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Ptolemy’s Harmonics

The Harmonics [12] is probably earlier than the Tetrabiblos.[13]  Although Ptolemy is well known for his astronomy and astrology, he was also one of the ancient sources for theories of harmony.
The work studies this universal field of knowledge for which music and geometry are considered subsets. Much of the material in Harmonics is exclusively concerned with music, including setting up exact number ratios for the different modes.

The Harmonics Book III applies his previous exposition of intervals, scales, and ratios to ethics and psychology, and to astronomy and astrology. Unfortunately, some of Book III has been lost to us. In Chapter 9 of Book III, Ptolemy presents the astrological aspects and their ratios to one another. Curiously, in this presentation Ptolemy does not use degree numbers for the aspects. Instead, he considers the twelve zoidia as discrete units and the whole number relationships between them that yield moriai and epimoriai. This discussion might have fit better into the Tetrabiblos Book I, within his treatment of the qualities and relationships between whole zoidia.

Ptolemy begins not with a line but the circle of twelve units for the twelve zoidia. Instead of using the line and 180°, Ptolemy considers the circle and the total of twelve zoidia that would make up the circle of the zodiac. I have provided a diagram below [14] 

Ptolemyharmonics_chartWithin this entire circle, A is a beginning of the zodiac and back again, and has the number 12 for the twelve zoidia. AB represents the diagonal line and 6 of the signs of the zodiac (the opposition), AC represents 4 zoidia and one third of the circle; AD is 3 zoidia and one quarter of the circle – this yields the diameter, triangle, and square of the circle. (The hexagon is not represented here.) Ptolemy proceeds to give many permutations of these numbers that yield the same ratios depicted in Tetrabiblos I Chapter 13. For our purposes here I will give the proportions most relevant to our discussion.

For three places together, ADB yields 9, ABC yields 8, and ADC yields 4. Ptolemy notes the distances that are double from one another (AB doubles AC, and the whole circle doubles AB), which give us the octave or diapason. He also notes the distances that give the fifth and the fourth, the diapente and diatesseron. For the former 3:2 ratio, Ptolemy notes that the whole circle (12) is 3:2 to ABC (8), as ABD (9) is to AB (6), and AB (6) to AC (3).

Ptolemy also repeats his finding from the Tetrabiblos: AB (the diagonal) is 3:2 to AD, For the latter 4:3 ratio, we note the ratio from the whole circle (12) to ABD (9), ABC (8) to AD (6), and AC (4) to AD (3). Ptolemy notes that AB (the diagonal) to AD is 3:2 (the square) or the diapente, and that AB is 4:3 or the diatesseron to AC (the triangle).

The difference between AD and AC is 1: one-twelfth of the circle. This gives an interval of 4:3 but spans one zoidion. This would correspond to the emmelic interval between the two tetrachords that constitute the diatonic scale in the Greater Perfect System, and is more of a transitional than a concordant interval according to that system.

Importantly, one cannot combine or subtract these segments of the circle to form five units or give a 5:12 ratio. This would be discordant, ekmelic, and conform to the astrological aspect of the quincunx.

Harmonic Scales and the Soul of the World

Plato’s Timaeus, concerned on its surface with cosmology and natural philosoply, is considered the most overtly “Pythagorean” dialogue in the Platonic corpus. Within a famous passage on the construction of the world, Plato (or Timaeus) depicts the Demiurge constructing the soul of the cosmos, within which time and motion and form could be realized sensibly, and matter could have a measure of intelligibility (35B-36B).
The cosmos’ soul will take the form of two bands constituting mixtures of Same and Difference that will eventually form the celestial sphere surrounding our earth. Prior to this, the Demiurge has to put together the material of the world’s soul and then sort it out according to specific quantities that conform to universal ratios. He begins by arranging two series of numbers.

One series uses the multiples of 2 to arrive at 1 – 2 – 4 – 8, etc. The other uses multiples of 3 to arrive at 1 – 3 – 9 – 27, etc. These numbers can continue indefinitely. The Demiurge then fills the intervals between them, using arithmetical and harmonic Means.[15]

Our result from 1 to 2 will be 1 – 4/3 – 3/2 – 2. This corresponds to the skeleton of the diatonic scale and the model Ptolemy uses to account for the astrological aspects.

However, by continuing exponential progressions indefinitely and including multiples and means related to the numbers 2 and 3, Plato expands his harmonics further than the realm of ordinary music.

But first it is necessary to explain more thoroughly the arithmetic and harmonic means that Plato employs.

The arithmetic mean exceeds the lower number by the same number as that number is less than the greater number. We all learned this in grade school and we still know how to do this calculation. This gives the same number between the lesser and greater numbers, by dividing the numbers of the two extremes. This gives us 3/2 between 1 and 2, 2 between 1 and 3, and 6 between 3 and 9. There are no surprises here. The harmonic mean is a more complex calculation and is more difficult to grasp. The harmonic mean exceeds the lower extreme by the same fraction, as the mean is less than the greater number.

  • Between 1 and 2, 4/3 exceeds 1, the lower number, by 1/3.
  • If you take the same fraction 4/3, this is less than the number 2 by 1/3 of 2 (converting 2 to 6/3), as 6/3 – 2/3 is 4/3.
  • We see the same pattern between 4 and 8, whereby 5 1/3 exceeds 4 by 1/3 of 4. 5 1/3 is less than 8 by 1/3 of 8. There are two ways to compute the harmonic mean between two numbers.

One is this formula below, A and B being the quantities of the two extremes. This will work perfectly to find this mean between any two numbers you choose.

2 AB)

A + B

The other way is more interesting but more specific to the multiples of 2 and 3 that Plato uses. It requires very simple calculation, and it shows the interdependence between the multiples and divisions of 2 and 3. For multiples of 2, we convert the extremes into thirds to obtain the harmonic mean.

Using 4 and 8 again, convert the former to 12/3 and the latter to 24/3. We are converting multiples of 2 to fractions with 3 as the denominator.

  • Add the lesser whole number 4 to 12 (the numerator of the lower number) and you get 16.
  • Subtract the higher number 8 from 24, the numerator of the greater number, and you get 16.
  • Therefore the harmonic interval between 4 and 8 is 16/3.
  • One will see that if the exponents of 3 are converted into halves, one easily arrives at a harmonic mean. If we calculate the harmonic mean of multiples of three, we convert the extremes to halves.
  • Between 1 and 3,

Convert 1 to 2/2 and 3 to 6/2.

  • The lesser whole number (1) plus its numerator (2) is 3.
  • The lesser whole number (3) from its numerator (6) is also 3.
  • This will give us 3/2.

What is the harmonic mean between 3 and 9?

  • Three is 6/2 and 9 is 18/2.
  • Add the lesser whole number 3 to the numerator 6 and the result is 9/2.
  • Subtract the greater whole number 9 from the numerator 18 and the result is also 9/2.

Importantly, this second procedure breaks down when attempting to find harmonic means between exponents of 5, 7, and so on.

In this way we can calculate the mean whereby adding the same fraction of the lesser number and subtracting the same fraction of the greater number gives a mean. We return to Plato’s Divine Worker. He combines the means between multiples of two and three into a single band. The series of the multiples of 2 and 3 are as follows. 1 – 4/3 - 3/2 – 2 – 8/3 – 3 – 4 – 16/3 – 6 – 8, 1 – 3/2 – 2 –3 – 9/2 – 6 – 9 – 27/2 – 18 – 27
They will make, 1 – 4/3 – 3/2 – 2 – 8/3 – 3 – 4 – 9/2 – 16/3 – 6 – 8 -- 9 – 27/2 – 18 – 27 And so on.

Plato’s Demiurge fills in numbers between by units of 9/8, corresponding to single tones in music, The amounts remaining he would fill in by units of 256/243, which are the semi-tones as represented in Greek musical theory. However, there is no limit to how far to take these numbers: the multiples simply continue. The reader probably knows the rest of the story. Having divided the main substance according to these proportions, the Demiurge fashions a very large circular band that he then cuts lengthwise into two and then brings them together in the form of a Chi. One becomes the Circle of the Same, our celestial equator, upon which the fixed stars move and which moves from east to west, the diurnal cycle. The other becomes the Circle of the Other; the ecliptic. This circle moves from west to east and divides itself further so that the seven planetary bodies can move along it. Hence time as ordered becomes possible.

Plato’s proportions show intimate relationships between number, music, and of the soul of the world. All this seems necessary the postulate how the world would need to be for true opinion to arise, and to account for true knowledge found reflected in it. This cosmological story brings us into the motif of the harmony of the planetary spheres, an idea that was pervasive in the Hellenic and Hellenistic worlds and lasted well into the Renaissance. Johann Kepler’s 1619 work Harmonice Mundi was probably the last full attempt to bring together the motions of the planets and harmonic ratios.

We return to astrology’s aspects, now using some of the proportions we see in the Timaeus. We begin with the cleanest example. 1 – 4/4 – 3/2 – 2

These correspond with the astrological aspects according to Ptolemy’s exposition in Tetrabiblos I Chapter 13, so that 4/3 gives us the hexagon or sextile and 3/2 gives us the square.

If one uses the numbers between 1 and 3 in the same way, however, you get something we haven’t seen before. 1 – 3/2 – 2 – 3 3/2 corresponds not to the sextile but the semi-square, an astrological aspect of 45°, half the square of 90°. This aspect requires using degree numbers, not whole zoidia.

The semi-square violates Ptolemy’s use of whole masculine and feminine zoidia to describe the effects of different aspects. Because Ptolemy uses specific numbers in Tetrabiblos I Chapter 13 for the astrological aspects, he expands the possibilities for aspects beyond those that are between whole zoidia. By employing multiples and means that Plato uses, the astrologer can finds himself or herself with a wide range of new possibilities.[16]

Conclusions

It is clear, from Ptolemy’s digression in Chapter 13 of Tetrabiblos I, that using degree numbers makes it possible for astrological aspects to imitate universal laws of harmony and thus account for their effects. Ptolemy presents a correspondence between aspects and musical harmony that allows us to see astrological “action at a distance” in a new and profound way. The modern mind may not consider the wider implications of these correspondences. How do the teachings on musical harmony help us account for the aspects of astrology?

Sitting down at a piano briefly supplies us with the answer. Sounding out harmonious tones (homophonous or consonant), they can be said to meet each other, to interact with each other. In music they act upon each other because of their distance along the scale. I know of no other phenomenon in nature in which interaction is based upon number ratios related to distance between two agents.

I also remind you of the contrasting experience that is quite familiar – the discordant and ugly result of striking the wrong note. This is the result of having accidentally come upon an unmelodic interval in the context of the harmonics established within the piece being played. An accomplished musician or composer may find a way to resolve the discord, but does so by finding a way back to the original harmonic intervals. The experience of accidentally unmelodic intervals may correspond to disharmony in the world, in the individual soul, and between planets affiliated by dispositorship but in disconnected zoidia.

Two or many harmonious tones played together also create a blend of sameness and difference that is analogous to the relationship between a visual perceiver and its objects of perception.

One can represent musical tones and intervals by numbers and ratio. Their arithmetical properties allow us to move from the aural sense perception of musical tones to an intellectual arena of harmony. This harmony may manifest in the soul of the world, the soul of the individual, and even given an account for the aspects of astrology.[17] The sensible model for principles of harmonics is music, not geometry, since parts of a geometrical figure do not interact with each other based upon the ratios of their distances.

Because divisions and multiples of 5 or 7 do not fit into these harmonic models, they cannot themselves form the basis for either musical harmonies or astrological aspects, if the correspondence between aspects and harmonic intervals is to be taken seriously.
Ptolemy’s argument in the first past of Chapter 13 is indeed a digression. The remainder of Tetrabiblos I uses the natural philosophy of his day to account for astrological effects in general. His argument for aspects, however, derives from Pythagorean and Platonic sources as evidenced in his earlier Harmonics. Yet his digression roots us in some of the basic principles of the western intellectual tradition.

REFERENCES

[11] In the Greater Perfect System, which was dominant in ancient harmonic theory, and generally covers two full octaves. Tetrachords where also cast in chromatic and enharmonic forms, although our interest here is in the diatonic. See R.P. Winningham- Ingram, cited above.
[12]  J. Solomon, Ptolemy Harmonics: Translation and Commentary.(2000) Leiden, Boston: Brill
[13]  See N.M. Swerdlow, “Ptolemy’s Harmonics and the ‘Tones of the Universe” in the Canobic Inscription” Charles Burnett, Jan P. Hogendijk, Kim Plofker, Michio Yano (edd.): Studiesin the History of the Exact Sciences in Honour of David Pingree, Leiden – Boston 2004 (Islamic Philosophy Theology and Science. Texts and Studies; Vol. 54).
[14]  N. Swerdlow, cited above. Pg. 154.
[15]  A fuller explanation is in D. Zehl. Plato’s Timeaus. (2000) Indianapolis, In.:Hackett Publishing, 2000) and F. Cornford, Plato’s Cosmology.(1935/1997) Indianapolis, In.:Hackett Publishing.
[16] Take the distance between two different numbers and superimpose that on the first 180° of the zodiac. It gives some intriguing possibilities. Using the sequence using numbers 1 through 9: 1 – 3/2 – 2 – 3 -9/2 – 6 – 9, if 1 is 0° Aries and 9 is 0°Libra. 3/2 is the semi-sextile of 30°; 2 corresponds to 40° which astrologers know as the novile, which divides the 360° circle into ninths and is the foundation of the modern Ninth Harmonic: 3 is the sextile, 9/2 is the square, and 6 is the trine. Modern astrologers who use Ninth Harmonic astrological charts may take comfort in this sequence. Now we take the sequence from 1 to 4: 1 – 4/3 – 3/2 – 2 – 8/3 – 3 – 4, if 1 is 0° Aries and 4 is 0° Libra 3/2 is slightly less than a sextile; 3/2 is a 67°30 aspect, which is a semi-square and another half of that; 2 is the square, 8/3 is slightly more than the trine; 3 is the sesequiquadrate of 135°, which is a square and a half-square. Plato’s expanded harmonics give possibilities to the modern astrologer that would be unavailable to Ptolemy, who restricted aspects to those whose aspecting zoidia have the same aspect. On the other hand, as we have seen with the sequence from 1 to 4, we sometimes only get approximations to aspects’ conventional degrees.
[17] The topic of the harmony of the soul is beyond the scope of this paper. It is suggestive that Plato’s Timaeus is supposed to have taken place the morning after the long discussion of the “just” – well proportioned – soul in the Republic. (Also see E. McClain, The Pythagorean Plato (1978) York Beach, Me.: Nicholas-Hays) In Ptolemy’s Harmonics III, Chapter 5, he brings together the harmonious activity of the soul as the integration of its parts resembling the familiar intervals of the diapason, diapente, and diatesseon. and haunts us with the possibility that the symbols of astrology have something to do with the nature of reality.

  • Ptolemy
  • Harmonics
  • Aspects
  • world soul
  • Hellenistic astrology
  • Tetrabiblos
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    joseph.crane2@gmail.com (Joseph Crane) Hellenistic Astrology Mon, 06 Jun 2011 16:15:48 +0000
    Ptolemy's Digression, Part 1 http://kepler.edu/home/index.php/news-mainmenu-139/articles-mainmenu-157/hellenistic-astrology/item/351-ptolemys-digression-part-1 http://kepler.edu/home/index.php/news-mainmenu-139/articles-mainmenu-157/hellenistic-astrology/item/351-ptolemys-digression-part-1 Ptolemy

    Ptolemy’s Digression: Astrology’s Aspects and Musical Intervals
    Statement of the Problem

    This essay addresses a problem in the development and continuity of astrology: how do astrologers, past and present, account for the astrological aspects? Aspects are the means by which a planet or position (Ascendant, Lot of Fortune) has contact from another planet or other planets. Once an astrologer has designated planet or a position to answer a question posed to the astrological chart, aspects to the designated position provide information to answer a question and determine an outcome.


    How is it that this connection by aspect occurs? Because aspects are based on the distances of two positions from each other along the zodiac, the solution is not obvious. Of course, one needn’t question aspects at all. The teachings on aspects from ancient India are very straightforward. All aspects are cast forward in the zodiac, and each planet aspects the house (and sign) opposite to it. Additionally, Mars aspects the fourth and eighth houses from itself, Jupiter aspects the fifth and ninth, and Saturn aspects the third and tenth from itself.[1]

    These rules are part of the astrological craft passed down from their tradition. Most ancient western astrologers also used aspects without questioning them, and were not different from their Indian cousins in this regard. However, the Hellenistic mind, and that of Ptolemy in Tetrabiblos I, sought to give astrology a coherent theoretical form, and integrate astrology more firmly with other fields of understanding. This attempt is, in large part, the reason why the Tetrabiblos is the most important single work in the history of western astrology.

    We will closely examine Tetrabiblos I Chapter 13[2]. Here Ptolemy gives two accounts for aspects. The first one, the subject of this paper, is based on what we might call “fractions” and “super-fractions,” moriai and eprmoraia. The second argument of Chapter 14 depicts sympathetic and unsympathetic aspects through the nature of the zodiacal signs or zoidia brought into aspect.

    Ptolemy’s first account in Chapter 13, although of a different style than the surrounding material, gives a provocative and critical account of aspects – one related to harmonics and the diatonic musical scale in particular. Later astrologers, from the Renaissance into the modern era, subordinated musical harmonics to an arithmetic template alone. They have brought more information to astrological analysis but perhaps with a shakier theoretical foundation.

    When, as a new astrology student, I first learned about them,the account given was wholly arithmetical and geometrical: because we can divide the whole circle by halves, quarters, thirds, and sixths, we can connect planets to one another by aspect. These relationships also give us the line, square, and what astrologers call the trine and sextile; they are geometrically the side of a triangle and hexagon within a circle. These aspects divide the whole circle into sections divisible by the twelve signs of the zodiac. We might also ask, however, why it is that we do not consider the dodecahedron, a twelve-sided figure, and thus an aspect of 30°? This should fit conveniently with the others, but this was not considered a true aspect in the ancient tradition, nor, indeed, by most modern astrologers.

    What is it about number relationships that empower planets to act upon one another, based on their distance from each other?

    The modern mind has an easier time comprehending action at a distance, because the physics of the modern era has given this understanding to us. And if we are not of a theoretical bent, we have our various remotes to unlock our cars, open the garage door, turn on and off our televisions and radios, and so on.

    Because of our background in popular science and technology, modern astrologers do not raise a skeptical eye to our understanding and use of the astrological aspects. This does not solve the problem, however, since neither gravity nor electromagnetic waves can account for their action.

    Most of the Greek words for aspects are those of seeing or looking. Directly or indirectly, aspects are acts of visual perception.

    Ancient astrologers indeed thought of aspects in terms of seeing and being seen. A planet “looks ahead” – epothoreõ -- to another planet, forward in the zodiac, to which it is in aspect. In return, the aspected planet “casts rays” – aktinoboleõ – back to the aspecting planet.[3] Additionally, an aspecting planet may “testify to” or “witness” – epimarturõ – another planet.

    Our English “aspect” is indeed such a word, as is the Indian word for aspects, drishtis, from the word “to gaze.”

    Two planets in the same sign or zoidion – the modern “conjunction” -- are not in aspect. Seeing words are not used for these relationships: instead, planets in the same zoidion are considered with one another. The planet must be outside its own immediate zodiacal environment than the other in order to see or be seen.

    Modern astrologers might note that vision is action at a distance, since we routinely see things distant from us. Our science books tell us that light waves are along a range of bandwidth of a vast vibratory spectrum that surrounds us. These waves allow an outside object to be represented to us, although an otter or a bat might “see” something quite different.

    Modern theories of vision hardly apply to astrology’s distinction between aspects that are based on the distance that are, in turn, based upon number. Nor are ancient theories of vision helpful to us. Nor perhaps to Ptolemy, who did not use visual perception to account for aspects, although most aspect words imply this kind of action.

    Ancient tradition gives a variety of accounts for visual perception. To our modern sensibilities, they range from relatively straightforward to rather weird. Aristotle’s De Anima takes not vision but touch as the most basic – and paradigmatic –sense faculty, and posits that vision, like the other sense faculties, has a medium (metaxu) by which the object carries itself to the perceiving subject. For vision, this medium is light. The object somehow alters the light by which the view of the object comes to the subject.

    “For what is to be colour is, as we say, just this, that it is capable of exciting change in the operantly (actual) transparent medium: and the actuality of the transparent is light.” (418b) [4]

    According to the Stoics, what binds together the object of perception and the subject is the tensing of pneuma. In the case of seeing, “the seeing-pneuma in the eye makes the object visible by ‘tensing’ the air-pneuma into a kind of illuminated cone with the object at base and eye at apex; the tension of this air is experienced as sight.”[5] The Epicurean school posited that images flow from the objects themselves in a constant manner and the eye picks them up.[6]

    Another possibility, of uncertain seriousness, is found in Plato’s Timaeus, 45 B-D. After commenting on the fact that the human body is well suited for the faculty of sight, especially to look up toward the heavens, he notes that vision occurs through means of the fire of daylight, another kind of fire in one’s eye faculty, and fire emanating from the object seen. As they connect, we see something.[7] In his astrological analysis, Ptolemy also uses words for seeing and looking (as well as witnessing or testifying) when referring to aspects. He does not use these words in Tetrabiblos Iwhen giving an account of the aspects themselves.[8] If the act of looking or seeing requires a medium to connect object of sight and subject, it is not at all clear what a medium would be that could carry the aspects of astrology. The medium must be the aspect intervals themselves, but how?

     


     

    Outside and Inside Ptolemy’s Digression

    Even in this theoretical section of Tetrabiblos I, Ptolemy’s allusion to musical intervals is out of place where it stands. Preceding and following Ptolemy’s accounting of aspects in Chapter 13 is material solely related to classification of zoidia as discrete units.[9]

    Chapter 11 discusses zoidia as cardinal (related to a solstice or an equinox), fixed, and mutable or double-bodied. Chapter 12 classifies them into masculine or feminine. The end of Chapter 13, continuing the topic of aspects, states that trines and sextile are sympathetic (sumphõnoi) because their genders agree and squares are unsympathetic (asumphõnoi) because their genders differ.

    Chapter 14 divides zoidia into commanding and obeying, based on their northern or southern declination respectively, symmetrical to the 0°Aries/0° Libra axis. (These particular zoidia are also symmetrical with respect to their rising times.) Chapter 15 concerns itself with the zoidia of “equal power,” symmetrical to the 0° Cancer/0° Capricorn axis, and spending equal amounts of time above the horizon. Chapter 16 takes up aspects again and tells us that zodiacal signs are averse (asundeta) when they are not familiar by aspect, nor in a relationship of commanding/obeying or equal power, i.e. symmetrical to the cardinal axes.

    Chapters 17-20 discuss the affiliations of the zodiacal signs to planets by means of domicile, triplicity, and exaltation. It is only when discussing the horia (translated as “terms” or “bounds”) does Ptolemy consider segments within the zoidia.

    Now we take up the material in the first part of Chapter 13.

    We have already noted the unusual vocabulary Ptolemy uses for aspects. A more important surprise is that he uses degree numbers for them. Many modern astrologers, I included, have read this passage sleepily, not noticing anything unusual -- yet in the context of his emphasis on whole zoidia in this part of Tetrabiblos I, the degree numbers are completely out of place.

    Using aspects from zoidion to zoidion, from one planet in Aries and another in Gemini, for example, planets could be in aspect to each other regardless of where in their respective zoidia they happen to be. Two planets in sextile by whole zoidia could be distant from each other anywhere from 31° (late Aries to early Gemini) to 89° (early Aries to late Gemini). As long as these planets are two zoidia from each other, they are in a hexagonal interval, i.e. in sextile. Why, then, does he include 180° for the diameter or opposition, 90° for a square, 60° for a hexagon, and 120° for a triangle?

    He continues.

    The diameter causes the zoidia – or the same degrees of opposite zoidia – to meet on a straight line. This is clear enough and also corresponds to the visible sky: if two planets are in opposition, one will be seen to rise at the same time as the other is seen to set.

    To derive the other aspects from the straight line, Ptolemy employs fractions (moriai) and “super-fractions” (Schmidt) or “super-particulars” (Robbins) – (epimoriai). First we discuss moriai. Bisecting the line above into two right angles gives us the square of 90°, and taking one-third of the line gives us the hexagon of the circle of 60°, and doubling the one-third mark gives us the triangular configuration of 120°. This gives us all the aspects by degrees that have come down to us as “Ptolemaic,” using what appears to be an arithmetical and geometrical account. The epimoriai, however, bring us into a different field.

    Ptolemy gives us two epimoriai, the hêmiolos and the epitriton, which correspond to one and a half (3/2) and one and a third (4/3) respectively. Ptolemy cites these aspect intervals in an interesting way. If you take these amounts related to one of the right angles, the hêmiolos (3/2) forms the square (90°) from the hexagon (60°), and the epitriton forms a triangle (120°) from the square (90°). In Latin these proportions are called the sesquialter and the sesquiterian.

    Here is the basic figure.

      Hexagon Square Triangle Opposition
      1/3 1/2 2/3 1/1
    0°___________ 60°___________ 90°___________ 120°___________ 180°___________
              3/2           4/3    


    Having mentioned these proportions at the beginning of Chapter 13, he drops the matter entirely. However, Ptolemy is nothing if not intentional and I cannot imagine that he would make a random point and just leave it. Yet he appears to do so.

     

    Both the moriai and the epimoriai he cites relate to ancient music and in particular the pentatonic musical scale.

    Perhaps a short introduction is in order here. Discovering that musical tones correspond to specific number ratios is a discovery attributed to Pythagoras and throughout history has been associated with the teachings of the Pythagoreans. Using a string or a wind musical instrument, a discrete tone arises from plucking or strumming a vibrating string or blowing air through a hollow of some kind. Other tones relating to this tone come from holding the vibrating string or containing the air somewhere up or down the length of the string or air current.

    If you sound out a vibrating string or air current and put your finger exactly halfway, you get another tone one octave higher –the same relative tone but at a higher pitch. Using the key of C, all the while keys on a modern piano, this is the interval from C to c. These two tones are homophonic.

    The beginning tone is given a ratio of 1:1. A note one octave higher gives a proportion of 2:1. This interval is the diapason. If you place your finger half again, that gives two octaves and a proportion of 4:1. Moving through many octaves, the ratios for intervals yield successive multiples of 2.

    If you take a string or an air current, and divide that into thirds and pluck the string or stop the air within the smaller segment, you get a tone between the higher and next higher octaves. If you drop this tone one octave you arrive at what is called the musical fifth, which gives a ratio of 3:2 to the fundamental of 1:1. Using the key of C, we have the note G. This interval is the diapente. The tones are not homophonic but consonant.

    If you take the original string or air current and lengthen it by half, you get a tone that is somewhat lower than the original tone but less than an octave below. If you raise this lower tone by an octave, you get what is called the musical fourth, which gives a ratio of 4:3. Using the fundamental C, we arrive at our F. This is the diatessaron. This interval also tones that are all harmonious.

    This yields the fixed tones of C – F – G – c for the key of C. The octave or diapason is from C to c, or, in the key of G, from G to g. The fifth or diapente is the interval from C to G and from F to c. The fourth or diatessaron is the interval from C to F and from G to c.

    Although there is much tradition about the ratios that make up all seven discrete tones of the diatonic scale, this scale also varied according to the modes of ancient Greek music, e.g. Lydian, Phrygian, Dorian, and so on.[10] We know far more about theory of Greek music than its practice.

    It is important to note that the dynamics of the diatonic scale does not conform precisely to Ptolemy’s rendition of the figure that yields astrology’s aspects, although both use the same ratios.

    Ptolemy’s use of the line for the 180° opposition gives exactly halves and thirds of 90° and 60° respectively, yielding the square and sextile. The diatonic scale spans eight notes, including the two homophonic notes one octave away. This scale traditionally consists of two tetrachords, consisting of two intervals of the fourth between the lower and upper notes, and a tone in between both tetrachords.[11] Again using the key of C, one tetrachord is between C and F, another between G and c, with a tone between F and G.

    Ptolemy supplies us with a rudimentary musical scale only. Perhaps this is sufficient.

    Part 2 in our next newsletter continues with an examination of Ptolemy’s Harmonics.



    References:

    [1] See J. Braha, Ancient Hindu Astrology for the Modern Astrologer. (1986) Hollywood, Fl.: Hermetician Press. pg. 55

    [2] This is according to the translation by F.E. Robbins (1994). This chapter appears as Chapter 14 in the translation by R. Schmidt (1994) The latter combines Chapters 10 and Chapter 11 of the Loeb edition.

    [3] Hephaistio, Book I Chapter 16; Tr. Schmidt,(1994) Cumberland, Md. Golden Hind Press. Antiochus, Chapter 20 Tr. Schmidt (1993) Cumberland, Md. Golden Hind Press

    [4]Translated by R.D. Hicks. See Aristotle’s De Anima In Focus, Ed. M. Durrant. London, Routledge, 1993.

    [5] J. Annas, Hellenistic Philosophy of Mind. (1992) Berkeley, Ca;: California University Press , pg. 72

    [6] ibid, pg. 158-159

    [7] F. Cornford, Plato’s Cosmology. (Hackett Publishing, 1997) pg. 152

    [8] In Tetrabiblos Books III and IV, Ptolemy presents astrological analyses of issues such as soul, profession, and parents. In this context, he also uses conventional terminology for aspects. In Tetrabiblos Book I, which is the more theoretical book that is the main concern of this article, he uses the words schêmatizô and suschêmatizô to refer to the aspects. These words refer to forming a figure or posture, as a group of dancers in an ensemble performance. It appears that instead of the planets looking to and back from each other, Ptolemy alludes to our perceptions when watching the planets in an arrangement with one another.

    [‘9] The numbers of the chapters are as they appear in the Robbins, not the Schmidt, translation.

    [10] The “species of the octave” preserved the fifth or diapente and fourth or diatessaron in Lydian, Phrgyian, and Hypophrygian modes, although they did not in the others. The Dorian and Hypodorian have a diminished diapente, and the Hypolydian and Mixolydian have a diminished diatessaron. See R.P. Winningham-Ingram, Ancient Greece. The New Grove Dictionary of Music and Musicians I., Ed. S. Sadie(1980) London, Macmillan. Pg. 665. Modes appear to have pervaded the practice of music, and theoreticians seem to have tried to make the best of them. Plato, in Republic (Book III, 398 C- 399 D), characterized their effects and banished most of them from an ideal state; Aristotle gave a more tolerant description of their effects. (Politics, Book VIII Chapter 5. 339-1342) If the general principles of harmony can make for the well-proportioned soul, the different musical modes seem to conform to discrete personality styles.

    • Hellenistic astrology
    • Aspects
    • history of astrology
    • Tetrabiblos
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      joseph.crane2@gmail.com (Joseph Crane) Hellenistic Astrology Sat, 05 Mar 2011 15:38:35 +0000