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

Composer, mathematician, visual artist. 
Seeks collaboration with performers and choreographers.

Please don't spam this account!

To all concerned: 
I am announcing publication of my new book, NOW AVAILABLE:
https://www.routledge.com/Maqam-Melodies-Pitches-Patterns-and-Developments-of-Music-in-the-Middle-East-and-other-Microtonal-Writings/Thoegersen/p/book/9789815129052
My book, Polytempic Polymicrotonal Music, is available on Amazon. 
https://www.routledge.com/Polytempic-Polymicrotonal-Music-The-Road-Less-Traveled/Thoegersen/p/book/9789814968294

I have a new album out through Magic&Unique Group Records on Spotify and other digital streaming platforms:
https://open.spotify.com/album/5qWHmcvyAemvVn2TJH1bio?si=OD5TmPptQ-ypr8WOI0EZKA&nd=1

https://whitecentipedenoise.com/products/peter-thoegersen-facebook-cantos-cdr-flea
My CD via Flea label has been out since April, 2021, featuring the wonderful Lore Lixenberg singing over my microtonal piano accompaniment
Hope you enjoy it!

https://www.newworldrecords.org/collections/vendors?q=Peter%20Thoegersen

From the liner notes on my CD now available on AMAZON and New World Records:
"The author of these notes has spent his life explaining radical music, and the music on this disc may be the most radical I've ever written about. Peter Thoegersen (b. 1967) is not yet a name known to the music world; not for any lack of connection to other famous musicians, but because he came to composition late, and because his artistic aims are so broad and complex that they have taken years to evolve. His aesthetic is well defined, and he is upfront about having a name for it: "Polytempic Polymicrotonal Music." That means he has musical layers simultaneously moving in different tempos, plus, even more radically, that he has different microtonal scales playing at the same time. Few composers have aimed at such fervent and heterogeneous multidimensionality. And yet, while Thoegersen's music can seem overwhelming at first, running through it is a level of charm and simplicity that belies the initial impression.

In total, these three pieces emphasize a particular side of Thoegersen's output. The image his music creates of ornately mathematical structures filled with intuitively friendly melodies maps on to the personality of the man himself: outspoken and acerbically ruthless in his musical opinions, Thoegersen is also marked by a jolly sense of humor and an inexhaustible love of cats and other animals that make him ultimately less imposing and more personable than he might seem at first. The same can be said for his music. And by going further than anyone else has gone in terms of this particularly American concept of fusing the polytempic with the polymicrotonal - extending and combining the conceptual worlds of Ives, Cowell, Nancarrow, Partch, Carter, even Frank Zappa and many others - he has created a special place for himself within American music."                                                             

-Kyle Gann (from the liner notes)

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

https://www.youtube.com/user/DRORELIMELECH120V

https://www.facebook.com/dror.elimelech.338/


ACOUSTIC AND   ELECTRO-ACOUSTIC  COMPOSER .
STUDIED  COMPOSITION WITH  SERGIU  NATRA .
ELECTRONIC AND  ELECTROACOUSTIC  MUSIC  WITH YIZHAK  SADAI .
PIANO  --  WITH  PNINA  SALZMAN  , BART  BERMAN  , AVI  SCHONFELD .
CONTEMPORARY CLASSIC  COMPOSER , ACOUSTIC ,
ELECTRONIC AND  ELECTROACOUSTIC   MUSIC , 
EXPERIMENTAL  MUSIC . SOUND  ARTIST .
IMPROVISATION AND INTUITIVE MUSIC .
PIANIST .
ACCORDION AND  RECORDER  PLAYER .
FOUNDED  AND  CURATED    2  SERIES  OF  CONTEMPORARY  CLASSIC  MUSIC   IN TEL AVIV  ,  ISRAEL  -- 
 " NIGHT  HAPPENING -  FORUM FOR EXPERIMENTAL MUSIC "    1986  -  2007
" OTHER  ISRAELI  MUSIC "      1985   -  1987
MANY COLLABORATIONS  WITH  POETS , ACTORS , DANCERS ,  PAINTERS ,
 VIDEO ARTISTS , PERFORMANCE  ARTISTS .
MORE THAN  120  PREMIERS  OF  NEW  CONTEMPORARY  CLASSIC MUSIC - 
ACOUSTIC , ELECTROACOUSTIC , MULTI-MEDIA  --  WERE  PRESENTED
IN  HIS  SERIES DURING  23  YEARS OF ACTIVITY .
HIS  MUSIC WAS  PERFORMED  IN  EUROPE , CANADA , USA .
WON  VARIOUS GRANTS AND PRIZES .

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

David Victor Feldman