Understanding Infinitesimal Calculus and Its Application in Higher Dimensions
The term ldquo;infinitesimal calculusrdquo; was historically used to refer to the mathematical field we now simply call ldquo;calculus.rdquo; The word ldquo;calculusrdquo; itself, up until the 19th century, referred to the broader body of knowledge about mathematical calculations. The qualifier ldquo;infinitesimalrdquo; became necessary to distinguish the specific type of calculation known as calculus, which involves the limits of expressions as a parameter approaches zero.
The Origins of Infinitesimal Calculus
Calculus, particularly in its early development, used geometrical figures to represent mathematical expressions. For derivatives, these were right triangles, and in integrals, trapezoids represented the sums of functions at various points as the spacing approached zero. Mathematicians would explain these figures using the concept of ldquo;infinitesimallyrdquo; small, indicating a degree of smallness beyond zero but not actually zero. This concept of infinitesimal, although not rigorously defined, laid the groundwork for the modern definitions of limits in calculus.
Dimensions and Spaces
An n-dimensional space, where n can vary from 3 to potentially an infinite number of dimensions, is a structure involving n mutually independent axes or variables. These can be rigorously defined in various ways, from continuous to discrete. A primary concept in this context is the potential use of these dimensions in whatever purpose one may require them. They may constitute a discrete Cartesian grid where variables are integers or rational numbers. They may be non-uniform and non-linear, or even non-geometric, if the axes do not represent geometric magnitudes such as lengths, areas, volumes, or angles.
Applications in Music Theory
A prime example of an n-dimensional space is found in the realm of musical intervals. Rational tonal musical intervals can be modeled as a structure in an infinite-dimensional space, with each axis representing prime numbers and their integer powers. For instance, the first axis might represent the prime number 2, with points at 2^-m, 2^-1, 2^1, 2^2, and so on. This structure is neither continuous, linear, nor geometric, but it is a vector space in the modern sense. However, it is not a classic literal vector space in the traditional sense since it lacks the concept of a carrier or vector.
These axes, which can represent such variables as prime integers, can create a grid where each original designated tonal musical interval is defined as the product of k coordinate points, one from each axis. This is essentially a prime-factor-analysis product of rational numbers, with both positive and negative powers allowed. The analytical image represents an n-dimensional grid where axes are integer powers of prime numbers. This space can be thought of as 4-dimensional, or potentially infinite-dimensional, depending on the prime numbers involved.
The practical application of this infinite-dimensional space in music theory is significant. It allows for the representation and analysis of complex musical intervals and structures, as seen in the use of Pythagorean, natural, and septimal scales. The inclusion of the prime number 11 opens the space to 5-dimensional considerations, reflecting the complexity of Balkan, Middle Eastern, and church chant music. This concept of higher-dimensional spaces in music theory provides a unique and profound framework for understanding the mathematical underpinnings of music.
As we delve deeper into these multidimensional spaces, it is crucial to maintain a balance between generalization and specificity. While expanding the scope to include more dimensions can be enlightening, it is also important to recognize the elementary aspects that have generated the initial concepts. Diluting or losing the fundamental components necessary for understanding and application can lead to conceptual blurs and impracticalities.