John Addey and the 15th harmonic

By way of providing evidence that James Wilson is far from being the only major astrological writer who has drawn attention to the 15th harmonic aspect of 24 degrees, which Wilson called the quindecile, I wish to point out that John Addey devotes almost three pages of his extremely well-known modern work 'Harmonics in Astrology' (1976) to expressly speaking of the importance of the 24 degree aspect in astrology. See pp. 172-174. I shall quote just the first paragraph but believe this piece of recent history penned by a widely respected leader in modern astrological research in the mid-late 20th century undermines any claim to the 165 degree aspect being regarded as 'the useful quindecile' or similar in modern practice.
One very important derivative of the quintile series which is almost entirely ignored, as a rule, is the third sub-harmonic of the quintile, which gives us the aspect of 24? and its multiples. This is the fifteenth harmonic, the third of the fifth or the fifth of the third. Figure 60 shows the aspect angles involved. It will be seen that this series includes the angles of 72?, 120? and 144? with which we are familiar but also the angles 24?, 48?, 96? and 168? which are not customarily used or understood but which are certainly important and can now be given quite a definite meaning in the horoscope.

Nomenclature is not an issue here. Addey does not go into names for the aspects much at all, preferring to refer to them in descriptive long-hand in relation to their place in harmonic theory. Otherwise he is only concerned with studying them in practice.
Last edited by Philip Graves on Mon Jan 04, 2010 4:45 am, edited 1 time in total.

Delphine Jay on the quindecile

I have just been refreshing my memory of the work of David Hamblin and Delphine Jay on the harmonics. They both treat of the fifteenth harmonic and accord it significance. Whereas Hamblin, like Addey before him, is content to give a purely narrative description of each aspect in terms of harmonic theory (eg the 'fifth harmonic trine'), Jay expressly names each aspect.

The source for Jay is her book 'Practical Harmonics' published by the American Federation of Astrologers, Inc., Tempe, Arizona, 1983.

See page 10, where she expressly names the 24? aspect the quindecile, just like Wilson 164 years before her.

Therefore my impression is that Noel Tyl (1996) and (following in his foosteps) Ricki Reeves (2001), in adopting the name 'quindecile' for the 165? aspect, have overridden the precedent set not only by James Wilson in 1819 but also Delphine Jay as recently as 1983. This is an unfortunate state of affairs in my opinion that can only serve to cause confusion.

In short I simply do not accept that 'the change has been made; it's no use crying over spilt milk'. It currently appears to me from the available evidence highly probable that the 'change' (which could equally be viewed as an unsatisfactory duplication of names for different concepts rather than a replacement) dates back only to 1996, and not to the work of Thomas Ring; while as recently as 1983 the term quindecile was still being used in print to refer to the 15th harmonic (24?) series of aspects, and all three of the major interpretative English-language books on harmonic theory of the 1970s and 1980s have expressly drawn attention to the importance of the 15th harmonic aspects in astrological interpretation. What I see as being more the historical reality here rather than a 'change' is that two concepts that have concurrently seen a large amount of use in modern astrology albeit from the points of view of different schools of thought have ended up being given the same name as each other, though the 15th harmonic aspect got the name first and had had it for the best part of two centuries before the 24th harmonic aspect was given it. This does not logically mean that the Tyl-defined (24th harmonic) quindecile has rendered the Wilson / Jay-defined (15th harmonic) quindecile invalid or superseded in astrological use. That such an impression may have been obtained, alas, is part of the inherent drawback of the duplication of names. So long as there are advocates of harmonic theory, the 15th harmonic aspect will be used, and the quindecile is the logical name for it.

I don't accept that the application of harmonic theory in modern astrological practice is dead or has died since 1983 either. These ideas were certainly still very current in the 1990s, and Addey's book for one has continued to be reprinted in the 2000s. The quindecile as used by all the adherents of this school needs a name too, and already has one that perfectly describes what it is, consistently with the traditional principles of aspect terminology, so let's keep using it.

If you are looking from this issue as a past student or Noel Tyl, then of course that will be your primary source of reference. But while I admire him very much in many ways, Noel Tyl is not representative of the only major current in modern astrological thinking with regard to the harmonics and minor aspects.

I do think however that there may be a case for adopting a term like 'Tyl / Reeves quindecile' to distinguish it from the traditional quindecile, among those who have got used to that name and do not wish it to be changed more radically to something more in tune with what Thomas Ring suggested in German (to judge by the extracts from his 1956 work I quoted directly in an earlier message), which would be in accordance with the tradition: something along the lines of quartvigintile for the 24th harmonic series that includes the 165? aspect - or in terms of harmonic theory, it has been described by some of the English authors I mentioned above as the 4th harmonic sextile.
Last edited by Philip Graves on Mon Jan 04, 2010 5:05 am, edited 5 times in total.

Llewellyn George, Alfred Pearce and Kepler on the quindecile

1. Another important 20th century source predating Delphine Jay by several decades that addresses the quindecile is:

Llewellyn George 'A to Z Horoscope Maker and Delineator'*

In the section headed 'Part I: the Natal Chart' (p. 93 in the 7th edition, 1939; p. 85 in the 4th edition, 1928; pp. 91-3 in the 29th edition, 1972), George writes under 'Minor Aspects':
Quindecile - 24 degrees apart, slightly good, has an orb of 2 degrees'

In the section headed 'Part V: Additional Studies', George further describes the quindecile thus (7th edition, 1939: p. 720; stated 3rd edition [but 2nd?], 1913: p. 353; 3rd edition, 1923: p. 409; 4th edition, 1928: p. 647; 29th edition, 1973: p. 784):
Another of Kepler's minor aspects; 24 degrees, slightly good
Llewellyn George's cited work above is probably one of the best-selling astrological text-books of the 20th century, having been through about thirty printings between 1910 and the late 1970s, prior to being reworked in a shorter format from the 1980s onwards. I believe this demonstrates that even if Wilson's Dictionary of Astrology was hard to obtain for many in the early-to-mid 20th century after the 1885 printing fell out of print and before the 1969 impression was produced, the traditional definition of the quindecile was still in common currency thanks to the oft-reprinted textbook of Llewellyn George, at least in the United States of America.

*1913 and 1923 editions are called 'A to Z Horoscope Delineator'; I do not have the 1910 first edition but this had the same name too

2. Further, the earlier continuity of the tradition at least on the level of nomenclature through the late 19th century in England is demonstrated in:

Alfred Pearce 'Text Book of Astrology' Volume 1 - Cousins & Co., Covent Garden, London, 1879

I quote from p. 39#
There are other aspects, viz., the vigintile (18?), the quindecile (24?), the decile (36?), the quintile (72?), the tredecile (108?), the biquintile (144?), and the quincunx (150?). All these aspects were invented by Kepler, and their influence is undeniable, with the exception of the vigintile, quindecile, and the tredecile, which require confirmation.
So Pearce, who was quite a traditionalist, was perhaps a natural sceptic towards novel ideas but nonetheless open-minded, whereas Llewellyn George seems for his part to have been more openly sold on the utility of the quindecile in interpretation.

# NB: In the 2nd edition of Pearce, published by Mackie & Co., Ld., Warrington and Fleet Street, London, and by Simpkin, Marshall, Hamilton, Kent & Co. Ld., 1911; also reprinted much later by the National Astrological Library, 1950s, and by the American Federation of Astrologers, 1970 and 2006, this quotation appears instead on pp. 30-1

3. I do not have all Kepler's works by any means, but am trying to find the possible source for the claim in James Wilson, Alfred Pearce and Llewellyn George that Kepler invented the quindecile (24?) aspect. In:

Johannes Kepler 'The Harmony of the World', translated by E. J. Aiton, A. M. Duncan and J. V. Field, American Philosophical Society, 1997:

Book IV, Proposition XV, pp. 347-8:

Kepler groups together the 15th and 20th harmonic astrological aspects as being, in the translators' words:
configurations which are on the borderline between influential and non-influential, namely the arc of 24? from the pentekaidecagon (fifteen-sided figure), and 18? from the icosagon

Kepler expressly includes the multiples of the 15th harmonic series, 48?, 96? and 168?, in his description (p. 348). He also includes two others in error, as the editors note in their footnote.

I do not have the presumed German original (unless it was Latin - I confess I do not know), but it seems clear that if the reference made was to a figure that the translators have called a pentekaidecagon, Wilson was not being unreasonable to transpose this in astrological lingo to quindecile, if indeed he is the earliest source of the English language version, which remains to be proven.
Last edited by Philip Graves on Mon Jan 04, 2010 10:34 am, edited 2 times in total.

Hello Tom,

Re: "Tyl, ever the dramatist..."

Thank you for my morning coffee spit-take.

I comment to thank you for that moment of levity, but also because your comments sparked an important thought. (to me anyways)

As you mentioned, Noel certainly has made much of the Italian pronunciation of quindecile in his presentation of it. As anyone and everyone could read "quindecile" and likely figure out how to pronounce it (in English), IT MAKES IT DRAMATICALLY TIED TO HIM by "enforcing" the (arbitrary, though interesting) Italian pronunciation.

Of course, Tom, as Americans, WE stand on shaky ground commenting on the pronunciation of anything. ;) How many Americans still pronounce Porsche as if it were Porsch? Or even more amusing how many pronounce the shoe brand, Adidas, as if it were a plural of "Adida", even though it is a shortening of the person's name, Adi Dassler. demonstrates how it should sound and how you will actually hear it said here (Germany). Since it is HIS name, it matters not if every single other person in the world said it differently, they would still be wrong.

It is quite easy for Americans to come across similarly as George Bush when he commented that "the problem with the French is that they have no word for entrepreneur."

In terms of the quindecile aspect, I honestly don't care WHAT we call it. I just hope that more will take notice and use it. I am USED to calling it quindecile...and as I mentioned, it HAS (in terms of those who actually USE it) already been "appropriated."

Thank you for the quote from Munkasey. I have much respect for him. His approach is normally quite, for him to also comment on the "quindecile" aspect and to have found similar results supports the ideas previously presented.

Re: Peregrine

I have commented to Tyl about this particular appropriation, because this word IS commonly used and has a useful function. My main problem was that there is already the word "unaspected" which has the EXACT MEANING that he is imparting to the word peregrine. Upon reflecting on your comments, Tom, perhaps he found unaspected as not dramatic enough. ;)

Take care,



Thank you for the time to look up and post those references.

Your posts always make interesting reading!

It is not necessary that I agree with everything said in order to still be entertained/amused by its presentation. ;)

One interesting point in the quindecile vs. quindecile ;) debate is that one presentation is basically based on the 15th DIVISION of the circle, giving the 24?; while the other is based on the 24th DIVISION, giving the 15? series of aspects. Now it should be fairly obvious that the 24th DIVISION would seem to be MUCH MORE LIKELY TO BE IMPORTANT since EVERY NORMAL ASPECT USED IN ASTROLOGY (conjunction 0/24ths, semi-sextile 2/24ths, semi-square 3/24ths, sextile 4/24ths, square 6/24ths, trine 8/24ths, sesquiquadrate 9/24ths, quincunx 10/24ths, opposition 12/24ths) [& "quindecile" 11/24ths] can be derived from particular fractional divisions of the circle by 24. (I obviously purposely didn't reduce the fractions in order to demonstrate my point...)

Using the 15th DIVISION, in terms of (somewhat) common aspects we get the quintile (3/15ths) and bi-quintile (6/15ths), and of course the trine (5/15ths). [Of course, the conjunction will show up in every division as the "0th" harmonic.]

Take care,



Hi Atlantean!

Interesting argument, but I'm afraid I don't agree!

On this basis, it could be argued that the 120th harmonic aspects (eg 3 degrees, 177 degrees) would be more important than either the 24th harmonic ones or the 15th harmonic ones, since they would include not only the conjunction, semi-sextile 10/120ths, semi-square 15/120ths, sextile 20/120ths, square 30/120ths, trine 40/120ths, sesquiquadrate 45/120ths, quincunx 50/120ths, opposition 60/120ths, and Tyl / Reeves 'quindecile' 55/120ths... but also the quintile 24/120ths, bi-quintile 48/120ths, decile 12/120ths, tridecile (a.k.a. sesquiquintile) 36/120ths, quindecile 8/120ths, bi-quindecile* 16/120ths, quadri-quindecile* 32/120ths, and septi-quindecile* 56/120ths, not forgetting the vigintile 6/120ths and its multiples....

This argument fails in my opinion, since the greater the circle division factor, the more major and minor aspects will fit into its pattern, but you are also introducing aspects that are more and more absurdly minor for each further division, and it should be the importance of these base aspects which cannot be derived from a simpler division of the circle that is the standard by which to judge the circle divisor, and not the number of multiples of these aspects that can equally be obtained readily from a simpler division of the circle.

NB: My calculations of fractions above are pure mental arithmetic on-the-fly so if I made any mistakes please forgive them - I'd be glad to put them right.

* for the sake of a working terminology that is more or less consistent with tradition - not definitive terms

To bring some additional elements of the "quindecile" aspects revival, I quote from Ricki Reeves book, The Quindecile: The Astrology & Psychology of Obsession from the 3rd Chapter, the History of the Quindecile:
The quindecile or 165? aspect, was first brought into astrological awareness by German astrologer Thomas Ring (1892-1983). He utilized it in his work on the horoscope of Leonardo da Vinci, noting it in the relationship of Leonardo's Moon in Pisces (the emotional need to merge sensitivity, intuitiveness, and working the intangible) and Neptune in Libra (seeking the illusive, imaginary, and ideal through all to which it relates). Thomas Ring termed this 165? spacial relationship "the separation aspect," denoting its characteristics to be those of disruption and upheaval that divorce the individual from the balance of his or her life, through the nature of the two planets involved.

The use of this aspect then faded into the background of astrology until it was rediscovered by international astrologer Noel Tyl, who came across the work done by Thomas Ring wile researching the horoscope of Leonardo da Vince for his book Astrology of the Famed. Noel Tyl did extensive research, through the use of over 600 horoscopes, and confirmed it to be a most prominant indicator of upheaval and separation in an individual's life. My Tyl included in his work the concept of obsession-compulsion as a factor in the manifestation of this aspect, denoting is as a response to upset and trauma in one's life.
She goes on to discuss Leonardo da Vinci specifically, commenting on his very precise and perceptive mind and then uses his "quindecile" aspect to explain why, even with all of his talents and mental abilities and drive to understand the world and how it functions, that still he was someone who never finished most of his works and how most of his inventive genius never got beyond the drawings and details from his many journals that he kept throughout his life.

Ricki was a student of Noel's so there may be a bias in the Tyl-heavy "history" of the aspect, HOWEVER, he really IS the person most responsible for "bringing the aspect back."



Hello :)

Yes, but by moving to the 120th harmonic, you have cut the 24th harmonic divisions EACH into 5 "pieces", many of which are nonsense aspects. The 24th Harmonic through using only simple fractions contains all the main aspects used in Astrology. (including the ever-more-popular-to-come, quindecile) ;)

From your comments, we should just go straight to the 21,600th harmonic which will include every division possible down to a minute of arc. ;)

The course continuous. The divisions that we use astrologically, extremely discrete for manageability.



Considering that Noel Tyl in Astrology of the Famed (1996) cites the delineation of Leonardo's horoscope from a book by Ring that was published in 1980, I personally feel that to say the aspect was lost then 'rediscovered' is stretching the truth a little. Ring's work has been popular in Germany continuously. What Tyl did was to discover it for himself, then conduct his own research, and re-present it to the English speaking marketplace. It had never been 'lost'.

This reminds me a bit of the hyperbole surrounding the supposed 'rediscovery' of the 'Law of Attraction' by Jerry and Esther Hicks and Rhonda Byrne, the latter marketing it as 'The Secret'. These ideas have been out there all along in books from the early 20th century. Some people just have a keen eye for marketing in my humble opinion.

I read the history section of Ricki Reeves' book you refer to late last night, hence my comments overnight regarding Tyl's sources in 'Astrology of the Famed' to which Reeves refers.

I still don't agree with your reasoning for the importance of the 24th harmonic, sorry to say. The only base aspects in the 24th harmonic that cannot be derived from simpler divisions of the circle than (2*2*2*3) are 15 degrees, 75 degrees, 105 degrees and 165 degrees. These and these alone are the aspects by which this division should be judged. The same argument you place for the 120th harmonic, that these might be 'nonsense aspects', could equally be applied here and for exactly the same reasons. The multiples of the same base aspect that can be derived more simply are completely immaterial in assessing the base aspect per se and its multiples that can only be derived from a minimum of the circle division factor that was applied (in this case 24) to obtain it.

To get the 45 degree and 135 degree aspects you do not need 2*2*2*3. You only need 2*2*2. Similarly, to get 30 degrees and 150 degrees you do not need 2*2*2*3. You only need 2*2*3.

For every additional divisor from the whole circle, an aspect becomes naturally more minor by its nature. Aspects derived from 2*2*2*3 will be more minor than those derived from 2*2*3 and those derived from 2*2*2.

In turn, those derived from 2*2*3 will be more minor than those derived from 2*2 and those derived from 2*3, and so on.

In the case of the traditional quindecile, the circle division by 3*5 is a lot simpler mathematically than that by 2*2*2*3. To demonstrate that the 24th harmonic aspects such as 15 degrees and 165 degrees are inherently more major than the 15th harmonic aspects, you are going to have to argue, in my opinion, that the division of the circle by 2*2*2 is a simpler, more natural operation than its division by 5, since both harmonics have in common a division by 3, and we can therefore factor the 3 out when comparing them. So if you believe that the quintile and biquintile are less important in astrological interpretation than the semisquare and sesquiquadrate, then I feel your argument at least becomes consistent according to the principles of how aspects are derived and how their power diminishes as they become more minor with additional divisors applied to the full 360? circle.

Kepler did not believe that the quintile and biquintile were less important than the semisquare and sesquiquare. He believed the quindecile (24?) series was of borderline influence (as per my citation above) and apparently (based on the same source) disregarded the quartvigintile (15?) altogether.

Before Ring (1956) I have found no reference to the use of the 15? aspect in astrology, unlike the several references to that of the 24? aspect. This no doubt in part because Kepler mentioned the latter but not the former.

This brings us into the later 20th century and the harmonic astrologers. Addey, Hamblin and Jay all advocate the importance of the 24? series. At least two of them also give some importance to the 15? series. They are not limited by the limitations imposed by Kepler, since they have taken harmonic theory to a new level of enquiry. Why can we not follow their example and accept both the 24? and 15? series as being of potential importance and study them both with equally open mind?

This brings me back to the whole point of my dissatisfaction with the appropriation of the term quindecile to have a new meaning. It is creating an artificial conflict between two different but theoretically comparably valid series of aspects. There should be no such conflict. Each should be free to be studied. But by giving one of them the name the other already had, Noel Tyl has generated in your mind, if you don't object to my making such an observation Atlantean, the idea that one or the other series of aspects must be cast aside and regarded as relatively insignificant. This need not be the case.

Re: "Considering that Noel Tyl in Astrology of the Famed (1996) cites the delineation of Leonardo's horoscope from a book by Ring that was published in 1980, I personally feel that to say the aspect was lost then 'rediscovered' is stretching the truth a little. Ring's work has been popular in Germany continuously. What Tyl did was to discover it for himself, then conduct his own research, and re-present it to the English speaking marketplace. It had never been 'lost'."

I will give you that one...and I must admit, quite enjoyably! ;)

Re: "Keen eye for marketing..."

Perceptive, though I realized this of course.

Rather than argue back and forth regarding the harmonics, let me present my "case" slightly differently. Peruse the following list and then you'll see my argument devoid of the mathematics that seem to keep leading us astray of each other, even though we both ride the same model high horse. ;)

0 Conjunction
2 Semisextile
3 Semisquare
4 Sextile
6 Square
8 Trine
9 Sesquiquadrate
10 Inconjunct
12 Opposition

As the above list could reasonably be said to contain the "common aspects", a point could certainly be made that inspection of "spaces" #1, 5, 7, and 11 MIGHT prove fruitful.

Further inspection of "slot" number 11, the "quindecile" has shown quite conclusively that there IS something going on with space number 11.

Though we don't agree perhaps on the name for the slot, let's not disagree that there is an obvious slot there. ;)



There is, but the slots are also demonstrably an artifact of your attempt to find the common harmonic that will incorporate both the eighth harmonic aspects (2*2*2) and the 12th harmonic aspects (2*2*3). The eighth harmonic aspects were introduced by Kepler and are non-zodiacal in their intervals. Yet their derivation is still simpler than those of the 24th harmonic because only three rather than four factors are needed.

Similarly, you could seek to find the common harmonic that will incorporate both the 12th harmonic aspects (2*2*3) and the fifth haronic aspects (5) and you will get (2*2*3*5) the 60th harmonic. Does this mean that the six-degree aspect is of importance just because slots appear at 6 degree intervals when we attempt this? I regret I believe not.

All the zodiacal aspects are obtainable by the 12th harmonic. To push beyond that is in my view to admit of harmonic theory as a rational basis for aspects, since the 15 degree aspect, as much as the 45 degree one, does not exist on a 12-fold zodiacal sign basis.

What Kepler did with the 15th harmonic (3*5) is really no different to what you are doing with the 24th harmonic in my view. It is an attempt to find the common harmonic between the third and the fifth, just as you were seeking to find the common harmonic between the eighth and the twelfth. Is one of these procedures really any more valid than other in terms of harmonic theory? Both involve looking for the common harmonic between a zodiacal one and a non-zodiacal one. In this case, the third harmonic is zodiacal; the fifth is not.

Thus similarly we get slots:

1 24?
2 48?
3 Quintile 72?
4. 96?
5 Trine 120?
6 Biquintile 144
7 168?

Does that mean that the missing aspects are valid? According to harmonic theory, yes, just as the slots in your list are. There are more slots in this case because there are no 2s in the circle division factor, and finding the common harmonic of larger prime numbers than 2, such as 5, leads to larger numbers of slots. I don't see any difference otherwise.

Must go now - food on, then work! Thanks for the chat!