Phi-Based Temperament (φ-TE)

This document records the construction of a unique temperament, φ-TE (Phi-Based Equal Temperament), based on the irrational number φ (the golden ratio), using the Terpstra Keyboard WebApp.
Each pitch interval in this scale is irrational, approximated by rational values, and measures approximately 741.6407865... cents.
Since just two steps exceed one octave (1200 cents), the scale theoretically extends infinitely.
This is a highly conceptual attempt to explore structural analogies between space and sound.

Reiji's Observations

1. Repeatedly dividing a golden rectangle always yields a 1:φ or φ:1 ratio.
2. By treating φ as the length of one octave, a scale was constructed accordingly.
3. This concept results in each step being approximately 741.64 cents (irrational).
4. Stacking intervals similarly to a circle of fifths yields steps of approximately 1483.28 cents.
5. This scale is constructed so that each pitch rises by approximately 741.6407865... cents — an irrational interval approximated by rational numbers — and since just two steps exceed one octave (1200 cents), the scale theoretically extends infinitely.
6. The fundamental frequency (244.756924741 Hz) was selected for its optimal overtone resonance.

Configuration Screen (Terpstra Keyboard WebApp)

Screenshot of tuning and color settings within the WebApp. The cent values of the φ-based scale are input into the Scala format field.

Keyboard Display

A visual representation of the φ-TE scale on the Terpstra keyboard. Colors and numerical labels illustrate the hierarchical structure of each pitch class.