# Tuning Reference (String Frequencies & Temperaments)

> String frequencies for guitar, bass, violin and more at any A4, in equal or historical temperaments. Look up the exact frequency of every string of an instrument tuning and of a whole octave of notes for a reference pitch between A4 = 415 and 466 Hz, in equal temperament, 5-limit just intonation, Pythagorean tuning, quarter-comma meantone, Werckmeister III or Vallotti on any tonic — with the deviation of each note from equal temperament in cents. The same tables as the tuner and drone in the Tonegrid app.

Part of usefulweb.app (https://usefulweb.app/). Part of [Tonegrid](https://tonegrid.usefulweb.app/) (ownerApp: tonegrid). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://tonegridtuningreference.usefulweb.app/ (interactive tool at https://tonegridtuningreference.usefulweb.app/tool)
- JSON API: POST https://tonegridtuningreference.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://tonegridtuningreference.usefulweb.app/api/v1/tool)
- OpenAPI: https://tonegridtuningreference.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://tonegridtuningreference.usefulweb.app/mcp - tool `tonegrid_tuning_reference`

## Inputs
- instrument (select): Instrument tuning
- a4 (number): Reference pitch A4 (Hz)
- temperament (select): Temperament
- tonic (select): Tonic (key the temperament is built on)
- octave (integer): Octave of the note table

## Method
- Equal temperament: f = A4 × 2^((n − 69) / 12) for MIDI note n. Other temperaments shift each pitch class by its deviation from equal temperament, with A held at 0 so that A4 always sounds at the reference pitch, as on tuners with temperament settings.
- Just intonation (5-limit, asymmetric 12-note scale) uses the ratios 1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8 above the tonic. Pythagorean, meantone, Werckmeister III and Vallotti are built from the chain of fifths E♭–G♯: pure fifths (3/2), fifths narrowed by ¼ syntonic comma (meantone), ¼ Pythagorean comma on C–G, G–D, D–A and B–F♯ (Werckmeister III), or ⅙ Pythagorean comma on F–C–G–D–A–E–B (Vallotti).
- Cents = 1200 × log2(f1 / f2). Frequencies are rounded to 0.001 Hz and cents to 0.01.

## Limits
- Reference values. Real instruments also need inharmonicity and string stretch taken into account (pianos are tuned with stretched octaves).