10/05/04. edited 02/10/08: The Golden Spiral or Golden Mean or PHI (1.618...) is a decimal approximation* to the ratio of the Fibonacci Series expressed in Whole Numbers. In order of increasing decimal equivalent value, as shown below, the music interval equivalent oscillates between Minor and Major Sixths. As the PHI ratio decreases a musical interval of Enharmonic Sixth is approached save there is no Enharmonic Sixth in standard music and as such cannot actually be reached any more than the approximation can resolve itself.
PHI is an irrational number just as PI is irrational; i.e., a reduced ratio of two unlike entities being a straight line and a curved line. Which is to say should never be reduced because to do so is dividing apples by oranges - a straight line (diameter) is not the same as a curved line (circumference). The same principle applies to PHI whole number ratios being composed of two distinct entities; i.e., a lower quantity and a higher quantity. I admit this sounds trivial but it is not as nature does not admit to a 0.9 pregnancy. Please review Hull's Quadrature of the Circle wherein he reviews John A. Parker's original squaring of the circle by NOT dividing unlike terms such as triangles, squares, circles and components thereof such as straight and curved lines.
| RATIO | DECIMAL APPROXIMATION | INTERVAL |
| 1:1 | 1 | Unison |
| 2:3 | 1.5 | Perfect Fifth |
| 5:8 | 1.6 | Minor Sixth |
| 13:21 | 1.61538... | Minor Sixth |
| - | - | Enharmonic Sixth |
| 21:34 | 1.61904... | Major Sixth |
| 8:13 | 1.625 | Major Sixth |
| 3:5 | 1.6667 | Major Sixth |
| 1:2 | 2 | Octave |
Nature, especially music, works on a natural basis. The human tendency to "round off" is never found in nature. The (in)human decimal (Metric) system has naught to do with naturally occurring phenomena which seems to prefer whole number ratios and proportions whether base 2, 3, 5, 7 or 12 or other. For a comprehensive discourse comparing the natural system (Egyptian and British) of metrology with the Napoleanic metric system see Smythe's The Great Pyramid.
- APPROXIMATION: something that is approximate especially a mathematical quantity that is close in value to but not the same as a desired quantity. [Merriam-Webster]
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See Also
3.04 - Power Accumulation via Fibonacci-like Patterns
12.19 - Fibonacci Relationships
12.21 - Fibonacci Whole Numbers v Irrational Decimal near Equivalents
Fibonacci Relationships
Fibonacci Series
Figure 12.12 - Russells Multiple Octave Waves as Fibonacci Spirals
Golden Mean
Golden Number
Golden Section
Interval
Inverse Square Law
Music
PHI
PI
Quadrature of the Circle
Square Law