I blame the Greeks. Pythagoras was the first to discover fundamental tones and the various harmonics that constitute the world of sound. These physical properties of acoustic wave motion lie unquestionably within the formal domain of mathematics. Yet, sadly, most musicians I know can tell you very little about the nature of sound.
Similarly, acoustic engineering, instrument making and repairing, cymatics, and other related and interesting disciplines are by-products of the world of music, and remain in the domain of science.
It is true that numbers are used to express rhythmic values and pitch relationships in western musical notation: half-note, quarter-note, thirds, fifths, etc. Rhythmic beats are often represented by fractions. In addition, time signatures are defined strictly by numbers, as are some tempo markings expressed as metronomic values.
It is letters, however, that describe the names of musical notes, the names of lines and spaces on the musical staff, and various clefs. Key signatures are also determined by letter-names.
Not surprisingly, both mathematics and music are combinatorial systems. Each uses a finite set of symbols, in combination, to build a nearly infinite number of relationships. The same is true for written language and visual art.
The various scales and modes that determine the fixed tonal relationships in music, including the melodic and harmonic pitch intervals and related keys, are certainly describable in numeric terms, but are expressed and explored by musicians purely within a musical context.
The twelve tones of the western musical scale, and their octave displacements, combine to form the melodic and harmonic foundations of western music, inclusive of all musical categories, genres and styles.
It is interesting to note that mathematics is strictly abstract. It uses a set of graphic symbols to represent various quantities and relationships. Using the symbols alone, mathematicians are able to describe the world. Music, also employing graphic symbols in the form of musical notation, is primarily expressed by the arrangement of the tones themselves. Music is also abstract, by definition. Musical tones, even noises, can be organized in a variety of ways to represent any form of reality.
In serial composition, twelve-tone music, atonal music, aleatoric and indeterminate music, numeric sequencing provides a set of important musical devices, including random distribution, combination, permutation, inversion, retrograde, mirroring, etc.
These sequencing techniques, though suggestive of mathematical methods, are related more to modern uses of written language, especially sound poetry.
Another area in which math plays an important role in music, but again not an essential one, is algorithmic or computer music. Here, various mathematical formulas and equations, in the form of algorithms, are used to simulate musical processes. But it is the focus on musical aspects of the program that determine its artistic success.
Both mathematics and music employ a separate language to represent abstract concepts. Both are identified by a unique set of rules, language, vocabulary, and by their use.
At the heart of mathematics are expressions, equations. Music is not formulaic in this way. Although musical structure is often complex, music does not require mathematical analysis. Musical relationships are not algebraic, or geometric in the sense of, say, architecture. Calculus is not necessary to divine a musical phrase.
Music is more like speech. It is constructed from a sequence of connected elements,
sounds, that are combined to form larger units of meaning, or phrases. Phrases are strung together to form longer musical sentences, which, taken together, form an overall structure. The texture of music involves elements of speed (fast-slow), dynamics (loud-soft), pitch (high-low), duration (long-short), etc., while elements of musical structure include formal relationships, repetition, variation, continuity, simultaneity.
At an instinctive level, the language of music represents our deepest concerns. Music models the world around us. The best of it touches our lives, flooding our experience with ideas, physical sensations, and powerful emotions.
The composing and performing of music is influenced by a wide variety of considerations, including aesthetic tendencies, cultural currents, psychological states, conceptual and philosophical constructs, the origins of music, socio-political concerns, semantics, religion and ethics.
It is my contention that music shares much more in common with the language of thought, with physical sensations, and with the psychology of chemical states, than with mathematics.