Harmonic Number Games
Mark Pottenger
The basic physical definition of a “harmonic” is an oscillation whose frequency is an integral multiple of a fundamental frequency. As used in astrology, the fundamental cycle is a full 360° circle. The term harmonic has come into astrology through the statistical work of John Addey in England. (The word is also used by John Nelson, but he uses it to refer to a series of aspects—something entirely different from the subject of this article.) Distribution of zodiacal positions in a sample can be graphed and analyzed like the physical waves to which harmonic terminology best applies. If there is only one peak, it is the fundamental cycle of 360°—which can also be called the first harmonic. Two peaks show the second harmonic, three the third, etc. (Assuming the distribution is evenlooking like a sine wave. Uneven intervals between peaks show combinations of harmonics.) The basic nature of harmonics is that they complete some whole number of cycles in the interval of the fundamental cycle being completed once.
As you can see from the drawings, harmonics can be presented in either linear or circular graphs. The harmonic number can be viewed either as the number of peaks (or troughs) or as the number of repetitions of the cycle. Cycle length is the inverse of harmonic number—the larger the harmonic number, the shorter the cycle, to fit more repetitions into the fundamental length.
Note: for the rest of the article I will give zodiacal positions in absolute longitude (up to 360°) rather than in terms of degree, sign and minute.
The basic phenomenon of cyclic repetition is used in a variety of ways in astrology. One method is to chop up your original 360° into equal segments and superimpose them. This is the approach of the German “dials” and American “sorts”. Whatever harmonic you are working with, determine the degree measure of one cycle. (E.g. 4th harmonic = 360°/4 = 90°.) Positions up to one harmonic cycle are left alone. Positions above the full cycle length are brought down into the desired range. This can be done either by subtracting the cycle length or starting to count from zero at the end of the cycle—the actions are really equivalent. Whenever you complete a cycle, you start counting from zero again or subtract another cycle length. When you finish, all your positions will be between zero and the length of one cycle in that harmonic. The resulting sort is an excellent tool for spotting aspects. Anything separated in the original 360° chart by the length of the harmonic cycle will be conjunct in the sort. Thus, in a 72° (5th harmonic) sort, all natal quintiles and biquintiles are conjunct.
If one does a lot of sorts (especially those with all midpoints) by hand, it helps to have a table giving the degree and minute at which each cycle ends (the amount to subtract from positions above that point). For example, the fifth harmonic completes cycles at 72°, 144°, 216°, 288° and 360°. To get the sort, positions less than 72° are left alone, positions from 72° to 144° have 72° subtracted, from 144° to 216° have 144° subtracted, and so on.
Sort positions can also be gotten with a calculator. Convert the natal longitude (L) to decimal if it is degree and minute. Divide 360° by the harmonic number to get the exact harmonic cycle length (360°/H=C). Divide the longitude by the cycle length (L/C=Q). Subtract out the integer (whole number) portion of the quotient (Qint[Q]=R). [Note: int(x) means “the integer portion of x”.] Multiply the remainder by the cycle length (R*C=S). The answer is the sort position.
If your calculator doesn’t have a memory to store the cycle length after calculating it, an alternate approach eliminates the need. Multiply natal longitude by harmonic number (L*H=A). Divide the answer by 360 (A/360=Q). Subtract out the integer portion of the answer (Qint[Q]=R). Divide the remainder by the harmonic number (R/H=B). Multiply this answer by 360 (B*360=S). The answer is the sort position.
The crucial part in both sequences above is subtracting out the integer portion of the quotient (Q). This is what gets rid of extra harmonic cycles and brings the position into the proper range. The rest of the operations are first to get that integer, then to undo the changes and get back to degrees.
To summarize the three methods of doing harmonic sorts. By hand: Longitude  nearest value below in table of cycle ends = sort position (LTV=S). Calculator with memory: 360/Harmonic number = Cycle length (store) (360/H=C). L/C= Quotient (Q). Q  int(Q) = Remainder (R). R*C=S. Calculator without memory: L*H= Answer (A). A/360=Q. Qint(Q)=R. R/H= Answer (B). B*360=S.
As you can see from the summary, using a table is the easiest way. A quick example: Richard Nixon’s 72° (5th harmonic) sort positions for Vesta and the south node of the Moon (by table): Vesta (19 Sag 0) 259°  216° = 43°. South Node (7 Lib 15) 187° 15’  144° = 43° 15’.
A natal quintile shows up as a 5th harmonic conjunction.
In addition to harmonic sorts, there is another whole approach to harmonics suggested by Addey. You overlay the positions, then you expand your sort to again fill 360°. Thus, your harmonic sort becomes a harmonic chart that you can read just like any other chart. If you have already done a harmonic sort, you get longitudes for a harmonic chart by multiplying sort positions by your harmonic number. (S*H=HL) (E.g. Nixon’s Vesta: 43° * 5 = 215° = 5 Scorpio.)
If you haven’t already done a sort, you can get harmonic longitudes directly without sorting. Multiply the natal longitude by the harmonic number to get the harmonic longitude (L*H=HL). If the HL is less than 360°, that is the final answer. If it is greater than 360°, subtract 360° (repeatedly, if necessary) so it will be less than 360°. If you are doing this by hand, a table of multiples of 360° is helpful. On a calculator a technique like that used in the sorts brings the answer into the proper range. HL/360= Temporary (T). Tint(T)= Remainder (R). R*360=HL. (E.g. Nixon’s South Node: 187.25*5 = 936.25  720 = 216.25 = 6 Scorpio 15.)
How to read the harmonic chart produced by these procedures is still very much a question.
Harmonic Cycle Starts/Ends & 360 Multiples
Harmonic 2 
Harmonic 3 
Harmonic 4 
0 0 
0 0 
0 0 
180 0 
120 0 
90 0 
360 0 
240 0 
180 0 

360 0 
270 0 


360 0 
Harmonic 5 
Harmonic 6 
Harmonic 7 
0 0 
0 0 
0 0 
72 0 
60 0 
51 26 
144 0 
120 0 
102 51 
216 0 
180 0 
154 17 
288 0 
240 0 
205 43 
360 0 
300 0 
257 9 

360 0 
308 34 


360 0 
Harmonic 8 
Harmonic 9 
Harmonic 10 
0 0 
0 0 
0 0 
45 0 
40 0 
36 0 
90 0 
80 0 
72 0 
135 0 
120 0 
108 0 
180 0 
160 0 
144 0 
225 0 
200 0 
180 0 
270 0 
240 0 
216 0 
315 0 
280 0 
252 0 
360 0 
320 0 
288 0 

360 0 
324 0 


360 0 
Harmonic 11 
Harmonic 12 
Harmonic 13 
0 0 
0 0 
0 0 
32 44 
30 0 
27 42 
65 27 
60 0 
55 23 
98 11 
90 0 
83 5 
130 55 
120 0 
110 46 
163 38 
150 0 
138 28 
196 22 
180 0 
166 9 
229 5 
210 0 
193 51 
261 49 
240 0 
221 32 
294 33 
270 0 
249 14 
327 16 
300 0 
276 55 
360 0 
330 0 
304 37 

360 0 
332 18 


360 0 
Harmonic 14 
Harmonic 15 
Harmonic 16 
0 0 
0 0 
0 0 
25 43 
24 0 
22 30 
51 26 
48 0 
45 0 
77 9 
72 0 
67 30 
102 51 
96 0 
90 0 
128 34 
120 0 
112 30 
154 17 
144 0 
135 0 
180 0 
168 0 
157 30 
205 43 
192 0 
180 0 
231 26 
216 0 
202 30 
257 9 
240 0 
225 0 
282 51 
264 0 
247 30 
308 34 
288 0 
270 0 
334 17 
312 0 
292 30 
360 0 
336 0 
315 0 

360 0 
337 30 


360 0 
Harmonic 17 
Harmonic 18 
Harmonic 19 
0 0 
0 0 
0 0 
21 11 
20 0 
18 57 
42 21 
40 0 
37 54 
63 32 
60 0 
56 51 
84 42 
80 0 
75 47 
105 53 
100 0 
94 44 
127 4 
120 0 
113 41 
148 14 
140 0 
132 38 
169 25 
160 0 
151 35 
190 35 
180 0 
170 32 
211 46 
200 0 
189 28 
232 56 
220 0 
208 25 
254 7 
240 0 
227 22 
275 18 
260 0 
246 19 
296 28 
280 0 
265 16 
317 39 
300 0 
284 13 
338 49 
320 0 
303 9 
360 0 
340 0 
322 6 

360 0 
341 3 


360 0 
Harmonic 20 
Harmonic 21 
Harmonic 22 
0 0 
0 0 
0 0 
18 0 
17 9 
16 22 
36 0 
34 17 
32 44 
54 0 
51 26 
49 5 
72 0 
68 34 
65 27 
90 0 
85 43 
81 49 
108 0 
102 51 
98 11 
126 0 
120 0 
114 33 
144 0 
137 9 
130 55 
162 0 
154 17 
147 16 
180 0 
171 26 
163 38 
198 0 
188 34 
180 0 
216 0 
205 43 
196 22 
234 0 
222 51 
212 44 
252 0 
240 0 
229 5 
270 0 
257 9 
245 27 
288 0 
274 17 
261 49 
306 0 
291 26 
278 11 
324 0 
308 34 
294 33 
342 0 
325 43 
310 55 
360 0 
342 51 
327 16 

360 0 
343 38 


360 0 
Harmonic 23 
Harmonic 24 
Harmonic 25 
0 0 
0 0 
0 0 
15 39 
15 0 
14 24 
31 18 
30 0 
28 48 
46 57 
45 0 
43 12 
62 37 
60 0 
57 36 
78 16 
75 0 
72 0 
93 55 
90 0 
86 24 
109 34 
105 0 
100 48 
125 13 
120 0 
115 12 
140 52 
135 0 
129 36 
156 31 
150 0 
144 0 
172 10 
165 0 
158 24 
187 50 
180 0 
172 48 
203 29 
195 0 
187 12 
219 8 
210 0 
201 36 
234 47 
225 0 
216 0 
250 26 
240 0 
230 24 
266 5 
255 0 
244 48 
281 44 
270 0 
259 12 
297 23 
285 0 
273 36 
313 3 
300 0 
288 0 
328 42 
315 0 
302 24 
344 21 
330 0 
316 48 
360 0 
345 0 
331 12 

360 0 
345 36 


360 0 
Multiples of 360 
360 
720 
1080 
1440 
1800 
2160 
2520 
2880 
3240 
3600 
3960 
4320 
4680 
5040 
5400 
5760 
6120 
6480 
6840 
7200 
7560 
7920 
8280 
8640 
9000 
A note on language: All of the math in this article works just as well for numbers that are not integers as for those that are. However, the results should not really be called harmonics when using nonintegers. Harmonics, by definition, involve integer relationships to the fundamental cycle—a whole number of smaller cycles completed for each main cycle. Thus while the math of harmonics produces results with nonintegers, we need a new name for those results. I have been using the term “expansion factor”, but it is awkward and I would welcome a better term. One form of noninteger “harmonic” now being tested is using the solar arc for an event as a “harmonic” number. Natal positions are multiplied by the solar arc instead of having it added. Opening up the area of noninteger “harmonics” increases the problem of proliferation of charts. True harmonics have already added dozens of new charts. Expansion factors can create an infinite number of new charts.
When doing a series of Addeytype harmonic charts for a single individual, it can be seen that positions change systematically from one harmonic to the next. I will describe the behavior of the numbers without offering any interpretations as to meanings.
When you increase the harmonic number by one, the harmonic position moves forward in the zodiac by the amount of the natal absolute longitude. For a natal (1st harmonic) position of 30°, the 2nd harmonic position is 60°, the 3rd is 90°, the 4th 120° and so on. This change from one harmonic to the next is really just breaking the multiplication that produces harmonics into successive additions.
As long as the natal longitude is between 0° and 180°, this addition will appear as increasing longitude with successive harmonics. However, if the natal longitude is between 180° and 360°, successive harmonic positions will show decreasing longitude. For example, a natal 330° becomes a 2nd harmonic 300° (660°), a 3rd harmonic 270° (990°) and so on. The numbers in parentheses are the raw answers from multiplying by the harmonic number (before subtracting extra sets of 360°)—you can see that those do increase. The harmonic positions decrease because the true increase each time is most of, but not quite all of, a full circle. The amount of apparent decrease is the distance the natal position falls short of the end of the zodiac.
The nearer a natal position is to the start or end of the zodiac, the less it will change from one harmonic to the next. Positions in the middle change the most. For example, a natal 180° becomes a 2nd harmonic 0°, a 3rd harmonic 180°, a 4th harmonic 0° and so on. It flips clear across the zodiac every time. In contrast, 0° stays 0° in every harmonic.
To summarize the apparent motions from one harmonic to the next: Longitudes less than 180° add the natal longitude each time. Longitudes greater than 180° subtract the natal distance short of 360° each time. Longitudes near 0° or 360° change slowly. Longitudes near 180° change rapidly.
The following short lists show natal (1st harmonic) through 5th harmonic positions starting from different parts of the zodiac: 1, 2, 3, 4, 5. 90, 180, 270, 360 (0), 90. 179, 358, 177, 356, 175. 181, 2, 183, 4, 185. 270, 180, 90, 0, 270. 359, 358, 357, 356, 355.
For noninteger expansion factors the position will change by the fraction of the natal longitude that your factor changes by. Increasing the factor by 0.1 will add 0.1 of the natal longitude to the expanded position. E.g. 30°*2.1 = 63°; 30°*2.2 = 66°. Because so many different decimal fractions are possible between any two integers, expansion factors can jump around quite dramatically between successive integer harmonics. For example, Nixon’s Sun for harmonics of 61, 62 and 63 is at 13 Aries 32, 2 Aquarius 57 and 22 Scorpio 21, while the solar arc of 62.229 for his resignation produces an expansion factor Sun of 9 Aries 12. The integer harmonic positions move evenly backward, while the expansion factor position jumps.
The angles between factors also change systematically between successive integer harmonics. Angular separations increase each time by the natal separation, just as positions increase by the natal longitude (in fact, because of the position changes). A natal separation of 1° becomes, in successive harmonics, 2°, 3°, 4°, etc. A natal conjunction stays a conjunction, but with an increasing orb. As with positions, natal separations near 0° change slowly, and those near 180° jump around.
If the natal separation is exactly 180° (an opposition), as in any angle axis or the Moon’s nodes, successive harmonics will be conjunct and opposite. On all even numbered harmonics, natal opposition axes become conjunctions. On all odd numbered harmonics, natal opposition axes are again oppositions. E.g., natal positions of 4° and 184° become 8° and 8° (368), 12° and 192° (552), 16° and 16° (736), and so on. If something is close but not exactly opposite, the behavior will be similar, but the positions will gradually separate.
I have tried to describe the antics of harmonics so some purely mathematical aspect of their behavior won’t be taken to be of great astrological significance. (“Look, his Sun is backing up!”, etc.)
That, then, is some of what is being done in the field of harmonics in astrology. With the techniques described, one can create a whole new set of charts to play with. However, that can use up a lot of time which can be better spent so I suggest that you let a computer do as much of the math as possible. Astro Computing Services, which has just moved from Pelham, NY to San Diego, CA, offers several harmonic services. They offer a harmonic sort of planets and midpoints for any integer divisor of the circle for $1.00. Positions are printed out with proportional spacing to make clumping easier to see. They offer a listing of positions for any 30 consecutive harmonics for $1.00. They offer any single harmonic chart in their great wheel for $1.00. They have a $1.00 per order handling charge. Their new address is Astro Computing Services; PO Box 16297; San Diego, CA 92116. They don’t currently offer anything for nonintegers because that is very new and experimental.
Have fun exploring this new area!