 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 play guitar chords along with a piano playing this piece.
Pythagoras and his followers and later Kepler were to consider that these musical relations or harmonies had wider application in the universe. This idea was almost forgotten or dismissed for many centuries. However I will hope to show you that there is much evidence that the universe is completely organised on a system of mathematical harmony 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 for corporations in New Zealand. In the course of doing this I found that many aspects of the economy showed quite clear cycles. After designing a method to search out the most consistent 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.
After a while I noticed that the periods that I was using were all very near exact fractions of 35.6 years. Also, other cycles existed at other fractions of this period such as ~12 years and a fraction under 4 years. The literature showed that there were other shorter cycles known as well as longer ones. I acquired some weekly data to look for shorter cycles and found 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 which is exactly a major chord in music. Also, the shorter cycles turned out to be exactly in the proportions of the just intonation musical scale plus a couple of back notes (E flat and B flat 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 254 is very near to 335.6.
There was of course the question “Why 35.6 years?” and the answer almost surely had something to do with causes from beyond the earth. For Jupiter’s orbital period is 11.86 years which is very close to 35.6/3 and the node of the moons orbit takes 8.85 years to travel 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 would think I was weird. However, I heard about a place called the Foundation for the Study of Cycles 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 one of his articles I was to find the following diagram. Dewey found many relationships with proportions 2 and 3 in cycle periods starting from a period of 17.75 years, in an enormous variety 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 \ /
---- ---- / \
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.
Interestingly Dewey, using data from different countries, different time periods and different fields of study had arrived at a table which included a very good match to my figures. 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 a clear 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, 6.0 years. However it doesn’t show the 11 and 22 year cycles and some others. To find these it is necessary to introduce a ratio of 5 just as was done by Galilei to Pythagoras’ music scale. For 22.2 years is 5 times 4.44 and 11.1 is 5 times 2.22 years. When the above periods are multiplied by 5 they also produce many other commonly reported cycles such as 178 years which is found in the alignment of the outer planets, in solar activity and in climatic 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, animal populations, social variables, stock and commodity prices. We literally live inside a giant musical instrument which is playing notes, chords and scales in such slow motion that only the Gods could hear it.
Dewey wrote a very touching piece late in his life where he likened himself to Tycho Brahe who gathered and catalogued the information about the planetary motions. He said that he had so wanted to solve the riddle but was then very old and knew that he was leaving 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;
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 same musical/harmonic relationships between the cycle periods to those that I had found I knew that it was real effect and not some delusion on my part. I found several other sources for similar observations and in all cases they fitted the same periods. The question 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 cause of it all was related to the formation of harmonics. The word “harmonics” has a slightly narrower meaning in physics than in music and means “frequencies which are a multiple of 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-linear because 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 full of ways for things to affect each other and I was not concerned with the details, just the broad idea. I quickly proved that an initial long cycle could only ever produce other cycles which were harmonics, that is, had multiples of the original frequency or fractions of its period. That was fine, but it could produce any harmonic, not just the observed favoured ones of 2, 3, 4, 6, 8, 12 etc.
Perhaps I should say something about the use of the word “frequency” for cycles. Usually we use “period” for long cycles (as in period of 5 years) and frequency for short ones (as in frequency 440 cycles per second) but it is equally valid to say a frequency of 0.2 cycles/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 of frequencies 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 of themselves which will be frequencies of:
1 --> 1 2 3 4 5 6 7 8 9 10 11 12 13 ...
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
Now what is immediately obvious here is that some frequencies are produced in many more ways than others; 4, 6, 8, and especially 12 are produced often while 11 and 13 aren’t.
The number of ways each number can be factorised is a measure of how much power we can 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 in the proportions of major chords in music, and moderately strong frequencies in exactly the proportion of the musical scale (the old just intonation scale, not the modern equitempered scale). An example is the range of harmonics from 48 to 96 shown below with 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
48 60 72 96 <--MAJOR CHORD
48 54 60 64 72 80 90 96 <--SCALE white
56 84 black
C D Eb E F G A Bb B C <--scale of C
There are some much nicer versions of this graph one showing the result of calculations for harmonics up to See 1,000,000 and one showing the detail of harmonics 20 to 320.
What shows up is that the strongest expected harmonics in the range 48 to 96 are 48:60:72:96 which is our old friend the major chord 4:5:6:8. Also the other strong harmonics match the other notes of the just intonation scale. I have labelled the white notes and the two black notes which are the same ones found in my cycles research.
There are some less important in between harmonics and these turn out to be in just the places where there have been disputes (Pythagoras’ 81/64 vs Galileo’s 5/4) and where the extra notes in Indian music are.
So let me stress what this proves. The pattern of cycles found in every field of study on earth, in astronomy and also in music are all explained by a simple rule that says that a single initial frequency will generate harmonics AND EACH OF THESE WILL DO THE SAME. 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 like 2300 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 also determine ice ages. But not so long ago someone reported a 27,000,000 year cycle in the extinction of species and geologists find even longer cycles.
This all seemed to be leading towards a conclusion which I initially joked about and then finally 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 go back 2 years in my research and do it again. However what came out of that is the realisation that there is an especially important harmonic which is 34,560. This number 34,560 is 22222222333*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 as the fundamental oscillation, the 34560th harmonic will be an especially important one. It also predicts that at further ratios of 34560 in size there will be important oscillations and sizes.
To understand how harmonics divide space as well as time, consider a stringed instrument. It can oscillate at a fundamental frequency which has just one wave in the string. It can also oscillate at the 2nd harmonic. In that case both the length of the string and 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 by 34560.
V V V V V V V V V V <--of 34560
A A A A A . . A A A <--things
Univ. Stars Moons Cell Nucleon observed
Galaxies Planets Atom
When we do the calculations from the size of the observable universe we find that the 34560 harmonic predicts the correct typical distance between galaxies. When we divide this by a further 34560 we get the typical distance between stars, then next time we get the distance between planets and so on. Eventually we get the typical distance between cells, atoms and nucleons (protons and neutrons). So the entire structure of the universe is predicted from this one simple principle. A table shows the values predicted by a repeated ratio of 34560.
Why do galaxies and stars form at these places? An analogous situation is to toss a handful of sand on a drum and then beat it (not at the centre) and you will find that the sand moves to certain places. These are the nodes of the standing waves in the drum. My 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 strong waves 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 waves and for the distances between the stars for example they are 4.45, 5.93, 8.9 and 11.86 light years. These are the same periods that were found by Dewey and I in cycles on earth. They are indeed “influenced by the stars” but not in the way that astrologers normally mean.
I reached this stage in about 1993. Since then I have found that the detailed predictions of the Harmonics theory are confirmed by observations in cosmology, geology, atomic physics, 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 the Harmonic 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 who finds it interesting can find a lot more material under the Harmonics theory. This part will just 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 fitted the just intonation musical scale and that the strongest of these; 48, 60, 72, 96; are a major chord (ratios 4:5:6:8). Further examples of major chords happen at other places in the harmonic structure.
There are also minor chords found. These happen in the transition zones between the places where the major chords are found. It is this transition which I believe gives the minor its quality. This graphic shows the harmonics from 20 to 360 and shows some of the 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 is 40,000 km. Therefore the predicted strong harmonics of this vibration should have frequencies of 7.5 Hz times the various harmonics numbers. Interestingly, the frequencies resulting 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
The modern standardised scale has A=440 Hz and the others adjusted according to the equitempered scale which does not quite fit this table. However the trend has been for A to increase with time and it had got to 450 Hz before the standard was set. Based on Indian music, the earth’s natural resonance, a study of the rhythm speed for great composers 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 see graphic.
Redshifts are what astronomers use to tell how far away galaxies are and are believed to be based on the velocities of galaxies relative to us and caused by the big bang. I don’t believe in the big bang or that redshifts are due to velocity. The harmonics theory predicts that 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
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 galaxies should come at these favoured redshifts. I knew that the 72 km/s value had been observed but not any of the others. Those observations were not taken seriously by most astronomers because they could not reconcile them with their beliefs in the big bang theory. As a result of that post, someone directed me to the work of W G Tifft who had observed 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 found them anyway.
If Tifft’s observations were really the results of “noise” in the data as most astronomers believe, or if my theory was not correct about the universe, then the chance of such a good match between the numbers would be 1 in about 1,000,000,000,000,000,000. In other words, most astronomers believe in something that is incredibly unlikely.
Similar calculations show that the stars should favour certain distances. The following histogram is based on the distances between all pairs among the nearby stars. Each “*” is one star distance.
Number Of Star Pairs At Distance *
** * *
** * ****
* * * ** ** * * ****
* * * * * * *** **** * *** ******
* * * **** * ***** **** ************** ***** ******
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 and that these distances in light years exactly match the period of common cycles on earth as reported by Dewey.
Likewise the distances of the planets favour multiples of two distances, near 10 and 0.35 astronomical units (1 a.u. = earth-sun distance). The accuracy of this agreement is shown in a graphic. These two distances correspond to waves that oscillate in 160 minutes and 5.8 minutes. The sun has a strong oscillation at 160 minutes and a set around 5 to 6 minutes. This shows that such waves exist in the solar system even though we cannot directly detect them.
The Harmonics theory also works at the atomic and sub-atomic scales. In 1994 at a lecture at Princeton I predicted that there should be a particle with a mass 68 times an electron or 1/27 of a proton. In 1995 just such a particle was discovered and it was unexpected and unpredicted by any other theory.
Last year while travelling by plane I noticed some very regular cloud formations, like ploughed fields. As near as I can estimate the distances between the rows were 1/34560 of the earth’s circumference, or about 1.16 km. More research is needed into regular structures 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 a previous post in reply to Mary Lynn Richardson, it seems that neolithic people and ancient greeks used a system of measurements which had many ratios of 2 and 3 and also of 12. Their units include feet, yards, chains and such and the entire pattern of these is extremely similar to the pattern of wave sizes predicted by the harmonics theory. After finding 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 naturally occurring dimensions which reflect the electromagnetic wave sizes in the universe.
After part 1, Andrew Green wrote:
I’m going to speak only of the classical Greek seven-stringed lyre – not of monochords, or mathematics. 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. That means that each string MUST be capable of sounding an harmonic which can be found on 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 also almost impossible to hear.
Having said that, one can treat the lyre as a 7 by 5 matrix which has (I think) a unique solution 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 15/8
2nd 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 scale is 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 less equally spaced parts – although only eleven of those notes are available on the one instrument. Interestingly, adding more strings (using similar principles) increases the number 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 the experiment involving the “blacksmith’s shop” does not work – Galilei, again, seems to have been the first to have published a refutation of that. What the lyre tuning adds to our knowledge 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 the lyre tuning may be derived from first principles. But that died out while we argued over whether we should discuss music at all. Maybe the time has arrived.
Ray Tomes wrote:
The difference between the ratios 27:8 and 10:3 is a ratio of 273:108 or 81:80 and this is the same difference as between Pythagoras’ 81/64 and Galilei’s 5/4. This 81/80 difference and another one of 64/63 are two common discrepancies which occur as we move 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 modulation occurs the small shaped area moves one position. If it moves one position to the left, then we 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 the context of the quotation, having only this, from George Leonard’s Silent Pulse. Do you know where it comes from?
Ray Tomes wrote:
Mary, I am quite ignorant about what Pythagoras said, but it certainly wouldn’t surprise me.
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 and rectangular tiles. After measuring quite a few, because they seemed to be of certain consistent sizes, I found that the common sizes were in proportions of 1:2 and 2:3 with each other.
Then it struck me that I was measuring in metric units and that if I converted to the old british 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 a span is 9″ while a foot is 12″. [Note: ” means inches] In other words the english units are in fact natural sizes.
The megalithic people used various measurements which it seems are related to the natural sizes of rock formations. This extends also to a common use of measurements which relate to the modern chain which is 22 yards or 66 feet. Many megalithic sites are multiples of 33 feet. I found that books about greek temples had pictures which showed the structures were based on units relating to this also.
Other people have reported a neolithic yard which is about 2.75 modern feet. This value is in fact 33 feet divided by 12 and so also connects to chains.
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 those proportions of 2 and 3 are there just the same. There is another possible interpretation of rocks 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 mathematicians using movable pebbles to calculate (caculus means pebble or stone, as in calx, stone used in gambling). In the Pythagorean tradition the tetraktys of the dekad was demonstrated with such pebbles as unit markers. Dried beans were also used as calc to calculate but as the Pythagoreans knew, if defrosted by digestion could lead to noisy flatus down below. Hence their taboo. Since the mathematicians were not prone to eat pebbles, 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. How else 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 Pythagoreans held preferable to endless tranmigrational becoming.
In other words, they were impatient to end the train of serialized karma. Eating the tabooed beans could only propitiate this rather than allow ontological permanence and the eternity of Being. These, of course, are also Orphic notions that are handed down in Christian theology and which we unfortunately have no way of unravelling from what we know 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 I entertain a notion of proto-Hebraic people originating in Northern Europe I cannot help notice the consistency of inferred correlation between “Pythagoreans,” “Orphics,” and Jews. Since we are ending the third millennial binarium (6000 years) this may have something to do with the event of the Holocaust, Germans, Jews and the legend of a lost tribe.
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 hopping around 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, for example. I am interested that you have referred to the Renaissance tuning, which Zarlino adopted and which was employed by the Renaissance architects. I was very interested in this some years ago, but particularly in the different modes and their qualities and reputed effects. I would be interested if you could tell us something about how these various modes (ancient Greek or the later Church modes) might relate to your investigations. You have mentioned in this message a distinction between the major and minor chords, so I wondered 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 that there were special church tuning systems and a little about the historic context, there are probably some present who know the history much better, so I will concentrate on the theory. 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 the following 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
This starts from C as being a power of 2 (for simplicity) and each perfect fifth is a ratio of 3/2. Therefore a second becomes 9/8 and a sixth 27/16. We know that this breaks down if we 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 thing is that C-G-D are correctly related, G-D-A are correctly related, D-A-E are correctly related and A-E-B are correctly related and yet C is incorrectly related to E and B. How does that happen?
Well as Galilei observed, C and E want to be in the proportion 4:5 (which is 64:80) and we have 64:81 so it just misses. The thing is that the right note depends on the key we are in. 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 music and take the key that it is in (which may be a modulation from another key) then the notes in 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 changing frequencies on us. This is no problem for voices of unfretted stringed instruments where they can accommodate naturally without thinking about it. It is only a problem for organs, pianos and such fixed note instruments.
The more keys that we travel around the worse this problem gets. But even if we stay in one key it won’t go away. In the key of C it turns out that D want’s to be different things depending 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 anyone knows better). It was easy to say that D# was different to Eb, and have the black key divided 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 for Automatic Just Intonation. I realised that an electronic keyboard doesn’t have to make compromises when selecting frequencies as there is no reason that a single key cannot produce different frequencies depending on the circumstances. The circumstances include the other notes played at the same time and the other notes played recently. So if we have been playing in C and play D with a G it should be 288 Hz to give a 3:4 ratio with G at 384 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 keyboard manufacturers. However if I wanted to keep up with it full patents would have cost in excess of $50,000 and so I decided to just make it public so that anyone can use it for free. 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 like 1, 2, 3, 4, 5, 6, 8, 9 and maybe 7, 10, 12, 14, 15 and 16. I have found one piece, “Concerto in C Major” by Mozart, that has two wonderfully bitter-sweet chords that I think are intended 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 that was indeed Mozart’s intention. In other words I am guessing that Mozart really intended a note that is not normally considered to be part of any scale. It would be interesting to have 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 of 1 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, although the chords that want ratios of 5 are not called “perfect” like the ones that have ratios of only 2 and 3. A ratio of 7 is a bad fit however and 11 and 13 even worse. They simply fall 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 a suspicion that Mozart knew exactly what he was doing with sounds and that if he does something harmonically unusual he has good reason. In some of his Masonic music there are very strange and potent harmonies. If you could send me privately a midi file of the chord sequence I could reproduce it on my computer sound card – if that is not too much trouble 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 a harmony.
One final question. Have you given any consideration to rhythmic ratios and what significance 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). I will 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 the just intonation scale will likely get a ratio of 45/4 rather than my 11 and something else for the 13. I don’t know whether you can produce the sound with a ratio of 11 on your MIDI instrument. Can you?
As I mentioned, equitempered scales don’t produce 7 at all accurately, but I believe that the meaning of a dominant 7th chord such as G-B-D-F-G is the ratios 4:5:6:7:8 but the correct frequency is between F and Fb.
In answer to the question “Have you given any consideration to rhythmic ratios and what significance 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 be considered as slow vibrations and could be in tune with the key. For example, if a piece of music is in A then as A=440 Hz it is also 220, 110, 55, 27.5, 13.75, 6.88, 3.44, 1.72 Hz. But 1.72 Hz is 1.72/sec*60sec/min = 103 beats/minute. Therefore if a piece is played at 103 bpm then its rhythm is in A.
My guess was that good music should have a match between the music key and the rhythm with possibly one ratio of 3 among the ratios of 2 used to get down from the note frequencies to the rhythm. It is only possible to compare when the tempo is accurately stated, 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. He is an especially interesting case because many pieces have accurately stated tempo and I found that he played them about 2 to 3% faster than would be expected. But adding this percentage to the standard A of 440 gives 450 Hz. I regard this as another good piece of evidence that the true and correct A is 450 Hz. This was in fact the extra evidence that I mentioned earlier.
I explained the above by email to a friend who is into drumming with others and was investigating putting different ratio rhythms together. He sent me an email about a week later 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 hear the oscillations as rhythm. The writer describes how many of the common rhythms are in fact simple chords played in slow motion.
The harmonics theory does also produce rhythm due to different cycles coming together at 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 more detailed structure. I have calculated the sum of thousands of harmonics in the correct proportions according to the Harmonics theory and produced a graph of the result. It ought 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 the numbers from 1 to 24 (or 48 or 60 or 144) and then make a bigger mark beside all the ones that have many factors then you get interesting rhythms. They do approximately repeat 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, 6 and 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 the rhythm] 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:
Thanks again for your illuminating message. Unfortunately I can’t reproduce MIDI files in anything but tempered tuning, but can get round this with ‘real’ instruments! All I have on my computer is a score writing programme which can play the score with synthetic noises, but I can play MIDI files separately. I have a rule when composing never to use sounds 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 with so many other things to do! A few years ago some friends used to meet to play string quartets (mostly Mozart) and we experimented a lot with the just tuning. We found that if we tuned the open strings to just tuning in C major (for works in C major) the harmony was very clear. This meant that the ratio between D and A was 40/27 and that the DFA chord 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 the instruments resonated better being tuned in this way. Modulating much beyond the relative 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 the theory side. I am interested to hear more if it is not too much like repeats for the others here.
I have sent by private email 2 GIFs of the Mozart piece and 1 GIF of the rhythm of the universe 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 9.4 billion years. Needless to say, when some of this big universal drum rolls come around the 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 laborious process. Automatic control of pitch is (in my experience and opinion) a blind alley. Rather, concentrate on Vincenzo Galilei (the father of Galileo). Changing “key” has nothing to do with Pythagoras – changing “mode” is everything. Vincenzo Galilei noted that “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 a most valuable, but quite different kind of music.
I’m really using this as no more than a placeholder. I’ve read what Ray has written, and also his web pages, and also many of the leads from there. A good deal of this is new material, 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 and play – relative to those indicated in the written music – together with opinions as to what sounds 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 Renaissance Poetics” (The Huntington Library, San Marino), and if you want stuff about how people perceive 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 there was a reprint in 1875).
And, finally, Mark Lindley’s “Lutes, Viols & Temperaments” (Cambridge) – which comes 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 – but might not have come as a surprise to some of them. How far must one go before one begins 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 reply no 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 be mailed 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 “classical guitar” you get on normal computer sound cards.
Ray Tomes wrote:
Andrew thanks for all the references. I can see that I am due to make a trip into the university music library. Like Joseph, this isn’t what I was intending to be doing, but it is a 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 that word has such importance. Harmonics is also related to astrology and I have found that my theory does produce the various aspects with exactly the degrees of importance attributed to them by many astrologers. Of course most of astrology as practiced is wrong but 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 by Stephen 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 conditions that stuff it up (as the 40/27 mentioned by Joseph). However some Yamaha instruments allow individual notes to be reset on the fly. Now it would be laborious to have to do this even in a MIDI file without a computer doing all the grunt for you. But with a somewhat clever program it could recognise that 40/27 and convert to 3/2 by retuning one of the two notes just before playing it. There is a bit of thought required to decide which one to retune 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 extra WWW pages.
Let me pose a question to you. Given a skilled string quartet do you agree that for any piece 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 possible that computers are too dumb for this but never underestimate what a cunning programmer can 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 Andrew recommended. 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.
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
Unfortunately not available, but there were 3 other books of the same. The best of these was edited by Diana Deutsh and had some very interesting stuff.
A graph showed the psychological judgement of consonance/dissonance for intervals up to an octave. The interval was continuously varied in the experiment, not just by semitones. The ratios that have definite peaks of consonance are 1, 6/5, 5/4, 4/3, 3/2, 5/3 and 2 though there are almost peaks at two other places. The text says that chords with low ratios are consonant and that ratios of 7 (e.g. 7/4 and 7/5) are on the borderline between 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 often found that around 50 years of age everything suddenly started to sound about 1 to 2 semitones 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 what that 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 as the 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:
About changing modes, Andy and I may have different answers. As I understand it, changing mode is changing the scale structure and creating a different tonal sense – as in changing from major to minor scales that Andy mentioned. Theoretically there are several ways to do this. One way is simply to change the place of the tonic and modify the new notes with accidentals, as in modulating to the relative harmonic minor. Another is to retain the same tonic but modify all the other notes to obtain a different mode. A third way, which I have explored in composition, is to retain all the notes of the major scale in just tuning, and shift the tonic to one of the Church modes (without any accidentals). Thus C, D, E, F, etc can become the tonal centre of different modes, giving seven modes in the major scale without employing accidentals. Andy can tell you about the Greek modes and how one changes from one to another.
As I mentioned before, this involves keeping the just tuning, including the 40/27 ratio D – A. I say this because there are two possible answers to the problem of tuning. One is to ‘correct’ the tuning of each chord so that they are in small whole number ratios. This is the usual remedy, easily done by ear with strings or voices. The other, which I find the more interesting, is to retain the tuning of each note to C and ‘permit’ the strange tunings that occur with certain harmonies. The theoretical reason for this is that the ‘centre’ of the harmony is the tonic, while ‘correcting’ the tuning of certain harmonies takes each harmony in isolation from the tonic, and so any modification changes the relation to the ‘silent’ tonic.
As I mentioned before, I was a member of a string quartet and did a lot of experimentation with tunings. Two of us were very interested in composing in the various Church modes. We always tuned the open strings to just tuning, tuning from C, not A. This means that A is slightly flat and the perfect fifth to E is true in relation to C, not sharp as it usually is in a string quartet. Quartet players get used to E being sharp and are quite happy to play the open E of the violin against the open C of the viola or cello. However, we got completely familiar with the just tuning without any correction.
This means that decisions have to be made about the harmonies of each mode. For example in the mode on D does one employ the chord D-F-A which is not a true minor chord? My answer is no, unless it occurs naturally as a dissonance. This means, in my view and working solely from the sounds themselves, rather than from theory, that there are certain harmonic restraints that belong naturally to each mode – especially if one thinks of harmony polyphonically rather than in terms of block chords or triads. I have no authority from history to back up my belief, only experience in composition in the modes in just tuning. My feeling that my instinct is right is confirmed by the fact that music composed this way sounds dreadful played in tempered tuning, which becomes all discords to my ear! But more significantly, it destroys the sense or ‘tonal feel’ of the mode.
Each mode (of the Church modes) has a unique quality and lends itself to certain types of music, but this can be discovered only through deep familiarity with the sounds of the modes and the way the tonic of each mode shapes it. From this basis, one can either compose music that belongs to a particular mode and stays in that mode, or one can compose music that moves through several modes but all keeping the identical tuning for each note. The possibilities here are immense. I have composed a lot of music this way without using a single accidental. Once your mind gets into this tuning, accidentals become ‘foreign’ sounds. Their use can only be to change the pitch of the modes – as when 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 is known is rather mysterious – it may be connected with your idea of the harmony of the universe.
As to the Galilie piece of music, I will have to ask a friend about that, who has it on an old record. It is a little piece for harp. I will get back to you on this. I hope this goes some way to answering your questions – and that it has not been too boring for others on this list. I am afraid that to us votaries of the Muses this is the most fascinating and important thing in the whole world!
Andrew Green wrote:
Regarding changing modes, I think Joseph explained admirably. I will probably add something though.
Vincenzo Galilei was associated with Mei and the Camerata in Florence, engaged in the resurrection of ancient Greek music. He published a number of books (available in a few libraries, and in facsimile), and engaged in arguments over the emerging iniquity of what would 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, worthy of a free man, and among the Greeks its masters and discoverers, like those of almost all the other sciences, were always in great esteem. And by the best legislators it was decreed that it must be taught, not only as a lifelong delight but as useful to virtue, to those who were born to acquire perfection and human happiness, which is the object of the 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 the Greeks, but they practiced chiefly that part appropriate to the theatres where tragedy and comedy were performed, without much prizing the part which is concerned with speculation; and being continually engaged in wars, they paid little attention even to the former part and thus easily forgot it. Later, after Italy had for a long period suffered great barbarian invasions, the light of every science was extinguished, and as if all men had been overcome by a heavy lethargy of ignorance, they lived without any desire for learning 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 heard or seen today the slightest sign of its accomplishing what ancient music accomplished, nor do we read that it accomplished it fifty or a hundred years ago when it was not so common and familiar to men.
Thus neither its novelty nor its excellence has ever had the power, with our modern musicians, 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 or human 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 are available. 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 the world? 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” of equal 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 one maha-yuga, or ‘great age.’ Two thousand maha-yugas equal one day and one night in the life 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 calculate by 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 two zeros added. For example, 4320, 8640, 12960, 17280 are all strong harmonics, being my 34560 number divided by 8, 4, 8/3 and 2 respectively. Generally the Vedic periods have more ratios of 10 (or 5) than I believe is correct.
About 18 months ago I had some email correspondence with a Hare Krishna chap who was able to answer some of my questions about the subject. My conclusion, as with several other aspects of ancient knowledge, is that originally the basic knowledge appears to be very similar to the Harmonics theory but that with the passage of time some of the meaning has been lost. I suspect that the use of 100 year multiples is a misinterpretation along 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 examples the 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 cycle there which is fascinating, because there is a 2300 year cycle in climate and in astronomy, but the astronomical one depends on knowing about Uranus and Neptune! It takes a large amount of recorded history to discover a 2300 year cycle or some good inspiration. 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) texts from africa and the middle east decoded by someone who realised that they were in fact astronomical 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 accurate predictions.
To return to Vedic knowledge and the musical relationships mentioned earlier. As well as the octaves (ratios of 2) they had up to eight ratios of 3, three ratios of 5 and one ratio of 7. In the harmonics theory, beginning from the fundamental, after about 19 ratios of 2 there should be eight of 3, three of 5 and one of 7 (although the next ratio of 7 would be nearly due). In other words, Indian music has the ratios in it to cover a range of frequencies of more than a million million times. As human hearing cover a range of about 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 my understanding is so limited compared to yours, but you have been exploring a number of ideas that have just started working on me in the last couple of years. For one thing, Ray and Andy have finally explained why it is that when I am tuning my harp by ear, the damned chords just don’t match up — if I get C, G, and Dm chords perfect off of each other, then the Am and F chords won’t fall in, and vice-versa. It’s so nice to know that the reason 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 to the 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 few things 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. 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 does affect how the piece is heard. A man’s voice and a woman’s singing the same piece will arouse different emotions. 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. 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. 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. But occasionally, a piano piece in a minor or dissonant mode will defy this rule . . . which I suspect but cannot substantiate might involve the destructive interference in dissonances.
Again, changes in tempo and rhythm can also affect the effect of the basic piano voice, or enhance it. A simple four/four beat in a moderately quick tempo again calls on the third chakra. But slow the beat or syncopate the rhythm, and the interactions become more complex.
One possible channel of interaction between music and human emotion and health could be the “pacing” effect well-known to hypnotists. The heart and breathing are subconsciously reactive to outside rhythms, and the body will attempt to “pace” the rhythms in its environment either exactly or at an achievable harmonic. This means that factory workers work faster when disco music is played over the loudspeakers, and slow music assists relaxation. But it also means that other endocrine reactions will occur in a cascade at the prompting of changes in heartbeat and respiration; these physical reactions are interpreted by the receiver as emotions and feed upon themselves in that sense, especially if there are words to go along with the music.
Volume controls the raw force with which the music and its rhythmic effects impinge on the human system. For this reason, I am appalled at the common tendency in this era of people to listen to rock music at high volumes. Frankly, the cardiovascular system can only tolerate so much jerking from one direction to another at high power before it collapses. However the emotional effects seem to involve generating an artificial sense of excitement and power; clearly a panacea for generalized alienation, sense of powerlessness, and massive boredom. I find it especially interesting that most of this raucous, overplayed music also stimulates the third chakra (4/4 beat, moderately quick tempo, major key, drums overwhelming both voice and guitar) — again propping up the ego-sense and self-satisfaction.
If my perceptions are correct, it is no wonder that a culture which lives in a constant matrix of such music at continual high volume would develop towards ever-increasing degrees of self-centered individualism.
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 murky theories, and would be interested if you have any connections or (sigh) corrections to make with them.
Curt Lang wrote:
I would like to hear more of your subjective thoughts on the physiological and emotional effects of music. They may not be easy to defend but they are interesting. Have you observed 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 chair attached to a machine which could vibrate the chair and person at various frequencies (I suppose that sound could be used directly instead) and they explored the effect of all the low frequency ranges (about 1 to 100 Hz). It listed the frequencies and which parts of the body had what sorts of feelings in them. Many or the responses are emotional as well as physical. Unfortunately, later when I went back to make a copy of the paper it was no longer 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 must travel in a bone for example and back again and so it will have a series of natural vibration modes just like a string. The other is similar, being natural resonances, but not necessarily sound. Nerve impulses have a very definite speed and so are capable of having resonances also. No doubt there are others.
It follows that because different people have different dimensions and possibly different nerve characteristics that there may be some variations between people in the responses that 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 earth supports a natural 7.5 Hz standing wave and harmonics of this. Because of the variations in the size of the zone that separates the earth’s e/m field from the solar wind (the earth has a sort of bag that sometimes has a cometary type tail) there are variations in the natural frequencies.
I believe that human brain waves are in the ~7.5 Hz range because of the effect of these waves around the earth. Experiments have shown that exposing people to slower or faster e/m fields (say 5 or 10 Hz) does either slow down or speed up their reaction times. In other words our brain clocks itself to the signal.
Exposure to slower or faster ELF waves also produces unpleasant or exciting feelings respectively. 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 does affect the ELF waves.
This is confirmed by another research (in Budapest) which found that when 3 Hz ELF waves are strong around the earth the number of accidents that people have is considerably increased. My interpretation is that the old clock is running too slow and we bang 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 other observations.
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 of emotional responses associated with it. I can’t find my book which lists the traditional associations 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 are a 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 F but 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 fundamental frequency is C3 because when C3 is taken as 1 the other notes are 2:3:5:6 in this case. In most music the fundamental note moves around very little.
What is the meaning of this fundamental note? It is the time interval over which the sound repeats. So, even though we play C4 G4 E5 G5, the sound is repeating over a time interval 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 even below the threshold of hearing. The fundamental frequency is usually below 100 Hz. For example in C, using the notes in the octave from middle C will give a fundamental 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 frequency is my own term, but it is probably known to other people also (though I haven’t seen references to it anywhere). It is important in my automatic just intonation invention as all the 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 is 1.72 beats per second. Although we cannot hear this frequency, we can work out its key by 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 minute then the rhythm is in tune with the key. The great composers get this right most of the time (I checked them out 🙂
Therefore if you switch this piece from A to C to accommodate someone’s singing range then you are either out of tune or need to sing it 19% slower (heaven forbid!) to have it be in 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 of all notes are much stronger when they fit evenly into the echo cavity of the instrument and 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 computer and analysing the overtones. The cavity is either the length of the instrument (for mouth organ) or the depth (for the others).
Therefore a piano has a characteristic timbre associated with its physical depth. For a one foot depth you get 550 Hz as sound travels 1100 feet per second there and back.
This is also the reason why many electronic instruments are no good. They make a G by playing a C 50% faster and it has the wrong overtone structure and sounds like a different instrument. It would be possible to do this right in an electronic instrument.
One possible channel of interaction between music and human emotion and
health could be the “pacing” effect well-known to hypnotists. The heart
and breathing are subconsciously reactive to outside rhythms, and the body
will attempt to “pace” the rhythms in its environment either exactly or at
an achievable harmonic. …
Literally, the composer is “playing” the audience. We are his instruments. Apparently there are strong ultra sounds that can cause involuntary responses in people including such things as bowel motions!
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 murky theories, and would be interested if
you have any connections or (sigh) corrections to make with them.
I am fascinated by your observations. It is interesting for me to see the relationships between the emotional and intuition side and the scientific side.
R Brzustowicz wrote:
This is curious enough to evoke a take-it-for-what-it’s-worth anecdotal account.
Some years ago,a friend of mine was very involved with importing various ch’i kung (or qigong) teachers from the PRC — though just that fact alone requires an article on the scientific qigong movement to give it a proper socio-religio-political context, and I’m not going to do it.
One of the groups that came through had developed a “qi machine” (or “qigong machine” — a Faraday cage in which one sat, while connected to a device that read what seemed to be changes in galvanic skin response from one’s arms and played them back (a) into the area of the head by means of EMF generated by a metallic mesh embedded in a rather dilapidated hat and (b) via a variable magnetic field generated in a cushion on which one sat.
The idea was that two important regions (and their associated acupuncture points and their associated meridians) would be activated, much as is supposed to happen in more traditional qigong meditations and exercises — and would be shaped by the machine into more salubrious patterns.
At a public presentation, the people who had developed the machine said that the head end of the body was more “yang,” and the perineum more “yin”, and thus the head got the electricity and the perineum the magnetism. I thought this was a bit reminiscent of 19th century European occultism (“electricity” and “magnetism” were seen as polarities — a notion 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 places to 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 tried reversing the two and found that putting the magnetic field near the head could produce short-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 until recently, when I came across work involving the use of highly focused magnetic fields as a 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 of actual 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 can follow part of what you say but I suffer from not being able to hear the notes and harmonies you describe and enumerate. Have you ever considered writing an essay on your 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 how big 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 just intonation idea. Unfortunately the programming effort was substantial to do it with computer wave sounds and it needs good equipment to do it with a keyboard. I have an old Kawai but it cannot vary its frequency on request. The people at Yamaha were interested 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 it never eventuated. I did some work on the Amiga that I had at that time but the quality of sound was not good enough to truly show the effects and programming in BASIC made it a bit slow.
If any of the people in the group with good yamaha MIDI systems (that can tune individual notes on the fly) are interested enough in this to do the work then I would be very happy to supply the details of the automatic just intonation idea and assist in the logic.
Actually I have a son who is a sound engineer but he keeps pretty busy doing jobs that he gets 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 person will 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? No promises 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 the instrument.
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 must be about what you need.
The bad news (if it is bad) is that my database of music only includes music for the Renaissance 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 my AJI (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 then nearly retyped them to fix the errors. There are also some associated graphics, some which 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.
One of the really nice things about the just intonation scale is that the beats between notes are in the scale, whereas for equitempered tuning they miss by quite a lot. I am using “beats” here a bit loosely because sometimes these beats are in the range of frequencies of notes, not the wa-wa-wa sound we associate with beats.
First the equitempered scale to show why it doesn't work in this way:
note do re mi fa so la ti do
Freq. 1.0000 1.1225 1.2599 1.3348 1.4983 1.6818 1.8877 2.0000
Beats 0.1225 0.1374 0.0749 0.1625 0.1835 0.2049 0.1123
Beats*8 0.980 1.099 1.198* 1.300 1.468 1.639 1.797*
[* means Beats*16]
The reason for *8 or *16 is to move the note up 3 or 4 octaves to see
where it falls in the scale.
In this case the beats are not generally in tune with the key. When beats are considered for notes that are 2 or more apart in the scale the situation is even worse.
Now the just intonation scale
note do re mi fa so la ti do
Freq. 1.0000 1.1250 1.2500 1.3333 1.5000 1.6667 1.8750 2.0000
Beats 0.1250 0.1250 0.0833 0.1667 0.1667 0.2083 0.1250
Beats*8 1.000 1.000 1.333* 1.333 1.333 1.667 1.000
Note do do fa fa fa la do
In this case all the beats are perfectly in tune with the scale itself. When notes 2 or more apart are considered then the situation is still very good. This is more easily expressed by using the ratios of frequencies as shown below:
note do re mi fa so la ti do re mi fa so la ti
Ratio 24 27 30 32 36 40 45 48 54 60 64 72 80 90
Beats-1 3 3 2 4 4 5 3 6 6 4 8 8 10
Beats-2 6 5 6 8 9 8 9 12 10 12 16 18
Beats-3 8 9 10 13 12 14 15 16 18 20 26
Beats-4 12 13 15 16 18 20 19 24 26 30
Beats-5 16 18 18 22 24 24 27 32 36
Beats-6 21 21 24 28 28 32 35 42
Beats-7 24 27 30 32 36 40 45
What is shown is the relative beat frequency between notes, for example in the “Beats-3” line the 10 is the beats (or difference) between mi=30 and la=40. The figures are always up the diagonals to the top line.
Now what is noticeable is that in most cases the beats are actually perfectly in the scale as follows:
note do re mi fa so la ti do re mi fa so la ti
Beats-1 do do fa fa fa la do ...
Beats-2 do la do fa so fa so ...
Beats-3 fa so la do# do mib mi ...
Beats-4 do do# mi fa so la so# ...
Beats-5 fa so so la# do do re ...
Beats-6 tib tib do mib mib fa sob ...
Where the … means that the cycle repeats for the next octave. All the notes are perfectly in the key except the ones labelled: do# mib sob so# la# tib. These all fall in the places where there are black notes and the “b” (flat) ones are correct in my view as to what the flat frequencies should be. The “#” (sharp) ones are unusual ratios but then again there is no 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 chord anyway, and others by the cases like re-la which is our old friend D-A when in C, that is 27-40, which wants to be 2-3.
I haven’t yet worked out whether the 11 and 13 ratios are consistent with the cases where I thought Mozart intended these ratios in his “Concerto in C Major”.