Pythagorean’s Lambdoma
Jill Mattson – Ancient tales suggest that the Lambdoma matrix pattern originated in Atlantis, later surfaced in ancient Egypt, and sparked interest later still in ancient Greece. The diagram is attributed to Pythagoras, who spent 20 years studying in ancient Egypt before starting his famed mystery school, who guarded its secrets.
The secrets of the Lambdoma, an ancient matrix of numbers, allured philosophers, geniuses and spiritual aspirants throughout the ages. The ancients prized this mathematical template and its significance was cloaked in secrecy. Currently the concept of the Lambdoma matrix is relatively unknown, and is not cited in most dictionaries.
At first glance the Lambdoma is a pattern that creates mathematical tables. One can insert any number in the corner of the Lambdoma, the 1/1 position and create a table according to its guidelines. The right axis represents 2N, 3N, and 4N… (Multiplication table) and the left axis represents N/2, N/3, and N/4… (Division table). The Lambdoma is a multiplication and division table made up of whole number ratios and whole numbers. Division and multiplication are reciprocal operations.
The Lambdoma has many unique features. The rows multiply and the columns divide or vise versus depending on the quadrant used. An odd number is always divided by an even or vise versus. Whatever is added, subtracted, multiplied or divided is found elsewhere on the table. If you were to take the square Lambdoma diagram and fold it diagonally along the center 1/1 line, you would see that each ratio that is on top of each other is a reciprocal relationship.
On the diagonal the numbers on the edges cancel each other out. Each square that touched each other has an inverse or reciprocal relationship. These same touching squares when multiplied equal one. Tones that have a reciprocal relationship can have a balancing effect. The Lambdoma is a table of numbers that mirror each other in this way.
In the Lambdoma each axis is an inverse proportion to each other, representing duality. Ratios that are less-than one are on the left axis and they progressively descend. Ratios that are more-than-1 are on the right axis of the diamond and ascend. Every equation on the right side may be reversed on the left axis.
Being mathematical, the chart can be translated to accommodate frequencies of audible sound. Any number representing the cycles per second of a frequency may be placed in the 1/1 position. One can then compute frequency values for each square. One axis then represents the harmonic cycle while the other suggests the notes of a scale.
If you insert a number into the 1/1 position, then all numbers generated in the table are in perfect harmonic proportions, making this table useful for architects.
Musically the first six Lambdoma positions mirror ratios of shapes in crystallography and manifest in infinity calculation of chemical elements, chromosome numbers and plant structure. [1]
At its elemental level everything vibrates and produces harmonics. That is indisputable science. The Lambdoma is a “transparency” diagram that the ancients believed could be placed over anything and reveal its harmonic patterns. This pattern correlates to everything vibrational and is a “one size fits all.”
The Lambdoma structure bears resemblance to ratios of the distance of some planets’ orbits. It resembles patterns found in nature, aromatics, chemistry, crystallography, cybernetics, art, geometry and music. Vibrations exist in relationships, brainwave patterns and ideas. The Lambdoma can be applied to many arenas.
Using the theory of sympathetic resonance or energy transference between octaves, the Lambdoma diagram displays relationships between all kinds of things, creating bizarre connections. For example it could show subtle energy relationships between a plant’s vibrations and intervals in a Byzantine hymn, or the relationship of one’s horoscope and colors.
Others have used sounds from harmonics to quicken the growth of lettuce, accelerate the speed of composting (low sounds), imitate dolphin sounds (high harmonic frequencies) and an amazing variety of applications. [2] The Lambdoma matrix chart also bears mathematical relationships to Diophantine Equations [3] and the Farey series, [4] as well as reflects the work of Isaac Newton and current day scientist, Georg Cantor. [5]
References
[1] Levarie and Levy. Tone: A study in Musical Acoustics, Oberlin Printing: USA, 1968. Pg. 30.
[2] Hero, Barbara.See Lambdoma.com for a fascinating adventures!
[3] An algebraic equation with two or more variables whose coefficients are integers, studied to determine all integral solutions.
[4] The Farey sequence of order n is the increasing sequence, from 0 to 1, of fractions whose denominator is equal to or less that n, with each fraction expressed in lowest terms.
[5] Quote from Lambdoma.com. Georg Ferdinand Ludwig Phillip Cantor (1845-1918) was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are “more numerous” than the natural numbers. Cantor’s theorem implies the existence of “infinity of infinities.”
SF Source Jill Mattson Jul 2025