Pythagorean Tuning is a tuning scheme
that is constructed by co-relating the ratios
obtained from a chain of perfect 5ths.
In other words, the notes in 1 octave are determined
by their perfect relative 5ths
from other octaves.
It's considered an un-tempered scale,
a natural system,
based on the interval of the Perfect 5th (or ratio 3:2).
After the span of 7 pure octaves & 12 pure 5ths a discrepancy develops,
known as the Pythagorean Comma.
The Pythagorean Comma (PC) represents a measurement of 'left-over space'.
It is also known as the Diatonic Comma,
as well as the ratio 531441:524288,
which is the numeric value of 1.0136433.
The PC is closely associated in the construction of today's Equal Temperment (ET) Tuning System.
* Pythagorean Tuning may be thought of as inclusive
because it bridges numeric gaps that the
divisive Just Intonation system does not blanket.
Thousands of years ago music was understood as a force of nature, represented through numbers and ratios instead of notes on paper. It wasn't a form of entertainment, instead it was used as a tool to bring harmony back into that which was in dis-ease. History provides us with the mathematical language of music, all that we need to do is to re-insert those numbers and harmonic concepts back into the science of today. By putting the concepts that govern harmony back into the scientific process & applying the results into the needs of daily life, as a society, we end up moving along With the flow of nature's current as opposed to against that current. This current of energy that effects every cell in our bodies & continually flows through the natural world is known as the frequency spectrum. This energy can be fashioned & directed, through the study of Music Theory and Harmony.
Musical relationships based on Astrological principles:
If you have no prior knowledge of music theory this is going to make absolutely no sense, I'm sorry. Those who have studied music will notice that some fundamental rules, that are taught in schools, are different. Academically altered for the practicality of musical performance ..in short, this type of music theory does not describe the laws of 'listenable music' but instead the motion of the natural force of electromagnetic & electroacoustic energy. The first relation of electromagnetic frequency as music theory is to be made from our solar system, sun, moon & stars.
Aries is traditionally the key signature of Db Major (Bb minor), Taurus is Eb Major (C minor), Gemini F# Major (Eb minor) and Cancer is Ab Major (F minor). The key signature of Bb Major (G minor) has historically been associated with the constellation of Leo the Lion, C Major (A minor) with Virgo, D Major (B minor) associates to Libra and E Major (C# minor) to Scorpio. The key signature of Sagitarius is F Major (D minor), Capricorn is G Major (E minor), Aquarius is A Major (F# minor) and Pisces is B Major (Ab minor).
*It's interesting to note that the scales of C Major/A minor are associated with the constellation Virgo. Virgo has been long associated with the notion of purity (Venus/Ishthar/Virgin Mother Mary) & historically the corresponding scales are constructed of pure intervals, lacking accidentals of sharps or flats.
The zodiacal key signatures are not the familiar key signatures of today's modern music theory. Much of the mythology remained intact while the actual logic & reasoning, which pushed the applicational development of modern music theory over time, subconsciously changed.
The principles that guide the naming of notes and scales used in music are quiet different from the natural tuning system that the universe offers us, but the construction of those scales is no different, it's all based on same mathematics, with only slightly altered algorithms. The universe is constructed upon principles of both the Just & Perfect intervals. The generation of Pythagorean tuning brings the concept of tones & scales closer to the functioning faculties of our biology.
In the quest to share written music around the world, and for the performers to reproduce those written notes in a musically pleasing way, the tuning system of Equal Tempernment (or equal distance) was adopted around the world & the piano keyboard became the main vehicle for musicians. This switch to piano was good for the concept of consistency of performance but bad in the fact that the natural meaning (& exactness of the desired pitch) within music was approximated and sacrificed ..Like the Tower of Babel, in trying to capture communication the original meaning behind the language was destroyed.
So the idea of pure intervals is not limited to just white or black keys on a keyboard but instead: pure intervals refer to numbers - specifically, whole numbers that represent distances involved in the construction of musical intervals. When combined together those intervals make scales & key signatures. Thus what we know & recognize today as C Major (c-d-e-f-g-a-b-c) in Equal Temperment is both an approxiamation to the natural frequency and inaccurate according to the laws of nature and the universe. The scales of music theory were fashioned over time for our aesthetic sensibilities as opposed to natural laws.
The notes representing the scale of Virgo (C Major/A minor) are naturally approximated by its particular function. For example: The sign of Virgo is an earth sign and alludes to the geometry of the triangle. The mathematical proportions of the triangle point to the interval of the Major/minor 3rd & 6th. The quality of Virgo is known as mutable (or adaptable) and speaks to the geometry of the square. The square is mathematically descriptive of the intervals of the 4th & 5th. From these geometric intervals it's possible to create Tetrachords which act as 'functions of the harmonic elements that radiate from the constellation Virgo'. Keeping in mind that harmony has a fluid-like motion & constantly fluctuates within itself, it's extremely plausible to map the route of that motion by combining the geometric distances to the corresponding Enharmonic spellings. This a very non-exact representation of the Virgo scale (based on note name instead of frequency): C-D-Eb-F-F#-G-A-B-C/C# ..Very different than the C Major scale I learned in music school.