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Musical Composition with Multiway Turing Machines
Musical Composition with Multiway Turing Machines
Including key topics in philosophy of music that can potentially be resolved with a mathematical approach
by Joel Dietz
Composition and philosophy of music evolved substantially in the 20st century with new spatial aspects of music composition, the advent of more complex tonal systems, and stochastic methods of composition. Multiway turing machines, especially those generated with Wolfram Language, may be used to simulate and expand upon of these major developments, potentially leading to a new standard for musical composition as well as providing a dynamic illustration of the principles of relativity.
Specifically:
[1] Turing machines can be used to represent a series of possible states each corresponding to the activity of a particular note (or, more specifically, the emission of a sound wave). Additionally, the set of possible states is not restricted to a particular tonal system and can be used to expand through all possible tonal systems.
[2] Multiple turing machines can be used to represent the activity of specific instruments leading to an orchestral effect
[3] Rulesets for the turing machines including postprocessing rules can be added to replicate known harmonic systems or to discover new tonal systems pleasing to the human ear
[4] Multiway turing machines can be used to reproduce nondeterministic music and various possible observer states
[5] Taking the foliations of the multiway turing machine can lead to a stochastic effect
[6] These different experiences of the music effectively represent different hyperplanes and are dynamic illustrations of the principle of relativity (i.e. each observer will perceive the music differently).
Specifically:
[1] Turing machines can be used to represent a series of possible states each corresponding to the activity of a particular note (or, more specifically, the emission of a sound wave). Additionally, the set of possible states is not restricted to a particular tonal system and can be used to expand through all possible tonal systems.
[2] Multiple turing machines can be used to represent the activity of specific instruments leading to an orchestral effect
[3] Rulesets for the turing machines including postprocessing rules can be added to replicate known harmonic systems or to discover new tonal systems pleasing to the human ear
[4] Multiway turing machines can be used to reproduce nondeterministic music and various possible observer states
[5] Taking the foliations of the multiway turing machine can lead to a stochastic effect
[6] These different experiences of the music effectively represent different hyperplanes and are dynamic illustrations of the principle of relativity (i.e. each observer will perceive the music differently).
History of spatialized and relativistic approaches to music
History of spatialized and relativistic approaches to music
Although musical notation as traditionally conceived does not easily accommodate the concept of spatialization, unique observers, or stochastic methods of composition a few notable attempts have been made in the last century. Iannis Xenakis (1963) (INART 55 Pithoprakta, Mm. 52-60 Iannis Xenakis)In musical composition, construction must stem from originality which can be defined in extreme (perhaps inhuman) cases as the creation of new rules or laws, as far as that is possible; as far as possible meaning original, not yet known or even forseeable. Construct laws therefore from nothing, since without any causality. But a construction from nothing, therefore totally engendered, totally original, would necessarily call upon an infinite mass of rules duly entangled. Such a mass would have to cover the laws of a universe different from our own. For example: rules for a tonal composition have been constructed. Such a composition therefore includes, a priori, the "tonal functions." It also includes a combinatory conception since it acts on entities, sounds, as defined by the instruments. (p. 256, "Concerning Time, Space and Music" in Formalized Music Thought and Mathematics in Composition)Xenakis is unique in including the documenting software behind his compositions and various stochastic elements which he discusses at length. Karlheinz Stockhausen (1982) “They have the effect of expanding the audience's perception of musical space outward from the point into the void, like Hawking radiation from a black hole... After dot, line, and plane, this is music expressing space; with the final unfolding of space, time is also redeemed, and Luzifer's spell is finally broken.” (Maconie, Robin. Other Planets (p. 406, 417). Rowman & Littlefield Publishers. Kindle Edition.)Stockhausen was notable for his experimental approach which contained many novel ways of spatializing the concert experience. Leonhard Euler and 3d representations of the Tonnetz (1739, 1998) At the age of 23 Euler wrote a vast treatise on music theory, including a ranking of different chords according to their degree of “pleasantness,” but “it contained too much geometry for musicians, and too much music for geometers.” (Maor, Eli. Music by the Numbers: From Pythagoras to Schoenberg (p. 49). Princeton University Press. Kindle Edition.)Nonetheless, this system of ratios has been additionally been spatialized on numerous occasions, most recently via toroidal structures. See https://tonnetz.eu/ and Edward Gollin, “Some Aspects of Three-Dimensional “Tonnetze,” Journal of Music Theory, Vol. 42, No. 2, Neo-Riemannian Theory (Autumn, 1998), pp. 195-206 (12 pages)
Conclusions
Conclusions
Despite some interesting and provocative approaches, none of the aforementioned have derived a satisfactory system for composing and spatializing music that builds off of the new understanding of math and physics in the last century.
Building a multiway music composer
Building a multiway music composer
Step 1: Importing music from various sources
Step 1: Importing music from various sources
In order to guarantee a baseline quality of output we have started by importing music from a variety of musical instruments, notably the handpan, shinobe, and wood flute, including our own recordings of such. These each form a list of sounds for what would amount to a scale on that particular instrument.
In[]:=
handPan=,,,,,,,,,;
In[]:=
shortFlutes=,,,,,;mediumShinobe=,,,,,,,,;longFlutes=,,,,,,,,,,;
Step 2 : Setting up initial turing machine states
Step 2 : Setting up initial turing machine states
The first Turing machine consists of a simple set of four possible states with simple rules for changing between them.
The first Turing machine consists of a simple set of four possible states with simple rules for changing between them.
Step 3 : Resolve turing machine states to sound
Step 3 : Resolve turing machine states to sound
This output must then be resolved to a choice of sounds to correspond to the place in the scale.
Step 4: Foliate results based on individual observers
Step 4: Foliate results based on individual observers
Step 4: Foliate results based on individual observers
Here we create the foliations that represent different possible results based on different observer coordinates.
Step 5: Merge audio stream into observer-specific symphonic output
Step 5: Merge audio stream into observer-specific symphonic output
The rules are used to select individual pieces of audio which then are merged into a single audio stream
Samples of additionally interesting results
Samples of additionally interesting results
Next Steps
Next Steps
Acknowledgements
Acknowledgements
This project would not have been possible without the framing by Stephen Wolfram and the code and editorial contributions by James Boyd and Nik Murzin.
Summary
Summary
In[]:=
We are able to use Wolfram Language to provide meaningful matching of instruments to scales and turing machines. This, via multiway turing machines, allows stochastic music generation that illustrates the principles of relativity. While this is in an early stage of development this may be both a useful tool for musical composition and, additionally, should be able to resolve several key philosophical questions regarding the cross-cultural emergence of harmonic systems.
Additionally potentially solvable research questions
Additionally potentially solvable research questions
What is the relationship between the arts (inclusive of scifi, music, poetry, and interactive experiences) and research (inclusive of mathematics, physics, and other sciences) ? If it can be termed ‘motivational,’ (i.e. a person discovers a particular topic via a scifi novel and studies it sufficiently much that this person makes a unique and valuable insight into the field), can this degree of motivation be aggregated and quantified? Additionally are there clear examples of an insight or key breakthrough in these fields being stimulated by artistic input (e.g. Einstein’s violin, Von Neumann and Faust, etc.) ? Is there any reliable rule that can used to map these categories of human experience to each other?
Is there one or more sets of harmonics that can be formalized and described mathematically that correspond to intelligence and, by extension, by the growth of complex forms? Can such be derived from physical matter, compositions, or other large data sets? Can such be used to procedurally generate the same?
Is there any formal heuristic that can be used to rank the quality of thoughts? What is the baseline meaningful concept of intuition with respect to selecting “better” thoughts? What makes them better? How did the folk definition of “intuition” evolve into its current format? What are the most useful historical versions of ‘intuition’ and what were their immediate precedents?
To what extent is the quality and meaningful nature of thought related to the complexity of the grammar used to produce it? Specifically does use of nested sub clauses and, by extension, recursion lead to higher quality thought? Can a similar analogy be used for music? Can these be mapped over a time axis such that complexity of grammar or musical phrases is illustrated? If so, can transformer sets or analogous be used to recreate and simulate various versions of the same?
Is there one or more sets of harmonics that can be formalized and described mathematically that correspond to intelligence and, by extension, by the growth of complex forms? Can such be derived from physical matter, compositions, or other large data sets? Can such be used to procedurally generate the same?
Is there any formal heuristic that can be used to rank the quality of thoughts? What is the baseline meaningful concept of intuition with respect to selecting “better” thoughts? What makes them better? How did the folk definition of “intuition” evolve into its current format? What are the most useful historical versions of ‘intuition’ and what were their immediate precedents?
To what extent is the quality and meaningful nature of thought related to the complexity of the grammar used to produce it? Specifically does use of nested sub clauses and, by extension, recursion lead to higher quality thought? Can a similar analogy be used for music? Can these be mapped over a time axis such that complexity of grammar or musical phrases is illustrated? If so, can transformer sets or analogous be used to recreate and simulate various versions of the same?