Precise Temperament Tuning: 432.081216 Hz, recalculated
Robert Edward Grant’s Precise Temperament Tuning, recalculated.
216 × 1.26³ = 7.56³ = 432.081216
With the exact cube root of 2, it would be 432.000000 Hz.
One number. Three major thirds. One rounding.
An octave is a doubling: 216 Hz becomes 432 Hz. If you divide it into three equal major-third steps, each needs a factor of about 1.259921. Grant uses 1.26 — very close, but not identical. The digits after the decimal point come from this rounding. More precisely: 1.26 = 63/50 is a rational choice that behaves like rounding 21/3 to two decimal places — and it is exactly what creates the excess.
Five terms, then it all makes sense
If you understand pitch, interval, octave, cents, and temperament, you understand the entire debate around 432.081216 Hz. No prior knowledge required.
Pitch & Frequency
A tone is a vibration. Frequency, measured in hertz, tells you how many cycles occur per second. 432 Hz means 432 cycles per second.
Interval
Not a distance in hertz, but a ratio. A just major third is 5/4 = 1.25; a perfect fifth is 3/2.
Octave
The ratio 2/1. The frequency doubles: 216 becomes 432; 432 becomes 864.
Cent
A logarithmic measure of pitch — not a percentage. An octave has 1,200 cents, and an equal-tempered semitone has 100 cents. It is better suited for comparison than hertz.
Temperament
A system of compromises. Just intervals sound beautiful, but they do not fit every key at once. Temperaments distribute the conflict.
Equal, Just, and Pythagorean compared +
Equal temperament divides the octave into twelve equal steps; the major third is 24/12 = 21/3 (400 cents). Just intonation uses simple ratios, with a just major third of 5/4 = 1.25 (386.3 cents), but three stacked just thirds remain below the octave. Pythagorean tuning is built on perfect fifths; its third, 81/64 = 1.265625 (407.8 cents), is clearly wider than 1.26.
The formulas behind cents +
Cent(r) = 1200 × log2(r) and r(c) = 2c/1200
This lets any frequency ratio be converted into cents, and any cent value back into a ratio.
Three paths. One octave.
If you stack three major thirds above 216 Hz, the chosen third determines where you land. Only the exact cube root of 2 closes the octave. Grant instead uses the rational 1.26 — and that creates the small excess. Choose a path.
0.081216 Hz above the octave
+0.325 cents relative to the octave
Grant uses the clean, rational approximation 1.26 = 63/50. The excess of 0.081216 Hz (0.325 cents, well below typical pitch-discrimination thresholds) is not a coincidence, but the direct mathematical consequence of this choice.
Independent reconstruction of a third-party system — no third-party graphics; all values calculated independently.
Where 2.000000 becomes 2.000376
Three major-third steps are meant to fill the octave, so the individual step r must satisfy the equation r3 = 2. This is exactly true for the cube root of 2. It is not quite true for 1.26.
1.263 yields
2.000 376
the exact cube would be
2.000 000
Because 216 = 63, we have 216 × 1.263 = (6 × 1.26)3 = 7.563. The number 432.081216 is therefore also the cube of 7.56. No separate frequency constant needs to be assumed — the digits after the decimal point come entirely from the choice of 1.26.
1.26 = 63/50 · a terminating rational number.
| 21/3 | 1.259921049894873 |
| 1.26 | 1.260000000000000 |
| 1.26 - 21/3 | 0.000078950105127 |
| Cent(1.26 / 21/3) | 0.108480 / major third |
| 1.263 | 2.000376 |
| Delta = 1.263/2 | 1.000188 |
| Cent(432.081216/432) | 0.325441 |
Side note: The trail of the digits 081216 +
The decimal digits 081216 are not an arbitrary interpretation; they arise exactly from the excess of 1.263 over 2: 0.081216 = 216 × (1.263 - 2). Since 216 = 63, 216 × 1.263 = 7.563 — remove the decimal point, and the number appears as a perfect cube: 7563 = 432081216 (viewed as an integer). This structure is mathematically sound.
Symbolic readings — for example, reading 08:12:16 as the ratio 2:3:4 — remain explicitly secondary and are not treated as proof.
From the number to a playable scale
Using the building blocks t = 1.26, q = 3/2, and Delta = t3/2, most scale degrees can be derived directly. The values match the practical implementation, except for rounding.
How to read the table: note on the left, frequency in the middle, deviation from equal temperament on the right — negative values are slightly lower, positive values slightly higher. Green = directly derivable; Yellow = reconstructed or close to the implementation.
Note: The frequencies follow the rounded cent values of the practical PTT implementation. The formula column gives the ideal generator or best reconstruction — small deviations in the thousandths-of-a-cent range are rounding effects only.
Full value table: PTT/ET cents, factor, formula +
| Note | PTT cents | ET cents | Deviation | Factor | Frequency (Hz) | Formula / status |
|---|---|---|---|---|---|---|
| A | 0.000 | 0 | +0.000 | 1.000000000 | 432.081216 | 1 |
| A# | 97.933 | 100 | -2.067 | 1.058198907 | 457.227871 | ~ 200/189 |
| B | 203.910 | 200 | +3.910 | 1.124999999 | 486.091368 | (3/2)²/2=9/8 |
| C | 302.170 | 300 | +2.170 | 1.190698651 | 514.478521 | q·t²/2 |
| C# | 400.107 | 400 | +0.107 | 1.259998923 | 544.421867 | 1.26 |
| D | 498.044 | 500 | -1.956 | 1.333332564 | 576.107956 | 4/3 |
| D# | 599.675 | 600 | -0.325 | 1.413948101 | 610.940415 | ~ 21/2/ Delta |
| E | 702.279 | 700 | +2.279 | 1.500280750 | 648.243131 | ~ (3/2)·Delta |
| F | 800.217 | 800 | +0.217 | 1.587600036 | 685.972154 | 1.26² |
| F# | 897.829 | 900 | -2.171 | 1.679685153 | 725.760403 | 4/(q·t²) |
| G | 1004.126 | 1000 | +4.126 | 1.786049007 | 771.718227 | q²·t²/2 |
| G# | 1102.062 | 1100 | +2.062 | 1.889998383 | 816.632800 | (3/2)·1.26=1.89 |
| A | 1200.000 | 1200 | +0.000 | 2.000000000 | 864.162432 | 2/1 |
Independent reconstruction. Where values come from Grant’s deck or from the practical implementation, the formula is stated. D# and E remain the most delicate points: D# lies just below 21/2, and the fifth appears slightly widened.
A5, A4, and MIDI 69: a notation issue +
Sources associated with Grant name the reference pitch A5 = 432.081216 Hz. In modern standard notation, the same pitch range is often interpreted in an A4 context, and ISO 16 defines A4 = 440 Hz. The calculation 216 × 1.263 does not depend on what the target pitch is called. It is a notation issue, not a mathematical one.
What you actually hear
The difference between 432 and 432.081216 Hz is 0.325 cents — as an isolated pitch shift, well below the conscious pitch-discrimination threshold for most listeners. That threshold varies with frequency, timbre, loudness, training, and context; when tones are played together, tiny differences are more noticeable as beats. In any case, the audible difference lies not in the reference pitch, but in the thirds.
The same major chord, four tunings. Listen to the third.
Live waveform when audio is playing.
Beating
When 432 and 432.081216 Hz are played together, a very slow beat occurs. This example shifts the tones two octaves higher so the pulse becomes audible within a few seconds. The difference that is barely perceived as pitch can be heard here as a slow volume pulse.
| 432 to 432.081216 | 0.325 cents | barely audible in isolation |
| 1.26 to 21/3 | 0.108 cents | very small; treating them as identical would be false |
| 1.26 to 5/4 | 13.795 cents | audibly relevant in practice |
| 81/64 to 5/4 | 21.506 cents | clearly relevant |
Synthetic listening examples for interval comparison. Not a composition and not evidence of any effect. BOHN AI’s audible Whisperer sounds are a separate matter: hand-composed, not generated.
What is exact, and what is interpretation
The greatest risk with this topic is mixing levels. A mathematical identity has a different status than a source statement, an implementation, or a symbolic interpretation. That is why we keep them separate.
432.081216 Hz is mathematically reproducible.
But it arises from 1.26 — not from the exact cube root of 2.
Grant provides a geometric-musical system with a clearly computable core number. BOHN AI makes this number transparent and separates mathematical identity from approximation, source statements from implementation, and sound aesthetics from claims of effect. That is precisely the value of this first study on Grant.
[G1] R. E. Grant: Precise Temperament Tuning, official topic page. 432.081 Hz, 1.26 rather than 5/4. robertedwardgrant.com/precise-temperament-tuning
[G2] R. E. Grant: PTT Overview. A5 = 432.081216 Hz, derivation from 3/2 and 21/3. .../precise-temperament-tuning-overview
[G3] Grant / Janover: World’s First Concert Piano tuned to 432.081 Hz. 1.263, A5. .../concert-piano-432-081hz [Original URL: "tenperament", sic]
[G4] Grant / Janover: See and Hear Differences… at 432 Hz. Comparison of Equal, Precise, Just, and Pythagorean. .../see-and-hear-differences
[G5] Grant / Janover: The Case for PTT in 432.081 Hz rather than 432 Hz. .../the-case-for-ptt [Original URL: "he-case", sic]
[G6] R. E. Grant: Presentation Deck (PDF), analyzed for the BOHN AI research edition. Frequency list, Cube of Delos, seed values. PDF
[G7] Grant / Janover: 24 Note Quartertone Relationship to Ancient Pyramids. Handled separately. .../24-note-quartertone
[M1] T. Mazzotti: How to create and use the 432 Hz Precise Temperament scale files. SCL/TUN/TXT files, cent values. tonimazzotti.com/how-to-create-and-use-the-precise-temperament-for-432-hz
[B1-B3] Encyclopaedia Britannica: Tuning and temperament (Equal, Just, Pythagorean). britannica.com/art/tuning-and-temperament
[I1] ISO 16:1975, Standard tuning frequency. A = 440 Hz (confirmed in 2022). iso.org/standard/3601
[P1-P2] Cent (music), Pitch (music): logarithmic unit and perceptual threshold. Cent · Pitch
From Grant’s number to our own research.
The same rigor in the Universal Frequency series: calculate first, interpret second. Geometry becomes number; number becomes sound — cleanly separated, boldly interpreted.
Discover the Universal Frequency