"Third Octave is Golden." Keely
"This belief in perfect geometry, natural order and predictability was central to the Pythagorean worldview, as it had been to civilizations long before the Greeks. So when it was discovered that a stack of five perfect 5ths does not close to form a regular pentagram at the third octave as expected – forming instead an open and warped pentagonal shape in logarithmic pitch space – this was taken as a profound error in nature.
"What Pythagoras found during his musical experiments was the last interval must be stretched up by a messy ratio of 128:81 (instead of the perfect 5th ratio of 3:2) to align with the third octave. In music lingo, the last interval must become an augmented 5th instead of a perfect 5th.5 While very close to creating a pentagram, it still fell short of the perfection expected.
"This imperfection is central to understanding the Greek worldview because it reveals a conflict, a paradox really, between the cyclic geometry of a regular pentagram and the Spiral of 5ths as it occurs naturally in sound. Philolaus (c470 BC – c385 BC), a “most ancient” follower of Pythagoras, referred to this paradox in the opening of his book Peri physeos, or On Nature:
“Nature in the cosmos is composed of a harmonia between the unlimited and the limited and so too is the whole cosmos and everything in it.
"This gap between the closed or “limited” octave cycle and the infinite or “unlimited” spiral of pitch was a major embarrassment to the Pythagoreans because it undermined the purity of their philosophy of numbers and simple proportions. Given the importance placed on numeric proportions by the Pythagoreans, we have to wonder how they might have reconciled this error within their belief system.
"With many early Pythagorean treatises lost or stolen, we are left with only the accounts of later Greek philosophers such as Philolaus, Nicomachus and Plato. From these accounts we know about Pythagoras’ theories of numerical proportion, his tuning methods, the Greek modes and the supreme importance of his adopted symbol the pentagram.
"As for the pentagram, Pythagoras appears to have first learned of it from his closest teacher, Pherekydes of Syros, who wrote a treatise entitled Pentemychos describing what he called the “five hidden cavities” of the soul. Of course, the notion that a geometrical shape could somehow be related to our “soul” sounds very mystical and unscientific to modern ears, but he was essentially correct about the pentagram playing a very important role in how nature organizes itself.
"Beyond the obvious organizing principle of the number 5 in such things as roses, starfish and the human anatomy, the pentagram contains a very special numerical proportion known as the “golden ratio.” If you have not heard of it before, the golden ratio is an infinite non-repeating proportion of about 1 to 0.6 usually represented by the Greek symbol Phi or “Φ” (pronounced either “fi” or “phee”). The most important thing about this ratio is that it is found approximated everywhere in nature" INTERFERENCE - A Grand Scientific Musical Theory, Richard Merrick, 2011, Third Edition
"The invisible space octaves beyond our range of present perception may be considered three and one half in number: if one considers 1) the last half of the ninth octave to be part of the first, invisible, space octaves, and 2) all of the third octave elements to be unknown, though only four elements are yet undiscovered. The last half of the ninth octave includes the elements tomion, alberton, blacton, and boston. The four yet keynote inert gas alphanon. See page 272. [The Secret of Light, page 292]
Law of Octave
Figure 11.01 - Octave composed of Equal Thirds and Triads
Figure 12.11 - Russells Locked Potential Full Ten Octave Gamut
Figure 12.12 - Russells Multiple Octave Waves as Fibonacci Spirals
Figure 12.12 - Russells Multiple Octave Waves as Fibonacci Spirals - See Also
Figure 17.03 - Analysis of the Octave Gravity Bar
Figure 7B.10 - Russells Periodic Chart of the first four octaves of proto-matter
Figure 9.16 - Russells 1-4 Octaves of Matter as Integrated Light - The Universal Constant
Figure 9.17 - Russells Ten Octaves of Matter as Integrated Light - The Universal Constant
Part 14 - Keelys Mysterious Thirds Sixths and Ninths
RULE OF THE OCTAVE
Scale of the Forces in Octaves
Table 1 - Relations of Thirds
The Russell Nine Octave Chart of the Elements
12.17 - Note about Octave Relationships in Russells System
12.18 - Multiple Octave Progression