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Divination By Mirrors: Too Much Fun with Music and Math



Music Geek Alert! Once upon a time I wrote a piece called Divination By Mirrors for Saw and Strings to feature the virtuosic musical saw playing of Dale Stuckenbruck (who is also a fine violinist). I decided to base it on the concept of a musical mirror - the idea is that if one musical line goes up, another one goes down, or when one is played forward, it later gets played backwards. Not being content with that level of complication, I added a concept borrowed from architecture: dynamic symmetry. Dynamic symmetry is spatial relationships based on the Golden Mean (and the related Fibonacci Series) explained in the notes for the piece below. Let's just say that at that point the musical mirrors became funhouse mirrors.

For those who are already lost, don't worry. The music is intended to stand on its own even if you don't understand any of this. But for the mathemusically inclined, here is more fun than ferrying a Rubik's Cube across a river with a fox, a goose, and a bushel of wheat.

The piece was first performed in 1998 at the Abraham Goodman House in the Lincoln Center complex in New York City. The ensemble was the New York Virtuosi, conducted by Kenneth Klein, featuring saw soloist Dale Stuckenbruck (pictured below playing with his father). The recording linked below was a studio performance by the Fibonacci players recorded and mixed by John Guth, again with Stuckenbruck as soloist.

Divination by Mirrors is a roughly ten-minute work featuring the bowed saw, accompanied by a string orchestra that is divided into two groups of players tuned 1/4 step apart and occupying opposite sides of the stage. The piece has eight Sections, each named for a different kind of divination – foretelling the future by means of the occult.

The unifying compositional idea is an exploration of the myriad possibilities of musical symmetry. These include pitch and time mirrors of various sorts, counter-balancing journeys into the two different pitch “universes” of the oppositely-tuned halves of the orchestra, and extensive use of the architectural concept of “dynamic symmetry” in defining durations of Sections, sub-sections, and tempi.

Because it’s so important to the structure of the work, I’d like to take a moment to explain the

concept of dynamic symmetry. Dynamic symmetry in art and architecture refers to the division of space by the ratio of the golden mean. The golden mean, or divine section (the ratio of roughly 1.618 to 1), when applied to physical distances, is thought to divide space in a natural, aesthetically balanced manner. Some experts believe the ancient Greeks made use of the divine

section in the construction of the Parthenon.

The golden mean, and its related concept the Fibonacci series, is also found in nature, from the arrangement and number of seeds in a sunflower, to the dimensions of the spiral arms of galaxies. It also occurs in the relationships of the parts of the human body, immortalized by Leonardo da Vinci in his famous anatomical drawing (see illus). The Fibonacci series is what mathematicians call a summation series. It begins with the members 1 and 1 which are added to produce its next member, 2. Each succeeding member is generated by adding the last two, hence: 1+2=3; 2+3=5; 3+5=8 ; 5+8=13; 8+13=21 and so on. As the series goes higher, the ratio of two succeeding members closes in on the golden mean.

The first four Sections of Divination by Mirrors total, ideally 350 seconds (5m 50s) in length while the last four Sections total 216 seconds (3m 36s) — a golden mean ratio. Similarly, the individual Sections are in golden mean relationship to each other: the length of the second and fourth Sections are 1.618 times the length of the first and third Sections. The second half is a mirror of the first with the length of fifth and seventh Sections being in a ratio of 1.618 to the sixth and eight Sections. However, each first-half Section is a golden mean ratio to its mirrored Section in the second half. (i.e. the first Section is 1.618 times the length of the eighth, the second is 1.618 times the length of the seventh, and so forth.)

The tempi reflect multiples of Fibonacci numbers in beats per minute. So sections have tempi of 50, 80, 130, or 160. (5 X 10, 8 X 10, 13 X 10, 8 X 20) Likewise, the number of beats in a given Section are determined by Fibonacci numbers as are, often, the number of beats in subsections and, on the micro scale, the durations of individual notes. The Fibonacci numbers 3,5,8,13, and 21 also appear in tuplets and metric groupings.

The reason for so much mathematical skullduggery is only partly because it is fun. The intent of the composer is to place the most exciting and inspiring events in the piece (and I do hope there are some) in the most effective manner.

So much for time. Let’s talk about pitch.

A great deal of the melodic and harmonic material of Divination by Mirrors is generated by the four superimposed triads — and their 1/4 step sharper counterparts — heard in the opening moments of the piece. From the bottom, the triads are C major, G# minor, D minor, and Gb major. The bottom two form an augmented scale (C,D#,E,G,G#,B) and the top two form a second augmented scale a step higher (D,F,Gb,A,Bb,Gd). Together they cover all twelve notes of the chromatic scale with no overlap — a tone row.

The saw’s theme, introduced in the second Section, is, with many octave displacements, a descending version of the first scale beginning on C (played in pairs of ascending major sevenths), followed by a descending version of the second scale, (played in pairs of ascending major sixths) beginning on Db.

There are other types of pitch symmetries employed as well including a 1/4 tone mirror in the third Section, a diatonic mirror in the fourth Section, and a hocketed modal mirror in the sixth Section (with parallel Ionian and Phrygian modes, which are mirrors of each other).

The sixth Section also has a hocketed canon between the two differently tuned groups.

There should be very short pauses between Sections.

Here is deeper analysis, Section by Section:

Section 1: Pyromancy (Divination By Sacrificial Fire)

50bpm; 55 beats plus pause; 67 sec (1m 7s) [for clarity it may be 100bpm/110 beats]. As noted above, much of the material of Divination by Mirrors is generated by the four super-imposed triads — and their 1/4 step sharper counterparts — heard in the opening moments of the piece. From the bottom, the triads are C major, G# minor, D minor, and Gb major. The rhythmic patterns in this Section are Fibonacci number groupings (8,5,3,2,1) which correspond with a gradual change of pitch range from extremities to the middle of the keyboard, ending with a single note. The saw is tacet.

The piece begins with string tremolos for 8 beats followed by 13 beats of 32nd notes. Then, beginning in 6:2 (bar 6, beat 2), 8 beats subdivided into 5 are followed by 5 sets of triplets. In addition, the various subdivided rhythms are related to each other. For example, at 11:3 (bar 11, beat 3) the 5 subdivision repeats 5 times; last time the 5 subdivision repeated 8 times (at 6:2). Similarly, the triplets at 12:4 repeat 3 times whereas at 8:2 they were repeated 5 times. All of these are Fibonacci relationships as are others later that derive from the sequence 1,1,2,3,5,8,13,21…

At bar 14, all parts arrive on a unison C (disregarding the tuning differences). The detuned parts play the C for 5 beats. Then, the detuned first violins enter on a Gb Major triad which is held for 21 beats. The second violins enter with a d minor triad, also held 21 beats, on the 13th beat of the first violin Gb chord. The detuned cellos and basses enter at 22:2 with a C Major chord. The section concludes with the violas playing a g# minor chord for 13 beats.

Section 2: Aeromancy (Divination By Appearances in the Air)

80bpm; 144 beats (plus 2 pickup beats); 108sec (1m 48s) The saw introduces the main theme, consisting of a row divided into two descending augmented scales. The first scale (C,B,G#,G,E,D#) is presented in pairs of ascending major sevenths, the second scale (Db,Bb,A,Gb,F,D) in pairs of ascending major sixths. At any given time, the accompaniment derives from which ever half-row is being played by the melody. The second, 1/4 step higher-pitched group begins playing on the “bridge” of the melody, which occurs 89 beats into the piece — a Fibonacci number and a golden mean division of 144, which is the total number of beats in the section.

Section 3: Psychomancy (Divination By Ghosts)

130bpm; 144 beats; 67 sec (1m 7s) This is a quartertone mirror with notes in Fibonacci-length groups (5,8,13,21). The phrase length and structure as well as the relationship between phrase groups all follow Fibonacci ratios. For example, the section begins with 5 long phrases. Each phrase consists of 3 bars of 5 beats followed by 1 bar of 8 beats. The phrase group is then concluded with a bar of 13. After two repetitions of the material, each an ascending another octave, the section concludes with 1 bar of 13 beats followed by a bar of 8 and a bar of 5.

The string parts of the section are derived from pitch mirrors that expand from middle C in opposite directions. The pitch clusters grow wider and denser as the section progresses.

The saw plays a much slower version of the mirror beginning with no vibrato and employing a multiphonic gliss at a key moment. This Section is divided into golden mean sections, with each of them subdivided by traditional (equal) symmetry.

Section 4: Oneiromancy (Divination By Dreams)

80bpm; 144 beats; 108 sec (1m 48s) A second theme is introduced with a cello solo. A diatonic mirror based on C major, it consists of a series of 7-6 suspensions — a nod toward the main theme’s use of 7ths and 6ths. It modulates up a quartertone at the golden mean division of the Section (89 beats).

Section 5: Sciomancy (Divination By Shadows)

130bpm; 144 beats; 67 sec (1m 7s) Pizzicato! Back to the augmented scales and their corresponding chords (Gb, dm, g#m, C), the saw plays a variation on the main theme. Symmetrical division of time.

Section 6: Meteoromancy (Divination By Winds)

160bpm; 110 beats; 41 sec Hocketed mirrors between high and low strings with upper playing C Phrygian and lower C Ionian modes. Phrases are Fibonacci groups of sixteenth notes beginning with 1 and 2 (1,2,3,5,8,13,21) followed, after the first note, by rests one Fibonacci member lower in duration (1,1,2,3,5,8,13). Later, the quartertone higher group fills in the rests with a quasi-canon. Saw plays melody based on opening triads.

Section 7: Catoptromancy (Divination By Mirrors)

Cadenza — it should ideally take 1m 7sec. Variations on the two themes and the quartertone mirror.

Section 8: Tephramancy (Divination By Writings in Ashes)

50bpm; 34 beats; 40 sec [for clarity it may be 100bpm/68 beats] The first Section in retrograde and compressed by a ratio of 5/8 in comparison with Section 1, with added saw playing the opening motif in retrograde.

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