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Microtone Frequency Calculator

Use this microtone frequency calculator to generate frequency tables for any N-tone equal temperament (N-TET) — 19, 22, 24, 31, 53, 72, or any custom N. Includes audio preview per step, CSV export, a 12-TET overlay comparison, and just-intonation approximation error metrics. To convert an individual cent offset to a precise Hz value, use the cent-to-Hz converter.

Input

TET
212315372
Hz
Common N-TET systems

Result

Step size in cents
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Steps per perfect 5th (best fit)
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Step ratio
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Notes per 12-TET semitone
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Best 3:2 fifth error
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Best 5:4 major-3rd error
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Octave closure error
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Formulas
Step ratio: r = 21/N
Step in cents: c = 1200 / N
Frequency at step k: fk = fref · 2k/N
Cents from reference: 1200·k/N
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Cents axis — N-TET pegs above baseline, 12-TET semitones below

Frequency table for one octave (step 0 → step N)

Step kFrequencyCents from refNearest 12-TET semitoneNearest notePlay

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About N-TET Tuning Systems

N-tone equal temperament divides the octave into N equal logarithmic steps. The step size is 21/N (a frequency ratio) or equivalently 1200/N cents. The familiar Western system is 12-TET (N=12, step ≈ 100 cents = one semitone). Other N values open up vastly different harmonic possibilities — some much closer to just intonation (small-integer frequency ratios), others genuinely alien to Western ears. Use this microtone calculator to inspect any N from 1 to 200 and evaluate each system's just-intonation approximation quality through its fifth and major-third error metrics; if you also want to hear named non-Western scale presets (Maqam, Slendro, Pelog, Bohlen-Pierce), the microtonal scale generator complements this tool with audio playback and Scala export.

Why try non-12 systems?

12-TET is a deliberate compromise: it closes the octave perfectly (2:1) but every other interval is slightly mistuned from its pure just intonation counterpart. The syntonic comma (81:80, about 21.5 cents) is the most famous source of this drift. 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 major third is the worst offender.
  • 19-TET: P5 off by −7¢, M3 off by −7¢. A meantone-friendly compromise — better major thirds than 12-TET at the cost of slightly worse fifths. Historically aligned with 1/3-comma meantone temperament.
  • 22-TET: P5 off by +7¢, M3 off by −13¢. Used in Indian music theory and superpyth temperament; distinct harmonic character from 19-TET.
  • 24-TET (quartertone): Just 12-TET plus an extra step halfway between each semitone. Used in modern Arabic, Persian, and Turkish maqam music for neutral intervals.
  • 31-TET: Excellent 5-limit approximation. P5 off by −5¢, M3 off by only −0.8¢. The Fokker 31-tone organ system; also naturally represents the septimal (7-limit) intervals.
  • 53-TET: Near-perfect Pythagorean. P5 off by less than 0.1¢ — virtually exact. Used in some Persian and historic theoretical work; also approximates 5-limit intervals nearly perfectly.
  • 72-TET: A superset of 12, 24, and 36. Provides extremely close approximations of 11-limit and 13-limit harmonics. Used by composers like Easley Blackwood and in the Boston Microtonal Society's notation system.

The 7-limit, 11-limit, and beyond

Higher-prime harmonic series ratios (7:6, 7:4, 11:8, 13:8, 17:16…) appear in overtone singing, brass instrument partials, and some world music traditions. 12-TET approximates none of them well — the 7th harmonic (7:4 ≈ 969¢) is 31 cents flat of 12-TET's minor seventh, an audible gap. Systems like 31, 41, and 72 do approximate these ratios — they're called "high-limit" or "xenharmonic" approximations and enable harmonies that 12-TET literally cannot produce. Use the frequency ratio calculator to check how closely any specific just ratio is approximated by an N-TET step.

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 with performers trained in standard notation: "play the 7th step of 24-TET, which is the 12-TET semitone 3 + 50¢" — i.e., a quartertone above E♭. This mapping is also how MIDI Tuning Standard (MTS) messages work: they express pitches as a semitone number plus a fractional cents offset.

Frequently Asked Questions

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.
How do I calculate the frequency of step k in any N-TET system?
The formula is fk = fref × 2k/N, where fref is your reference frequency (step 0), k is the step number, and N is the number of divisions per octave. For example, step 7 of 19-TET rooted at A4 (440 Hz) gives 440 × 27/19 ≈ 567.4 Hz. This tool computes and displays that value for every step in the table automatically.
What is the difference between N-TET and EDO?
N-TET (N-tone equal temperament) and N-EDO (N equal divisions of the octave) are functionally the same thing when the period is an octave. Both terms describe a system of N equally spaced logarithmic steps that span exactly 1200 cents. The distinction matters only for non-octave systems like Bohlen-Pierce (which divides the 3:1 tritave, not the octave), where "EDO" becomes technically misleading and "equal temperament" is more accurate. For any system on this calculator (all octave-based), the terms are interchangeable.
Which N-TET system is best for approximating the harmonic series?
There is no single best answer — it depends which prime limits matter to you. For 5-limit (ratios involving only 2, 3, 5): 31-TET is excellent. For 7-limit (adding the 7th harmonic): 31-TET and 41-TET do well. For 11-limit (adding the 11th harmonic): 72-TET is considered near-ideal. For sheer Pythagorean (3-limit) accuracy: 53-TET is hard to beat. The "best 3:2 fifth error" and "best 5:4 major-3rd error" metrics shown by this tool are a quick proxy for evaluating any N you type in.

Related Tools

Frequency Ratio Calculator
JI vs 12-TET interval matching — see how N-TET intervals compare
Cent to Hz Converter
Apply any cents offset to a reference frequency
Octave Frequency Calculator
Octave shifts and ISO band lookup — the 12-TET reference for this tool
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