Romanticism, a 19th-century intellectual and artistic movement, placed great importance on the emotional and subjective aspects of human experience, including music. It viewed music as a powerful means of expressing and evoking feelings, rather than as a purely rational or objective art form.

Pythagoras believed that the mathematical ratios and proportions found in music, particularly the intervals between notes, corresponded to the underlying harmony and structure of the universe. His famous theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, is said to have been inspired by his studies of musical harmony.

Immanuel Kant, in his influential work 'Critique of Judgment,' posited that music has the ability to communicate emotions and ideas directly to the soul, bypassing the need for words or concepts. He believed that music's universal appeal and emotional power made it a unique and profound form of human expression.

Mimesis, derived from the Greek word 'mimesis,' refers to the imitation or representation of reality in art, including music. In the context of music and philosophy, mimesis explores how music can imitate or reflect the emotions, thoughts, and experiences of human beings, creating a sense of connection and empathy between the listener and the music.

Theodor W. Adorno, a prominent member of the Frankfurt School, developed the concept of 'ontological aesthetics.' He argued that music has an intrinsic value and existence independent of human perception or interpretation. Adorno believed that music's unique ability to express emotions and ideas that cannot be conveyed through language gives it an ontological status that transcends its physical form.

Marxism, a socio-political philosophy developed by Karl Marx, examines the relationship between music and the economic, social, and political structures of society. It explores how music can reflect and shape cultural values, beliefs, and social hierarchies, and how it can be used as a tool for social change or resistance.

Phenomenology, a philosophical movement founded by Edmund Husserl, focuses on the study of conscious experience and subjective perception. In the context of music and philosophy, phenomenology explores how individuals experience and interpret music, emphasizing the role of personal and subjective factors in shaping musical understanding and appreciation.

Transcendence, in the context of music and philosophy, refers to the idea that music has the ability to transcend the limitations of time and space. It can connect people across generations and cultures, allowing them to share emotions, ideas, and experiences through the universal language of music.

Rationalism, a philosophical school of thought that emphasizes the importance of reason and logic, has influenced the way some philosophers and musicologists approach the study of music. Rationalists believe that music can be understood and evaluated through rational analysis of its structure, form, and content, rather than relying solely on subjective experiences or emotions.

Catharsis, a concept originating in ancient Greek philosophy, refers to the process of emotional release and purification through art, particularly tragedy. In the context of music and philosophy, catharsis explores how music can evoke strong emotions, allowing listeners to experience and release pent-up feelings, leading to a sense of emotional cleansing and renewal.

Postmodernism, a philosophical and cultural movement that emerged in the late 20th century, challenges traditional notions of truth, objectivity, and universal values. In the context of music and philosophy, postmodernism emphasizes the importance of context and social factors in shaping the meaning and significance of music. It explores how music is influenced by and reflects the cultural, historical, and political circumstances in which it is created and experienced.

Ineffability, in the context of music and philosophy, refers to the idea that music has the ability to express or evoke emotions and ideas that cannot be adequately described or conveyed through language. It explores the unique power of music to communicate feelings, experiences, and concepts that transcend verbal expression.

Existentialism, a philosophical movement that emphasizes the importance of individual existence, freedom, and choice, has influenced the way some philosophers and musicologists approach the study of music. Existentialists believe that music is a deeply personal and subjective experience, and that its meaning and significance are shaped by the individual's unique perspective and life experiences.

Transcendentalism, a philosophical and literary movement that originated in the 19th century, emphasizes the importance of intuition, experience, and the individual's connection to nature and the universe. In the context of music and philosophy, transcendentalism explores the idea that music can provide a glimpse into the nature of reality and the universe, allowing listeners to transcend the limitations of everyday experience and connect with something greater than themselves.

Aesthetics, a branch of philosophy that focuses on the nature of beauty and aesthetic experience, has a significant influence on the way philosophers and musicologists approach the study of music. Aestheticians explore the role of beauty, form, and expression in music, and how these elements contribute to the overall aesthetic experience and appreciation of music.