By: Joseph Crane

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]

Within 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.