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Mathematics of the MML Functional Quantizer Modules for VCV Rack Software Synthesizer

Core Concepts
The Mathematics and Music Lab (MML) at Michigan Technological University has developed a line of "functional quantizer" modules for the VCV Rack software modular synthesizer platform, allowing synthesizer players to tune oscillators to new musical scales based on mathematical functions.
The content provides a detailed mathematical formulation of the MML functional quantizer modules for the VCV Rack software synthesizer. Key points: The MML has developed a line of functional quantizer modules that allow synthesizer players to play new musical scales tuned to mathematical functions. The first module, the MML Logarithmic Quantizer (LOG QNT), modulates control voltage (CV) signals to produce a non-Pythagorean scale based on logarithms. The mathematical definition of a functionally quantized (FQ) musical scale with T tones per octave is provided, where the nth pitch Fn is defined by Fn = F0 · f(n/T), with f(x) a strictly increasing function such that f(0) = 1 and f(1) = 2. The formulas for computing the FQ output voltage Vout from the input voltage Vin are derived, including for the LOG QNT module. Other functional quantizer modules planned for release in 2024 include the Square Root Quantizer (SQT QNT), Sine Quantizer (SIN QNT), and two Power Quantizer (POW QNT) modules. The development of these modules involved collaboration between the Departments of Visual and Performing Arts and Mathematical Sciences at Michigan Technological University, as well as undergraduate research contributions.
Fn = F0 · 2^(Vout/Vref) Vout = Vref · {⌊Vin/Vref⌋+ log2 f (⌊T frac (Vin/Vref)⌋)} Vout = ⌊Vin⌋+ log2 f (⌊T frac(Vin)⌋) (volt-per-octave case) Vout = Vref · {⌊Vin/Vref⌋-1 + log2 log2 (4 + ⌊12 frac (Vin/Vref)⌋)} (LOG QNT) Vout = ⌊Vin⌋-1 + log2 log2 (4 + ⌊12 frac(Vin)⌋) (LOG QNT, volt-per-octave case)
"For 0 ≤x ≤1, let f(x) be a strictly increasing function such that f(0) = 1, f(1) = 2, and let F0 denote an arbitrary base frequency in hertz (Hz) that serves as the root note in the scale." "Twelve tone equal temperament in music theory is the prototype for FQ scales: set f(x) = 2^x, T = 12, such that Fn = F0 ·2^(n/12)." "In the logarithmic non-Pythagorean musical scale defined in [2], the fifth author uses T = 12 tones playable on a piano keyboard, along with the function f(x) = 1/2 log2(4 + 12x), x = 0, 1/12, 2/12, 3/12, ..., 11/12, 1."

Deeper Inquiries

How can these functional quantizer modules be used to create new and innovative musical compositions?

The functional quantizer modules developed by the Mathematics and Music Lab (MML) at Michigan Technological University for the VCV Rack software synthesizer platform offer a unique way to explore and create new musical compositions. By tuning synthesizer oscillators to mathematical functions, musicians can access a wide range of non-traditional scales and tones that go beyond the conventional chromatic scale. These modules allow for the generation of musical scales based on functions like logarithms, square roots, sine functions, and power functions, providing endless possibilities for creating innovative and experimental music. Musicians can use these modules to experiment with different mathematical functions, adjusting parameters to create custom scales that suit their artistic vision. The ability to modulate control voltage signals to produce specific scales opens up avenues for exploring unconventional harmonies and melodies. By incorporating these functional quantizers into their music production workflow, artists can introduce fresh and unique sounds that challenge traditional musical conventions.

What are the potential limitations or drawbacks of using mathematical functions to define musical scales?

While using mathematical functions to define musical scales offers a novel approach to music composition, there are potential limitations and drawbacks to consider. One limitation is the complexity of understanding and implementing mathematical functions in a musical context. Not all musicians may have a strong background in mathematics, which could pose challenges in effectively utilizing these functional quantizer modules to their full potential. Another drawback is the potential for creating scales that may not resonate with a broader audience. Traditional musical scales have been ingrained in cultural and historical contexts, providing familiarity and comfort to listeners. Introducing highly abstract or mathematically derived scales may result in compositions that are perceived as too avant-garde or inaccessible to some listeners. Additionally, the use of mathematical functions to define musical scales may lead to compositions that lack emotional depth or human expression. Music is often valued for its ability to evoke feelings and connect with audiences on a visceral level. Relying solely on mathematical formulas to generate musical content could risk sacrificing the emotional impact and authenticity that comes from human creativity and intuition.

How might these modules inspire further interdisciplinary collaboration between music and mathematics?

The development of functional quantizer modules that bridge the gap between music and mathematics presents exciting opportunities for interdisciplinary collaboration. These modules serve as a meeting point for musicians, composers, mathematicians, and software developers to explore the intersection of art and science. By working together to create and refine these tools, experts from different fields can leverage their unique perspectives and expertise to push the boundaries of music composition and technology. Collaboration between music and mathematics can lead to the discovery of new musical structures, harmonies, and scales that would not have been possible through traditional means. Mathematicians can contribute their knowledge of mathematical functions and algorithms to create innovative scales, while musicians can provide insights into the practical application and artistic implications of these scales in compositions. Furthermore, interdisciplinary collaboration can inspire creativity and innovation by encouraging individuals to think outside their respective disciplines. By fostering a collaborative environment where ideas are shared and explored across disciplines, these functional quantizer modules can spark new avenues of research, experimentation, and artistic expression at the intersection of music and mathematics.