Rhythmic Undertone Series
The mathematical mirror
The Rhythmic Undertone Series is the mathematical mirror to the Rhythmic Overtone Series. Where the overtone series is generated by dividing a master beat—creating accelerating rhythms and producing $d:1$ ratios—the undertone series is generated by multiplying the period. This results in decelerating rhythms, producing ratios of $1:d$ relative to the fundamental period.
Core mechanism
Instead of an inter-onset ratio of $d:1$ (representing $d$ events in the space of one master beat), the undertone series uses a ratio of $1:d$, representing one event spanning $d$ master beats. This expands the inter-onset interval rather than subdividing it.
The inter-onset ratios for the first few beats map as follows:
- $1:1$ = Unison (the fundamental period)
- $1:2$ = The octave below
- $1:3$ = The Perfect 5th descending
- $3:4$ = The Perfect 4th descending to So below Do
Pitch-class circle mapping
This arithmetic inversion has a direct geometric counterpart. Just as the rhythmic overtone series maps to clockwise movement on a pitch-class circle, the rhythmic undertone series visually and mathematically maps to counter-clockwise movement.
See also
Knowledge Graph
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