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Piano Stretch Tuning Calculator

Visualize and calculate the piano stretch tuning curve for any piano type. See how inharmonicity — caused by string stiffness and scaling — causes real pianos to be tuned sharp in the treble and flat in the bass compared to theoretical 12-TET equal temperament. Choose between concert grand, baby grand, upright, and spinet profiles to explore how string speaking length affects the degree of stretching required, and use the interactive stretch tuning chart to compare each key's deviation in cents.

Long strings, lowest inharmonicity. The reference for professional tuning.
Stretch: 100%
0%100%150%
100% = standard Railsback curve for this piano type
Hz
415440450
A4 = 440.0 Hz

Railsback Curve

Stretch offset (cents) 0-cent baseline (12-TET)

All 88 Keys — Stretch Tuning Data

Key # Note 12-TET Hz Stretched Hz Cents Offset Compare

Stretch Tuning Explained

Inharmonicity Coefficient
B = π3 d4 Q / (64 L2 T)
Where d is string diameter, Q is Young's modulus, L is string length, and T is tension. Shorter, thicker strings (as in uprights and spinets) have higher B values, causing more inharmonicity.
Partial Frequency
fn = n · f1 · √(1 + B · n2)
Real piano string partials are stretched sharp compared to perfect harmonics. The nth partial is sharper by a factor of √(1 + B·n²). This is why octaves on a piano must be tuned wider than a perfect 2:1 ratio.
Railsback Curve
offset(cents) = f(key, pianoType, stretch%)
The Railsback curve plots how much each piano key deviates from mathematical 12-TET. Treble notes are tuned sharp (+cents) and bass notes flat (−cents). The shape depends on the piano's string scaling and inharmonicity profile.
Stretched Frequency
fstretched = f12-TET × 2(cents / 1200)
Once the stretch offset in cents is determined, the actual tuned frequency is calculated by applying the cent offset to the theoretical 12-TET frequency. One cent equals 1/1200 of an octave.

How to Use This Calculator

  1. Select your piano type — Choose Concert Grand, Baby Grand, Upright, or Spinet. Each type has a different inharmonicity profile due to string length and construction. Shorter pianos need more stretch.
  2. Adjust the stretch amount — The default 100% represents a standard Railsback curve for the selected piano type. Reduce to 0% to see pure 12-TET, or increase beyond 100% for exaggerated stretch. Use this to experiment or match a specific tuning style.
  3. Set the A4 reference — Default is 440 Hz. Adjust for alternate concert pitch (e.g., 442 Hz for European orchestras, 432 Hz for Verdi tuning). All frequencies recalculate instantly.
  4. Read the Railsback curve — The graph shows cents deviation from 12-TET for each of the 88 keys. Positive values mean the key is tuned sharp; negative means flat. Hover over the curve to see exact values.
  5. Compare with audio — Click the A/B buttons on any key row to hear the difference between pure 12-TET and the stretched tuning. This demonstrates why stretch tuning sounds more "in tune" on a real piano.
  6. Export your data — Copy the table to clipboard, export as CSV for spreadsheet analysis, or use the print view for a clean reference sheet.

Understanding stretch tuning also deepens your grasp of why pianos sound different from purely synthesized tones. Because real piano partials are stretched sharp relative to the harmonic series, a piano octave tuned to a perfect 2:1 frequency ratio would sound narrow and "closed." Concert grand tuning typically shows the least deviation from 12-TET, while upright piano tuning requires noticeably more stretch because the shorter strings are more inharmonic. You can explore the pure harmonic series in the harmonic series calculator and see how the ideal ratios compare to what a piano tuner achieves on an acoustic instrument. For a foundation in how equal temperament intervals are derived, the interval calculator shows frequency ratios and cent values for every diatonic and chromatic interval.

Why Is Piano Tuning Stretched? Inharmonicity and the Railsback Curve

Real piano strings are not perfectly flexible — they have stiffness, which causes their overtones (partials) to land slightly sharp of exact integer harmonics. This effect is called inharmonicity, and its magnitude depends on string geometry: thick, short strings (like those in the bass section of an upright or spinet) are far more inharmonic than the long, relatively thin strings of a concert grand. The consequence is that when a tuner matches octaves by ear, they are aligning stretched partials — so each octave must be widened slightly beyond the theoretical 2:1 ratio to sound beatless.

Summed across all 88 keys, those widened octaves produce the characteristic S-shaped deviation graph first documented by O.L. Railsback in 1938: treble notes end up tuned progressively sharp and bass notes progressively flat relative to equal temperament, while the middle of the keyboard sits closest to 12-TET. The degree of curvature tracks the piano's size — a spinet needs more stretch than a concert grand because its shorter strings are more inharmonic. You can hear this directly using the piano note frequency finder to compare theoretical equal-temperament pitches against what a well-tuned acoustic instrument actually produces, or verify your own instrument's pitch against the pure-tone reference in the chromatic tuner.

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Frequently Asked Questions

What is stretch tuning on a piano?
Stretch tuning is the practice of tuning a piano's treble notes slightly sharp and bass notes slightly flat compared to mathematically perfect equal temperament (12-TET). This compensates for inharmonicity — the physical phenomenon where real piano strings produce overtones that are sharper than perfect integer multiples of the fundamental frequency. Without stretch tuning, octaves would sound "dead" or "narrow" because the partials wouldn't align.
What is the Railsback curve?
The Railsback curve, first documented by O.L. Railsback in 1938, is a graph showing the deviation of a well-tuned piano from theoretical equal temperament. It plots cents offset (vertical axis) against piano key number (horizontal axis). The characteristic S-shape shows bass notes tuned flat and treble notes tuned sharp, with the middle range closest to 12-TET. Every piano has a unique Railsback curve determined by its string scaling.
Why do different piano types need different amounts of stretch?
Inharmonicity is inversely proportional to string length squared. Concert grands (7-9 feet) have long strings with low inharmonicity, requiring modest stretch. Baby grands (5-6 feet) need slightly more. Uprights (3.5-5 feet) have significantly shorter strings and need more stretch. Spinets (under 3.5 feet) have the shortest strings and highest inharmonicity, requiring the most stretch tuning of any piano type.
What is inharmonicity in piano strings?
Inharmonicity is the degree to which a vibrating string's overtones (partials) deviate from perfect integer multiples of the fundamental frequency. In an ideal string with zero stiffness, the 2nd partial would be exactly 2x the fundamental. Real piano strings have stiffness, causing each partial to be progressively sharper. The inharmonicity coefficient B depends on string diameter, length, tension, and material properties.
How much stretch is typical for a concert grand?
A well-tuned concert grand typically shows about +20 to +30 cents of stretch at the highest treble notes (key 88, C8) and about -10 to -20 cents at the lowest bass notes (key 1, A0). The middle octaves (around A4, key 49) show near-zero deviation from 12-TET. The exact values depend on the specific instrument's string scaling, condition, and the tuner's preferences.
Can I use this calculator to tune my piano?
This calculator provides a theoretical reference based on typical inharmonicity profiles for each piano type. Real piano tuning requires measuring the actual inharmonicity of each individual string, which varies from instrument to instrument. Professional piano tuners use electronic tuning devices (ETDs) that measure inharmonicity and calculate custom stretch curves. This tool is best used for education, understanding the concept, and as a starting reference point.
Does a digital piano need stretch tuning?
Most digital pianos include a built-in stretch tuning option that simulates the Railsback curve of an acoustic piano, because listeners have been trained to expect the characteristic stretched sound. When stretch tuning is disabled, a digital piano plays in pure 12-TET — which can sound unnaturally flat in the bass and narrow in the treble when compared with an acoustic instrument. High-end stage pianos and sample libraries typically offer adjustable stretch curves matching different acoustic models.
What is "octave stretching" and why does it happen?
Octave stretching refers to the fact that a piano tuner must widen octaves beyond the theoretical 2:1 frequency ratio to make them sound in tune. When a tuner matches the 2nd partial of the lower note to the fundamental of the upper note (a common aural tuning technique), the octave ends up slightly wider than 2:1 in Hz. The higher the inharmonicity of the strings involved, the wider the stretched octave must be. This is the principal reason bass notes are tuned flat and treble notes sharp — the cumulative effect of stretching every octave up the scale.
How does piano string scaling affect inharmonicity?
String scaling — the engineering of string length, diameter, and tension across the keyboard — directly determines each string's inharmonicity coefficient B. Longer, thinner strings (as in concert grands) have lower B values and produce overtones that are closer to true harmonic ratios, requiring less stretch tuning. Shorter, thicker strings (common in bass sections and spinets) have higher B values, causing overtones to diverge sharply from integer multiples. Wound bass strings use a copper winding over a steel core to increase mass without adding stiffness, partially reducing inharmonicity in the lowest registers.

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