One Point Six
The observation of a particular ratio, the Golden Ratio, in nature (and duplicated by the human) is not just coincidental. The Golden Ratio is equal to 1.618... and is used frequently in geometry. Phidias (circa 450 BC) the great painter and sculptor used the golden ratio in his work. Fibonacci (1170–1250) created the numerical series now named after him; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically. The golden ratio is used in architecture (Parthenon's facade), painting (Leonardo da Vinci's illustrations of polyhedra in De divina proportione), book design, finance (trading algorithms), music (Bartok, Chopin, Wolfe, etc...), Egyptian pyramids, and observed in nature (atomic scale in the magnetic resonance of spins in cobalt niobate crystals).
I thought it might be interesting to relate the Golden Ratio to the tonal frequency, not just the ratio of the intervals between notes or regions of a motif in a piece. The following table displays the note and its associated frequency:
Scientific Name | Frequency (Hz) | F*1.618 (Hz) | Golden Note |
G3 | 195.998 | 317.125 | D#4 |
G#3 | 207.652 | 335.980 | E4 |
A3 | 220.000 | 355.960 | F4 |
A#3 | 233.082 | 377.127 | F#4 |
B3 | 246.942 | 399.552 | G4 |
C4 | 261.626 | 423.311 | G#4 |
C#4 | 277.183 | 448.482 | A4 |
D4 | 293.665 | 475.150 | A#4 |
D#4 | 311.127 | 503.404 | B4 |
E4 | 329.628 | 533.338 | C5 |
F4 | 349.228 | 565.051 | C#5 |
F#4 | 369.994 | 598.650 | D5 |
Notice that the "Golden Note" does not exactly match the calculation, for example: (195.998 Hz)(1.618) = 317.125, yet D#4 that has a frequency of 311.127, is associated with G3. Since the human ear can not detect a difference of frequency between the interval of about 5 - 10 Hz, then this calculation is still deemed valid.
The next step is to determine some golden-chords, or chords made of minor sixth intervals, so that I can create some three-note chords to manipulate using the twelve-tone technique. These particular three-note chords will be derived by constraining the intervals betweens tone to be that of the Golden Ratio. The chord is derived by taking the ratio 1.618 and multiplying it by the tone frequency. For example A is associated with F, which in turn is associated with C#. The following table displays these chords:
Chord |
A F C# |
B G D# |
C G# E |
D A# F# |
E C G# |
F C# A |
G D# B |
Note that every interval is now the Golden Ratio equivalent and can now be used to compose a piece of music. If you take any one of these chords and try to build each chord by continuing the 1.6 rule, it will become apparent that the chord actually has a periodic structure to it. For example, if the chord, A F C#, is to be made into a tetra-chord by adding another tone, it will eventually wrap around on itself: "A" is to "F" and "F" is to "C#", but "C#" is to "A", and "A" is to "F". As you can see, this is the case for all these chords. This is quite interesting, and represents a sort-of symmetry structure to these chords.
I decided to calculate the primes, inverses, retrogrades, and retrograde inverses from the following set:
A F C# B G D# C G# E D A# F# |
I choose to use only the prime and retrograde inverse to derive the following piece. There are obviously many variations that can be produced.
Play this piece slow and gracefully, yet with some intensity.
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