Organa Music Group
Musing The Golden Ratio
Nothing moves when divided into equal parts. Things are divided slightly ‘off the center cut’ in nature and in art... so that each part may experience - or be experienced as a 'difference' in relation to its whole... so that each part may move and grow as a whole unto itself. This natural phenomenon can be expressed in mathematics and may be seen and heard everywhere in the natural world. It is beautiful and fearful symmetry. I try to touch it often... but I've learned that only intelligent fools try to hold it down with their finger or put it under their thumb... --Ben Higgins
Symmetric Modal Fusion is a theoretical observation. The pupose of its description is to provide ideas to train our ears to an enhanced perception (and meaning) of all 12 tones (and all 12 Keys) -- against (or in relation to) one single (central) tone. This tone is identified at the 12 o'clock position in any 'Harmonic' Circle Of Fifths.
As an integral part of this introduction - there are occasional attempts to convince you that the study of mathematics is the axis of true love -- but as our interest compels us further... we'll remember that the study of harmonic movement is something entirely different.... something more like a specialized experience of calculous.
As an integral part of this introduction - there are occasional attempts to convince you that the study of mathematics is the axis of true love -- but as our interest compels us further... we'll remember that the study of harmonic movement is something entirely different.... something more like a specialized experience of calculous.
The following 'studies' represent just a few of the ideas you will encounter here... along with my attempt to describe the reasoning used to create them.
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Harmony & The 'Golden Ratio'
The unique ratio that expresses a small part standing to a larger part in the same proportion as the larger part stands to the whole of both measurements -- is represented by the Greek letter "phi" in mathematics. It's closely related to the famous 'Fibonacci Series' and is the core mathematical principle expressed in nature. It's often referred to as the 'golden ratio' (sometimes - the 'divine ratio') - with its simplest expression as: a:b = b:(a+b). When the number 1 is substituted for the large part; the small part always balances to a range between (Approx.) 0.618 - 0.668.
What does this have to do with music?
The crux of all the wonderful mystery propagated in the sounds of Western Music; and the bewildered amazement that can occur in the brightest musical minds that study tonal harmony is derived from the fact that the tone "So" can be heard; physically, as the "dominant overtone" of its relative "Do." This means that the ear can and will 'mistake' "So" for "Do" and vice versa. We hear music in two keys at once! Let me say that again: "We hear music in two keys at once!"
When we divide the frequency of any "Do" by the frequency of its "perfect 5th above" ("So"); the resulting ratio between the frequencies of "Do" and "So" is always in the range of the golden ratio. It is 0.666 in 'Pythagorean Resonance' -- and it may be 0.667 - 0.668 in the 'equal tempered' system commonly heard in Western Music.
What does this have to do with music?
The crux of all the wonderful mystery propagated in the sounds of Western Music; and the bewildered amazement that can occur in the brightest musical minds that study tonal harmony is derived from the fact that the tone "So" can be heard; physically, as the "dominant overtone" of its relative "Do." This means that the ear can and will 'mistake' "So" for "Do" and vice versa. We hear music in two keys at once! Let me say that again: "We hear music in two keys at once!"
When we divide the frequency of any "Do" by the frequency of its "perfect 5th above" ("So"); the resulting ratio between the frequencies of "Do" and "So" is always in the range of the golden ratio. It is 0.666 in 'Pythagorean Resonance' -- and it may be 0.667 - 0.668 in the 'equal tempered' system commonly heard in Western Music.
In The Interest Of Precision - There Is Another 'Golden Interval'
The interval of the Perfect Fifth can be represented (mathematically) as a ratio of 0.666 : 1 (Pythagorean Resonance) - and this interval is a primary (and primal) interval for students of music theory to study. 0.666 : 1 = 2/3 : 1 - and is the result of simple divisions within a vibrating string into its integral parts (1/2, 1/3, 1/4, etc.). These are the physics of harmonics.
Though we have previously defined the Golden Ratio as a 'range' between 0.618 & 0.668; thereby including the Perfect 5th interval -- the true (mathematically correct) number for this exact ratio is at the lower end of the range -- 0.618. The difference in this range between the physical (0.666 = Perfect Fifth) and the mystical/mathematically-precise (0.618 = Minor 6th) is approximately one half-step... a rather large half-step, actually.
It turns out that this particular half-step is the one we find ourselves listening to (and playing around with) in those moments when we hear magic in our melodies... as its 'function' shifts through the roots of chord progressions that relate to a 'home' key center. It is the reason behind why the 'major 3rd' and the 'minor 3rd' (of a single key center) can sometimes be heard at the same time... in harmony.
Though we have previously defined the Golden Ratio as a 'range' between 0.618 & 0.668; thereby including the Perfect 5th interval -- the true (mathematically correct) number for this exact ratio is at the lower end of the range -- 0.618. The difference in this range between the physical (0.666 = Perfect Fifth) and the mystical/mathematically-precise (0.618 = Minor 6th) is approximately one half-step... a rather large half-step, actually.
It turns out that this particular half-step is the one we find ourselves listening to (and playing around with) in those moments when we hear magic in our melodies... as its 'function' shifts through the roots of chord progressions that relate to a 'home' key center. It is the reason behind why the 'major 3rd' and the 'minor 3rd' (of a single key center) can sometimes be heard at the same time... in harmony.
Real Quick:
... for those that wish a better understanding of this calculation: as we know - the larger the bottom number (of a ratio), the smaller that ratio:
'C' (262hz) divided by 'G' (393hz) = 0.6666667 = Perfect Fifth
'C' (262hz) divided by 'Ab' (424hz) = 0.61792453 = Minor 6th
'C' (262hz) divided by 'G' (393hz) = 0.6666667 = Perfect Fifth
'C' (262hz) divided by 'Ab' (424hz) = 0.61792453 = Minor 6th
Organa:
It is demonstrated in nature that enduring systems of knowledge derive their lasting authenticity and truth through the harnessed balance of energies expressed in this Golden Ratio (phi).
Organa Music Group strives to learn, live, and publish from the heart of this creative principle -- this rational organon -- this language of phi.
Organa Music Group strives to learn, live, and publish from the heart of this creative principle -- this rational organon -- this language of phi.