Divergent Astrology: The true Quindecile (24°) and The Overall 15th Harmonic Aspects

15th harmonic aspects are the 24 degree series of aspects:

1/15 24° - Quindecile
2/15 48° - BiQuindecile
3/15 72° - Quintile aka TriQuindecile
4/15 96° - QuadriQuindecile
5/15 120° - Trine aka QuinqueQuindecille
6/15 144° - BiQuintile aka SexQuindecile
7/15 168° - SeptQuindecile

15th Harmonic Aspect will appear as Conjunction in 15th Harmonic Chart, Trine in 5th Harmonic Chart, and 5th Harmonic Aspect in 3rd Harmonic Chart.
If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the 15th Harmonic aspects have a maximum orb of 1 degree.

Quindecile

A minor 24 degree aspect. Somewhat favourable. Quindecim means 15 in Latin and 24x15=360, the number of degrees in a circle, and so the zodiac. Influential contemporary astrologer Noel Tyl calls the 165° aspect quindecile, because he says that all the other 15° increments are already named, except the 105° interval which is insignificant. He says the 24th harmonic is what we are talking about, rather than the 24° interval. This is a moot point, as yet undetermined in practice, though Tyl says that in some 900 charts it indicates passionate obsession, unrelenting motivation, determination, upheaval, disruption, even separation. Maybe not so favourable, then? Or maybe more research is needed...

https://astrologycom.com/glossq.html#A

I can't buy into what Noel Tyl had been teaching about what's a Quindecile.

I don't know why Noel Tyl claimed that Quindecile is referring to the 24th Harmonic and the Quindecile aspect is the 11/24th Harmonic aspect (165°) and not the 1/15th Harmonic aspect (24°).

Septile is the 7th Harmonic. Septile (51°25'42'') is the 1/7th Harmonic aspect, BiSeptile (102°51'25'') is the 2/7th Harmonic aspect. TriSeptile (154°17'08'') is the 3/7th Harmonic aspect. If using harmonic proportionate orb based on The 1st Harmonic Aspect Conjunction (0°) having a maximum orb of 15 degrees, the maximum orb for 7th Harmonic aspects is 2 degrees, 8 minutes.

Octile is the 8th Harmonic. Octile (45°) is the 1/8th Harmonic aspect, and it is also known as the SemiSquare. The BiOctile (90°) is the 2/8th Harmonic aspect, and it is also known as the 1/4th Harmonic aspect Square. The TriOctile (135°) is the 3/8th Harmonic aspect, and is also known as SesquiSquare. It is also known as SesquiQuadrate. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the maximum orb is 1 degree, 53 minutes.

Novile is the 9th Harmonic. Novile (40°) is the 1/9th Harmonic aspect, BiNovile (80°) is the 2/9th Harmonic aspect. Trinovile (120°) is the 3/9th Harmonic aspect, and it also known as the 1/3rd Harmonic aspect Trine which is also the 1/3rd Harmonic aspect and the 6/1 QuinqueQuindecile QuadriNovile (160°) is the 4/9th Harmonic aspect. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the maximum orb is 1 degree, 40 minutes.

Decile is the 10th Harmonic. Decile (36°) is the 1/10th Harmonic aspect and is also known as SemiQuintile. BiDecile (72°) is the 2/10th Harmonic aspect, and it is the same as the Quintile. TreDecile (108°) is the 3/10th Harmonic aspect, and it is also known as the SesquiQuintile which is what Solar Fire has it named. If using harmonic proportionate orb based on Conjunction having a maximum orb of 15 degrees, the maximum orb is 1 degree, 30 minutes.

Undecile is 11th Harmonic. Undecile (32°43'37'') is the 1/11th Harmonic aspect. BiUndecile (65°27'16'') is the 2/11th Harmonic aspect. TriUndecile (98°10'54'') is the 3/11th Harmonic aspect, and QuadriUndecile (130°54'32'') is the 4/11th Harmonic aspect. QuinqueUndecile (163°38'10'') is the 5/11th Harmonic aspect. If using harmonic proportionate orb based on Conjunction (0°) having a maximum orb of 15 degrees, the maximum orb is 1 degree, 22 minutes.

Considering these facts, there is logic in thinking that Quindecile is referring to the 15th Harmonic and not the 24th Harmonic. The Quindecile is the 1/15th Harmonic aspect (24°). It is not the 11/24th aspect (165°) which has been referred to as the Johndro. Robert Blaschke referred to it as the Contraquindecile in his book Sabian Aspect Orbs.

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), Llewellyn George wrote 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"

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

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."

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 includes the multiples of the 15th harmonic series, 48°, 96° and 168°, in his description (p. 348).

Delphine Jay names the 24° aspect Quindecile on page 10 of her book 'Practical Harmonics' published by the American Federation of Astrologers, Inc., Tempe, Arizona, 1983.

The following is pages 126- 27 of from Harmonics in Astrology by John Addey:

"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.

These aspects are indicators of the enjoyment of and facility in some form of activity as shown by the planets involved. In other words they show us what kind of power or what art or activities (five) a person delights (three) in exercising. One can think of them as trines in the 5th harmonic map and interpret them in the light was said about that chart. This must be seen in a wide context, for example a 24° aspect between Moon and Venus might indicate someone who enjoys cultivating the social graces, the art of the hostess, the art of forming sympathetic and agreeable relationships, understanding people and generally making the wheels of life turn smoothly. This is a most important art."

The 15th harmonic indicates a facility and easy in expressing oneself creatively (Cochrane)

http://www.astrosoftware.com/harmonicfirst32.pdf

Harmonic 15: A need to play and have enjoyable times. A strong need to work in way that is not overly tedious and onerous. Enjoying surprises and discoveries, and an ability to fully enjoy life by keeping alive a childlike sense of wonder and not feeling overly burdened by responsibilities.

http://avalonastrology.com/cheatsheet.pdf

15: A magic number according to the old masters of numerology. It rules magicians, alchemy, black magic, want of erotic experiences, actors and eloquent speakers. In my personal experience this is the actor’s number. It is a “tricky” number, not always lucky, and sometimes it can be associated with illness.

https://star4cast.com/harmonics-the-spiritual-essence-of-numbers/#

15 was Inanna/Ishtar's sacred number

http://www.mesopotamia.co.uk/gods/explore/ishtar.html#:~:text=Sacred%20number%3A%2015%20Ishtar%20is%20the%20morning%20and,standing%20on%20a%20lion%2C%20generally%20holding%20several%20weapons.

In my natal chart,

Mercury BiQuintile Eris - 5 min

Mercury BiQuindecile Pluto - 30 min

Pluto SeptQuindecile Eris - 36 min

15th Harmonic Triangle

I have Mercury in 15th Harmonic Triangle with two transneptunian dwarf planets which would indicate evolutionary intensified mental processes.

I believe that they factor into the neurodivergence (Dyslexia, Dyspraxia, ADHD).

Venus trine Orcus - 40 min

Venus quadriundecile Makemake - 32 min

Orcus quindecile Orcus - 8 min

5th Harmonic Triangle

I have Venus in 15th Harmonic Triangle with a transneptunian dwarf planet (Makemake) and a transneptunian dwarf planet candidate (Orcus) which would indicate a evolutionary intensified love nature.

Sun septquindecile Sedna - 20 min in 3D

"3D aspects" refers to aspect calculations that take planetary latitude into account. When calculating the angle and orb of aspects, in classic astrology only the zodiac position longitude of a planet is considered, and the planet's latitude is ignored. However, it's possible for aspect calculations to take the latitude of planets into account too. In other words, the aspect angle between two planets is determined by the 3D great circle distance between them on the celestial sphere, and not just the 2D difference between their zodiac position longitudes.

http://www.astrolog.org/astrolog/ast3d.htm

My Sun is in 15th harmonic aspect to a transneptunian dwarf planet candidate can indicate evolutionary intensified vitality.

I have a more significant aspect of Sun-Sedna opposition in Right Ascension (Equatorial Longitude) with an orb of 43 minutes.

Considering that I share a view about the importance of Harmonics and Geometry far more than zodiac signs and the knowledge of having a trine between the asteroids Kepler and Urania with an orb of well under 1 degree, I checked to see if my Mercury is in a 15th Harmonic aspect with those asteroids. I also checked Coelestina and Mesopotamia because of my strong interest in Mesopotamian Astrology. They're in a conjunction with my Imum Coeli.

Mercury biquintile Eris - 5 min

Mercury quindecile Coelestina - 0 min (1 min in 3D)

Mercury quindecile Mesopotamia - 45 min

Mercury septquindecile Urania - 17 min (0 min in 3D)

Mercury biquindecile Kepler - 8 min (2 min in 3D)

Mercury biquindecile Pluto - 30 min

Kepler quintile Coelestina - 7 min (5 min in 3D)

Kepler quintile Mesopotamia - 53 min

Kepler septquindecile Eris - 13 min

Colestina quintile Pluto - 29 min (16 min in 3D)

Mesopotamia conjunct Coelestina - 47 min

Mesopotamia trine Eris - 40 min

Kepler conjunct Pluto - 13 min

Pluto quintile Coelestina - 29 min

Urania trine Pluto - 47 min

Urania trine Kepler - 25 min (2 min in 3D)

Urania septquindecile Coelestina - 17 min (48 min in 3D)

Urania biquindecile Eris - 11 min

Pluto septquindecile Eris - 36 min

Note how small the orb is for my Mercury biquintile Eris with 5 minute orb.

The thing with small orb aspect like that is that it's not only conjunct in its particular harmonic but also conjunct in other harmonics that are multiples of the number of that harmonic.

Using proportionate orb based on Conjunction having an orb of 15 degrees, Mercury and Eris are conjunct in 5-multiple harmonics up to the 150th harmonic.

If using my usual orb of 3 degrees for transneptunian dwarf planet/candidates. Then Mercury and Eris are conjunct in 5-multiple harmonics up to the 30th harmonic. That's a total of six 5-multiple harmonic charts, and that's still a lot.

Mercury-Eris conjunctions in harmonics

5th harmonic - 28 minutes

10th harmonic - 57 minutes

15th harmonic - 1 degree 25 minutes

20th harmonic - 1 degree 54 minutes

25th harmonic - 2 degree 22 minutes

30th harmonic - 2 degrees, 51 minutes

With Eris being a transneptunian dwarf planet/candidate of Pluto, it is has to do with Evolutionary Intensification at the least.

I have the view that Eris is ideology that unites/divides leading to inclusion/exclusion. It is also about the dialectical process as well as controversy and discord.

Harmonic 5: A fascination or interest in creative forms, fluid structures, puzzles, games, and entertainments of all kinds. A childlike enjoyment of life, and playfulness, and a distaste for rote learning, unnecessary repetition, tediousness, useless habits and customs.

Harmonic 20: A need to create things and to work in a creative way. The 20th harmonic aspects can become easily bored by repetitive tasks. Note that the 20th harmonic is regarded as inclining to be creative because the playfulness of 5 and the need to achieve of 4 combine together as the need to achieve something playful, interesting, or fascinating. The need to achieve and accomplish puts an emphasis on doing and accomplishing and this is likely to manifest as accomplishing something that is entertaining or fascinating in some way, and thus can be referred to as creative.

Harmonic 25: An exceptionally creative, inventive, or playful tendency. The person sees things in new ways and feels bored by doing things the same old way.

Harmonic 30: Cleverness and imagination. Facility for keeping others entertained and interested in something. Ability to synthesize different elements in an interesting and harmonious way.

http://avalonastrology.com/cheatsheet.pdf

5th Harmonic - 5 multiplied by 1

10th Harmonic - 5 multipled by 2

15th Harmonic - 5 multiplied by 3

20th Harmonic - 5 multipled by 4

25th Harmonic - 5 multiplied by 5

30th Harmonic - 5 multiplied by 6

A 15th harmonic aspect that with a narrow orb will appear as a conjunction in 30th Harmonic Chart. 30th Harmonic - 15 multiplied by 2. My Mercury biquindecile Pluto with 29 minute orb doesn't appear as a conjunction in the 30th Harmonic chart. Mercury and Pluto has a separation of over 15 degrees in the 30th Harmonic chart.

15th Harmonic Triangle of Mercury, Pluto, Eris