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.
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?
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.
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. 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. 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.