Greek multiplication and division table named the Lambdoma* because the Greek letter lambda has the shape of an inverted "V". The shape of the Lambdoma results when one either drawing lines or colors on each ratio position according to its musical tone. Pythagoras used the Lambdoma with his theorem of The Music Of The Spheres.
MUSICAL SCALES**
Among the many styles of music, a scale using seven tones (notes) is the basis of Western music. A scale is a method of organizing sound into music and can be based on any number of notes. The Chinese use five-note scales and some Vedic music uses 22 notes. A diatonic scale is keyed to a series of major or minor notes. The principle of a diatonic scale can be demonstrated using a single string of any length subdivided into seven notes.
1. The basic division of a musical is created by dividing the string in half, or into two parts.
2. The pitch of the string divided into two parts will be higher than the entire string before it was divided.
3. The relation of the two pitches - the whole string before it was divided and the divided string - is an octave. The same eight notes used in the Western musical scale are heard in the divided and undivided string.
The Greeks called the interval formed between the whole and the halved string the "diapason", which literally means "through all" or "through the whole". Sigmund Levarie and Ernst Levy call this concept "the entirety of the tonal space", saying that all tonal relations can be found within the diapason, or within the "entirety". This is an absolute principle recognized in all civilizations at all times.
Division of the interval into eight notes is an arbitrary convention of Western music, so we call tonal space in which eight notes are identified an octave based on the Greek word meaning "eight".
4. Each half of the divided string is divided into eight notes that will sound at a different pitch because the divided string is shorter than the undivided. e. The tonal space of a musical interval is always multiplied or divided by two. If the ratio of the interval is 1:2, the same interval will sound whether "the absolute measurements are 1 and 2, or 10 and 20, or 20 and 40, or 100 and 200". The relationship of each pair remains constant.
5. Vibration is the source of all sound, which is to say that all sound results from vibration. The number of vibrations per second is the "frequency", and is measured in terms of cycles per second (cps). The human ear generally hears sounds that vibrate between the rate of 16 and 20,000 cycles per second. Vibrations lower than 16 cps or higher than 20,000 cps are beyond the range of human hearing.
6. Musical tones are identified by their rate of vibration. If middle C, for instance, is identified as vibrating at 256 cycles per second, the same tone will sound one octave higher by doubling the rate of vibration to 512 cps for the tone of C-2. The same tone can be found one octave lower by dividing by two, for 128 cycles per second to produce the tone of C+2.
7. The ratios of major and minor series of notes in diatonic scales are defined by a Pythagorean table of ratios, the Lambdoma.
* LAMBDOMA UNVEILED: The Theory of Relationships by Barbara Hero, Strawberry Hill Farm Studioes Press **TONE: A Study in Musical Acoustics by Siegmund Levarie and Ernst Levy, published by The Kent State University Press.