This mathematical abstraction allows theorists to calculate intervals and relationships regardless of enharmonic spelling. A C-sharp and a D-flat are functionally identical in this system (both equal 1), which simplifies the analysis of non-tonal harmonies.
In tonal music, a diminished 7th chord is always dissonant. In post-tonal music, any interval can be consonant or dissonant depending on context. For example, in Bartók’s "Mikrokosmos," a perfect 5th (traditionally the most consonant interval) is treated as a harsh, open dissonance. materials and techniques of post-tonal music pdf
: Examines melody and voice leading through the lens of motivic cells and scalar transformations. In post-tonal music, any interval can be consonant
Most advanced PDFs end with a composition prompt. Force yourself to write an 8-bar piece using only Set 4-16 (0147). You will understand the "materials" only when you struggle to use them. Most advanced PDFs end with a composition prompt
Most resources utilize integer notation to represent the twelve pitches of the chromatic scale.