These works are structured on nested golden proportions and their inversions.

# Golden Proportions

There are a number of synonyms for this mathematical ratio: *Golden Mean, Golden Ratio, Golden Proportion*, and *Golden Section*. It is denoted by the Greek symbol 𝛗 (phi, pronounced fee).

The golden ratio can be found in the structure of the Egyptian pyramids, the balance of the human form in De Vinci's paintings of the human body, and in nature. The ratio is an irrational number, approximately 0.618, somewhat less than 5/8.

It has a unique mathematically recursive property stemming from the fact that:

1/𝛗 = 1+𝛗.

1-𝛗 = 𝛗^{2}

The Fibonacci number series converges to 𝛗 as it approaches infinity along the progression when taking the penultimate value divided by the last value of the series. This progression was developed in *Liber Abaci*, which Leonardo Fibonacci of Pisa wrote in 1202. The series is generated by taking the sum of the two final numbers in the series as the next value in the series. The relevant Fibonacci series values I use musically are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610.

The golden ratio's recursive feature provides the potential to nest golden sections. That is, one can take the golden section of the golden section, which is the overall length minus the original golden section. Depending on the length of the overall work, one can nest golden sections down to the pulse level of composition.

Use of the golden ratio in music composition has a history going back at least to the 16th century, so my use of it is not new. It is a formal structure, and like any other formal structure, it lends itself as a compositional guide. It can be likened to sonata form, providing a structural framework within which musical material can be composed. It does not provide or suggest content, only structure. It is also possible to superimpose traditional forms (like rondo or sonata allegro) onto the golden ratio. This is explored in my *Five Meditations*.

I have written over 20 compositions using the Golden Ratio and the Fibonacci series.

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### Golden Proportions #1

**2000**

This work is my first use of the golden proportion and Fibonacci series to structure the music rhythm, meter, and tempi.

## Golden Proportions #1

Christina Davis Sheets, a member of a family of musicians dear to me, asked me if I knew if the golden proportion was found in examples of music; she was writing a research paper on the topic of the golden ratio. I told her about a shakuhachi piece I had analyzed in which I discovered that a very magical musical event occurred at the golden mean. What amazed me was that the piece is not notated, and it is a meditation piece with arhythmic phrases determined by breathing.

I had been interested in the golden ratio and Fibonacci series but hadn’t really thought about applying it compositionally. Christina’s request inspired me to derive a rhythmic structure that nested the ratio and its inversion eight levels deep. I then composed a piece using this structure.

** Golden Proportions #1** is scored for the instruments of the Davis family, who performed this studio recording with Alan, french horn; Christina, flute; Rebecca, violin; Sarah, cello; and Susan, flute.

##### Performance by the Davis Family (2000)

##### Performance by Computer (2002)

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### Golden Proportions #2

Whatever happened to this piece?

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### Golden Proportions #3

**2002**

This is a hybrid piece in which Part 1 is scored for trumpet and piano. Parts 2 & 3 can be performed with trumpet and piano, but the preferred accompaniment is synthesized.

This is a studio recording with Chris Dolske, trumpet. Part 1 is a live recording with piano. Parts 2 & 3 use synthesized accompaniment.

##### Part 1

##### Part 2

##### Part 3

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### Five Meditations

**2001**

These * Five Meditations* were originally for organ but then scored for chamber ensemble as part of a performance art work with cell phones. The recordings here are set without the cell phone component.

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### Six Minute-Tours

**2002**

Six (okay, seven) one-minute piano pieces. There were originally six. For reasons I have forgotten, I added *Octatonic Express* after the fact.

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### More Golden Pieces

- When All Paths are Peace
- Northern Lights
- Derivation of Chaos
- Golden Age
- Golden Spring