N-tone equal temperament divides the octave into N equal logarithmic steps. The step size is 21/N (a ratio) or equivalently 1200/N cents. The familiar Western system is 12-TET (N=12, step ≈ 100 cents). Other N values open up vastly different harmonic possibilities — some closer to just intonation, some genuinely alien.
Why try non-12 systems?
12-TET is a compromise: it gets the octave perfect (2:1) but every other interval is slightly off from its just-intonation counterpart. Other N values can approximate JI more closely at the cost of more notes per octave. The trade-off:
- 12-TET: P5 off by +2¢, M3 off by +14¢. Acceptable in keyboard music; the M3 is the worst offender.
- 19-TET: P5 off by −7¢, M3 off by −7¢. A meantone-friendly compromise — better 3rds than 12-TET at the cost of slightly worse 5ths.
- 24-TET (quartertone): Just 12-TET plus an extra step halfway between each semitone. Used in modern Arabic, Persian, and Turkish maqam music.
- 31-TET: Excellent 5-limit approximation. P5 off by −5¢, M3 off by only −0.8¢. The Fokker 31-tone organ system.
- 53-TET: Near-perfect Pythagorean. P5 off by less than 0.1¢ — virtually exact. Used in some Persian and historic theoretical work.
- 72-TET: A subset of useful systems (12, 24, 36 are subsets). Used by Easley Blackwood and others for microtonal composition.
The 13-limit and beyond
Higher-prime ratios (7:6, 11:8, 13:8, 17:16…) appear in some musical traditions. 12-TET doesn't approximate any of them well. Systems like 31, 41, and 72 do — they're called "high-limit" approximations and enable music that 12-TET literally cannot play.
Mapping N-TET back to 12-TET
For each N-TET step, this tool shows its position relative to the nearest 12-TET semitone (in semitones + cents deviation). This is how microtonal composers communicate: "play the 7th step of 24-TET, which is the 12-TET semitone 3 + 50¢", i.e., a quartertone above E♭.
Why is 19-TET considered "meantone-friendly"?
Meantone temperament (popular before equal temperament) tunes the major third 5:4 as the priority and lets fifths drift slightly flat. 19-TET coincidentally provides almost-exactly the same compromise: M3 at 379¢ vs just 386¢ (−7¢), P5 at 695¢ vs just 702¢ (−7¢). Both intervals are equally compromised. In meantone, this is called "1/3-comma meantone" — and 19-TET is its equal-tempered shadow. Composers like Easley Blackwood and Vincentino explored music in 19-TET extensively.
What's "quartertone" music?
Music using quarter-tone intervals (50 cents = half a semitone). 24-TET = 12-TET + a quartertone step between each semitone. Microtonal Western composers (Hába, Wyschnegradsky, Carrillo) used it; it's foundational to Arabic and Turkish maqam systems. The maqamat use specific quartertones (often closer to 1/3 or 2/3 of a semitone than exact quarters) — 24-TET is an idealized notation, not a literal recipe.
Why does 53-TET get the Pythagorean fifth so close?
53-TET's step is 1200/53 ≈ 22.64¢. The just Pythagorean fifth (3:2) is 701.955¢. The closest 53-TET interval is 31 steps × 22.64¢ = 701.89¢ — off by 0.07¢, completely imperceptible. This is because 53 is a near-perfect approximation of log₂(3/2) × N = N when N = 53 (the "Pythagorean comma" is suppressed). Persian theoretical works from the 13th century proposed 53-tone divisions for exactly this reason.
Can I export the frequencies to use in a synth?
Yes — use the "Export CSV" button. The CSV has columns step, frequency_Hz, cents_from_ref, nearest_12tet_semitone, deviation_from_12tet_cents, nearest_note. Most modern synths and DAWs can load tunings via .tun, .scl, or MTS-ESP formats; the CSV is for manual workflows (Excel, Python, raw frequency lists). For .scl format (Scala), the lines are just the cents values from step 1 to step N, no header — easy to derive from this CSV.
What's the "octave closure error"?
It's how close N steps get to a perfect 2:1 octave (1200 cents). For equal temperament, this is ALWAYS exactly 0 — the system is designed to close at the octave. The metric is more meaningful for non-octave systems (Bohlen-Pierce uses 13 equal divisions of the 3:1 tritave instead of the octave). For pure N-TET, expect 0¢; if you see anything else here it indicates a floating-point precision issue.
Why does the audio sound like beeping rather than music?
This tool plays a simple sine wave at the chosen frequency. Real instruments produce complex harmonic spectra that interact with each other to produce a sense of consonance/dissonance. To evaluate a microtonal system musically, load the CSV into a synthesizer that supports microtonal tuning and play actual music. Web-based players that support custom tunings: VCV Rack, Surge XT, Pianoteq, MTS-ESP-compatible plugins.