How often the MC falls in the whole-sign 10th house, by latitude and rising sign

Key findings

• At the equator the MC is in the whole-sign 10th house in 90.5% of charts. The share falls steadily with latitude: 70.9% at 30° N, 48.8% at 45° N, 38.0% at 52° N (London, Berlin), 29.5% at 60° N (Helsinki, Oslo) and 23.7% at 65° N.
• From 45° N northward the MC is more often outside the whole-sign 10th than inside it. In whole-sign charts from most of Europe and Canada an 11th- or 9th-house Midheaven is the normal case, not an anomaly.
• The rising sign matters more than the latitude. In the tropical zodiac, with Aries or Pisces rising the MC is in the 10th sign in every chart at every latitude in the range. With Taurus, Gemini, Capricorn or Aquarius rising it never is from 50° N onward (Gemini and Capricorn already from 48° N). Where it goes instead depends on the sign: with Taurus or Aquarius rising it is always the neighbouring 9th or 11th, while with Gemini or Capricorn rising it drifts one sign further — at 60° N 60.2% of Gemini-rising charts have the MC in the 8th sign, and at 65° N 100.0% do.
• Sidereal (Lahiri) charts give almost the same totals: 89.6% at the equator, 36.4% at 52° N and 21.9% at 65° N. The ayanamsa shifts the sign boundaries but not the geometry, so the per-sign picture changes: only Pisces rising keeps the MC in the 10th sign at every latitude, and with Aries rising it is there in 87.5% of charts at the equator and 30.5% at 45° N.
• In Porphyry houses a chart has intercepted signs if and only if the MC is outside the whole-sign 10th house. So every share of charts with the MC in the 10th sign on this page is also the share of Porphyry charts with no interceptions. Verified on all 2,376,000 charts with 0 exceptions, and proved below.

Method

For house purposes a chart is fixed by two numbers: the geographic latitude and the local sidereal time, expressed as the right ascension of the Midheaven (ARMC). Over a full year the ARMC at any fixed clock time sweeps the whole 360° circle, so across a population of births the ARMC is close to uniformly distributed even though births cluster by hour of the day (the residual seasonality of births leaves a density deviation of a few percent, far below the effects tabulated here). A uniform sweep of the ARMC is therefore a fair model of “all charts” at a latitude.

• ARMC swept from 0° to 360° in steps of 0.01°, offset by half a step so no cusp lands exactly on a sign boundary: 36,000 charts per latitude, at every integer latitude from 0° to 65° N.
• Cusps from swisseph.houses_armc (Swiss Ephemeris 2.10.03): pure spherical geometry, no ephemeris files, no planets.
• Obliquity of the ecliptic: the true obliquity on 2026-01-01, 23.4381°. Shifting it by ±0.05° (more than it changes in a lifetime) moves the headline shares by at most 0.13 percentage points.
• Sidereal comparison: the same cusps with the Lahiri ayanamsa of 2026-01-01 (24.2203°) subtracted from every longitude.
• “MC in the whole-sign 10th” means the MC lies in the tenth sign counting the rising sign as the first. “Rising sign” is the sign of the Ascendant. An intercepted sign is a sign containing no house cusp.
• Grid check: below the polar circle the MC and the Ascendant both move monotonically with the ARMC, so the answer changes only at the 24 sidereal times where one of them crosses a sign boundary, and the exact share is a sum of interval lengths with no grid at all. The grid reproduces it to within 0.01 points: 0°: 90.51% on the grid, 90.5105% exact; 45°: 48.79% on the grid, 48.7994% exact; 52°: 37.96% on the grid, 37.9614% exact; 60°: 29.51% on the grid, 29.5074% exact.

The script that produced every number on this page is published next to it as mc_whole_sign_10th.py. It needs Python 3 and pyswisseph, runs in about ten seconds and rewrites the data files this page and the CSV are built from; a downloaded copy writes them next to itself.

Share of charts with the MC in the whole-sign 10th, by latitude

All rising signs together. The last column is the share of Porphyry charts without intercepted signs, counted independently: by the theorem below it must equal the tropical column, and it does.

By rising sign

Tropical zodiac. Each cell is the share of charts with that rising sign whose MC is in the whole-sign 10th. The table is symmetric about the 0° Aries–0° Libra axis: Aries pairs with Pisces, Taurus with Aquarius, Gemini with Capricorn, Cancer with Sagittarius, Leo with Scorpio, Virgo with Libra.

What the table shows

• Aries and Pisces rising: always. These are the signs of shortest ascension in the north. At every latitude 0° Aries rises at the very moment 0° Capricorn culminates, and Aries rises in less sidereal time than the MC takes to cross Capricorn, so the MC is still in Capricorn, the tenth sign, when the last degree of Aries rises. Pisces mirrors this: it finishes rising just as the MC leaves Sagittarius.
• Taurus, Gemini, Capricorn and Aquarius rising: never, from mid-latitudes up. The MC last reaches the 10th sign at 47° N for Gemini and Capricorn and at 49° N for Taurus and Aquarius; Cancer and Sagittarius follow at 55° N. Where the MC goes instead is not the same for the four: with Taurus or Aquarius rising it is the 9th or 11th sign at every latitude in the range, but with Gemini or Capricorn rising it drifts one sign further as you go north — at 52° N a Gemini-rising chart has its MC 8.9% of the time in the 8th and 91.0% of the time in the 9th; at 60° N 60.2% of the time in the 8th and 39.8% of the time in the 9th; at 65° N 100.0% of the time in the 8th. Capricorn mirrors this into the 11th and 12th.
• Virgo and Libra rising: the count is constant from 21° N up. Libra starts rising at the very moment 0° Cancer culminates, at every latitude. From 21° N it is still rising when 0° Leo culminates, so the Libra-rising charts with the MC in Cancer fill exactly the sidereal-time interval between those two culminations, which does not depend on latitude; the share still drops with latitude because Libra takes longer to rise the further north you go, so the same number of charts is spread over more Libra-rising charts. Below 21° N Libra rises for less time than the MC spends in Cancer, so the end of that interval falls to Scorpio rising instead and every Libra-rising chart has its MC in Cancer — which is why the Libra cell reads 100% at the equator.
• Southern hemisphere. At latitude −φ the totals are identical to +φ and the rising-sign rows swap with their opposite signs. Computed at 52° S: total 38.0% against 38.0% at 52° N, and Aries rising 74.9% = Libra rising 74.9% in the north, Aquarius rising 44.6% = Leo rising 44.6% in the north, Pisces rising 74.9% = Virgo rising 74.9% in the north.

Where the MC goes when it leaves the 10th sign

Whole-sign house of the MC across all charts, tropical. Up to 47° N the MC only ever steps into the neighbouring 9th or 11th sign; from 48° N the 8th and 12th appear, first with Cancer or Sagittarius rising, and from 63° N the MC can land in the 7th or 1st, only with Cancer or Sagittarius rising.

Porphyry houses: intercepted signs appear exactly when the MC leaves the 10th sign

Theorem. Below the polar circle, a Porphyry chart has intercepted signs if and only if the MC is not in the whole-sign 10th house. The “if” half holds for every quadrant house system; the “only if” half is specific to Porphyry.

Proof

Let the MC be at longitude m and the Ascendant at m + q with 0° < q < 180° (true below the polar circle). The half-circle from the MC to the IC contains exactly six sign boundaries, because its ends are 180° apart. Porphyry trisects each quadrant: houses 10, 11, 12 are each q/3 long and houses 1, 2, 3 are each (180° − q)/3 long. A sign is intercepted exactly when both of its boundaries fall inside one house.

If. Suppose the MC is outside the whole-sign 10th. Then the arc from the MC to the Ascendant crosses k ≠ 3 boundaries, and the other quadrant crosses 6 − k, so one quadrant crosses at least four. Four boundaries in three houses: some house contains two consecutive boundaries, so it swallows the sign between them. This half uses only the fact that a quadrant is split into three houses, so it holds for Placidus, Koch, Regiomontanus, Campanus, Alcabitius and every other quadrant system.

Only if. Suppose the MC is in the whole-sign 10th. Then each quadrant crosses exactly three boundaries, 30° apart, and its three houses are equal; write L for their length and x for the distance from the start of the quadrant to its first boundary. Then 0 < x < 30° (the quadrant starts inside a sign) and x > 3L − 90° (a fourth boundary at x + 90° would lie outside the quadrant). A house can hold two boundaries only if L > 30°, and then the first boundary, at x < 30° < L, lies in the first house, so the second and third houses could only hold the pair at x + 30° and x + 60°. For the first house that needs x + 30° < L, which with x > 3L − 90° gives L < 30°. For the second house it needs x + 60° < 2L, again giving L < 30°. For the third it needs x + 30° > 2L, which with x < 30° gives L < 30°. Each case contradicts L > 30°, so no house of the quadrant contains a whole sign, and the opposite quadrants are the same houses turned through 180°. This half needs the houses to be equal thirds, which is why it fails for the other quadrant systems. ∎

Numerical check

The script tests both directions on all 2,376,000 charts of the main sweep (66 latitudes × 36,000 sidereal times) and finds 0 mismatches. It also checks the “if” direction for five other quadrant systems on a coarser grid (70 latitude–system pairs, step 0.1°) and finds 0 violations. The table shows how far those systems are from the “only if” half: the share of charts with the MC in the whole-sign 10th that still have intercepted signs.

At the equator the five other quadrant systems give the same figure. That is not a copy: at latitude 0° the horizon passes through the celestial poles and cuts every diurnal circle in half, and Placidus, Koch, Regiomontanus, Campanus and Alcabitius all collapse onto the same cusps.

Corollary for practice: if you read whole-sign houses and someone asks why a Porphyry chart of the same birth shows an interception, the answer is the same fact seen from two sides. At 52° N, 62.0% of Porphyry charts have intercepted signs.

Limits of applicability

• Latitudes 0–65° only. Above the polar circle (66.56°) part of the ecliptic never rises and the opposite part never sets: the Ascendant, taken as the eastern intersection of ecliptic and horizon, jumps by about 180° twice a day and runs backwards in between (the rising-sign study tabulates this). Placidus and Koch are not defined there; Swiss Ephemeris switches to Porphyry and reports it only as an error code, which chart software may not pass on.
• The sweep weights every sidereal time equally. That is what a year of births does, even if they all happen at the same hour of the day; only a sample narrow in both season and hour — births at one clock hour within a single month — would not be uniform.
• Shares are per chart, not per person: nothing here uses real birth records. Real populations are not spread evenly across latitudes.
• The obliquity is fixed at its 2026-01-01 value; the sensitivity test above bounds the effect of that choice.
• The sidereal figures use the Lahiri ayanamsa of 2026-01-01, applied to every chart. Taking it for 1950-01-01 instead (23.1587°) moves the sidereal totals by at most 0.43 points (at 61° N) but the shares by rising sign by up to 12.85 points (Aquarius rising at 61° N): the ayanamsa moves the sign boundaries, and with them the charts each rising sign holds. The other common ayanamsas differ from Lahiri by up to about 1.5° (Raman; Fagan–Bradley 0.9°, Krishnamurti 0.1°), which moves the numbers by amounts of the same order.