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 75/8The 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 27*3:10*8 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 "http://www.vive.com/connect/universe/in_music.gif".
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 is at "http://www.vive.com/connect/universe/rocks2b.gif".
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 B 486This 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"
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.
>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
>sounds best.
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.