Organizing Just Intonation Pitches through Xenakis' Sieves and Prime Decomposition

Abstract

In this article, we will present how we can create, organize and understand pitches of Just Intonation using 1) the theory of sieves (XENAKIS, 1992) to filter the numbers of the harmonic series, 2) the decomposition by prime-numbers (EXARCHOS et al., 2011) applied to the numbers resulted from the sieve, 3) the concept of harmonic identities by Johnston (2006), and finally, 4) the Combinations-Product Set (CPS) by Erv Wilson (NARUSHIMA, 2018), that allows us to see the relationships between harmonic identities. When we understand that only prime numbers generate new harmonic identities in the interval sense (JOHNSTON, 2006), and odd-numbers create just a new pitch-class from staking prime numbers, we can apply the prime-decomposition for all the numbers of one sieve and claim that: all the non-prime numbers can be understood how intersections between distinct harmonics series. Based on Exarchos' (2011) analysis of sieves, this idea can help us find a way to use the sieves of Xenakis in the Just Intonation context. Complementary with this compositional idea, we will present two OpenMusic libraries developed to help the compositional process.

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                
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limitidentity Diamond Theory
identity
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identityprime number
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

    
                 



Figure 1 - OM patch to reproduce the microtonal system of String Quartet no. 2 and no. 3 of Ben Johnston.
identity
              

Figure 2: Hexany by Erv Wilson (Narushima, 2017, p. 153).
          
     





    



Figure 3 – The structure of harmonic series in Wilson (NEIMOG, 2021, p. 60).

 

      

  



          




             

⋂

      


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

                 






⋂⋃⋂
      

prime-decomposition

Figure 4 - Examples of the sieve analysis using the object prime-decomposition.

⋂
⋃  ⋂
 

diamond-identity






Figure 5 - Patch example to create melodic contours with sieves.
          

Figure 6 - Patch that creates a modulation between two different JI systems.
                
interval-sob          perfil   



Figure 7 - This algorithm is best introduced in Neimog et al. (2022).


           



  


             
  



 

cribles
et al.
3.1. OM-JI: Partch, Johnston e Wilson theories


1. rt->mc
2. range-reduce  rt-octave         

3. filter-ac-inst
               
Techniques of flute playing

4. Modulation-notes modulation-notes-fund 

          




5. Ji-change-notes  sdif 
et al. 



          

3.2. OM-Sieves


crible sieve
1. 
Figure 8 - Comparison between the same procedure in MathTools and OM-Sieves.
2.               
u' 
i24@23 | 30@3 | 104@70 | 0@0

 
     

Figure 9 - Example of the creation of sieves using Ariza's (2005) syntax.
3. 
s-decompose


Figure 10 – The patch exemplifies the decomposition of one sieve in unions.
4. 
perfil
Figure 11 - It uses s-symmetry-perfil to find symmetrical sieves et al..
5. 
Figure 12 - The decomposition of non-prime modules proposed by Exarchos et al. (2011).
             
        

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
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     
Computer Music Journal
     


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
Proceedings of the Xenakis International Symposium    



Maximum Clarity

Microtonality and the Tuning Systems of Erv WilsonMicrotonality and the Tuning
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   Afinação Justa, Crivos e (as)Simetrias: Estratégias Composicionais Com
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Genesis of a Music
Composing Electronic Music
Formalized Music