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Dr. Alan C. Bowen reveals how Archytas developed these means from Pythagorean harmonics in his article “The Minor Sixth (8:5) in Early Greek Harmonic Science,” The American Journal of Philology, 1978: “For it was during this time that scales of a double octave magnitude, i.e. the Greater Perfect System, were constructed to facilitate the analysis of melody.”
Now, using the Greater Perfect System, the middle of the octave, the geometric mean as the irrational square root of two or 9/8 cubed, could be expanded and turned into just 2:1 as the middle of the double octave.
So Archytas then took the double octave, the “Greater Perfect System,” as four but it is also the square of the octave used by Pythagorean number theory. The result is that four times or the square of the weight to stretch the string now makes twice the frequency, instead of half the frequency at double the string length. This was the “bait and switch” trick that was the direct inspiration, later, for Newton’s inverse square law of gravity!242
242 James H. Bunn, Wave Forms: A Natural Syntax for Rhythmic Languages (Stanford University Press, 2002) citing “Newton and the Pipes of Pan” from Notes and Records of the Royal Society of London, 1966.
Originally posted by fulllotusqigong
So that is from my 2007 article How the West Lost Alchemy
Instead of the above system, the alchemical Pythagorean Tetrad relies on complimentary opposite harmonics so that an equilateral triangle of geometric points equals the continued proportion 1:2:3:4 as the octave, perfect fifth and perfect fourth music intervals. In “orthodox” Pythagorean harmonics this was known as the “subcontrary mean” whereby the complimentary opposites of the Tetrad were maintained in violation of “divide and average” mathematics. So for the Tetrad A:B is 2:3 and B:A is 3:4 against the commutative property, A x B = B x A.
The reciprocal of the arithmetic mean of the reciprocals of a specified set of numbers.
The American Heritage® Dictionary of the English Language, Fourth Edition copyright ©2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.
(Mathematics) the reciprocal of the arithmetic mean of the reciprocals of a set of specified numbers: the harmonic mean of 2, 3, and 4 is 3(½ + ⅓ + ¼)-1 = 36/13
Collins English Dictionary – Complete and Unabridged © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003
The reciprocal of the arithmetic mean of the reciprocals of a specified set of numbers.
The American Heritage® Science Dictionary Copyright © 2005 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.
However, Fragment 2 of Archytas is the only relevant text, and the most natural reading of Fragment 2 is that Archytas' predecessors, including Philolaus, called the third mean "subcontrary," while Archytas tells us that he himself calls it "harmonic."
This Hertz issue was returned to after I had come across the power chord distortion creating subharmonics from the Perfect Fifth.
So the 200/300 Hertz interval is 66 Hertz while the 300/200 hertz interval is 150 hertz. So that 66 Hertz is F to 100 Hertz as C -- a Perfect Fifth subharmonic. 150 Hertz is the Perfect Fifth as C to G. The same C and so it's non-commutative.
Now your question was is there a doubling of octaves or not with frequency.
If the natural harmonics are used then the octave doubling does not line up with the Perfect Fifth/Perfect Fourth doubling. Is the octave not connected to the fifth?
So then UncleV you said that actually I'm just trying to make a different scale than equal-tempered. Actually in order to have the Perfect Fifth line up with the Octave then it has to go against the natural harmonics.
DenyO asked me earlier if I thought that the square root of two was an infinite number
Is the infinite length of a diagonal of a square whose sides are one unit in length an example of this infinity contained by materialist geometry?
I will try it like this, is the square root of two an example of infinity contained by materialist geometry?
Philip Hugly and Charles Sayward Abstract A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that proposition.
They have been likened to the Free Masons, in that they served as a kind of Council of Foreign Relations or New World Order…. Archytas developed the musical scale into a political metaphor for the scales of justice. What gave music this imagery of social balance and just proportion was the ability of its mathematics of harmonic (“geometric”) proportions to serve as an analogy for how inequities of wealth and status rendered truly superior men equal in proportion to their virtue — which tended to reflect their wealth. By this circular logic the wealthy were enabled to rationalize their hereditary dominance over the rest of the population.
Any number that ends in a decimal that continues forever is called "irrational." The mathematics books on numbers claim that irrationals can also be located on the line and hence are in the set of real numbers. I don't think this is true. I will tell you why.
Irrationals are becoming numbers but they will never, for all of eternity, be numbers. Think about this: (s rt) 2 = 1.4142135... and the integers continue on forever approaching some value but never ever reaching it. Since (sq rt) 2 will never be a number because it is in continual motion moving towards some number, it is a non-number. Since it is a non-number, it cannot be located on the number line. I conclude then that only rational numbers have a location on the line and hence only rational numbers are real numbers.
The problem here, as I see it, is we have switched from a scale consisting of points on the line to a scale with lengths on the line.
Applying the Pythagorean theorem, 1(sq) + 1(sq) = 2 and the square root of 2 is an irrational of 1.4142135... which is a non-number. But, how can this be since we can measure the diagonal's length with a ruler and get a definite answer? The length of the diagonal is not becoming a number;
Mystical Intercession And Practical Prevention: Devotional Music And Healing In Badakhshan, Tajikistan Dissertation Benjamin D. Koen My Dissertation builds upon research completed for my master's thesis"Efficacy of Music-Prayer Dynamics In Healing and Health Maintenance: A Literature Review and Pilot Study." Both projects are part of a broader, long-term interdisciplinary collaboration aimed at investigating the relationship between music, prayer, and healing in both traditional and clinical contexts. I recently completed a field research project in Tajikistan, during which I collected both ethnomusicological and physiological data that form the basis of the dissertation. My dissertation introduces music-prayer dynamics as a concept in order to model the affective use of music and prayer in healing and in daily human experience. The dissertation will examine the music-prayer dynamics among the Pamiri people of Badakhshan, and analyze madoh devotional music as performed by heretofore-unrecorded master musicians and traditional healers. For several decades prior to the independence of Tajikistan in 1992, significant political changes forced the performance of madoh and traditional healing practices to go underground. While considering musical performance in a broader context of the local music-culture, the dissertation explores the recent practices of music and prayer in the daily life of the Pamiri people as well as in the main hospital of the region. Thus, traditional madoh is reconceptualized in the dissertation according to its current purpose and function as a genre. The dissertation proposes a culturally sensitive approach and builds a model of music-prayer dynamics that can be used in professional health care environments, and by individuals. Hence, the results of the dissertation also contribute to and fit within the framework of recent biomedical research concerning complementary and alternative therapies in medicine.
Koen returns, in the last chapter, to the ontological principle of wholeness and introduces what he calls the Human Certainty Principle (HCP), which is a certainty or knowing that emerges in human consciousness from an unknowable and hence uncertain dimension, and which underlies, accompanies, or facilitates the experience of healing. It is a quality of calm certitude, he asserts, that is borne of the higher self, which by definition is linked to and expressive of unity and wholeness, which, in the author's words, "here has been described in part, by the term tâwhid, vahdate vojud, the matrix of all matter." Experiencing and conceptualizing music, prayer-meditation, and healing in the context of his study, Koen found out that balance equals health, imbalance equals illness and reestablishing a balance equals healing.
So you apparently haven't noticed me citing several scholars on this issue?
Their answer is that the square root of two is illogical because it confuses infinite distance with geometric length.
The results showed the presence of delta activity in human hippocampal spontaneous EEG also during wakefulness. The activity in the delta range exhibited a peculiar bimodal distribution, namely a low frequency non-oscillatory activity (up to 2 Hz) synchronized between hemispheres mainly during wake and REM sleep, and a faster oscillatory rhythm (2-4 Hz). The latter was less synchronized between the hippocampi and seemed reminiscent of animal RSA (rhythmic slow activity). Notably, the low-delta activity showed high inter-hemispheric hippocampal coherence during REM sleep and, to a lesser extent, during wakefulness, paralleled by a (unexpected) decrease of coherence during NREM sleep. Therefore, low-delta hippocampal state-dependent synchronization starkly contrasts with neocortical behavior in the same frequency range. Further studies might shed light on the role of these low frequency rhythms in the encoding processes during wakefulness and in the consolidation processes during subsequent sleep.
So then G to C as 4/3 can not be used as the Perfect Fourth. Why? Because it is inconsistent with the "root" frequency as 1 or 100 hertz.
In other words in the same scale if G to C is 4/3 then why is G to C as 4/3 in the harmonic series not allowed for defining the Perfect Fourth? Earlier in the thread this was brought up.
So instead the subharmonic of 1 as the Harmonic Series or 100 Hertz is used as 2/3x or C to F in the opposite direction -- a longer wavelength.
So you're saying we can just ignore that it is C to F as the Perfect Fifth in one direction while it's C to G as the perfect Fifth in the other direction?
You asked me if this meant that the subharmonic was a subset of 3/2x and I answered yes. Why? Because harmonics are perceived as pitch intervals and not just Hertz frequency.
132 plus 66 = 198 Hertz. Yep it's almost the same as 200 Hertz. So you have chosen to wave off this difference as inconsequential
and the next is the fifth of 198hz, which is not perfectly C but then we're dealing with nice round numbers for simplicity's sake.
give or take a couple hertz as I had to round them down in reducing the freqs to lower octaves for easier workability
the magnitudes (properties) of classical physics can be determined at any time with any required precision. On the other hand, quantum-mechanical entities are complementary in the sense that at a given time they are able to possess only some of their possible properties. Now classical mechanics is a special case of quantum mechanics, which means that all the objects of the macroscopic level obey the laws of quantum mechanics. Hence, we must re-interpret the signs of classical physics as designating properties, which apply to their objects (the objects of the macroscopic level) in almost all circumstances (whereas according to classical physics they apply strictly in all circumstances). This means that having adopted quantum mechanics we must drop the classical interpretation of classical physics (Feyerabend 1975)
It's censorship to demand someone to only answer "yes" or "no" when obviously you are asking a question that you think you already know the answer to.
Genetic Determinants of Time Perception Mediated by the ... www.plosone.org/article/info:doi/10.1371/journal.pone.0012650 by OV Sysoeva - 2010 - Cited by 15 - Related articles Genetic Determinants of Time Perception Mediated by the Serotonergic System ... variant of MAOA VNTR gene compared with 'high expression' variant, and 3)
the representation of longer durations is more dependent upon other cognitive processes, such as attention and memory, and therefore influenced by many other substances. There are also quite consistent findings that the representation of intervals s is associated with serotonergic (5-HT) activity –.
Changes of the neurotransmitter serotonin but not of hormones during short time music perception
The 5-HT content of platelets, however, was higher during the perception of pleasant music as compared to the perception of unpleasant music indicating an increased release of 5-HT during unpleasant music (748 mg/109 platelets vs. 699 ng/109 platelets; p < 0.014). The difference of the 5-HT level was significantly correlated to the score of unpleasantness as rated by the subjects. Our data suggest that perception of unpleasant music induces increased release and decreased peripheral and possibly also central intracellular content of 5-HT.
Demkov (22), succeeded in avoiding the deficiencies in the commutation rules of the quantum mechanical ladder operators for the anisotropic oscillator — in particular for the case of a 2:1 frequency ratio — by dividing the states into two groups (of even and odd total energy, respectively) each one of which belonged to a unitary unimodular symmetry group.