There is a wonderful paradox in Kepler's method. He approaches everything knowing what he is looking for, and finds it; but at the same time, makes lawful, yet unexpected discoveries. He knew, that in discovering that the principle of non-constantly changing curvature, was the actual "characteristic" of the solar system, he had negated not only the Ptolemaic, but Copernican universes. What new "characteristic", or "curvature", would he discover in music? After examining the intervals formed by the ratios of the aphelial, and perihelial motions of individual planets, Kepler then examines the extreme intervals of different planets with one another. He finds that the intervals between the planets form two scales, both running from G to g. It is notable that Kepler does not try to find all the modes accepted at that time, but only two. (5) These are usually translated as major, and minor. - although Kepler's terms, hard and soft, are better (6). His emphasis on these two types can be found in Chapter Six of Book Three, and as early as Chapter Twelve of Mysterium Cosmographicum. This author has long been intrigued by the revolutionary change from the modal system, to that of 24 keys These are two lattices that are completely different. The problem is, that since they are formal "fixed systems", you can't find the revolution by comparing the two lattices, since both are relatively untrue! You need to find the DISCOVERIES of physical principle, in order to find the revolution. As far as I know, no-one else besides Kepler examines only two scales that are minor and major scales of the SAME OCTAVE. (The "lattice" of the 24-key system, considers relative major and minor scales that share the same key signature to be the most closely related, such as C major, and A minor, where inversion is not as much a question). This change, from the "modal" to the "tonal" universe is usually seen as taking place gradually, over almost two centuries (7), but discoveries are made by MINDS, and the principle of inversion, along with the related Lydian principle is, I believe, the key to this change. Kepler's soft scale, corresponds to what we would today call the G "natural" minor scale. However, his hard scale differs from the G major scale, in that it features an F natural rather than F# (7). Here is the most important point in this entire writing.This might seem like a deficiency, until you realize that these two scales are INVERSIONS OF ONE ANOTHER! Try it! The hard scale is the old Mixolydian mode, but is it not also C major, from G to G, which is the inversion of G minor? (G A Bb C D Eb F G inverts onto G F E D C B A G). This is unique to Kepler! I certainly don't know everything being discussed in Kepler's time, but I can think of no "theorist" who does this. They all tend to see scales as fixed, a priori existences, rather than arising from a form of CHANGE, such as inversion. Inversion is important for all of Kepler's work, (8) and he places much importance on these two types of scale. From Chapter Five of Book Five: "Accordingly, the musical scale or system of one octave with all its pitches, by means of which natural song is transposed in music, has been expressed in the heavens by a twofold way, and in two, as it were, modes of song." In chapter nine, when discussing that none of these harmonies can occur by chance: "[L]east of all that very subtle business of the distinction of the celestial consonances into two modes, hard and soft, should be the outcome of chance, without the special attention of the Artisan: "Accordingly you won't wonder any more that a very excellent order of scales has been set up by men, since you see that they are doing nothing but aping God the Creator and to act out, as it were, a certain drama of the ordination of the celestial movements. "But there still remains another way whereby we may understand the two-fold musical scale in the heavens, where one and the same system but a twofold tuning is embraced, one at the aphelial movement of Venus, the other at the perihelial." Kepler associates these two scales, with his discovery of the cause of the intervals in his second great archetype. In chapter nine, propositions 24-35, he locates a major (hard) sixth between the aphelia of the Earth and Venus, and a minor (soft) sixth between their perihelia. The two movements differ then by a diesis (the smallest kind of half-tone). He says that the two planets that SHIFT the genus of harmony SHOULD differ by this interval. Thus the half-tone is the characteristic interval of inversion. Although both scales use perihelial, and aphelial motions, he says that "THE HARD SCALE IS DESIGNATED PROPERLY ONLY IN THE APHELIAL MOVEMNETS, THE SOFT, ONLY IN THE PERIHELIAL" (Ch 9-prop 33). He also says that "Saturn and the Earth embrace the hard scale more closely, Jupiter and Venus, the soft" (prop 35) (9) So, the question is, is Kepler really the discoverer of Bach's principle of inversion, and, for that matter, the 24-key system? First, keep in mind that we speak of the PRINCIPLE of inversion, which means that it is more than just the mechanical act of inversion, which is much more ancient than Kepler, or Bach. Remember the example from "Grammar and Science": "Please: feed the cat" "To what?" "The meaning of either of the two statements is located in the Bach-like mental act of inversion." Focus on the words, "mental act". Don't look for letters turned upside down, or words running backwards and forwards at the same time. Praxiteles' sculpture, is like polyphony, not because of resemblances in the art forms themselves, but because both create paradoxes that must be resolved, in the mind. The principle of inversion, also addresses the question of discoveries of PHYSICAL principle, that negate the Ptolemaic systems, whether astronomical, or musical; as both Kepler's and Bach's discoveries do. So, Kepler's discovery of the role of inversion in the solar system, points towards exactly such a principle, as the basis of a necessary revolution in music. This will have the musical priesthood howling. Such people always wish to believe that music is a hermetically sealed tuna sandwich, growing into whatever it does, as a purely internal matter. But, when a scientific discovery about the universe is made, art must change itself, according to that discovery. Thus, Kepler, and Leonardo, revolutionize music as much, or more, than any composer. We can only imagine what future works of art will come out of Lyndon LaRouche's discoveries! - Kepler's Little Joke on the Lydian Interval? - In Book Three, there is little or no mention of the Lydian principle. In Book Five, there are hints of it. The most well-known reference is the question of the destroyed planet, but there are others. Chapter Six of Book Five, ends in an unusual way. Kepler finds, between the aphelial, and perihelial motions of the Earth, a half- tone, to which he assigns the tones G Ab. The last of his marginalia reads thus: "The Earth sings MI, FA, MI so that you may infer even from the syllables that in this our domicile MIsery and FAmine obtain" In Guido's hexachord system, Mi-Fa always refers to a half-tone within a single hexachord, but across hexachords, it refers to the "devil's interval." If you remember a previous pedagogical entitled "Bach's Little Joke on the Lydian interval", the pedantic verse prohibiting the Lydian interval was: "Fa contra mi- mi contra fa est Diabolus in Musica" Bach's answer was: "Fa-mi mi-fa est tota Musica" This proscription was very well-known. Kepler refers to Mi-Fa as a half-tone in Book Three, but he could not have been unaware of the "Lydian" implication of Mi-Fa, as his pun on the "dissonance" of misery and famine, attests. (10) Perhaps making fun of this devilish ban was common to great Humanist thinkers. Notes (5) In Chapter Fourteen of Book Three, Kepler makes his own attempt to determine the modes. At first, he contemplates twelve modes, based on twelve tones, but ends up with variants of the old church modes. But, in Book Five, Kepler is only concerned with two modes. (6) Hard and soft, which are used in German today, seem to correspond more to the inversion intervals of arithmetic-harmonic mean (which Kepler discusses in the Third Book), than do major and minor. In the hexachord system, C is the natural hexachord, G (the arithmetic mean of the C octave) hard, and F (the harmonic mean) soft- yet; they are not referred to as major and minor. Kepler refers to major and minor thirds as hard and soft. The major third is the arithmetic mean of the fifth, and the minor third, the harmonic. Take the fifths above the three first tones of the natural, hard, and soft hexachords, C-G, G-D, and F-C. Divide them at the arithmetic (hard) mean, or major third: G D C E B A C G F And you will have all the tones of the C major scale. Then divide the same fifths at the harmonic (soft) mean, or minor third: G D C Eb Bb Ab C G F And you have all the tones of the "natural" minor scale. Here you have both scales derived as inversions of the division of the octave and the fifth. (7) It is usually seen as being being finalized with Bach's 1722 Well-tempered Clavier, which is passed off as "demonstrating that one could play in all 24 keys on the same instrument, if it is well-tempered." A more interesting question is: did anyone ever compose in keys such as Gb major, or G# minor before? (8) In chapter five, through an adjustment in the orbit of Mars, he does add the F# to the hard scale. The same scale is a kind of transposed Lydian scale, because he adds a C# from the aphelial motion of Mercury. G A B C C# D E F# G (9) (insert footnote on Kepler's discoveries of non-linearity of inversion from "Riemann for anti-dummies") (10) Kepler finds, in the aphelial motion of Saturn to the perihelial, the ratio of 4:5, a hard third, but in Jupiter 5:6, a soft third. The two compound into a fifth, 3:2. (Ch 9-prop 11). G B D is an ascending major third and minor third, compounding to a fifth. D Bb G is a descending major third, and minor third. Kepler finds Saturn at G B, and Jupiter at G Bb. (11) Although stressing the role of consonance, Kepler throughout, underlines the role of dissonance. In Chapter nine, prop twelve, he cites the cube, and octahedron as being defined by the octave and double octave, but sees the sphere between them as 1: the square root of two, the ratio of the Lydian interval. Thus he sees this ratio as being between the solids, which is of interest, since the Lydian interval often lies between the keys. In the early part of Book Five, he says that some solids are wed to each other through rational proportions, and others divine proportions (Golden Section), such as the wedding of the dodecahedron, and the icosahedron, which are between the Earth, and it's neighbours. When considering merely the perihelial, and aphelial proportions, he finds the Lydian interval between Earth and each of its neighbours. He rejects that determination , though, for the ratios of the motions. In reading these hints at the Lydian principle, I keep thinking of Kepler's own footnotes to the Mysterium Cosmographicum, written twenty-two years later, and how his own thinking had developed. What might he have written twenty-five years after Harmonice Mundi?
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