The Well-Tempered System: Kepler vs Ptolemy

by Fred Haight

Some of this material was presented in a recent cadre school, and some in a previous pedagogical: at this time I wish to emphasize a particular point. There are still many gaps to be filled in, and questions to be asked, about the history identified here, but I am convinced that I am on the right track, and that Kepler is identifying the right problem.

Lyndon LaRouche has always stressed the importance of Kepler for our music work, but in the past, two problems arose:

1. Professional musicians resented such “outside intrusions” into “their turf”.

2. For a while, only Book Five was translated. You cannot “look at the back of the book”, and expect to find the answer. You have to read the entire work, and pay special attention to the relation between Book Three, where Kepler lays out his own revolutionary musical ideas(fn1), and Book Five; a relation which Kepler himself cites in the Introduction to Book Five:

“I found it truer than I had even hoped, and I discovered among the celestial movements the full nature of harmony, in due measure, together with all its parts unfolded in Book Three – not in that mode wherein I had conceived it in my mind (this is not last in my joy) but in a very different mode which is also very excellent and very perfect.”

I am putting forth the contention, that Kepler, without having composed a single measure of music, may be the greatest musical revolutionary, and that Bach’s breakthroughs, would not have been politically possible, without Kepler.

No great discovery has ever been made without attacking lies, falsehood, and stupidity. Kepler’s discoveries, from Mysterium Cosmographicum on, were all inseparable from his attacks on the method of Ptolemy, Aristotle, Tycho Brahe , Copernicus(fn2) et al. A recent 21st Century article, reprints Kepler’s argument that Aristotle lied, and reinstated the idea of an earth-centered solar system, when he knew that the Pythagoreans had known its true Heliocentric nature much earlier. This, and the revival of the “flat earth” theory, after Erasthosthenes’ discoveries, set science back for centuries.

Humanity lost 17 CENTURIES between the Rata-Maui expeditions, and Columbus, Magellan, et al. SEVENTEEN CENTURIES, because of politically imposed, in fact, REINSTATED, false axiomatic assumptions! As late as 1616, the Counter-reformation, once again, condemned the heliocentric system. Even today, fundamentalist “Christians” will sometimes use the Bible to “prove” that the sun rotates around the earth.(fn3) Mankind must be freed of such arbitrary, but popular opinions, before progress may take place.

Think about the following quote from “Economics: At The End of a Delusion”:

“Kepler was the founder of the first successful effort to establish a comprehensive form of mathematical physics, the first to establish a method which freed science from the ivory tower mathematician’s blackboard, and to civilize mathematics by bringing it into the real world, the world of universal physical principles, rather than the purely imaginary world of abstract ivory-tower mathematical speculations.”

Kepler did the same for music, which had been held back for Centuries, by a similar Ptolemaic system. Out of the thousands of years of mankind’s existence, the period of great Classical masterworks, from Bach to Brahms, lasts just under two hundred years! In the Twentieth Century, under the evil influence of the Frankfurt School, humanity allowed its greatest gifts to be stolen, again.


In order to examine the problem that held back musical progress, we must examine a few things that Kepler understood, but are not present in his Harmonice Mundi. We shall do this through the posing of a paradox. What I shall present here, are sometimes known, as the discoveries of Pythagoras, and his school, but I suspect that they may have suffered the same sort of rewriting, as Aristotle did to Astronomy.

Boethius tells the story, that Pythagoras was walking by a blacksmith shop one day, and “noticed”, that the different sizes of hammers hitting anvils produced different tones. This sounds too much like Newton “noticing” getting hit on the head to me, and besides, I think it would be the size on the anvils that made the difference.

Anyhow, Pythagoras was said to have investigated this, and, supposedly, moved quickly to investigating string lengths on an instrument called the monochord. This is a box with two strings of the same length tuned to the same tone. You produce different tones by dividing the second string into different lengths, and comparing the tones produced, to the sound of the open, first string.

You can approximate the experiment yourself using a Cello, and substituting your finger for the bridge that was used to divide the monochord. We shall use modern terms like “fifth”, rather than “diapente”, etc. Tune the second string of the Cello down to C, so that it is in unison with the first string.

First ,divide the second string in half. If you place your finger so as to divide the sounding portion (from the scroll to the bridge) of the second string in half, and compare it to the open first string, the interval should be an octave. You have blocked off the upper half of the second string with your finger, so that only the lower half of the string is sounding. So, the string length is half of the string, but the sound is twice as high (C at 64 becomes C at 128), so the ratio of the interval is 2/1. The string lengths and frequencies are in inverted ratios.

Next, try a string length of 2/3 (blocking off one third with your finger, and letting two thirds sound). This approximates the fifth. The ratio of the interval is 3/2, so multiply 64 times 1.5 to get G at 96. Then, continue by dividing the second string into three parts, four parts, five parts and 6 parts. The sounding portions will be 3/4, 4/5, and 5/6 of the string, and ratios for the intervals will be, 4/3, 5/4 and 6/5. (Don’t skip the experiment – you will undermine the discovery).

These were said to correspond to: String length: 1/2 2/3 3/4 4/5 5/6 Intervallic 2/1 3/2 4/3 5/4 6/5 ratio: Interval: octave fifth fourth major third minor third

Now, isn’t this beautiful? Here you have an interesting ordering of number (as in the sequence of numerators and denominators), a series of arithmetic and harmonic means (fifth and fourth as those two means of the octave, and major third, and minor third, as the same means of the fifth), inversion of a sort, and musical intervals derived from a physical process.

So, what is wrong with this? Think from the standpoint of method, and take a few minutes from your busy schedule before proceeding.

Did you do what I asked, or are you like those clients, who, when challenged to think, say “I’m sure you’re going to tell me, so let’s get to the bottom line”? Go back!

Three things, all interrelated, stand out. Perhaps you will find more:

1. If there are three types of successively higher-order physical processes, non-living, living, and cognitive; then this determination is from the lowest level, non-living, which might reflect higher order processes, projected downwards, but as through a glass, darkly.

2. If you try to determine planetary orbits individually, they won’t fit together as a solar system. Kepler, in the Mysterium Cosmographicum, starts by seeking the highest, top-down, ordering principle for the entire solar system. We shall see how this problem arises in these musical intervals shortly, in the paradox of the comma.

3. This is not as obvious, but the axiomatic PREJUDICE, that intervals could only be represented by rational numbers (fractions), set music back for centuries; much as the prejudice, that planetary orbits could only be perfect circles, did Astronomy. Organizers can do the same thing. “This way is the best, because we have been doing it this way for centuries.”

Kepler recognizes how long this prejudice held court. From the introduction to Book Three of Harmonice Mundi:

“Having discovered definite proportions,” or “the fact that,” it remained to track down the causes as well or “the reason why” some proportions marked out consonant intervals, and others dissonant.

And in the course of two thousand years the opinion had been reached that the causes are to be looked for in the proportions themselves, as they are contained within the bounds of a discrete quantity, that is to say, of Numbers.(fn4)

Question: Is the monochord experiment itself, a tautology?

Boethius, in his fifth century “De institutione musica”, states that intervals can only be represented by rational numbers, as they are the best. How could an irrational number, which is not precise, represent something as specific as an interval, he asks? Boethius was considered THE AUTHORITY for a thousand years.

Even in the debates over tempering, it was often insisted that the “pure fifth” 3/2 was the best, the closest to perfection, and should be used whenever possible, or come as close to as possible.

Kepler, on the other hand, goes for the throat on this point. He acknowledges the use of incommensurables, preferring the Greek term, {alogoi}, which he translates as “inexpressible”, to the Latin term irrational (which CAN mean without reason, as well as without ratio). In Book One he elaborates their “degrees of knowability”. Everything beyond the third degree of knowledge is an “inexpressible”. What a beautiful concept: the incommensurable is ordered, in a knowable way! From Book One:

“People are always molesting inexpressibles, by trying to express them – as numbers!”

Let’s look at the problems that arise from this fixation on rational numbers:

1. The Lydian interval, even on a monochord, is represented by the square root of two, so it would be have to be banned (it was banned on its own merits as the Devil’s interval).

2. Since half-tones and tones cannot even be approximated as fractions, the system begins to break down at their determination. They invented different sizes of them: 9/8 was a major whole-tone, 10/9 a minor whole-tone – half-tones were at 16/15 (wide semitone), 18/17, 25/24 (diesis), even 256/243 (narrow semitone)! Since they were trying to add these intervals up, to the pure ratios of the above mentioned consonant intervals, they had to invent certain critters to fill the gaps, such as the same diesis, the limma (135/128), etc. Does not all this remind you of the way quantum mechanics sometimes makes up particles, to force experimental evidence to conform to a faulty theory? Doesn’t it remind you of the “made-up” epicycles in the planetary orbits? Imagine trying to teach singers to sing all these!

3. Supposedly, Pythagoras himself developed the paradox of the “comma”. If he did, one would have to admire a man who challenged his own system, but I’m not sure that was the way it worked.

I will demonstrate the “comma” from our modern terminology of tones, intervals and frequencies.

Take C at 256. Go down three octaves to C at 32. This is the lowest C on the piano. Play, on the piano, a series of 12 fifths (with one “enharmonic”): C G D A E B F#(or Gb) Db Ab Eb Bb F C. This takes you up 7 octaves to C at 4096, the highest C on the piano. However, if you take C at 32, and multiply by 3/2, or 1.5, the ratio of the “pure fifth, you will get 48 for G. Keep doing this twelve times, and instead of an octave of C, 4096, you will get 4151. The fifths are a little too large! If you have a diagram of the circle of fifths, imagine if it did not meet at the top, C, but there was a slight gap. This gap was dubbed the comma, and it had a precise measurement.

There is an abbreviated version of this. Three major thirds should comprise an octave: C E G# B#(C). An octave of C at 256 should be 512.

Multiply 256 by the ratio of the major third, 5/4, or 1.25, three times, and you will get, 256, 320, 400, 500. So the major thirds are too small! (If you wish to argue that the Greeks did not know frequencies, you can multiply the ratios and find the same problem. If you accept the octave, or diapason, as having a ratio of 2/1, then multiply 1 by 1.25 three times, and you will obtain,1, 1.25, 1.5625, 1.953125. Again, it falls short of an octave).

So, think back to point number two, the problem that arises when you try to determine the system as a whole, rather than one interval, or orbit, at a time. Kepler has great fun pointing out, that even if one rejects incommensurables for rational numbers, as the only representations of intervals, these rational numbers are themselves incommensurable – with one another!

Kepler, in Book Three, is polemical about the need to destroy this prejudice. Not only are rational numbers not the best; they are not a cause at all.

From the introduction to Book Three:

“… the causes of intervals have remained unknown to men…. I shall be the first, unless I am mistaken, to reveal them with such accuracy.”

Also from Book Three, next to a margin entitled “His error (Ptolemy’s) in treating a non-cause as a cause”:

“… since the terms of the consonant intervals are continuous quantities, the causes which set them apart from the discords must also be sought among the family of continuous quantities, and not abstract numbers, that is in discrete quantity; and since it is the Mind which shaped human intellects in such a way that they would delight in such an interval….the causes of such intervals being harmonious, should also have a mental and intellectual essence….

“if the cause was sought in abstract numbers. Yet it would still not be very clear why the numbers 1,2,3,4,5,6,etc conform with musical intervals but 7,11,13, and the like do not conform.”

In Chapter One of Book Three, Kepler states that he is using a geometrical method (the inscription of plane figures in a circular string), as a “substitute for the Pythagorean abstract numbers, which have been repudiated.”

Kepler was more opposed to the numbers being seen as a cause in themselves, than the division of strings; his division, however, is a very different, geometrical one. He inscribes the plane figures in a circular string, and orders the intervals according to the same degrees of knowability that he laid out in Book One. This is still not his highest determination.

Throughout Harmonice Mundi, he consciously UPLIFTS the cause of intervals to a cognitive one – from Chapter Sixteen of Book Three:

“The theme of that book (Five) is the sole object which I intend in this whole work. For, being an astronomer, just as I argue about the regular figures not so much geometrically….as astronomically and metaphysically, so also I write about the ratios of melodies not so much musically as geometrically, physically, and lastly, as before, astronomically and metaphysically.”


1. In the title page to the entire work, Kepler’s description of Book Three includes:

“…and on the nature and distinguishing features of matters relating to music, contrary to the ancients;”

I.e., he is refuting the ancients. The translators, Duncan and Field, insist that the only “real discovery” in the entire work, is the so-called, Third Law.

But, Kepler challenged future musicians to act on his discoveries. He sought his Bach, as well as his Leibniz. The introduction to Book Five reveals his sense of what a revolution he was unleashing:

“I am free to taunt the mortals with the frank confession that I am stealing the golden vessels of the Egyptians, in order to build of them a temple for my God, far from the territory of Egypt. If you pardon me, I shall rejoice; if you are enraged, I shall bear up. The die is cast, and I am writing this book- whether to be read by my contemporaries or not. Let it await its reader for a hundred years, if God himself has been ready for His contemplator for six thousand years.”

2. Bruce Director, in a conference presentation, quoted Ptolemy on how, of the Theological, Physical, and Mathematical causes of something, mankind could only know the Mathematical “cause”. Years ago, Bruce had a pedagogical, quoting Copernicus on how it didn’t really matter, if your mathematical model corresponded to physical reality, only if it described it. If that quote is reliable, then that, plus his continued insistence on perfect circles, would tend to put Copernicus in the Ptolemaic camp, despite his acknowledgement of the Heliocentric nature of the Solar system.

3. Harmonice Mundi is completed in 1619, at the beginning of the German part of the Thirty Years War, and the same year as Kepler’s works were put on the Index of Prohibited Books.

4. Supposedly there was a difference between Ptolemy and what was represented as the Pythagorean view, on whether it even mattered how the intervals sounded, or whether the ratios alone determined consonance or dissonance. Kepler doesn’t think there is much difference. The two Venetians, Galilei, and Zarlino, took side in this matter. 5. The translator of Boethius into English, says that his work is basically just a translation of Ptolemy. I have to check this out. Ask yourself, what is the axiomatic prejudice built into the monochord experiment?


by Fred Haight

Is the question of tempering then, just a question of finding then right ratios, and correcting the errors, or is it something far more important?

Equal tempering arose, supposedly, as early as Aristoxenus, a pupil of Aristotle, as a mechanistic procedure of simply dividing the octave into a “chromatic scale”of twelve equal tones, based only on what sounded good, in disregard to any physical cause (I’m not sure that I am giving fair due to him that’s something I have to look into more closely). A kind of gang-countergang debate sprung up, between those who said that this was best, because the ear was the ultimate guide, and those who said that you cannot abandon the physical cause of the intervals, for what seems merely sensuously pleasing. This allowed Boethius to make a phony distinction between practicing musicians (whom he considered vulgar), and the superior, theoretical musicians, who only contemplated the beauties of the ratios! Centuries later the Venetian Vincenzo Galilei, in a phony debate with his deceased Venetian predecessor, Zarlino, proposed to divide the octave into twelve equal tones by the ratio 18/17. Kepler, who otherwise speaks positively of tempered intervals, rejects Galilei’s determination as “mechanistic”(6) ( he also points out that it doesn’t work-it generates a comma). The well-tempered system is not a matter of finding the right ratios, but of CHANGING YOUR THINKING, and starting from the TOP down, in terms of the actual processes governing the universe, as do LaRouche, and Kepler. Look back at the previous quotes from Kepler, on lifting the investigation of melodies to an astronomical, and metaphysical level, and on the cognitive nature of the causes of the intervals, as communicated from the Mind, to our minds.

An academic reader would have a hard time identifying Kepler as founder of the well- tempered system after all he keeps using these so-called “pure”ratios to represent the intervals, even in the Fifth Book.

But, in the Fifth Book, he does something different. The ratio between the aphelial, and perihelial angular velocities within a single orbit, he refers to as being like ancient plainchant, a single, primitive melody.

The ratios between planets though, he refers to in terms of polyphony, which blossomed in the Rennaissance, with the development of bel-canto. In Chapter Five, he sets up two scales, one from the set of convergent ratios, i.e., from the aphelion of the lower planet (farthest out from the sun), to the perihelion of the upper ( e.g. Saturn to Jupiter; Jupiter to Mars etc.) The other is the set of divergent ratios, where he inverts the process, by starting with the perihelion of the lower planet, and the aphelion of the upper. These two scales, which he calls hard, and soft, are not exactly our major and minor: the hard scale differs from the major by one tone, but that is enough to make the two scales inversions of one another!

In Chapter Seven of Book Five, Kepler examines the possibility of several planets being in tune, at these extreme ratios, at the same time, which he compares to four-part harmony. Here, he says that a “certain latitude of tuning,”is not only acceptable, but necessary. In his charts, in this chapter, he identifies the highest and lowest possible tunings for each of these measurements. (This latitude of tuning is not an arbitrary variance, as in equal tempering, but comes from different means of measuring these physical ratios, of perihelial, and aphelial angular velocities).

After the chart on the possibility of five planets being in tune, he states: “Here at the lowest tuning, Saturn and the Earth coincide at their aphelia; At the mean tuning, Saturn joins in at its perihelion, Jupiter at its aphelion; At the highest tuning, Jupiter joins in at its perihelion.”

Even a single pair of planets being located strictly at the “pure ratio”can exclude other planets from being “in tune” at all.

The same problem arises in polyphony. So-called “just intonation”was an attempt to construct a scale with as many “pure”fifths and thirds as possible. This doesn’t even work within a single scale (how many fifths are there in any “diatonic”scale? How many major, and minor thirds? Can they all correspond to the “pure ratios”)? In both cases, bel-canto polyphony, and the solar system, tempering arises not from some pragmatic evening out of the scale, but new discoveries of physical principle in the universe; and, in both cases, tuning is determined, not “at the blackboard,”but by the composition itself, whether it be by a human artist, or The Divine Architect himself! (see footnote 10)

Lyndon LaRouche has always insisted that music originates in the polyphonic, bel- canto vocalization of sung poetry (which itself “contains a score”in its prosodic elements such as the ordering of vowels, meter, etc.) Is bel-canto vocal registration living, or cognitive; or perhaps living participating in the higher level? The discovery of solutions to paradoxical problems: to ironic, polyphonic “dissonances”through inversion etc, is cognitive, and parallels the discovery of new physical principles as solutions to paradoxes in physical science, but, such cognitive ironies, are usually expressed, as ironies in the living harmonics of VOCAL REGISTRATION, and can only exist in the “physics”of actual polyphonic musical composition, not the “classroom mathematics”of formal systems of scales, keys etc. In other words, cognitive musical ideas do not exist as disembodied notes, as Heinrich Schenker, or a counterpoint text would imagine: but, perhaps we could follow Vernadsky and LaRouche, in saying that cognitive discoveries in music, create ironies in voice registration, as “natural products,” in the “living processes”of bel-canto, much as the biosphere creates “natural products”such as soil, water cycles etc.; and as cognitive processes, generate increased relative potential population density, as a natural product, in the biosphere.

Wait a minute! Does not all this sound like what has been presented in Volume One of the Music Manual (or projected for Volume Two?) But step back a bit. Did not Lyn, in the Music Manual, revolutionize musical theory, by finding the origins of music in the highest cognitive, and living levels of physical processes? Compare this to oligarchical theories, of music originating in, “bird songs,” “the dance,””hammers hitting anvils,”etc. They all wish to eliminate the idea that human cognitive activity originates in anything human! Now, you can begin to appreciate what Kepler actually did. (7)


As discussed in a previous pedagogical, but worth repeating, human voice registration produces an entirely different set of harmonics than a mere vibrating string, characterized by a series of Lydian intervals, and a chromatic scale of register shifts (when the down shifts are considered as well as the up). (8)

Now look at the Lydian intervals in the six voice species:


Soprano-Tenor C F# B F

Mezzo-Baritone Bb E A Eb

Bass Ab D G C#

Here we have all six Lydian intervals organized in a series of descending half-steps (the next one would be F#-C), in a form where each of the two tones comprises the main register shifts for a specific voice type. C F# B F BbE A Eb Ab D G C#

This is , of course, not all that could be said about the harmonic ordering of the human voice, but here, in this series determined by voice register shifts, we have a “chromatic scale” of twelve half-tones, but from the true physical cause. Not only is every tone a register shift (as Eliane Magnan used to demand), they are not all equal. As Lyndon LaRouche first pointed out in “Beethoven as a Physical Scientist,”tones are not “point frequencies,”but more like regions of negative curvature. They can occupy an area, and move (as can, and must, Brunelleschi’s dome), according to the Analysis situs of actual composition (for this reason, well-tempering could not be derived from the keyboard, which is a FIXED tuning). (9) So, Gb can differ from F#, as Pablo Casals clearly understood, but it will differ more or less, as the composition itself, requires. (10)

Ironically though, there are still only twelve tones; all attempts to create quarter-tones etc, fail.

One of the most exciting ideas ever presented by Lyndon LaRouche on music, was in the famous footnote 65 to “The Becoming Death of Systems Analysis,”which applies perfectly to Mozart’s K.475: “The pivot of the entire composition so unfolding, is a conflict in tonality, derived lawfully from those simpler ironies of well-tempered counterpoint, but expressing a clash of ironies equivalent to an ontological paradox in physical science. Thus, it is a physical reality, as represented by the natural (i.e., bel-canto) composition of the natural-determined division of the human singing-voice . which imposes naturally generated ironies and paradoxes upon the formalist’s musical scale.”

Think of that idea: of the cognitive activity, of playfully imposing the Lydian centered harmonic series of the human voice, on a lower order, non-living, more formal harmonic species, and generating ironies and paradoxes throughout; much as the collapse of the real, physical economy, such as U. S. steel production is posing such paradoxes for the formal, utopian schemes of the Globalists, now. K 475 , by including the F# in the opening Bach statement, and the pedal point series, generates new modalities, unthinkable in, say, a pre-1782 Mozart Sonata.

So, the well-tempered system, is inseparable from human bel-canto voice registration, from the Classical principle in Art, and from the moral intent of actual classical musical composition, whereas equal tempering, implies nothing for composition. Not surprisingly, Schoenberg seized on it as the basis for his so-called twelve-tone system, which throws out the voice, the mind, etc.

Could the Greeks have known this? Could equal tempering have occurred on its own, or only as a Delphic operation against a real discovery? Polyphony is certainly natural (despite textbooks that say it wasn’t thought of until the 11th century A.D., and then as an annoying drone)! But, I’ll bet that human beings were born with natural bel-canto voices then, even as they are now. It’s true it has to be developed, but in the last century, some great singers were “naturals”for whom the “voice” was there. The nature of the best of Greek culture, and science, would suggest that they would investigate the right areas, and if you’re looking in the right place, it should not be that hard to find.

There is a lot more work to be done here, but let me ask another question: How many people, through a revolution in method, bring about such a fundamental change in science, and art? How many Leonardos, Keplers, and LaRouches are there in history? In our new Century, great works of art shall be made, by artists who absorb LaRouche’s ideas, as a whole.

Lastly, let me leave you with a beautiful thought by Kepler on tempering:

“There is an absurd arithmetic equality at banquets if everybody is seated indiscriminately, with no account taken of sex, condition, or age. On the other hand mere geometric similarity is insipid. For if the learned are put only next to the learned, what good will they do to the unenlightened? If women only next to women, what pleasure will there be? If the rowdy next to the rowdy, who will instill good behavior into them? But if you admit neither blind equality, nor peevish similarity, the proportion will be harmonic. For you will bring it out that the old rejoice to see the young,the men to see the women, the young are ruled by the wisdom of the old,the women by the authority of the men (sic), the sociable stimulate the unsociable …this is not a combination of intact kinds, but to a certain extent an infringement of them, to set up a harmonic proportion. Friendships are given life by harmonic tempering. For what concord is to proportion, that love, which is the foundation of friendship, is to the whole compass of human life.” Footnotes:

6. In the Nineteenth Century, Helmholtz divided the octave into twelve equal parts by the twelfth root of two. He follows Mersennes and Rameau in basing musical theory on the overtone series of a vibrating string, which is somewhat worse than a monochord (Mersennes and Descartes also had a phony debate over tempering going on, with Descartes taking the side against temperament).

In the late 1600s, Werckmeister, wrote that he had created one diatonic-chromatic-enharmonic scale. Diatonic vs Chromatic music was another phony debate. (Read “the Case against Rock,” and other writings from that time, and you will see how we fell into that trap). These three “genera,”chromatic, diatonic, and enharmonic, were considered incommensurable; the well-tempered system, created a Gauss-like congruence, and thus integrated them 7. Thought processes themselves are highly musical. This is an area that requires a lot of investigation, but, rather than the so-called Mozart effect, I find it very interesting that professionals who work with Alzheimer’s patients, and stroke victims, find that the musical memory persists, even when memory is otherwise impaired, or gone. 8. The F# centered voice register series is what determines C at 256. Without that, you have no defense against the arguments of “relative pitch,”which does exist. It is the intersection of the fixed, voice register values, with the “transposable”keys, which gives each key its unique “color,”and protects us against random transposition.

9. For this reason, Werckmeister had at least three tunings for his well-tempered Clavier. Though Kepler identified the difference between living and non-living processes in his “Snowflake”paper, it remained for Bach to discover all these questions of the living bel-canto voice. His son, Emmanuel, makes it clear that bel- canto was the basis, even of Bach’s keyboard technique.

10. Plato’s, and Kepler’s, Composer of the Universe, requires the same quality of change. In the Seventh Chapter, of Book Five, of Harmonice Mundi, Kepler must temper the intervals differently for the hard, and soft scales. Thus, the well-tempered system, is neither a fixed tuning, or a series of fixed tunings, but requires constant change, as generated by the composition itself!