The 3D lambdoma frequency-ratio matrix is a tool used in harmonics—the study of ratio and proportion, quality, and quantity. This lambdoma reveals a union of arithmetic rows expanding from front to back, harmonic columns contracting from left to right, and geometric diagonals expanding from smallest center to largest top. The lambdoma is limited here to an index 16 that can grow astronomically; harmonic practice requires that any distant ratio be reduced or expanded to a lower index for comparison. The visible 1/1 diagonal separates ratios greater than 1 and less than 1. The exchange of all ratios on either side of this 1/1 diagonal produces a string length lambdoma because frequency is inversely proportional to string length. The object of this Demonstration is primarily visual, yet it may be useful to note that lambdoma frequency ratios are constructed from rows of the overtone series (1/1, 2/1, 3/1, 4/1, ...) combined with columns of the undertone series (1/1, 1/2, 1/3, 1/4, ...).
Manipulate controls: The sliders create weighted perspectives that are for the most part incomplete or partial lambdomas. When the sliders are equal, true lambdomas are displayed. Rotate to display face views of the classical arithmetic, harmonic, and geometric proportions.