Pythagorean tuning frequency
From A at 440 Hz, one pure fifth up lands on 660 Hz. Stack fifths up or down, shift by octaves, and read the frequency, ratio and cents from your reference for early music, instrument building or theory homework. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Reference frequency
- 440
- Fifths
- 1
- Octave shift
- 0
- Fifth ratio
- 3/2
- Fifth ratio decimal
- 1.5
- Octave ratio
- 2
A Pythagorean tuning frequency calculator turns a reference pitch and an integer stack of pure perfect fifths into the sounding frequency of that step in the Pythagorean lattice. Unlike twelve-tone equal temperament, which divides the octave into twelve equal logarithmic semitones, Pythagorean tuning builds every interval by stacking the pure fifth of ratio three to two and reducing or raising by pure octaves of ratio two when needed. Instrument makers, early-music ensembles, theory teachers, and synthesis patches that target historical intonation need those frequencies in hertz without hand-computing fractional powers. This capability accepts a positive reference frequency, a signed fifth count, and an optional octave shift, multiplies by one point five to the power of the fifth count and by two to the power of the octave shift, and returns the frequency, the ratio to the reference, and the cents offset from that reference so you can compare against equal temperament or another temperament table.
What Pythagorean tuning means for a single pitch
Pythagorean tuning is the classical system in which every musical interval is generated from pure octaves and pure perfect fifths. A pure perfect fifth multiplies frequency by three over two; a pure octave multiplies by two. Starting from a chosen reference pitch such as A four at four hundred forty hertz, stacking one fifth upward yields six hundred sixty hertz, the Pythagorean E above that A when no extra octave fold is applied.
Stacking two fifths multiplies by nine over four; stacking downward multiplies by two over three at each step. Over a long chain the system is not circular: twelve pure fifths do not land exactly on seven pure octaves, and the residual is the Pythagorean comma that equal temperament spreads across the scale. For a single target pitch you usually care about a modest stack count—zero through seven or so in either direction—plus an explicit octave shift so the sounding note sits in the register you want.
This calculator encodes that model as a closed form f equals f zero times three over two to the fifths times two to the octave shift. Positive fifths climb the chain; negative fifths descend; octave shift is optional and defaults to zero so a pure fifth stack alone is one field away from a homework answer.
What you can do with it
Theory class and early-music drills
Students enter a reference pitch and a fifth count to verify hand-computed pure-fifth frequencies and cents offsets against the closed-form Pythagorean stack.
Instrument scaling and monochord layouts
Builders and luthiers generate exact 3/2-chain frequencies from a chosen reference when marking frets, bridges, or digital historical-instrument patches.
FAQ
How does this differ from equal-temperament note frequency?
Equal temperament uses f = A4 · 2^(semitones/12). Pythagorean tuning stacks pure fifths of ratio exactly 3/2 (and optional pure octaves of 2). The same letter name can differ by cents between the two systems.
What happens if I stack twelve fifths?
You get a valid frequency, but it will not match seven pure octaves from the same reference—the residual is the Pythagorean comma. This tool reports the pure stack result; it does not auto-temper the comma.
How do I calculate a Pythagorean frequency from a reference pitch?
Multiply the reference by 3/2 once for each fifth and by 2 once for each octave shift. Starting at 440 Hz, a single fifth up gives 660 Hz.
How do I go down a fifth instead of up?
Enter a negative number of fifths. Each step down multiplies the frequency by 2/3, so the pitch falls by a pure fifth rather than rising.
How do I bring a stacked note back into the right octave?
Use the octave shift. Each positive step doubles the frequency and each negative step halves it, so a note reached by many fifths can sit in the register you need.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/optics/pythagorean-tuning-frequency \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"reference_frequency":440,"fifths":1}'const res = await fetch("https://api.kit.forhosting.com/optics/pythagorean-tuning-frequency", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"reference_frequency": 440,
"fifths": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/pythagorean-tuning-frequency",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"reference_frequency": 440,
"fifths": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/pythagorean-tuning-frequency", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"reference_frequency":440,"fifths":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"reference_frequency":440,"fifths":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/pythagorean-tuning-frequency", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"reference_frequency": 440,
"fifths": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.pythagorean_tuning_frequency",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
What Pythagorean tuning means for a single pitch
Because the model is exact rational arithmetic realized in floating point with fixed rounding, the free browser path and the API return the same rounded frequency, ratio, and cents fields every time you send the same numbers.
Input contract, response shape, and error behaviour
A non-positive reference frequency is rejected because a pitch frequency must be strictly greater than zero. POST a JSON body with two required fields and one optional field. Required: reference_frequency, a finite positive number in hertz or any consistent inverse-time unit, with aliases f0, reference, ref_hz, f_ref, and ref_frequency; and fifths, an integer count of pure fifths in the inclusive range negative forty-eight through forty-eight, with aliases n, stack, fifth_count, and perfect_fifths. Optional: octave_shift, an integer pure-octave adjustment in the inclusive range negative twenty-four through twenty-four, defaulting to zero when omitted, with aliases octaves, octave, m, and octave_adjust. Numeric strings that match a strict finite pattern are accepted; empty strings, non-finite values, and non-integers for stack fields fail with invalid_input naming the field. Stack counts outside the published bounds are rejected so intermediate powers stay finite on IEEE double precision in the Worker and the browser. Pricing is a flat $0.002 per successful request; validation failures are not billed. Pair the response with your own note-name map if you need letter names: this endpoint computes frequency from stack geometry only and does not invent enharmonic spellings.
Where a Pythagorean frequency API earns its keep
Early-music workshops and conservatory theory labs use a Pythagorean tuning frequency calculator so students can hear or check the pure fifth chain without building a full temperament table by hand. Instrument builders and fret calculators need exact three-to-two stack frequencies when laying out monochords, harpsichord scalings, or digital recreations of historical instruments that were never fretted in equal temperament. Content sites that rank for Pythagorean tuning frequency calculator, pure fifth stack, or three over two pitch calculator benefit from a live free browser tool whose arithmetic matches the prepaid automation path byte for byte after fixed twelve-decimal rounding.
Integration use cases
Synth and sample pipelines
Automation scripts call the API as a golden oracle for Pythagorean mode tables, asserting ratio, frequency_hz, and cents_from_reference in continuous integration.
Technical questions
How is Pythagorean frequency computed from a reference?
Send a positive reference_frequency and an integer fifths count. The API returns f = f0 · (3/2)^fifths · 2^octave_shift, plus the ratio and cents from the reference. Octave shift defaults to zero when omitted.
Which inputs are rejected?
Missing or non-finite fields, non-integer fifths or octave_shift, fifths outside −48…48, octave_shift outside −24…24, and non-positive reference_frequency all fail with invalid_input and are not billed.
What does the Pythagorean tuning frequency calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.