​Geometrics is a song cycle consisting of four pieces, each revealing its respective title’s geometric shape through various devices and techniques. One of these devices is the shape of the song taking on the shape of its title. In other words, the form of each song is a projection of the geometric shape suggested by its title. Meter and rhythm are also used to reference the geometric shape. In the text, the shapes are used as symbols of metaphorical imagery. These and other devices are discussed more specifically in the notes below for each piece.

​The form in Circles is essentially ABCCBA—a palindrome. As such, the first half of the piece arches toward C, while the second half, in a mirror of the first, arches back toward A, alluding to the imagery of a circle. Repetition is another device used to suggest a circle’s unending curve. At A—the verse—both the strings and the voices repeat a two-bar motif, and at B—the chorus—the vocal line is repeated. Also at B, the second violin plays an arpeggiando, a bowing technique in which the player’s bow repeatedly rocks over the strings in a circular fashion, evoking the imagery of a circle both aurally and visually. The allegorical imagery of the text is also circular in concept: “I stand alone at noontide waiting for the sunrise; I stand alone at midnight waiting for the sunset.” At the second verse, this line is sung as a round.
            The A section—especially the closing instrumental A section—was influenced by, of all things, a mechanical noise coming from one of the attractions at Disneyland. While standing in line with my family, I noticed rhythmic sounds, resembling a cello and percussion, coming from a rotating mechanism along the path of the line. With nothing better to do, I decided to pass the time by transcribing the pattern in my head (it’s amazing what form of entertainment monotony will sometimes compel us to create). I became interested in the musical results from the seemingly random interaction between the cello-like and percussive sounds, and was inspired to consider composing a piece using some of the musical ideas. What the strings play here in Circles is not a literal transcription; rather, it is the essence of what I heard.

​The text in Square Dance expresses alienation—the idea of feeling like a square peg in a round hole (or, more specifically here, a triangular peg in a square hole). Musically, this idea is projected by creating a rhythmic friction of three against four between the melody and quartet. I have set the vocal melody and strings in two different time signatures here so that on each beat, the melody emphasizes three and the strings emphasize four. Thus, the melody is “waltzing on squares”—square dancing. From this perspective, Square Dance alludes to the imagery of two geometric shapes: a square and a triangle.
            The form—AAAA—represents the four equal sides of a square. All four sections are identical in length and harmonic progression. The form and genre of the twelve-bar blues seemed a perfect mold to project this geometric structure. The repetition of A is consistent with the twelve-bar blues form. Also typical in the blues, the twelve bars are divided into three four-bar phrases, which here is another subtle reference to the idea of three against four. Finally, the vernacular style of the blues seemed appropriate to use as an allegorical reference to the text’s “4/4” character.

​Triangle is a vocal duet written for my wife for our seventh wedding anniversary. Before our wedding, an old friend told us that our marriage might be viewed as a triangle—the two vertical sides of the triangle representing each of us, with God at the pinnacle. The idea was that as we individually drew our hearts toward God, we would, subsequently, be drawn closer to each other. I’ve attempted to musically project this concept in this piece.
            There are two melodies in Triangle, each representing a lover. These two melodies and their accompaniment are directly related to each other in that they both flow out of one heart, so to speak. The melody and harmonic progression in the soprano melody is literally a backwards performance of the melody and harmonic progression in the tenor melody. In other words, the last melody note and chord of the tenor melody are the first note and chord of the soprano melody. Through this structural symmetry, both sections literally point toward each other, manifesting both the symbolism and imagery of the triangular metaphor.
            As with the previous two pieces, the form of this piece projects the form of its title. Predictably, Triangle has three distinct sections, each representing one side of the triangle. The meter is 9/8, which divides into three both the pulse and subdivision of that pulse. Additionally, the strings play three-note chords and their harmonic progression moves in intervals of a third. Finally, it is no coincidence that Triangle is placed third in the cycle and is exactly three minutes long.

Straight Line, written for my daughter, Ariana, depicts the geometric figure of a straight, unwavering line. The pitch—a’* (representing Ariana)—is sustained throughout the entire piece. However, instead of being presented overtly as one long pedal note sustained by a single instrument, a’ is obscured behind rhythmic and articulated variety and delegated among the strings and voice.
            I’ve inserted personal references to Ariana into this piece. At the first instrumental bridge, the strings play Hillside Lullaby, Ariana’s first composition, written when she was only six years old. The second instrumental section is an arrangement of a lullaby written for and sung to Ariana by my wife. The melody is distributed among the strings: it begins with the first violin, transfers to the second, and, finally, concludes in the viola. While the first and second violin play this melody, the viola plays fragments of the melody in diminution before it takes over the melody from the second violin.
            * a’ is the A above middle C.
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​The Shore is a point of departure. The meaning of that is very different and personal for each of us. For one person, it may be a physical place. For another, it may symbolize a relationship or a bitter chapter in life. Regardless of how it is manifested, the Shore holds very personal experiences, emotions and memories. Departure’s face also wears many expressions—a withdrawal, a turning, an escape, or a reluctant and mournful farewell. No matter what your Shore or exodus looks like, allow The Shore to represent it.
            The string writing here is programmatic, which is to say that it paints with sound the imagery depicted in the text. As a piece that speaks of departure, it seemed appropriate to close this collection with The Shore. 

The score below is in the key of B. A version in the key of A is also available. Contact me if you are interested in that.

The opening section in Family Tree was influenced by a short film I saw years ago entitled Tango, by director Zbigniew Rybczynaki. In this film, brief, silent scenes by individual characters are successively introduced, repeated, and layered upon one another all in the same, small room. Following the introduction of each character, what becomes more intriguing than the individual scenes is how the characters interact with each other in that small space. It is this same concept and technique that is applied to the opening section here in Family Tree. The guitar, strings, and four voices are successively introduced and repeated, with each new “character” being layered upon the previous. As in Tango, the increasing juxtaposition of independent lines confined to a short space in time (four to eight bars) creates a vertically growing texture that, to me, provokes as much interest as the individual parts.

The melody in the verses, sung in Latin, is an adaptation of the chant with the same title as this piece.* Other than being transposed up a fifth, the pitch material of the original chant is virtually unchanged. The rhythm of the original chant had to be modified somewhat to match the meter of the piece.
            At the chorus, the tenor sings a paraphrased English translation of the Latin text sung in the verses. I was intrigued by the concept of “a rhythmic symphony,” a phrase in that translation. I’ve projected my interpretation of this idea through a dense, contrapuntal, and polyrhythmic fabric. Of the latter, throughout the piece, individual lines can often be perceived as belonging to more than one meter. Listen carefully to the opening guitar line for an example of this.
            During the verse before the second chorus, the second violin plays the melody of another lullaby written by my wife for my daughter. Throughout the piece, there are other brief and fragmented quotes from historical musical literature.
            * “Laude jocunda,” Paris, Bibl. nat., lat. 3719, fol. 53.
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If harp in unavailable, I have a setting of this piece for piano, which replaces the harp.  Additionally, I have a version with one soprano instead of two. Contact me if either of these versions interest you, and I will be glad to send them to you.

​I wanted to include in this collection an instrumental piece for solo violin and guitar. I decided to insert this piece, which I wrote one day in early June a few years back. It was intended to be simple and light, and I think that for the listener, it is. However, there are some fast passages and rhythmic challenges that lie embedded in some fairly complex meters, which, along with the fast tempo, might provoke the violinist to disagree with the “simple” part of that description.

Meditations on the Crucifixus  is a theme and variations for chamber orchestra based on the Crucifixus from the Mass in B Minor of Johann Sebastian Bach. The theme, an eleven-pitched melody built on a succession of minor second and tritone intervals, is taken from the initial antecedent phrase between the soprano and alto (mm. 5-7) of Bach's work. The theme also, as a two-part, chromatically-descending ostinato, quotes the  ground bass of the Crucifixus.    
    While the Crucifixus laments over the suffering and death of Christ portrayed in the portion of the Credo which states, "...and was crucified for us under Pontius Pilate, suffered and was buried", each variation reflects the composer's personal contemplation upon the significance and relevance of these words to mankind.      
    The pitch, G#, is absent from the theme and is withheld from the piece until the final cadence. The special treatment of G# then emphasizes and magnifies the work's finality.        
    Using a rotational sequence in which pitches are successively and systematically extracted from the theme, tetrachords or trichords are built upon each of the theme's eleven pitches. When all eleven subsequent chords have been successively stated, each voice of these harmonies will have quoted the theme. ​

I included this movement on my Geometrics CD, and stated the following in the liner notes:
In contrast to the more dissonant and complex language of the outer movements, I thought the aesthetic of this movement was a good match for the rest of the collection here. Most of the material throughout Compliments is derived from the first four bars of the opening violin line, reiterating it in just about every conceivable manner—backwards, fast forward, and harmonically. Toward the end of the piece (at precisely the Golden Mean*), there is a fragmented presentation of Amazing Grace, hidden within the texture of the violin solo.
            A consistent reaction from those who have heard Compliments is that the piece is very visual. Some even heard (and saw) it as part of a movie score. Whereas I did not write this piece with any program in mind, it is satisfying to hear that for some, the music manifests itself in new ways.
            * The Golden Mean is a mathematical expression of proportion—a ratio, which the ancient Greeks observed to exist in nature (i.e., the shape of a nautilus shell). For centuries, artists, architects, and engineers have used it for design. When applied to structure, this “Golden Ratio” (.618:1) becomes the focal point, or, in the case of art forms that move in time—films, poetry, and music, for example—the climax. For instance, you might take the total time or the total number of measures in a piece of music, divide it by .618, and you would find the Golden Mean.