written in spring 2020 by Wilma Pistorius
performed by Pau Sola Masafrets

Intermezzo in green

for solo oboe, 2020
Performed by Cèlia Tort Pujol, 24 July 2021, Singelkerk, Amsterdam

From Between The Cracks

Music for Marimbas and Harps

The prepared piano becomes a celestial canvas as magnetic whispers guide the notes, weaving a love story through cosmic vibrations.

Aphrodite on the Moon

Composed in honor of Tom Johnson's 60th birthday and premiered in Paris and Amsterdam by ENSEMBLE DÉCADANSE at a two-part festival in celebration thereof.  

Music based on a finite-state automaton devised by the composer;  Tom Johnson himself was pursuing musical applications of finite-state automata around that time.

T'homage (1999) for flute, violin and harp (for Tom Johnson on his 60th birthday)

12  = 
   12 = 11+1 = 10+2 = 9+3 = 9+2+1 = 8+4 = 8+3+1 = 7+5 = 7+4+1 = 7+3+2 = 6+5+1 = 6+4+2 = 6+3+2+1 = 5+4+3 = 5+4+2+1
Here each sum accumulates mutually different magnitudes; these 15 exhaust all the possibilities (when the summand order doesn't count). 

12  = 
  11+1 = 9+3 = 9+1+1+1 = 7+5 = 7+3+1+1 = 7+1+1+1+1+1 = 5+5+1 = 5+3+3+1 = 5+3+1+1+1+1 = 5+1+1+1+1+1+1+1
   = 3+3+3+3 = 3+3+3+1+1+1 = 3+3+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1
Now each sum accumulates odd magnitudes, and likewise these 15 exhaust all possibilities. 

15 possibilities both times; this is not a coincidence nor is it a special property of 12.  Start with any whole number and it turns out that the two tasks always lead to the same numbers of solutions.  

Now a claim like that requires proof; just testing finitely many cases does not suffice.  One sort of proof involves inventing some way of seeing an instance of one sort sum as a secret code for an instance of the other sort of sum (and vice-versa).  Then the implicit codebook matches up the two sets of solutions.  

Suprisingly, mathematicians know more than one recipe here for doing the coding and decoding.

And starting specifically with 12, coding by one recipe and decoding by the other happens to walk you back and forth between the all the solutions of both problems.  The structure of this piece is that process, with all the intermediate steps. The rhythms alone sonify the sums, but the pitches will guide you through the execution of the coding.

Different and Odd, for alto[aka tenor] horn and piano

Realization #82 of an open form work. 

Details for those who want to know *what* they're hearing:

Some unheard but endlessly repeating  rhythmicized tone-row girds each realization of this work.

Any tone-row contains 220 trichords.  Graphed in time and interval space, each trichord  (as a triangle now) has a geometrical (signed) area.  While the area of any one triangle depends on the units chosen for both time and interval width, the ratio of two triangle areas is invariant of the units.  So ordering the 220 trichords by increasing area makes intrinsic sense.  
It also produces a distinct musical effect, you will hear it, suggestive of careful and gradual development.  The reasons why the mathematical strategy has this musical effect are complex and perhaps hard to foresee in advance.

Now transposing the row up a semitone but also forcing all the notes back  into the original octave changes the area of 1/4 of the trichord/triangles. This shuffles the trichord order and produces a kind of variation.  This work, as a whole, proceeds through all 12 of these transposition-derived variations.

Well that describe the timpani part anyway.  

The flute doubles the timpani always at some number of octave, and climbs in register during the stretch of each variation, flute register roughly correlated with trichord/triangle area.  Octave transpositions always apply to all the notes in a trichord in order to preserve the shape, which of course is the chief musical determinant here.

The flute part can be performed alone.  It can also be performed with the timpani part on fixed media.  But live performance of the timpani part would require 12 drums, and perhaps two players, and perhaps some technology for coordination (since the rhythms will be hard to count in the usual way).

If there's interest, I'll upload other realizations.  All realizations are both recognizably "the same piece" yet also reflective of the idiosyncrasies of the basic rhythmicized tone-row.

Sharp Shadows Shadowed, for Flute and Timpani

The last two movements of the Symphony for the Waters are... 

4th movement, Andante, "Frog Meditation"

5th movement, Caccia, "Thus Spake Hippopotamus"

16 1/2 minutes 

In these two movements the symphony orchestra is accompanying frogs and hippo's, with porpoise and other marine flora and fauna also assisting. The two movements flow attacca one into the other, without pause.

Symphony for the Waters, 4th and 5th Movements ("Frog Meditation" and "Thus Spake Hippopotamus")

méson kaí ákron

The music can speak for itself, sonically, if you prefer.

But if you care to understand what's going on, here it is in brief.  The durations in the cello are 2, 4, and 8 8th notes; the durations in the bass line of the piano part are 3,5,6, and 7 8th notes.  Any rhythm of any duration of the former sort admits a translation into a rhythm of the latter sort, and vice-versa, indeed a translation that preserves the overall duration (to within a fixed few beats anyway).   I expect that this fact hardly seems obvious!  But the violin part is the intermediary, along with the other voices in the piano part.   Attend to the relationship between cello and violin; and between piano and violin, and, with some concentration, the logic indeed reveals itself.

Elegy for Piano Trio (High Phi II)

This piece is simply a love song to a tone row that enjoys a unique mathematical property. Let me put that in some context.

When people think of "mathematical music," three names tend to come up: Milton Babbitt, Iannis Xenakis, and Tom Johnson. For the point I want to make, let me concentrate on Babbitt and Johnson.

With his musical career well underway, Tom fell in love with mathematics writ large, despite having no formal training in the subject. He relied on professional mathematicians (myself included) to tell him about significant ideas, and he was a genius at finding ways to project mathematical concepts through music in the most natural possible ways.

In the preface to the score of Networks, Tom writes: “Composers should not tell their music what to do.” (This reflects Tom’s willingness to call himself a “minimalist,” even as the most famous minimalists rejected the label.)

I suspect Milton would have dismissed Tom’s sentiment as nonsense, holding that telling music what to do is composition perforce.

Milton did have mathematical training, and it expresses itself in his work as the rigor with which he approaches the intrinsically musical problems posed by twelve-tone practice. And unlike Tom, Milton actually fathered some original, albeit modest, mathematical discoveries. Still, his relation to mathematics resembled that of a typical engineer: formal methods helped him do what he wanted—and needed—to do, but his interest didn’t extend much beyond that.

Tom, for his part, had a studied disinterest in twelve-tone music, precisely because its practitioners often buried structural principles under richly “improvised” surfaces that attracted listeners’ attention. The common notion was that the underlying theory—the mathematics—was the exclusive domain of experts.

My piece here is twelve-tone music, but philosophically close to Tom’s minimalist spirit.

Indeed, I recently discovered this twelve-tone row:

0  1  4  8  3  6  5  11  9  7  10  2

It enjoys a remarkable property that makes it essentially unique.

By way of explanation: only about 1.5% of all possible tone rows allow you to combine the row with two rotations of two other forms (of the same row) so that the resulting three rows, when sounded together, produce twelve distinct trichords that exhaust all twelve trichord classes.

That said, most rows that allow this at all allow it in just a single way. But this row allows four ways, yielding four essentially different cycles of trichords.

I have never met two stools that I didn’t want to fall between. There is some simple surface detail here, and I can imagine Tom urging me to eliminate any competing idea. But that said, I am neither “improvising” here nor treating the row as mere “pre-compositional” material.

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I have chosen a tempo here for the web.  In performance, the player decides, and I enjoy the piece even at half the speed.

Let Me Make This As Clear As I Can, for solo piano

Composer, guitarist, momentarily writing orchestral pieces the "Human Rights' Oratorio" and the "Symphony for the Waters". And rock-CD "True Emotions".

Symphony for the Waters, 4th and 5th Movements ("Frog Meditation" and "Thus Spake Hippopotamus")

Anders Jallén