Musical Magic Squares
Magic Squares are a mathematical phenomenon, which consists of rising integers placed in an equal sided grid, such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number.
A Musical Magic Square is a composition that is formed from individual phrases arranged in terms of their number of beats and placed in an equal sided grid according to the numeric placement of a Magic Square, such that the sum of the beats in any horizontal, vertical, or main diagonal line is always the same number.
Though many variations of Magic Squares exist, I was drawn to the Astrological Squares associated with the numerologist Pivari. He associates Magic Squares with Saturn/Lead, Jupiter/Tin, Mars/Iron, the Moon/Silver, Venus/Copper, Mercury/Mercury, and the Sun/Gold.
Saturn/Lead seemed the simplest with which to test the idea, being a three by three grid using the numbers one to nine. I also wanted to expand the theory to a more complex Magic Square and the obvious choice was Sun/Gold.
The goal was to form a composition that would develop the idea of a Musical Magical Square and explore the idea of transmuting Lead into Gold. The name “Khemia” refers to the Greek name for the ancient Egyptian art of alchemy and the origin of modern chemistry.
The Magic Square of Lead
Key: D sharp / E flat
Numbers: 3, 9, 15, 45
Arrangement: 9 sequential integers in a three by three grid.
Sum of all numbers: 45
Sum of each line: 15
Number of Combinations of 15 : 8
If each bar has three beats, there are five bars per line. If there are eight combinational lines, there will be 40 bars. Notes: 4 is a starting phrase and 6 is an ending phrase, the 5 phrase is a central pivot.
The Magic Square of Gold
Key: A sharp / B flat
Numbers: 6, 36, 111, 666
Arrangement: 36 sequential integers in a six by six grid.
Sum of all numbers: 666
Sum of each line: 111
Number of Combinations of 111: 14
If each bar has three beats, there are 37 bars per line. If there are fourteen combinational lines, there will be 518 bars. Notes: 6 is a starting phrase and 31 is an ending phrase.
Forming Phrases of Lead
I decided upon some rules to adhere to in the process of forming musical phrases to fit within select numbers of beats.
The phrases would be formed for Lead first.
Each number of the Magic Square is representative of a musical phrase of a certain number of beats.
Each phrase is unique.
Each phrase ends on an extended note, this cap should act as an audible division between each phrase.
The phrase for “1” is the essence and its chords should be the basis for expansions, the phrase for “2” is an elongation of “1”, the phrase for “3” should then be a final extension of “1”.
The phrase for “9” should be the developed expression of the initial one beat phrase, as a seed for this expression I took a seven-note sequence from Holst’s “The Planets Suite - Saturn: the Bringer of Old Age”.
Forming Phrases of Gold
The phrases of Gold would be firmly based on an extrapolation of Lead. The first nine phrases are exact imitations of the nine phrases of Lead. The remaining phrases would be variations on these first nine, to do this, I needed a numerological connection between the mathematics of the two Magic Squares. By combining digits of a number, it can be reduced to a single integer between one and nine, linking together each phrase from ten to thirty-six with a Lead phrase.
1 1, 10, 19, 28
2 2, 11, 20, 29
3 3, 12, 21, 30
4 4, 13, 22, 31
5 5, 14, 23, 32
6 6, 15, 24, 33
7 7, 16, 25, 34
8 8, 17, 26, 35
9 9, 18, 27, 36
In the grid below the blue numbers are the irreducible nine phrases of Lead. Each quarter of the reduced Magic Square of the Sun now adds to 45, the sum of Lead. Furthermore, by dividing the Sun Square into nine, each line will add up to 60 in either vertical or horizontal and the sum of both diagonals also equals 60. This method maintains its state as a Magic Square.
6 5 3 7 8 1
7 2 9 1 8 3
1 5 7 6 5 6
9 2 4 3 8 4
7 2 1 9 8 3
9 5 6 4 2 4
Lead is the Prima Materia for transmutation and should be the basis from which Gold is formed, so that the nine phrases of Lead go through several transmutations before becoming Gold.
The art of alchemical transmutation takes place in seven stages; therefore, the transmutation of the music should also be complete in seven stages of change. There are four Phases of Alchemy, being Black, Yellow, White and Purple. The Black Phase has four stages: Calcination, shown in alchemical transcripts as being linked with Lead; Dissolution; Separation and Conjunction. The Yellow Phases consists of Fermentation; the White Phase consists of Distillation and the final Purple Stage consists of Coagulation, which is linked with Gold.
By examining the astrological characteristics of Saturn/Lead and the Sun/Gold, I can find an associated Key for each. Saturn, and therefore Lead, correspond to D sharp / E flat; whereas Gold and the Sun are linked to A sharp / B flat. This forms a separation of a perfect fifth.
Each stage of transmutation will change Key, in a sequence that moves from E flat to B flat. The best method for this is to use the Planetary Heptachord of esoteric Greek music theory, the seven-pointed star used as the symbol for the seven days, seven metals and seven planets. When overlaid onto the Chromatic Circle, containing equal divisions into a twelve-tone scale, the Septile Scale is formed from the closest harmonic pitches to the stars seven points.
This new Scale Pattern corresponds to the Scale of C sharp Major, if we use the Key of Lead (E flat) as a starting point, then it is from Supertonic to Supertonic and thus is Dorian C sharp Scale. The connections between each note in the scale is in a circle of fifths but progresses in perfect fourths; i.e. E flat, A flat, D flat, G flat, C, F and B flat.
Creating the Structure
To separate the forty bars of Lead and five hundred and eighteen bars of Gold across this form, I chose to mark a difference between the vertical, the horizontal and the diagonals; with each being an individual movement. The horizontal passes move through all the phrases in turn, the vertical passes move through the same phrases but in a new order and the diagonal phrases only cover a part of the series. Each movement will then be set in one of the Septile Keys.
Once the phrases for a Musical Magical Square are set down, the composition is formed at the performer's discretion by connecting lines of phrases together. As I am attempting a cogent process, I will specify a “correct” order for my interpretation of Alchemic Transmutation.
- Lead (E flat Minor) - A single chord.
- Black Phase: Calcination (E flat Minor) - The horizontal lines of Lead.
- Black Phase: Dissolution (A flat Minor) - The vertical lines of Lead.
- Black Phase: Separation (D flat Minor) - The diagonal lines of Lead.
- Black Phase: Conjunction (G flat Minor) - The diagonal lines of Lead and Gold.
- Yellow Phase: Fermentation (C Major) - The diagonal lines of Gold.
- White Phase: Distillation (F Major) - The vertical lines of Gold.
- Purple Phase: Coagulation (B flat Major) - The horizontal lines of Gold.
- Gold (B flat Minor) - A single chord.
I began creating the phrases with a quartet in mind, once the individual phrases were complete, I added them together to form the different movements. As a complete work Khemia Transmutation may still stand alone as a quartet but I found that the variations became too recognisable despite the reordering and key change. Also, as a quartet there is no acoustic development from the sound of Lead into Gold; it merely represents the pure mathematics of the Musical Magical Square.
To create a musical representation of Lead and Gold I focused on using more instruments in order to use a collective timbre to change from dull to bright and to use the timbres of different instruments in order to alter the perception of the same musical phrase.
I began by organising instruments by their tessitura and timbre, then arranged them into seven groups of four, also including an overlapping number of instruments that would be present in adjacent movements.
Once the instruments were chosen, I did not want to stray too far from the absolute mathematical nature of the final composition. The instruments were to double up the initial four voices; thus, maintaining a steady sound of the string section throughout but using the other instruments to create a changing quality to the musical process.
The final layer involved the alteration of the other instruments lines to shift the emphasis of the individual phrases within a movement.
The final mix used at present is a mix of Korg SP-200 sounds and some General Midi where specific instruments were required. This mix is not as it is scored but serves to represent the overall idea as clearly as possible.
With this composition I wanted to restrict myself to the piano to make the most out its simplicity. The main factors I considered were basic ideas of transformation that would lead from one state to a very different state in a smooth rising slope of change. I began to think in terms of moving from Minor to Major, from slow to fast, from simple to complex, elongated to condensed and used this idea as the basic structure.
Upon this I wanted to create a theme that would be introduced slowly over the first three bars and then create variations by adding in notes and breaking up the pattern every further three bars. This format would play out on the right hand whilst the left formed rhythmic structures that would shift and change along their own trajectory.
By a complex method of deconstruction based upon the seven stages of alchemical transmutation from lead into gold, previously mentioned, I arrived at a formal structure that would move through seven key changes; ranging from Eb Minor to Bb Major. However, this structure was only used to direct my intuitive musical direction and hopefully the musical qualities of the work are at the forefront of the experience.
The music begins in a timeless state and gradually becomes more and more frenetic, attempting to rise and mutate into a higher state only to crash to a halt, returning to the original key and theme.