Q: Pythagoras discovered that to create the interval of a octave by stretching out two strings you need to play the second string using a ratio of 21?

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A perfect octave

23 is prime. No strings.

What are the factor strings of 120

Factor strings are multiplication sentences. 3 x 3 is a factor string for 9.

Every bit can either be a 0 or 1. So to find the amount of bit strings of length either, you do 2length to find the amount of bit strings there are of a given length.

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The first musical scale was likely developed by the ancient Greeks, specifically by Pythagoras. Pythagoras discovered the mathematical relationships between vibrating strings that relate to musical intervals. This mathematical understanding paved the way for the development of musical scales.

2:1

Tension force is present in stretched strings, ropes, and cables. This force is directed along the length of the material and arises due to the stretching or pulling of the material.

That is called "tension".

It is highly probable that the Greek initiates gained their knowledge of the philosophic and therapeutic aspects of music from the Egyptians, who, in turn, considered Hermes the founder of the art. According to one legend, this god constructed the first lyre by stretching strings across the concavity of a turtle shell. Both Isis and Osiris were patrons of music and poetry. Plato, in describing the antiquity of these arts among the Egyptians, declared that songs and poetry had existed in Egypt for at least ten thousand years, and that these were of such an exalted and inspiring nature that only gods or godlike men could have composed them. In the Mysteries the lyre was regarded as the secret symbol of the human constitution, the body of the instrument representing the physical form, the strings the nerves, and the musician the spirit. Playing upon the nerves, the spirit thus created the harmonies of normal functioning, which, however, became discords if the nature of man were defiled. While the early Chinese, Hindus, Persians, Egyptians, Israelites, and Greeks employed both vocal and instrumental music in their religious ceremonials, also to complement their poetry and drama, it remained for Pythagoras to raise the art to its true dignity by demonstrating its mathematical foundation. Although it is said that he himself was not a musician, Pythagoras is now generally credited with the discovery of the diatonic scale. Having first learned the divine theory of music from the priests of the various Mysteries into which he had been accepted, Pythagoras pondered for several years upon the laws governing consonance and dissonance. How he actually solved the problem is unknown, but the following explanation has been invented. One day while meditating upon the problem of harmony, Pythagoras chanced to pass a brazier's shop where workmen were pounding out a piece of metal upon an anvil. By noting the variances in pitch between the sounds made by large hammers and those made by smaller implements, and carefully estimating the harmonies and discords resulting from combinations of these sounds, he gained his first clue to the musical intervals of the diatonic scale. He entered the shop, and after carefully examining the tools and making mental note of their weights, returned to his own house and constructed an arm of wood so that it: extended out from the wall of his room. At regular intervals along this arm he attached four cords, all of like composition, size, and weight. To the first of these he attached a twelve-pound weight, to the second a nine-pound weight, to the third an eight-pound weight, and to the fourth a six-pound weight. These different weights corresponded to the sizes of the braziers' hammers. Pythagoras thereupon discovered that the first and fourth strings when sounded together produced the harmonic interval of the octave, for doubling the weight had the same effect as halving the string. The tension of the first string being twice that of the fourth string, their ratio was said to be 2:1, or duple. By similar experimentation he ascertained that the first and third string produced the harmony of the diapente, or the interval of the fifth. The tension of the first string being half again as much as that of the third string, their ratio was said to be 3:2, or sesquialter. Likewise the second and fourth strings, having the same ratio as the first and third strings, yielded a diapente harmony. Continuing his investigation, Pythagoras discovered that the first and second strings produced the harmony of the diatessaron, or the interval of the third; and the tension of the first string being a third greater than that of the second string, their ratio was said to be 4:3, or sesquitercian. The third and fourth strings, having the same ratio as the first and second strings, produced another harmony of the diatessaron. According to Iamblichus, the second and third strings had the ratio of 8:9, or epogdoan. The key to harmonic ratios is hidden in the famous Pythagorean tetractys, or pyramid of dots. The tetractys is made up of the first four numbers--1, 2, 3, and 4--which in their proportions reveal the intervals of the octave, the diapente, and the diatessaron. While the law of harmonic intervals as set forth above is true, it has been subsequently proved that hammers striking metal in the manner