Week 11 · Unit 1 · Ashtakavarga and Additional Classical Methods
Milestone 1 of 5: Cleaning the numbers
Cleaning the Numbers
The three reductions — Trikoṇa, Ekādhipatya and Piṇḍa — and what each one is correcting for.
Assumes you have done: Week 10 unit 4 — you need a bhinnāṣṭakavarga to reduce
Why this matters
This is where people abandon Aṣṭakavarga. The arithmetic is fiddly and the purpose is almost never explained, so the reductions look like ritual. They are not. Each one removes a specific kind of double-counting, and once you can say which, the whole system stops being a black box that produces numbers and becomes something you can reason about.
Learn
This is where most people give up on Aṣṭakavarga, and the reason is almost always the same: the arithmetic gets taught and the purpose does not.
What the reductions are for
You have a bhinnāṣṭakavarga: twelve signs, each with a count from 0 to 8. Why not just read it?
Because the raw counts contain two kinds of double-counting, and until those are removed the numbers are not comparable with each other.
Trikoṇa Śodhana — the common base
Signs in mutual trine — Aries, Leo and Sagittarius; Taurus, Virgo and Capricorn — share an element and a great deal of structural similarity. So a portion of whatever support they receive is support they receive together, simply by being that kind of sign.
The reduction strips it. Take each trine, find the lowest of the three counts, and subtract it from all three.
Work an example: raw counts of 5, 3 and 7 become 2, 0 and 4.
Notice two things. The sign that was lowest becomes zero — correctly, since it had no distinctive support relative to its partners. And the differences are preserved exactly: the spread between the three signs is unchanged.
What survives is the differential — what makes one trinal sign better supported than the others. That is a more informative quantity than the raw total, and it is why the operation exists.
Ekādhipatya Śodhana — the lordship duplication
Seven grahas rule twelve signs, which means five planets rule two signs each.
Support that flows through a lordship is therefore liable to be counted twice — once for each of the ruler’s signs. Ekādhipatya, “single lordship”, corrects for it.
Note that this applies only to the five double-rulers. The Sun and Moon rule one sign each and are unaffected.
Piṇḍa Śodhana — consolidation
The final step, producing the figure that Chapter 72 actually interprets.
Why the purpose matters more than the steps
One manual pass
Do all three reductions by hand, once, on the bhinnāṣṭakavarga you built last week.
The test the course has applied throughout: does performing the calculation change your understanding of the concept? Here it does. You watch the common base being stripped and the differential emerging, and the numbers stop being arbitrary.
Doing it a second time would add nothing, which is why the instruction is once.
Two practical benefits
You can recognise a reduced table on sight. Reduced figures are systematically lower, more spread out, and contain zeros. Raw figures do not. Software configurations differ on whether reductions are applied by default, which is a genuine source of confusion when two people compare figures for the same chart — and now you can tell which is which in a glance.
You can check the source’s own working. Chapters 68 to 74 carry a single continuous worked example. Follow it through the reductions and see whether you reproduce the text’s figures.
That last point is worth taking seriously. The Aṣṭakavarga block is seven chapters of interlocking numeric tables, and a damaged figure anywhere in it propagates silently through the rest. Being able to follow the worked example and confirm it holds together is the only real defence a reader has, and you can only do it if you understand the operations.
Next: reading what the numbers say.
Read BPHS
This is the source itself. The teaching above prepares you for it; it does not replace it.
एवं सलग्नखेटानां विधायाष्टकवर्गकम् ।
त्रिकोणशोधनं कुर्यादादौ सर्वेषु राशिषु ॥ १ ॥
त्रिकोणं कथ्यते विप्र ! मेष-सिंह- धनूंष्यथ ।
वृष- कन्या - मृगास्याश्च युग्म तौलि घटास्तथा ॥ २ ॥
कर्क- वृश्चिक- मीनाश्च त्रिकोणाः स्युः परस्परम् ।
1-2 : Maharishi Parasara said - O Vipra ! After making ready the format of the Ashtakavarga of all the planets alongwith the Ascendant, in the described manner (in the foregoing Adhyaya), Trikona shodhana (Triangular Reduction) has to be carried out for all the signs. Three signs mutually located at an equidistant are known as Trikona Rashi (Triangular signs). For example, Aries, Leo and Sagittarius; Taurus, Virgo, Capricon; Gemini, Libra, Aquarius and Cancer, Scorpio, Pisces are the triads of the Trikona signs.
अधोऽधः सर्वराशीनामष्टवर्गफलं न्यसेत् ॥ ३ ॥
त्रिकोणेषु च न्न्यूनं तत्तुल्यं त्रिषु शोधयेत् ।
एकस्मिन् भवने शून्यं तत् त्रिकोणं न शोधयेत् ॥ ४ ॥
समत्वे सर्वगेहेषु सर्वं संशोधयेद् बुधः ।
पश्चात् विपश्चिता कार्यमेकाधिपतिशोधनम् ॥ ५ ॥
3-5 : The number of Rekhas obtained in the Ashtakavarga of the planets, the Sun, the Moon etc., be written below the signs of Aries, Taurus etc. Amongst the triad of Trikona signs, whichever sign has lesser number of Rekhas, the same number be deducted from the number of Rekhas of all the three signs and the remainder be written under each of them. In the Trikona signs, if any sign is devoid of any Rekha, no Shodhana would be necessary. If all the three signs are having equal number of Rekhas, the Shodhana of all the three signs should be carried out. This should be followed by the Ekadhipatya Shodhana (Reductions).
पूर्वं त्रिकोणं संशोध्य राशीनां स्थापयेत् फलम् ।
पृथक् पृथक् ततः कुर्यादिकाधिपतिशोधनम् ॥ १ ॥
क्षेत्रद्वये फलानि स्युस्तदा संशोधयेद् यथा ।
क्षीणेन सह चाऽन्यस्मिन् शोधयेद् ग्रहवर्जिते ॥ २ ॥
उभयोर्ग्रहसंयोगे न संशोध्यः कदाचन् ।
ग्रहयुक्ते फलैर्हीने ग्रहाभावे फलाधिके ॥ ३ ॥
ऊनेन सममन्यस्मिन् शोधयेद् ग्रहवर्जिते ।
फलाधिके ग्रहैर्युक्ते चाऽन्यस्मिन् सर्वमुत्सृजेत् ॥ ४ ॥
उभयत्र ग्रहाभावे समत्वे सकलं त्यजेत् ।
सग्रहाग्रहयोस्तुल्ये सर्वं संशोध्यमग्रहे ॥ ५ ॥
कुलीरसिंहयो राश्योः पृथक् क्षेत्रं पृथक् फलम् ।
1-5½ : Maharishi Parasara said - O Vipra! First of all the signs after the Trikona Shodhana (Triagular reductions) should be written down separately and thereafter Ekadhipatya Shodhana (reduction) should be carried out. This reduction is applied if both the signs (of a planet) are left with some number (after the Trikona reduction). Further, if two signs have one lord and after the Trikona reduction, one sign is left with some number while the other does not contain any number, the Ekadhipatya reduction is not applicable.
If both the signs of a planet are devoid of any planet and the numbers in both these signs are dissimilar, make the larger number equal to the smaller number.
If there are planets in both the signs owned be a planet, no Ekadhipatya reduction is called for.
If one of the two signs owned by the same planet is occupied and the other is not, and the number in the sign with the planets is lesser than the sign without a planet after the Trikona reductions, the lesser number be reduced from the greater number and the lesser number is kept unaltered.
On the contrary, if the sign occupied by the planet is left with greater number' after the Trikona reduction than the sign without a planet, the number in the unoccupied sign be totally eleminated while the number of the occupied sign be kept intact.
If both the signs of a planet are unoccupied and are having an identental number after the Trikona reduction, eliminate the number from both the signs.
If one of the two signs owned by the same planet is occupied and the other is unoccupied and both these signs are left with equal number after the Trikona Shodhana, eliminate the number from the unoccupied sign.
The signs Cancer and Leo are not subjected to any kind of reduction and the number so obtained after the Trikona Shodhana is kept unchanged. (This is possibily so because their lords own only one sign).
संशोध्यैकाधिपत्यं हि ततः पिण्डं प्रसाधयेत् ॥ ६ ॥
Verse 66 : On completion of Ekadhipatya Shodhana, the Pinda Shodhana be carried out.
एवं शोध्यावशेषांक राशिमानेन वर्द्धयेत् ।
ग्रहयुक्ते च तद्राशौ ग्रहमानेन वर्द्धयेत् ॥ १ ॥
सर्वेषां च पुनर्योगः पिण्डाख्यः कथ्यते द्विज ! ।
गोसिंहौ दशभिर्गुण्यौ वसुभिर्युग्मवृश्चिकौ ॥ २ ॥
सप्तभिस्तुलमेषौ च मृगकन्ये च पञ्चभिः ।
शेषाः स्वसंख्यया गुण्या ग्रहमानमथोच्यते ॥ ३ ॥
जीवारशुक्रसौम्यानां दशाष्टनगसायकाः ।
पञ्च शेषग्रहाणां च मानं प्रोक्तमिदं क्रमात् ॥ ४ ॥
1-4: Maharishi Parasara said - O Vipra ! The numbers duly rectified (after Trikona and Ekadhipalya Shodhanas) be multiplied by the ascribed value of the signs (Rashimana - राशिमान). If there is any planet in the sign, these numbers should also be multiplied by the value (Zodiacal factor) of such planet. The sum total of these products (multiplied figures) is called Pinda.
Multiply signs Taurus and Leo by 10, Gemini, Scorpio by 8, Libra and Aries by 7, Capricorn and Virgo by 5 and the rest of the signs viz., Cancer, Sagittarius, Aquarius and Pisces be multiplied by 4,9,11 and 12 respectively. (These are the value of the signs or the multipliers of the signs).
Now, I will explain the values of the planets. Jupiter has 10, Mars-8, Venus-7, Mercury-5 and the rest of the planets viz., the Sun, the Moon and Saturn have identical value as 5. These are the multipliers (गुणक) for the planets.
The total Shodaya Pinda (क्षोद्य पिंड) for the Ashtakavaraga for the Sun will be:
| Rashi Pinda | = | 65 |
| Graha Pinda | = | 58 |
| Shodaya Pinda | 123 |
The Shodaya Pinda is very important and useful because specific results of the Ashtakavarga are delineated on this basis.
The three types of reductions namely Trikona, Ekadhipalya and Pinda are applied in the manner described in this as well as, in the preceding two chapters. Just as, the Shodaya Pinda for the Sun has been worked out, the Shodaya Pinda for the other planets may also be computed from their respective Ashtakavarga Charts before the delineation of their results are attempted.
Practice
What does Trikoṇa Śodhana remove?
Write your answer down before opening the explanation.
Show the answer and why
Answer: The support common to all three signs of a trine — leaving only the support that distinguishes them from one another.
Signs in mutual trine share an element and a great deal of structural similarity, so a portion of any support they receive is support they all receive together. Subtracting the lowest of the three removes that common base and leaves the differential. What survives is what makes one trinal sign better supported than its partners, which is a more informative quantity than the raw total.
Source: BPHS 69.1–5
What does Ekādhipatya Śodhana remove?
Write your answer down before opening the explanation.
Show the answer and why
Answer: Double-counting arising from one planet ruling two signs — where support credited to a lordship gets counted once for each of the ruler's signs.
Seven grahas rule twelve signs, so five planets rule two each. Support that flows through a lordship is therefore liable to be registered twice. Ekādhipatya — literally "single lordship" — corrects for it. Note that this only applies to the five double-rulers; the Sun and Moon, ruling one sign each, are unaffected.
Source: BPHS 70
Why does explaining the purpose of the reductions matter more than teaching the arithmetic?
Write your answer down before opening the explanation.
Show the answer and why
Answer: Because the arithmetic is done by software and the purpose is what tells you whether a figure means anything.
A practitioner who knows the reductions remove common support and lordship duplication can look at a reduced figure and say what it represents: the distinctive, non-duplicated support a sign carries. A practitioner who has only memorised the steps has a number with no interpretation attached, and that is precisely how Aṣṭakavarga acquires its reputation as arbitrary.
A trine of three signs has raw counts of 5, 3 and 7. What do they become after Trikoṇa Śodhana?
Write your answer down before opening the explanation.
Show the answer and why
Answer: 2, 0 and 4.
Subtract the lowest — 3 — from all three. The sign that was lowest becomes zero, which is the rule working as intended: it had no distinctive support relative to its trinal partners. Note also what this does to the spread — 5, 3, 7 becomes 2, 0, 4, so the differences are preserved exactly while the common base is stripped. That is the whole operation.
Source: BPHS 69.3–5
How would you notice that a printed Aṣṭakavarga had skipped a reduction?
Write your answer down before opening the explanation.
Show the answer and why
Answer: The figures would be uniformly higher than expected, and no sign would sit at zero where the reductions would have driven one there.
Reduced figures are systematically lower and more spread out than raw ones, and zeros appear. Software configurations differ on whether reductions are applied by default, which is a real source of confusion when two people compare figures for the same chart. Knowing what a reduced table looks like lets you tell immediately which you are looking at.
Apply it to a chart
Chart B — Albert Einstein, the synthesis chart — A whole documented life, where the evidence pulls in more than one direction. · about 24 min
| Graha | Sign | House | Degree | Nakshatra | Condition |
|---|---|---|---|---|---|
| Sun | Pisces | 10 | 1°20'30" | Purva Bhadrapada 4 | neutral |
| Moon | Scorpio | 6 | 21°47'54" | Jyeshtha 2 | debilitated |
| Mars | Capricorn | 8 | 4°44'52" | Uttara Ashadha 3 | exalted |
| Mercury | Pisces | 10 | 10°58'43" | Uttara Bhadrapada 3 | debilitated |
| Jupiter | Aquarius | 9 | 5°19'05" | Dhanishtha 4 | neutral |
| Venus | Pisces | 10 | 24°49'11" | Revati 3 | exalted |
| Saturn | Pisces | 10 | 12°01'26" | Uttara Bhadrapada 3 | neutral |
| Rahu | Capricorn | 8 | 9°18'36" | Uttara Ashadha 4 | not assigned in BPHS |
| Ketu | Cancer | 2 | 9°18'36" | Pushya 2 | not assigned in BPHS |
| Lagna | Gemini | 1 | 19°38'02" | Ardra | lord Mercury |
One manual pass through all three reductions. It is the only time you will do this by hand and the point is understanding rather than the result.
- Take the Sun's bhinnāṣṭakavarga you built for Chart B last week.
- Apply Trikoṇa Śodhana: for each set of three signs in mutual trine — Aries/Leo/Sagittarius, Taurus/Virgo/Capricorn, and so on — find the lowest count among the three and subtract it from all three. Where a sign has zero, the rule handles it differently; read verses 3–5 carefully.
- Apply Ekādhipatya Śodhana: where one planet rules two signs, compare their reduced counts and apply the rule of Chapter 70.
- Apply Piṇḍa Śodhana to arrive at the final figure.
- Now the question that matters: compare the raw counts with the final ones. Which signs changed most? Write one sentence on what the reductions did to this particular chart — not to charts in general.
Check your understanding
Answer from memory first. Looking back at the reading before you try turns a memory test into a copying exercise.
Name the three reductions in order and what each corrects.
Write your answer down before opening the explanation.
Show the answer and why
Answer: Trikoṇa Śodhana — removes support common to signs in mutual trine. Ekādhipatya Śodhana — removes duplication from a planet ruling two signs. Piṇḍa Śodhana — consolidates to the final interpretive figure.
Three chapters, 69 to 71, and only about 3,600 words between them. Short, procedurally dense, and conceptually simple once the purposes are stated.
Why does the course ask for one manual pass rather than none?
Write your answer down before opening the explanation.
Show the answer and why
Answer: Because doing it once makes the reductions concrete, and because it lets you recognise a reduced table on sight.
The same test the course has applied throughout: does performing this calculation change your understanding of the concept? Here it does — you see the common base being stripped and the differential emerging. Doing it twice would not add anything, which is why the instruction is once.
Chapters 68 to 74 carry one worked example throughout. What does that mean for reading the reduction chapters?
Write your answer down before opening the explanation.
Show the answer and why
Answer: That the figures in Chapters 69–71 are continuous with those in Chapter 68 — so you can follow the example through and check your understanding against the source's own numbers.
This is unusually helpful and worth using. Follow the worked example through the reductions and see whether you can reproduce the text's figures. If you can, you have understood the operations. It is also the only real defence against a conversion error in the tables, which is the documented risk across this block.
What you carry forward
Two things grow across the whole course. This unit adds to both.
To your chart-analysis process
- **Step 8 (extended) — assess support.** Use the reduced Aṣṭakavarga figures, not the raw ones, and know which you are looking at.
To your field manual
- Trikoṇa Śodhana (Ch. 69): for each trine of signs, subtract the lowest count from all three. Removes support common to the trine, leaving the differential.
- Ekādhipatya Śodhana (Ch. 70): corrects double-counting where one planet rules two signs. Applies only to the five double-rulers.
- Piṇḍa Śodhana (Ch. 71): final consolidation to the figure Ch. 72 interprets.
- Reduced figures are systematically lower and more spread than raw ones, and zeros appear. That is how you tell which table you are looking at.
- Ch. 68–74 carry one continuous worked example. Follow it through the reductions to check your own working.
By now you should be able to
- State what each of the three reductions removes
- Carry one bhinnāṣṭakavarga through all three by hand, once
- Say why the reduced figure is the one Chapter 72 interprets
- Recognise when a printed Aṣṭakavarga has skipped a reduction