Subdividing the 15/4 Interval: Microtonal Notation and Performance-Tunable Deviations

This project explores a notational and structural approach to dividing the 15/4 just interval into equal steps of 15, 10, and 5, inspired by Bohlen–Pierce concepts but diverging from it in tuning and application. Each subdivision is notated with microtonal accidentals and includes cent deviation annotations below each note. These annotations serve as performer instructions for interpreting pitches with ±10 cent accuracy.

The design aims to capture the full spectrum between conventional tones, embracing ambiguity and performer agency. It also proposes a flexible tuning strategy that bridges microtonality and perceptual harmony across several expressive resolutions.

Reiji's Observations

This is a notated exploration inspired by Bohlen-Pierce, dividing the interval of \(\frac{15}{4}\) from a reference tone into 15, 10, and 5 equal parts from the left.
The number of cents each pitch increases by is calculated as follows:

• \(\sqrt[15]{\frac{15}{4}} \approx 153\text{¢}\)
• \(\sqrt[10]{\frac{15}{4}} \approx 229\text{¢}\)
• \(\sqrt[5]{\frac{15}{4}} \approx 458\text{¢}\)

These resulting tones are notated using microtonal accidentals, approximated within ±10 cents.

Since microtonal symbols are not assigned in 1-cent steps, the cents of deviation are written below each note to allow performers to adjust during interpretation.

Tones enclosed in Gaussian brackets like \(\lfloor \cdot \rfloor\) indicate equivalence—such as treating C and D𝄫 as close enough to be considered the same.

This notation spans from C to B approximately one octave higher, producing a different sonic texture from Bohlen-Pierce.

“Since it’s in the middle of everything, I think it holds a bit of the image of every piece. So I thought it might be good to use when you want to combine all those images into a single piece.”

Notated Scale Using Subdivided 15/4

Each pitch step approximates a division of the 15/4 interval (major 13th), subdivided into 15, 10, and 5 equal parts. Microtonal deviations are annotated in cents beneath each note.

Root-Based Interval Calculations

Square roots and nth roots of the 15/4 interval are calculated to determine equal divisions (e.g., 153¢, 229¢, 458¢), used for tuning microtonal steps in a mathematically consistent way.