the problem with intensity is the difficulty of measuring it.
Pythagoras is creditied in making the connection between notes in a scale and ratios . These ratios are arithmoi.
Before Pythagoras the quanities or magnitudes of intensity were not specified, but a skilled artisan exposited these values as words , speeches, insights and oracular utterances. They were perceived as the inspiration of the Muses.
These performances were thus called music, and involved all manner of instruments and materials, thus Pythagoras is credited with the insight that yhese instruments had sympathetic vibrations witheach other, that the banging of an anvil could cause resonance in other objects and in particular in the strings of a lyre. He thus experimented with different lengths of the string of a lyre. As an experimental platform he set up a a mono chord and found that the notes could be related to the length of the strings. He realised that the lengths in ratio gave him a precise way to specify these notes. By applying the arithmos net to one string he began to explore the ratios, and their relationship to the notes. The arithmos gave him an analytical tool to measure the "frequency " of a note. The intensity was yet to follow.
Pythagoras set up the mono chord as an experimental platform. By weighting the "catgut" lyre string, he found a relationship between weights and pitch. Thus he established ratios of weights to pitch, which are in modern terms tension ratios.. Then using the monad analysis tool, he began to establish an arithmos net for a given weight. by dividing the string, Thus monas was structured into parts that a whole fit(harmonias), and the pitch was determined by these ratios. As Pythagoras learned more he realised that he could alter the ratios to produce another type of harmony based on the monas, which also harmonised the ratio forms.
The ratio forms were written in letters not numerals, and thus the knowledge was encoded in ordinal letters. For Pythagoras it was not the ordinal letter relationships that were important, it was the arithmoi analytical tool. With this tool he was able to posit the Harmony of the Spheres, the music of the planetary orbits. Poetical as it sounds, he was communicating that there was an arithmoi that describes the motion of the planets . It was not until Tycho Brae and Kepler that these ratios were expressed in numeral relationships.
The School of Pythagoras at Croton was run like a monastery, He Himself (Pythagoras the Master) set the tone, the curriculum and individual lessons. His students were committed to learning with and through him, some were titled as Akousmaticoi (auditors), others were Mathematicoi (graduate students), and the instruction was directed through verbal rapport with the Master. The organization and purpose of this Croton Monastery is so similar to the monastic schools which were organized in China and then in Japan after the tenth century, that a general awareness of what the Zen scholars were about can help us to understand more about the very fragmented history of the Croton school.
-the Akousmaticoi were in fact the chanters who lead the meditations and the rote rehearsals of the principles, whereas the Mathematicoi were the thinkers and astrologers who calculated the "tymes" of the heavenly gods from which(whom) all knowledge was derived, organised/conventionalised(sunthemata) and symbolised(sumbola)
Pythagoras was a mystic, and believed in reincarnation, and in previous lives. He therefore believed he had prior knowledges and knew things by the combined thought processes of many personas. Thus his insights are often those of a seer, like Nostrodamus, and encoded in ordinal letters as well as words and analogies. Nevertheless he made it his aim to be as plain as possible and consequently had great support amongst his followers. His enemies however were many and varied and his teaching of the common peoples created problems for vested interests. Despite his ground breaking philosophical ideas he was killed in the wars of the cities and states as a nuisance.
The ordinal letter convention imposes order on relationships of magnitude in a way numerals do not. Thus rules of grammar and alphabet impose on such notation in a way that they do not on numerals. Thus the potential for meditative reflection and analogy is vastly increased by this form of symbolising magnitude than by the numeral conventions. Thus gematria is far more rich in symbology than its sub relative numerology, which eventually gave way to cardinal number terms and numerals. Though quicker in arriving at numeral elationships than the ordinal letter convention, it was similarly bereft of much of the inherent meanings within Gematria. That these would be abstruse and difficult to comprehend goes without saying, but for the devotee it would be their life work to attain to that level of mastery. For others,they would take what they could apply and move on.
When the indians came up with the decimal modulo structure for Numerals, they revolutionised Numerology because they now imposed an ordinal letter type system on to the numerals. The ordinal letter type system is based on the alphabetic conventions of the day. Contrast this with the Babylonian sexagesimal modulo system which again is clearly ordinal but very specialist and scribal, and requires one to learn 60 ordinal distinctions based on the Assyrian syllabry. Their Gematria was consequently very rich and detailed and would perhaps encode vast amounts of insights and distinctions. The Greeks seemed to profit from labouring over these Babylonian/Assyrian Texts.
The use of music in healing and meditation allied with the insight of the arithmos for a given string tension , and indeed all the arithmoi for strings, meant he knew he could find the notes that harmonised or sympathetically resonated with a persons mental state. Complex as that sounds he was proved right. A persons dimensions were susceptible to these ratios!
Movement and flow are visual concepts and thus Arithmos is a fractal web of dynamic spatial relationships derived from a monad. These relationships to magnitudes of space can be multiples of the monad, or parts of the monad or a continuum of boh parts and multiples. This is why the concept of fractal is so applicable
The essential thing is that the Babylonian had so fully analyzed the speech-sounds that he felt entire confidence in them, and having selected a sufficient number of conventional characters – each made up of wedge-shaped lines – to represent all the phonetic sounds of his language, spelled the words out in syllables and to some extent dispensed with the determinative signs which, as we have seen, played so prominent a part in the Egyptian writing. His cousins the Assyrians used habitually a system of writing the foundation of which was an elaborate phonetic syllabary; a system, therefore, far removed from the old crude pictograph, and in some respects much more developed than the complicated Egyptian method; yet, after all, a system that stopped short of perfection by the wide gap that separates the syllabary from the true alphabet.
Byblos is an ancient Phoenician city along the coast of modern day Lebanon. Its name was the origin of the Greek word "biblion" which means "book", hence "bibliography" and "Bible". In short Byblos is nearly synonymous with writing.
Ironically, Byblos was also home to a still poorly understood script during roughly the middle of the second millenium BCE. There are only a few short examples of this script, mainly on stone or metal. This script contains roughly 100 signs, which fits with the number of signs necessary for a syllabary.
The following is an example of the Byblos script.
Nunerals are special scripts that developed out of the dual use of syllabries both to encode phonemes and to order or rank things. They are quite distinct from magnitude which was usually apprehended in many different ways, including one to one correspondence of object used to denote unit. The coalescence of unit, magnitude and numeral takes place over many centuries as the languagees developed in descriptive force, grammatical structural conventions and alphabet- phoneme convergence.
It recently became clear to me that the concept of arithmos was in fact mostly Platonic, and that in fact Pythagoras as a mythology is largely Platonic. Thus many notions ascribed to Pythagoras cannot be independently verified.
The notion of the Pythagorean arithmoi similarly cannot be independently distinguished from the platonic one. Nevertheless, the Platonic notion of arithmos is as I have described it.
I have frequently referred to the net as a referent of the notion of an arithmos, but realise now that Plato had a concrete referent for arithmos, and that was simply Mosaic! What we call a mosaic was what Plato called an arithmos. It seems so simple now I have said it , but it is not so simple to establish.
However it can be established through the etymology of the word mosaic as it leaves the Greek culture into the Latin culture. Museion is the late Greek word associated with a decorated shrine to the Muses(mussai). These shrines were decorated with rich and detailed mosaics.
Why are these shrines important? Because in every way they fulfilled what we now see as the role and function of a museum, again derived from the same root Greek notion. Thus the shrines had a significant cultural and educational role.
The research into the Muses indicates this significant role since they popped into literary form in Homers writings. Thus whatever the relationship and family history of the Muses in relation to the pantheon, they conveyed a exemplar model of inspirational education of humanity. Thus they formed the paradigm for Plato's founding and development of The Academy..
What is unclear is whether this development of shrines to Muses was so widespread, that it was synonymous with founding a school or a monastery to teach esoteric and philosophical insights, by koans. Thus gradually opening up students to the insights from the Muse; or whether it was a special religios cultural act of an established community.
Decorating such shrines with mosaics is clearly a rough and ready act of beautification, less expensive and time consuming than sculpture, but more enduring than al fresco paintings. Thus a proposed cheap form of decoration became a mystical symbol of thr Muses, and going deeper a symbol of a deeper atomic reality. Those "atoms" jbeing the monads selected for the mosaic, and by which inspired artistry any scene in this world and the world of the gods could be de pixeled!(depicted, but rather mosaicised).
It seems entirely reasonable to assert that when Plato wrote arithmoi he thought of mosaics of such monads, beautifully and dynamically describing and delivering the forms and ideas of our reality. In no way could he distinguish this experience from the agency and actions of the Muses.
In this regard I may point out that Euclid's Stoikeioon was therefore an introduction to Plato's theory of form or idea, and that was everyway related to The grand Mosaics in the academy shrines and all the shrines to the Muses.
As I have discussed before arithmos is derived from "arhea" a stylised form of communication, and thus through kairos, fitting to describe these nets or collections of Monads. It is also fitting that they would be called mousaion, that is inspired by the Muses. The name mosaic therefore brings together a substantial history of philosophy and conceptualising of the experience of reality.