31 TET
Thirty-one equal steps. An octave of possibilities.
ROOT · ROOT′ · THIRD′ · FIFTH′
0 · 31 · 41 · 49 steps
All roots lie in the C2 octave.
Hold a key · Play several with touch · Scroll sideways on small screens
Description
Explore 31-tone equal temperament on a synthesized organ and a complete circle of fifths. Every playable pitch comes directly from an integer on the same 31-step grid. C♯ and D♭ are different keys and different frequencies. The sound uses harmonic organ synthesis, without detuning or vibrato.
Instructions
- Organ: hold a key with your mouse or finger to sustain it. Touch several keys for chords. The three-octave range is C2 through C5, inclusive: 94 distinct pitches. The Fokker layout repeats pitches in different ranks for alternate fingerings; all copies light together.
- Circle: click any of the 31 roots to hear a major chord for 2.4 seconds. The bass root is in the C2 octave; the upper voices are root, major third, and fifth in the next octave. Clicking another root replaces the previous chord.
- Keyboard: Tab to the organ, then Left/Right changes pitch by one step and Up/Down by one octave. Hold Enter or Space to sound the focused key. In the circle, arrow keys move around the ring; Enter or Space plays a chord. Escape or Stop sound releases everything.
- Labels: “+” and “−” mean exactly one 31-TET step above or below a natural, not a quarter-tone. The small number is the pitch class 0–30. Frequencies below the instrument show the last played note or chord.
The exact tuning and layout
One frequency rule
f(n) = 440 × 2^((n − 85)/31) Hz
Here n = 0 is C2 and n = 85 is A4 at 440 Hz. Moving by k steps multiplies frequency by 2^(k/31). No pitch is rounded to a 12-tone MIDI note. Displayed hertz are rounded to five decimals; synthesis uses the unrounded calculation, subject to normal browser audio and hardware precision.
The major third is 10 steps (387.096774… cents) and the fifth is 18 steps (696.774194… cents). These are equal-tempered intervals, not the exact just ratios 5:4 and 3:2.
Voicing and the circle
For root r, the chord uses n = r + [0, 31, 41, 49]. At C this is C2, C3, E3, G3. At other roots the entire voicing moves by the same number of 31-TET steps. Pitch names use a consistent compact spelling, so a chord tone may appear under an enharmonic name.
Each clockwise move is +18 modulo 31. Because 18 and 31 are coprime, the ring contains every pitch class exactly once. After 31 fifths it closes at 18 octaves.
The Fokker lattice
Keys in one horizontal rank are five steps apart. Diagonally toward the back, right is +3 and left is −2. Eleven overlapping ranks provide duplicate fingerings. White keys are naturals, dark keys are sharps and flats, and blue keys are one-step alterations. This is one three-octave manual with a synthesized organ voice, rather than a sample reproduction of the historical instrument’s stops.
Layout and interval background: A. D. Fokker on the thirty-one-keyed organ; Ere Lievonen’s Fokker keyboard charts. This instrument uses A4 = 440 Hz.