The Fibonacci Series
Leonardo Fibonacci originally from Pisa, Italy developed a mathematic theory in 1202 that constructs a series of numbers. This sequence of numbers is extremely simple but has had dramatic impact on mathematics, music, architecture, and virtually every other mathematic discipline - including psychology, neurology, and a bunch of others if we look deep enough into each area.
In other words, we can find aspects in our every day life which, whether we know it or not, are related either directly to this sequence of numbers or indirectly to the impact that this theory has had on cognitive thought through the ages.
Here's how to construct the sequence: The first two numbers in the series are one (1) and one (1). Each subsequent number of the series is the sum of the two numbers preceding it.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.
- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5
- 3 + 5 = 8
- 5 + 8 = 13
- 8 + 13 = 21
- 13 + 21 = 34
- 21 + 34 = 55
- 34 + 55 = 89
- etc.
So, what does this have to do with music?! Well, consider our "octave":
- It's made up of 12 + 1 (13 including the octave) chromatic tones.
- We consider the semi-tone (1) and whole tone (2 semi-tones) to be building blocks of diatonic scales.
- Pentatonic scales are 5 tones.
- Diatonic scales are 8 tones.
- The 1st, 3rd, and 5th tones in the diatonic scale are the building blocks of root chords.
I could go on to demonstrate the influence that the Fibonacci series has had in music...but it's likely better (if you're not convinced) to seek other information specific to the Fibonacci series. I suggest the following:
- The Fibonacci Calculator
- The Fibonacci Spiral
- This web site explains how the Fibonacci numbers are used to create geometric shapes. It's an example of how the Fibonacci series is related to form and design - in graphic art in this case, but also in music, architecture, sculpture, etc.
- Fibonacci Numbers in Nature
- This is a beautifully written page which shows evidence that the Fibonacci sequence can be found in our natural world (i.e. plants, etc.)
Introducing Phi (What is Phi?)
Φ
Phi is an irrational number (which means that we cannot comprehend it in its entirety; which means that if we try to express it as a number it will have an infinite amount of decimal places.) Phi has to be represented as a ratio -- this is difficult because we can't really comprehend the ratio! But that's where the Fibonacci series becomes useful in calculating Phi.
Phi is equal to the ratio of any number in the Fibonacci series to the preceding number in the same series. Furthermore, our understanding of Phi becomes more and more accurate the deeper into the Fibonacci series that we go. For example, the first two numbers in the series are one (1) and one (1); therefore we could start by saying that Phi is 1:1 (but that's quite inaccurate). The second and third numbers in the series are one (1) and two (2) and the ratio would be 2:1 (but that's still innacurate). We can jump ahead a little now to, let's say, the 5th and 6th numbers in the series: five (5) and eight (8). We can now say that Phi is equal to the ratio 8:5...
Eight (8) divided by five (5) is equal to 1.6 (and we're getting a whole lot more accurate! ...but on to the next numbers in the series)...
13 divided by 8 = 1.625 ...then on to the next
21 divided by 13 = 1.6154
...skip a few...
89 divided by 55 = 1.618182
If we were to continue on like this for another hundred iterations, then our ratio will more accurately describe the value of Phi. For example, the 99th and 100th numbers in the Fibonacci series are:
- 218,922,995,834,555,000,000.00
- 354,224,848,179,262,000,000.00
And the ratio between those two numbers (represented by 32 decimal places) is:
Φ = 1.61803398874989484820458683436564
And the reciprocal of Phi is:
| 1 |
= 0.618033988749894848204586834365882 |
| Φ |
Presumably, we could calculate into the nth numbers of the series (perhaps the millionth number) and get a little closer to defining Phi...but it's likely that our brain would be dried up long before we succeed.
Golden Section
"Golden Section" is a concept based on Phi and can be defined like this:
Any linear expression (a line, or a duration in time) divided by Phi will effectively demonstrate "Golden Section".
Huh?!...
"Golden Section" is also referred to by these terms:
- Golden Mean
- Golden Ratio
- Divine Proportion
These terms describe a phenomenon which can be observed in virtually every natural element in our surround.
- "Golden Section" can describe our anatomy such as the proportionate sizes of our arms and legs, our torso, the location of our elbows and knees within their limbs.
- It can describe the relative size of a tree trunk to the size of its branches. Further, it can describe the size and inner designs of tree leaves.
- It can describe our aesthetic such as the proportions of sculpture and architecture and the durations of songs and verses, choruses, or the structure of symphonic masterpieces.
GoldenNumber.net is a very good resource if you're interested to know more about "Golden Section".
Ever since Ptolomy started thinking about the universe we've been developing mathematics to explain our existence, and we've adopted a belief that our natural environment can be rationalized via patterns and formulae (because we're convinced that this natural phenomenon is created via patterns and formulaic processes). If art reflects nature, then art can be created of divine influence. Ptolomy himself, after all, first envisioned a divine music that he referred to as "music of the spheres".
Whether art reflects nature or vice versa there certainly is a lot of evidence of one in the other. For some visual proof of this, I suggest you visit Fibonacci Numbers in Nature, then refer to Fibonacci vs. da Vinci which displays how both the Mona Lisa and The Vitruvian Man conform to the basic principles of geometry related to the Fibonacci series.
While Fibonacci's geometry is visually pleasing in and of itself, his peculiar sequence of numbers is represented in literally thousands of pieces of art by way of formal design. Although the Fibonacci series may be represented abstractly in a piece of music by such things as instrumentation or harmonic texture, it is perhaps easiest to identify this sequence of numbers according to the linear timeline of a piece of music. For example, where does the climax of a piece of music occur within the overall form? Where are significant changes in key or chord structure placed throughout a piece?
If you happen to have on hand a score of Bartok's "Music for Strings Percussion and Celeste", I suggest you mark the measures that correspond to the Fibonacci series (mark measures 1, 2, 3, 5, 8, 13, 21, 34, 55, and 89). You'll notice that these measures mark significant "moments" in the piece and correspond with a change of key, or a change in instrumentation and texture, or change in dynamics. Some evidence suggests that Bartok may have done this consciously -- certainly he had opportunity to study number theory and may have applied it in this composition. In the same manner we can analyze the music of Mozart, Beethoven, Debussy, etc. and find the presence of the Fibonacci sequence.
As well, if you consider that the recipricol of Phi (0.61803) is roughly equal to 2/3, then you'll find it uncanny how many pieces of music seem to climax approximately 2/3 through.
More About Bartok
I mentioned Bartok's "Music for Strings Percussion and Celeste" above -- briefly. I'd like to add more to the discussion of Bartok.
In 2004, a visitor of this website (Frank) contacted me about the information I had written in this page. In a previous reversion of this page, I had implied that Bartok was cognisant of the influence the Fibonacci series had in his music and may have consciously employed the number theory and "Golden Section" in his composition.
Frank has studied the Fibonacci number theory for many years and wrote a paper regarding Bartok's use of the Fibonacci series. Frank told me that he could find "no conclusive evidence that he actually used it consciously".
Below I will copy the email correspondence that I had with Frank in 2004:
From Frank to me:
Hi,
On your website you state, regarding Bartok's use of the Fibonacci Series, that "We know from Barok's letters and journals that he did this consciously."
I can not find any source material that confirms this assertion. If you could give me the references for this I would greatly appreciate it.
Thanks, Frank
From Frank to me:
Hi,
I recently wrote a paper re Bartok's use of the Fibonacci series and could find no conclusive evidence that he actually used it consciously. I would greatly appreciate it if you could give me the references in his letters and diaries which indicate his intentional incorporation of the Fibonacci series in his works.
Hope to hear from you.
Thanks, Frank
At this point, I had to admit my ignorance!
From me to Frank:
Hello Frank,
I'm afraid I cannot produce exact references – perhaps I misspoke and/or didn't believe that anybody would actually read my website.
I suppose I might have said, "Considering Bartok's education, it's possible to assume he had studied number theory and may therefore have consciously used the Fibonacci series to create coherent musical forms."
Or I might have said, "Bartok's composition so effectively pronounces the Fibonacci series that one might conclude that he had used this number theory consciously."
The fact of the matter is that you have written a paper re Bartok's use of the Fibonacci series – and others have too. These gestures may themselves be testament to the notion that he used this number theory purposefully in his compositions. I believe that it's an unlikely coincidence.
In any case, I'd be very interested to read your paper on the subject. You have likely gained more authority on the subject than I, and perhaps I should tap your expertise to make appropriate revisions to my web page.
Regards, Dave Sabine
And Frank responded graciously:
From Frank to me:
David,
Thanks so much for your response.
In researching my paper on Bartok I never got around to examining his diaries and letters and that's why I was so curious about the statement on your website. I did go through the biographies of his children, Peter and Bela, and since there was so much information revealed as to his interests and habits the absence of any mention of the Golden Mean seemed to hint that it was not a preoccupation of his. Also, since a number of Bartok scholars, including, Elliott Antokoletz, David Cooper, Malcolm Giles and Halsey Stevens do not cite his diaries or his letters in discussing this matter I tended to assume such evidence of his conscious use of the Fibonacci series did not exist.
The main proponent of this idea was Erno Lendvai, and even he contradicts himself on this issue.
According to Giles, Lendvai practiced "a good deal of doubtful arithmetic manipulation." Also, Bartok himself states that "all my music is determined by instinct and sensibility."
Personally I think there is a good possibility that he did employ the series consciously, as you said, his education makes it extremely likely that he encountered it, and I will still follow through on his diaries and letters to see if something supporting that guess comes up. Of course, he may have used it consciously and decided to keep it to himself. There is also a strong possibility that the series emerged naturally as a by product of Bartok's use of the pentatonic scale and pentatonic forms (Arch form), and his natural sense of proportion.
Thanks again for your reply. If you would like I can attach a copy of my paper for your perusal.
Frank
Unfortunatly, I never did receive Frank's article -- perhaps I will someday and I hope all is well with Frank.