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Magic Squares


I took my original 5-note sequence (E, G#, D, F#, A#) and extended it into a 6x6 magic square following Maxwell Davies' approach of (a) ensuring the sequence goes out of phase from line to line (because sequence of 5 pitches along lines 6 units long), and (b) transposing each subsequent sequence. The transpositions aren't always the same, because I was getting repetitions; because going up in minor-3rds ends up in the same place after four cycles (12 chromatics divided by 3 semitones = 4), so after the 4th cycle (when it should return to E) I transposed up a further semitone to push the whole cycle of four sequences a semitone higher than the first four. The square is (mostly) transposed by minor 3rds because the original 5-note sequence is whole-tone (the two possible whole-tone sets are six notes each), so minor 3rds give pitches not in the original whole-tone set. [commentary point] This solves the whole-tone problem of only having six notes to work with (or two sets of six), but still retains the whole-tone 'flavour' of the original sequence.

E G# D F# A# G
B F A C# A# D
G# C E C# F B
D# G F A D# G
B G# C F# A# D
B D# A C# F A#
Ex.1: magic square, showing different transposed 5-note sequences

In the piece, I use this square in several ways to do different things:
  • generating melodic sequences for the flutes in [bars (?)] by reading the whole square backwards; starting at the bottom-right corner and reading across right-to-left line-by-line. I read backwards because my flute idea was descending arpeggios but reading the square forward had too many large intervals, backwards gave the inverse (smaller) intervals more often than not.
  • chords for brass/winds in [bars (?)] by taking each horizontal line as a chord for my six instruments (clar, clar, sax, tpt, cor, euph). I mostly kept the chord voicing the same as the order in the square, but switched the pitch order in some cases because I preferred that voicing [intuitive change to the process]. 
    • Orchestration: unfortunately six instruments and six pitches gives no opportunity for doubling any pitches for emphasis. So to give the sequence more consistency I try to split the chord into a brass trichord* and a reed trichord. 
    • * a 'trichord' is a three-note chord that isn't a triad, a triad is specifically a tonal chord of two stacked 3rds (incl. aug/dim chords). The term 'trichord' avoids tonal implications: of course if you want to make tonal references then 'triad' might be appropriate.  See also tetrachord, pentachord, hexachord etc. in post-tonal theory.
Ex.2: 6x6 square

In [bars (?)] I specifically wanted to find a way to give each line a different set of durations (to avoid simultaneous attacks: see also Ligeti's micro-polyphony). The magic square was perfect for what I wanted because each line adds up to the same number, so I can have multiple overlapping lines that start and end together, but whose internal parts never overlap. I tried to superimpose a 6x6 numeric magic square for durations on my pitch square above. Or at least I intended to, but when I did some research here I found that 6x6 squares are problematic: Euler (in 1789) conjectured that "there were no Graeco-Latin squares of orders 2, 6, 10, 14, etc". It turns out that it is possible to generate 6x6 squares (see that link for cool construction method), but the numbers involved aren't suitable for what I want, they're too big to be usable durations (see ex.2). I need several pitches per bar, but using cell-numbers in the 20s would mean I'd have to subdivide my 4/4 bars into at least demi-semi-quavers, which is messy; especially when combined with irrationals (triplets, quintuplets etc). Lower-order versions of 6x6 don't seem to be possible, so I turned to 5x5 which has low-order versions; these have much smaller numbers, easier to apply to my problem of durations.

I considered solving this by using two low-order 5x5 squares superimposed and offset by one diagonally to cover the full 6x6, but how would I combine and read them... A more simple solution was to use one 5x5 square and simply add the same number to the end of each horizontal, then offsetting that extra number differently for each horizontal; so they still add up horizontally at least. I took the low-order square from here. In ex.3 I get the right 5x5 by adding two to each cell, so the totals remain the same but I have bigger durations, and avoid having cells with 0.

Ex.3 low-order 5x5, then with 2 added to each cell
In ex.4 I added in the extra digit, then added 2 (as above), then redistributed the extra digit to avoid them all being at the end: if they were all at the end, the final note of my sequence would all be aligned, which I wanted to avoid.

Ex.4: extra digit, then adding to get higher values, then redistributing the final digit.

This section arpeggiates the magic square chords from [bars (?)] in a staggered way. To read my duration square, the initial plan was to treat each number as the duration of the next note in the arpeggio. However, this seemed to foursquare a rhythm, so instead I made a rhythmic grid of irrationals (triplets, quintuplets etc) that the duration square could be applied to. Ex.5 shows the grid:

Ex.5: rhythmic grid of irrationals; b3 permutes the previous bar.
In ex.5, the beats of the bar are each subdivided by a different prime number, to get a general slowing/speeding of the basic pulse. Then in subsequent bars the different irrationals can be permuted to keep it varied. Similarly, each instrument will also be permuted so they don't line up. Once this grid is in place, the square will be used to apply note durations by counting "rhythmic units", not beats of the bar. In ex.7 you can see that rhythmic units change across the bar. The first number from the square (a "4") counts two quaver units and triplet units.

Ex.7: counting a line from a hypothetical duration square of [4,4,5,5] against the grid
of irrational rhythmic-units, then showing the rationalised version of this.
The next important decision is that I want to avoid the bars aligning, so as the process unfolds I'll always tie-over the notes at starts and ends of bars. Of course, any decision taken opens up the possibility of reversal at some strategic point. I'll use the tied-over strategy mostly in this section, but sometimes I'll leave them untied and emphasise the bar alignment (also accenting them to make more of this alignment). This then ties in with the saxophone and percussion duet which happens over this. The sax and percussion will use the same duration square but instead of counting rhythmic-units of irrationals, they'll count beats of the bar, so they change much more slowly. The slower change is because the material for both is slow; saxophone multiphonics, and bowed cymbals. See later blog post


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