I'm sure you've heard of the Fibonacci Sequence where we take two numbers and them together to get the next number in the sequence and then take that sun and larger of the two we added together to get the next number and so on. The sequence looks like this:

1,1,2,3,5,8,13,21,34,55,89,144, etc. The closer to infinity we go, the closer any two adjecent numbers will divide out to 0.618 (or phi) or 1.618 (or Phi). Phi represents the the most economical packing that nature has to offer and it turns up in countless ways. You've heard of the Golden Mean, the Golden Section, the Golden Spiral, etc. All based on phi. One odd thing is that the 11th term in the sequence, 89, has a reciprocal of 0.11235955...which is a close but slightly distorted Fibonacci Sequence which the ancients used to demonstrate macrocosm and microcosm. The macrocosm is contained within the microcosm but imperfectly--the holographic universe where every part contains the whole but, like a holograph, the smaller the microcosm, the more distorted image it contains of the macrocosm.

In a musical scale—easiest to see on a keyboard—there are 12 half-steps in an octave which consists of 8 notes (white keys) separated by 5 black keys (designated sharps or flats as the case may be) and 8+5=13. The 5 black keys are also separated in a group of 2 and a group of 3! Moreover 3 and 2 form the ratio for the Perfect 5th interval which consists of 5 notes covered in 7 half-steps. A string plucked "open" will play a Perfect 5th higher when shortened by 2/3rds and both notes will be in beautiful harmony.

Musical scales also follow phi. The smallest scale is two notes tuned a perfect 5th apart. To be able to complete the octave, we next need five notes—a pentatonic scale. The next scale in the sequence would be the diatonic scale of seven notes. It is obtained by adding the two-note scale and pentatonic scale together. However, we can only play a diatonic scale in one key unless we temper it. When we temper, however, we lose tonic relationships in that scale since the notes are equally spaced apart musically. So we must next add the diatonic and pentatonic scales together to obtain a chromatic scale of twelve notes which enables us to play the diatonic 7-note scale in 12 different keys.

What would the next scale be? We would add the chromatic and diatonic scales together to obtain a 19-note scale. Some musicologists have developed such a scale as well as 19-tone instruments. The perfect 5th in this scale however does not sound consonant. So we would have to resort to a 31-note scale but such instruments would be quite difficult to play. But notice the sequence of scale progression: 2, 5, 7, 12, 19, 31, etc. Again, the Fibonacci Sequence. While it is different from the one Fibonacci proposed, the scale progression sequence also has the property where two adjacent numbers in the sequence dividing out to phi the closer the sequence approaches infinity because any sequence that involves adding the last two numbers to obtain the next number in the sequence will produce phi in the manner described.