The Minor Aspects
Semi-Square
The Semi-Square is an 8th Harmonic Aspect - the division of the circle by 8, resulting in a 45 degree aspect (or half of a square). The 8th harmonic, like the 4th harmonic is also action oriented, and operates on the material plane. However, because of the 2nd harmonic aspect (2 x 4th harmonic), the 8th harmonic also introduces an element of balance and rest. Semi-Squares occur in signs that share a common polarity (masculine (Fire and Air) or feminine (Earth and Water)), and therefore there is the potential for harmony between them. This is still a "hard" aspect in that it can be uncomfortable and force us to take some action in order to resolve the discomfort; however, Semi-Squares can be resolved more quickly than squares because of the ability to find that point of balance.
Sesquiquadrate
The Sesquiquadrate is also an 8th Harmonic Aspect - it is 3/8 of the circle or 135 degrees. Sesquiquadrates tend to be slightly less action-oriented than the Semi-Squares. They still represent a feeling of tension, and a need for a release, of course. However, planets that are Sesquiquadrate each other, are usually 5 signs apart, and are therefore in signs that are Quincunx each other and have nothing in common. Sesquiquadrates are often best handled through humor - often being able to joke about the associated situations can release the tension.
Semi-Sextile
The Semi-Sextile is a 12th harmonic aspect - it is 1/12 of the circle or 30 degrees. Planets that are Semi-Sextile to each other are in signs that are adjacent to each other. Like the Quincunx, the Semi-Sextile signs have nothing in common by either polarity, modality or element. However, unlike the Quincunx the Semi-Sextile is usually an easier aspect to work with because each sign naturally evolves to the next sign. The Semi-Sextile is usually considered to be a weakly harmonious influence, one that requires even more effort to activate and take advantage of than the Sextile.
Quintile
The Quintile is a 5th harmonic aspect - it is 1/5 of the circle or 72 degrees. The Quintile is not an easy aspect to see directly, and this is one of the reasons that it is not often used by many astrologers. The 5th harmonic relates to creativity in action, and tends to operate on the mental plane rather than the physical plane. Quintiles have been linked to great creativity, and, being a 5th harmonic, also to a need for freedom and change.
Septile
The Septile is a 7th harmonic aspect - it is 1/7 of the circle, and the first harmonic that does not evenly divide the circle. Septiles are 51 1/2 degree aspects, and are extremely difficult to see in the chart without the aid of a computer program or an aspect dial. The 7th harmonic has been associated with matters of fate and karma, and also with spirituality and our connections to our higher self. In numerology, the number 7 is the first number that we can not grasp by just looking at it (we can tell how many objects there are up to six without having to count or break them down into smaller groups). The number 7 is just beyond our ability to comprehend. Personally, I feel that Septiles relate more to our spiritual lessons than anything else, and that they require a conscious effort in order to experience or work with them.
Novile
The Novile is a 9th harmonic aspect - it is 1/9 of the circle, or 40 degrees. In numerology, the number 9 relates to the end of a cycle; there is an element of completion and perfection here, as well as an element of being tested before being allowed to move on to the next cycle. Noviles (also called Noniles) have been linked with perfection and completion, and they have also been linked (somewhat less appropriately, I feel) with imprisonment and difficulty. Again, I feel that Noviles tend to operate on a more spiritual level than on a mundane level, and that they do indicate that we will be tested on our skills in working with the two planets involved; they also indicate that we have reached a point with these two planets where we have adequate resources and experience to pass the test - all we need to is integrate this and let ourselves be guided by spirit. In Jyotish (Vedic) astrology, the 9th harmonic chart is the chart of Marriage.