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Piano Stretch Tuning Calculator
Visualize and calculate the Railsback curve for any piano type. See how inharmonicity causes real pianos to be tuned sharp in the treble and flat in the bass compared to theoretical equal temperament.
Long strings, lowest inharmonicity. The reference for professional tuning.
Stretch: 100%
100% = standard Railsback curve for this piano type
Hz
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
- 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.
- 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.
- 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.
- 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.
- 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.
- Export your data — Copy the table to clipboard, export as CSV for spreadsheet analysis, or use the print view for a clean reference sheet.
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.