9th Harmonic Chart = Navamsa? Harmonics vs Vargas, Checked

Two families of derived chart look as if they should be related. Jyotish has its vargas, the divisional charts, which cut each 30° sign into n parts and hand every part a sign by a rule. Some Western astrologers use harmonic charts, which stretch the whole zodiac n times. Wikipedia’s article on aspects ties modern aspect work to astrological harmonics, says Kepler argued for that approach in Harmonice Mundi (1619), and names John Addey as a major proponent. This page puts the two side by side and checks, over every point of the circle, where they give the same sign and where they do not.

The Two Rules Side by Side

The harmonic chart computed here takes each longitude λ, counted from 0° Aries, and keeps only what is left of n×λ after removing whole turns:

harmonic-n longitude = n × λ, minus 360° as many times as it fits

That rule is this page’s arithmetic, not a quotation from either source. A varga works sign by sign instead. The navamsa (D9) rule that Brihat Parashara Hora Shastra gives, and the site’s divisional-charts guide tables, cuts each sign into nine parts of 3°20′. The first part of a movable sign is the sign itself; for a fixed sign the count begins at the ninth sign from it, for a dual sign at the fifth. Each later part is the next sign in order. So Aries’s first navamsa is Aries, Taurus’s first navamsa is Capricorn, Gemini’s first navamsa is Libra.

Why Multiplying by 9 Gives the Navamsa

Write λ as 30×S + D, with S the sign’s place counted from Aries as 0 and D the degree inside it. Then 9λ = 270×S + 9×D. The second term walks through one whole sign for every 3°20′ of D, so each navamsa part maps onto one sign and the parts advance one sign apiece. The first term places the start: 270° is nine signs, so each step from one sign to the next moves the start on by nine. Counted from Aries that start is the sign itself, then the ninth, then the fifth, and the pattern repeats every three signs, which is the movable, fixed and dual rule. Put another way, the 108 navamsa parts of the zodiac run Aries to Pisces nine times without a break: the last part of Aries is Sagittarius, the first of Taurus Capricorn.

The lib behind this page tests the equality at 2,16,000 points, one every 1/600 of a degree, and at both ends of all 108 parts, and refuses to build if a single point disagrees. Three worked examples, one in each kind of sign:

The two agree on the sign, but they do not deliver the same thing. The varga rule ends at a sign, while the harmonic also gives a degree inside it, nine times the distance into the navamsa part. This page does not say what any school does with that degree. The D9 navamsa page covers the chart itself.

All Sixteen Vargas Against Their Harmonics

The same argument settles every equal-part varga: the varga of order n is the n-th harmonic exactly when each sign’s first part is the sign n×S places from Aries and later parts step one sign at a time. The lib checks that rule and an exact census over the circle against each other (summing the arcs between every part edge, not sampling), and they agree. The result for the sixteen charts of the Parashari list:

Six charts, D1, D7, D9, D16, D20 and D27, are their harmonic outright (D1 trivially, since one times a longitude is itself). Each of their start rules turns out to be n×S in disguise. The other ten are not, and the closest of them, the D10, matches on only 25% of the circle; the D24 never does. The D2 and D30 are not n equal parts in a row at all (the Parashari hora gives only Leo and Cancer, and the trimshamsha has five unequal parts), so no harmonic could reproduce them.

D12, the 12th Harmonic and the Dodecatemoria

The D12 dwadashamsa cuts a sign into twelve parts of 2°30′ and starts the count from the sign itself. Twelve times a sign is 360°, so the 12th harmonic always starts its count from Aries, whatever the sign. The two therefore agree only in Aries, 8.33% of the circle; elsewhere the D12 sign is the 12th-harmonic sign moved on by as many signs as the point’s own sign is from Aries. The D60 behaves the same way: parts 0°30′ wide, counted from the sign itself, agreeing with the 60th harmonic in Aries only (8.33%).

The ancient twelve-part division called the dodecatemoria follows the D12, not the harmonic. Wikipedia’s article on dodecatemoria describes them as twelve 2°30′ parts in each sign, beginning with the sign itself, says the system apparently originated in Babylonian astrology and was taken up in Hellenistic astrology; Manilius and Firmicus describe the calculation and Ptolemy, it adds, dismissed it as illogical. It gives two computations, Hellenistic and Babylonian, and reports a third scheme from Abraham Ibn Ezra; each is a single line of arithmetic:

• Hellenistic: add 12×D to the start of the sign. The article’s own example, Capricorn 17°00′, gives 12×17 = 204° past 0° Capricorn, which is Cancer 24°00′; the lib reproduces it. The sign it lands in is the D12 sign at every point of the circle (100%). The 12th harmonic of the same point is Libra 24°00′: the dodecatemorion is the 12th harmonic moved on by as many signs as the point’s sign is from Aries, and the two coincide only in Aries (8.33%).
• Babylonian, which the article calls seemingly older: add 12×D to the degree itself. Since 30×S + 13×D is exactly 13λ less whole turns, this dodecatemorion is the 13th harmonic at every point (100%), which fits the article’s description of a micro-zodiac of thirteen sections. For Capricorn 17°00′ it gives Leo 11°00′. Its sign is the D12 sign on 50% of the circle.
• Abraham Ibn Ezra’s second system, as the article summarises it, gives each whole degree one sign, starting with the sign itself, so that Libra returns at the 13th and 25th degree of Libra. Reading the first degree as 0°–1°, its sign matches the D12 on 10% of the circle.

So “D12 vs dodecatemoria” has a plain answer at sign level: the Hellenistic dodecatemorion sign and the Parashari D12 sign are the same rule. Only the Hellenistic version also carries a degree. Whether anyone historically connected the two traditions is a question this arithmetic cannot settle. The site’s antiscia page works through another Western derived position, the mirror degree across the solstice axis.

Tropical or Sidereal: Which Longitude Goes In

The equality holds for any number put into it, so a navamsa computed from a Lahiri longitude is the 9th harmonic of that Lahiri longitude, and a navamsa computed from a tropical longitude is the 9th harmonic of the tropical one (100% in the census). What does not carry over is the zodiac. On 2026-10-28 the Lahiri ayanamsa in the site’s formula is 24.23°, and multiplying by 9 multiplies it too: 218.08°, or seven signs and 8.08°. The ayanamsa page covers what the value is.

A shift of between one and eleven whole signs cannot leave any sign in place, so “D9 vs 9th harmonic” comparisons between a jyotish chart and a tropical harmonic chart compare different signs at every point of the zodiac on that date. The ayanamsa grows by about 50.3″ a year in that formula, which the 9th harmonic turns into 7′33″.

An Error Grows n Times

Multiplying a longitude by n multiplies any error in it by n. An error of 1′ (one sixtieth of a degree) in the Moon becomes 9′ in its 9th harmonic, and the arc of real zodiac that holds one harmonic sign shrinks to 30°/n:

The D60 is not the 60th harmonic, but its parts are the same 0°30′ wide, so the same factor of sixty applies to how close a point can sit to a part edge. How that plays out for the ascendant is measured on the site’s page on how long the lagna holds a sign in each varga.

What This Page Does Not Give

This page gives no meaning for a 5th, 7th, 9th or any other harmonic, and no reading of a varga or dodecatemorion placement for anyone. The multiplication can be checked to the last arc-second; no multiplication can check an interpretation. Harmonic aspects, harmonic charts of two people compared, and the half-degree subdivisions Manilius adds to the dodecatemoria are left out as well.

In Brief

• Harmonic chart: n × longitude, less whole turns of 360° (the arithmetic used here)
• 9th harmonic: its sign is the Parashari navamsa sign at every point of the zodiac
• Other vargas: D1, D7, D9, D16, D20 and D27 equal their harmonic; the other ten do not, D24 at no point
• D12: agrees with the 12th harmonic only in Aries, and with the Hellenistic dodecatemorion sign everywhere
• Babylonian dodecatemorion: exactly the 13th harmonic
• Zodiac: on 2026-10-28 a tropical 9th harmonic and a Lahiri navamsa never share a sign