On an upright piano, the inharmonicity coefficient B of the lowest strings can reach 10⁻³, while on a large grand piano it stays near 10⁻⁴, about a tenfold difference (Dalmont, Acta Acustica 2021). It is one of the clearest measured differences between small and large pianos. The ear adds its own bias: 36 musicians set the octave 7.835 cents wide at A4 and 23.920 cents wide at A6 (Jaatinen, Pätynen and Alho, JASA 2019), and inharmonic partials made low piano tones sound about two and a half semitones higher to a 21-listener jury (Anderson and Strong, JASA 2005). Tuning does not stay put either: a year of concerts moved one grand’s top octave by up to 10 cents (Plath, ISMA 2019). This roundup covers the physics of the strings, not the business of pianos, which our piano market statistics roundup tracks. We aggregated data from the Journal of the Acoustical Society of America, Acta Acustica, the Royal Swedish Academy of Music lecture series on piano acoustics, the Piano Technicians Guild, the USDA Forest Products Laboratory, and peer-reviewed perception studies in Communications Psychology and Hearing Research.
A few plain-language terms first. A partial is one of the overtones a string produces above its lowest frequency (the fundamental). Inharmonicity means those partials run progressively sharp of exact whole-number multiples, because steel wire is stiff rather than perfectly flexible. The inharmonicity coefficient B measures how strong this effect is for one string: in the standard model, partial n sits at n times the fundamental multiplied by the square root of (1 + Bn²) (Gràcia and Sanz-Perela, Reports@SCM 2017). A cent is one hundredth of a semitone (Hinrichsen, Revista Brasileira de Ensino de Física 2012). Stretch tuning means tuning the bass slightly flat and the treble slightly sharp of mathematical equal temperament.
TL;DR
- B is near 10⁻⁴ for the lowest strings of a large grand but can reach 10⁻³ on an upright (Dalmont, Acta Acustica 2021).
- Inharmonicity is least in the two octaves below middle C (Schuck and Young, JASA 1943).
- B changes by 5% to 18% between most neighbouring strings in the high treble range studied (Applied Sciences, Automatic Tuning of High Piano Tones 2020).
- Musicians widened the octave by 7.835 cents at A4 and 23.920 cents at A6 (Jaatinen, Pätynen and Alho, JASA 2019).
- Inharmonic partials raised the perceived pitch of low piano tones by about two and a half semitones (Anderson and Strong, JASA 2005).
- A compressed mistuning was detected better than a stretched one by 101 listeners, F1,100 = 228.87, p < 0.001 (Madsen and Oxenham, Communications Psychology 2024).
- Longitudinal string vibrations were audible in blind tests up to note C5 (Bank and Lehtonen, JASA 2010).
- A modern concert grand carries about 210,000 N of total string load (Conklin, Five Lectures on the Acoustics of the Piano 1990).
- Steel piano wire has shown no significant tensile strength gain since the late 1800s (Giordano, ASA 161st Meeting 2011).
- One concert grand’s highest octave moved up to 10 cents in a year of about 40 concerts (Plath, ISMA 2019).
- Wood holds 4.5% moisture at 20% relative humidity and 16.0% at 80%, at 70 °F (USDA Forest Products Laboratory, Wood Handbook 2021).
1. Inharmonicity Across the Keyboard
The coefficient B is not a single property of “the piano”: it spans two orders of magnitude, from about 10⁻⁴ in the middle of the keyboard to about 10⁻² for the highest notes. The quiet zone is the tenor: Schuck and Young found inharmonicity is least in the two octaves below middle C, and that the sharpening of partials grows roughly with the square of the partial number (Schuck and Young, JASA 1943).
To show what the square law means in practice, here is our own calculation from the standard formula: with B = 10⁻⁴, the 10th partial is sharp by a factor of the square root of 1.01, which is 1200 × log2(√1.01) ≈ 8.6 cents. With B = 10⁻³ the same partial is sharp by 1200 × log2(√1.1) ≈ 82.5 cents, close to a semitone. That gap is the raw material tuners have to work around.
| Metric | Value | Source |
|---|---|---|
| Inharmonicity coefficient B, strings in the middle of the keyboard vs the highest notes | Order of 10⁻⁴ (0.0001) vs order of 10⁻² (0.01) | Dalmont, Acta Acustica 2021 |
| Register with the least inharmonicity | The two octaves below middle C | Schuck and Young, JASA 1943 |
| How the sharpening of partials grows with partial number | Approximately with the square of the mode number | Schuck and Young, JASA 1943 |
| Largest partial shift reported in that study (one string) | The 15th partial at about the frequency of a 16th harmonic | Schuck and Young, JASA 1943 |
| Change in B between neighbouring strings, most string pairs in the high treble range studied | 5% to 18% | Applied Sciences, Automatic Tuning of High Piano Tones 2020 |
| Change in B produced by a 3% change in string frequency | 6% | Applied Sciences, Automatic Tuning of High Piano Tones 2020 |
The 5% to 18% step between neighbours matters for automatic tuning: the treble cannot be tuned from one global B value (Applied Sciences, Automatic Tuning of High Piano Tones 2020). Most recent available data: the Schuck and Young measurements date from 1943 and remain the reference for the register pattern.
2. Grand vs Upright: Bass Strings and Piano Size
Size shows up first in the bass. The lowest strings of an upright can reach B = 10⁻³, against roughly 10⁻⁴ on a large grand (Dalmont, Acta Acustica 2021), and the paper notes the problem is particularly marked on uprights with strings around 1 m long. On the Yamaha C109 upright measured in that study, B fell steadily from 4.5×10⁻⁴ at the paper’s Bb−1 to 3.2×10⁻⁴ at its G0. The paper uses French octave numbering, in which octave numbers run one lower than in the scientific system, so Bb−1 is the second key from the bottom.
The same study shows the bass problem can be engineered down: small masses fixed near the end of each wound bass string brought the mean B to 1.85×10⁻⁴, a value closer to a large grand. Yet size is not the whole story. Wogram and Mori found that the three strings of one note can differ in inharmonicity more than a grand differs from an upright (Wogram and Mori, ASA 146th Meeting 2003).
| Metric | Value | Source |
|---|---|---|
| B of the lowest strings, large grand piano vs upright piano | Near 10⁻⁴ vs up to 10⁻³ | Dalmont, Acta Acustica 2021 |
| Measured B, Yamaha C109 upright, from the paper’s Bb−1 to its G0 | Decreases from 4.5×10⁻⁴ to 3.2×10⁻⁴ | Dalmont, Acta Acustica 2021 |
| Mean B of the same bass strings after small masses were fixed near their ends | 1.85×10⁻⁴ | Dalmont, Acta Acustica 2021 |
| Inharmonicity difference between the 3 strings of one note vs between a grand and an upright | Can be larger within the note than between the 2 instruments | Wogram and Mori, ASA 146th Meeting 2003 |
| Total sound level, grand piano vs upright piano in the same comparison | Grand about 2 dB higher | Wogram and Mori, ASA 146th Meeting 2003 |
| Range of B used for piano strings in a published tuning model | 0.0004 to 0.002 | Gràcia and Sanz-Perela, Reports@SCM 2017 |
Context note: Gràcia and Sanz-Perela describe B as being “about 10⁻³” for piano strings in general (Reports@SCM 2017), a figure that matches the bass of smaller instruments rather than the middle of a grand when set against the measured values above. Most recent available data: Acta Acustica 2021.
3. The Railsback Stretch and the Subjective Octave
The Railsback curve is the average stretch that skilled technicians produce, and Giordano showed that a sensory dissonance model fed with measured piano spectra predicts it (Giordano, JASA 2015). A newer test went further: 15 piano tuners and 18 orchestra musicians tuned synthesized tones that kept a Steinway D’s real inharmonicity, and the resulting keyboard curve resembled the Railsback curve, with a remaining difference of F(1,16) = 16.05, p < 0.01, largest below A1 (Jaatinen and Pätynen, JASA 2022).
Stretch is not only a piano habit. With orchestral instrument tones instead of piano tones, 36 musicians still widened the octave, and the widening grew in the treble, from 7.835 cents at A4 to 23.920 cents at A6 (Jaatinen, Pätynen and Alho, JASA 2019). The piano’s physics and the ear’s preference point the same way, which is why tuners rarely fight the stretch.
| Metric | Value | Source |
|---|---|---|
| Octave widening chosen by listeners beyond an exact 2:1 ratio, at A4 vs at A6 | 7.835 cents vs 23.920 cents (both p < 0.001) | Jaatinen, Pätynen and Alho, JASA 2019 |
| Average stretching slope in the middle register | 7.82 cents per octave | Jaatinen, Pätynen and Alho, JASA 2019 |
| Difference between the tuning curve from real Steinway D tones and the Railsback curve | F(1,16) = 16.05, p < 0.01, largest below A1 | Jaatinen and Pätynen, JASA 2022 |
| Smallest pitch difference listeners can notice (difference limen), depending on frequency and loudness | As small as 10 cents | Gràcia and Sanz-Perela, Reports@SCM 2017 |
| Pitch steps used by an entropy-based automatic tuning method | ±1 cent per step, on spectra binned in 1-cent units | Hinrichsen, Revista Brasileira de Ensino de Física 2012 |
| Instruments the entropy method had been tested on at publication | 1 upright piano | Hinrichsen, Revista Brasileira de Ensino de Física 2012 |
Outlier note: the orchestral study used flutes, reeds, brass and string instruments rather than piano, so its numbers describe the ear’s preference, not piano inharmonicity. Its authors note their curve differs slightly from the Railsback curve and typical Steinway D tuning but follows the same trend. Hinrichsen’s entropy method reproduced the stretch of an aural tuning on one upright (Hinrichsen 2012), so treat it as a proof of concept.
4. How Listeners Hear Inharmonicity and Mistuning
Inharmonicity does not only change tuning, it changes perceived pitch. A jury of 21 listeners heard inharmonic low piano tones as about two and a half semitones higher than harmonic versions, shrinking to about an eighth of a semitone at higher fundamentals (Anderson and Strong, JASA 2005). A tuner hears the real string, not the textbook one.
The ear is also asymmetric. Across 101 participants, a 50-cent mistuning in the compressed direction was detected significantly better than the same mistuning in the stretched direction (Madsen and Oxenham, Communications Psychology 2024). That fits the stretch: errors on the wide side are the harder ones to notice. Hearing matters more than age here, as a 91-person follow-up found (Madsen et al., Hearing Research 2025), a point that connects to our musician hearing loss statistics. For how precisely singers and synthesis models hit pitch in cents, see our singing voice synthesis pitch accuracy statistics.
| Metric | Value | Source |
|---|---|---|
| Pitch rise caused by inharmonic partials, lowest fundamentals tested (21 listeners) | About two and a half semitones | Anderson and Strong, JASA 2005 |
| Pitch rise caused by inharmonic partials, higher fundamentals tested | About an eighth of a semitone | Anderson and Strong, JASA 2005 |
| Detection of a 50-cent compressed vs a 50-cent stretched mistuning (101 participants) | Compressed detected significantly better, F1,100 = 228.87, p < 0.001 | Madsen and Oxenham, Communications Psychology 2024 |
| Largest single experiment in that mistuning series | 160 participants | Madsen and Oxenham, Communications Psychology 2024 |
| Factor linked to poorer mistuning perception (91 participants) | Hearing loss, but not age | Madsen et al., Hearing Research 2025 |
| Highest note at which longitudinal string vibrations were audible in ABX tests (two grands, one upright) | C5 | Bank and Lehtonen, JASA 2010 |
| Pitch change caused only by changing the starting phases of partials, A0 bass tones (27.5 Hz, 100 partials) | An octave, a fifth or a major third | Galembo, Askenfelt, Cuddy and Russo, JASA 2001 |
Context note: longitudinal vibrations are stretching motions along the string’s length, not the side-to-side motion that sets the pitch, and an ABX test is a blind comparison in which listeners say which of two versions a third sample matches. Bank and Lehtonen found the effect subtle between A3 and C5 (Bank and Lehtonen, JASA 2010), while in the deep bass the phase relations alone can move perceived pitch by large intervals (Galembo et al., JASA 2001).
5. String Tension, Wire, and Scale Design
In piano building, the “scale” is not a musical scale but the design table of string lengths, diameters and tensions. Its central number is pull: a contemporary piano string pulls about 830 N on average, nearly 12 times the 70 N of a string on the 1720 Cristofori piano (our ratio: 830 ÷ 70 ≈ 11.9), and the total load of a 9-ft (274 cm) concert grand is about 210,000 N (Conklin, The cast-iron plate, Five Lectures on the Acoustics of the Piano 1990). The iron plate that resists that load weighs 160 to 180 kg on a concert grand.
The wire sets the ceiling on tension. Steel music wire gained about 30% in tensile strength between 1850 and 1900 and has shown no significant improvement since the late 1800s, so the limits a designer faced in 2011 were those of 1900 (Giordano, ASA 161st Meeting 2011).
| Metric | Value | Source |
|---|---|---|
| Average pull per string, 1720 Cristofori piano vs contemporary piano | About 70 N vs about 830 N | Conklin, Five Lectures on the Acoustics of the Piano 1990 |
| Total string load, 1720 Cristofori vs modern concert grand | About 7,500 N vs about 210,000 N (47,000 lbf) | Conklin, Five Lectures on the Acoustics of the Piano 1990 |
| Weight of a concert-grand cast-iron plate | 160 to 180 kg | Conklin, Five Lectures on the Acoustics of the Piano 1990 |
| Tension of bass strings on keys 11 to 20 of a 161 cm grand, across three experimental string sets | 831 N to 1,131 N | Choi, Yoon, Kim and Sung, Seoul National University 2002 |
| Tensile strength gain of steel piano wire, 1850 to 1900 vs since the late 1800s | About 30% vs no significant improvement | Giordano, ASA 161st Meeting 2011 |
| Number of strings in a typical piano | Approximately 220 | Piano Technicians Guild, Technical Bulletin 1 (revised 1994) |
| Lowest soundboard resonance of a test piano, strings at pitch vs tension fully released | Shifts from 115 Hz to 90 Hz | Wogram, Five Lectures on the Acoustics of the Piano 1990 |
Context note: the Choi et al. tensions come from one 161 cm grand fitted with three different bass string sets for an inharmonicity experiment (Choi et al. 2002). Wogram found resonances above 200 Hz essentially unaffected by string tension (Wogram 1990). Most recent available data: 1990 to 2011.
6. Tuning Stability, Humidity, and Drift
The soundboard is a thin wooden panel with a slight upward arch, called its crown, and humidity changes that arch. When the board swells, the crown pushes the bridge harder against the strings and pitch rises, most in the middle octaves (Piano Technicians Guild, Humidity Control). The guild’s target for in-piano climate systems is 42% relative humidity. The wood numbers show why: at 70 °F, wood settles at 4.5% moisture content in 20% relative humidity and at 16.0% in 80% (USDA Forest Products Laboratory, Wood Handbook 2021), and Sitka spruce shrinks 7.5% tangentially but 4.3% radially as it dries out completely.
Measured drift on one concert instrument was modest. After one year and about 40 concerts, one concert grand’s tuning had moved about 2.5 cents in the mid range and up to 10 cents in the highest octave, while bass keys 1 to 20 showed no measurable change apart from three outliers (Plath, ISMA 2019).
| Metric | Value | Source |
|---|---|---|
| Tuning deviation after one year and about 40 concerts, mid range | About 2.5 cents | Plath, ISMA 2019 |
| Tuning deviation after one year and about 40 concerts, highest octave | Up to 10 cents | Plath, ISMA 2019 |
| Tuning deviation after one year, bass keys 1 to 20 | No measurable difference, apart from three outliers | Plath, ISMA 2019 |
| Soundboard crown difference between the two measurement dates | 1 ± 0.5 mm | Plath, ISMA 2019 |
| Recommended relative humidity inside the piano for climate control systems | 42% | Piano Technicians Guild, Humidity Control |
| Recommended tuning frequency | 3 to 4 tunings in the first year, at least 2 per year after that | Piano Technicians Guild, Technical Bulletin 1 (revised 1994) |
| Equilibrium moisture content of wood at 70 °F, 20% vs 80% relative humidity | 4.5% vs 16.0% | USDA Forest Products Laboratory, Wood Handbook 2021 |
| Shrinkage of Sitka spruce from green to oven-dry, radial vs tangential | 4.3% vs 7.5% | USDA Forest Products Laboratory, Wood Handbook 2021 |
Context note: the concert hall in that study is heated in autumn and winter, which drops the relative humidity to 20% at the extreme, and both measurements were taken in winter, so the crown’s yearly swing of up to several millimetres was not captured. The guild’s tuning advice comes from its pitch-raising bulletin (PTG Technical Bulletin 1). Equilibrium moisture content is the moisture level at which wood neither gains nor loses water at a given humidity. Most recent available data: the 2019 drift study covers a single instrument.
Summary: Piano String Inharmonicity and Tuning Stretch by the Numbers
| Metric | Value | Source |
|---|---|---|
| B of the lowest strings, large grand vs upright | Near 10⁻⁴ vs up to 10⁻³ | Dalmont, Acta Acustica 2021 |
| Register with the least inharmonicity | The two octaves below middle C | Schuck and Young, JASA 1943 |
| Change in B between neighbouring high-treble strings | 5% to 18% | Applied Sciences 2020 |
| Inharmonicity spread within the 3 strings of one note | Can exceed the grand vs upright difference | Wogram and Mori, ASA 2003 |
| Octave widening chosen by listeners, A4 vs A6 | 7.835 cents vs 23.920 cents | Jaatinen, Pätynen and Alho, JASA 2019 |
| Real-inharmonicity tuning curve vs Railsback curve | F(1,16) = 16.05, p < 0.01 | Jaatinen and Pätynen, JASA 2022 |
| Smallest noticeable pitch difference | As small as 10 cents | Gràcia and Sanz-Perela, Reports@SCM 2017 |
| Pitch rise from inharmonic partials, lowest tones | About two and a half semitones | Anderson and Strong, JASA 2005 |
| 50-cent compressed vs stretched mistuning detection | Compressed detected better, F1,100 = 228.87, p < 0.001 | Madsen and Oxenham, Communications Psychology 2024 |
| Mistuning perception, 91 participants | Linked to hearing loss, not age | Madsen et al., Hearing Research 2025 |
| Longitudinal string vibrations audible | Up to C5 | Bank and Lehtonen, JASA 2010 |
| Pitch change from partial phases alone, A0 tones | An octave, a fifth or a major third | Galembo et al., JASA 2001 |
| Total string load, modern concert grand | About 210,000 N | Conklin 1990 |
| Bass string tension, 161 cm grand | 831 N to 1,131 N | Choi et al. 2002 |
| Steel wire tensile strength gain, 1850 to 1900 | About 30% | Giordano, ASA 2011 |
| Tuning drift after one year of concerts, mid range vs top octave | About 2.5 cents vs up to 10 cents | Plath, ISMA 2019 |
| Recommended relative humidity inside the piano | 42% | Piano Technicians Guild |
| Wood moisture content at 20% vs 80% relative humidity (70 °F) | 4.5% vs 16.0% | USDA Wood Handbook 2021 |
Methodology and Sources
We ran targeted searches across inharmonicity measurement, the Railsback stretch, octave perception, string tension and scale design, and humidity-driven drift, then read each figure in the primary document (paper, lecture, professional guidance or government handbook). Search-snippet figures, forum posts and technician blogs were excluded. This is a measurement topic, so most data are older than three years: the physics of steel strings does not age, but each section flags its most recent available data.
- Dalmont, Piano bass strings with reduced inharmonicity: theory and experiments, Acta Acustica vol. 5, 2021
- Schuck and Young, Observations on the vibrations of piano strings, JASA 15, 1943 (abstract record, Technical University of Munich)
- Automatic Tuning of High Piano Tones, Applied Sciences 10(6), 2020
- Wogram and Mori, Is a grand piano always better than an upright piano?, ASA 146th Meeting 2003
- Gràcia and Sanz-Perela, The wave equation for stiff strings and piano tuning, Reports@SCM vol. 3, 2017
- Giordano, Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance, JASA 138(4), 2015
- Jaatinen and Pätynen, Effect of inharmonicity on pitch perception and subjective tuning of piano tones, JASA 152(2), 2022
- Jaatinen, Pätynen and Alho, Octave stretching phenomenon with complex tones of orchestral instruments, JASA 146(5), 2019
- Hinrichsen, Entropy-based tuning of musical instruments, Revista Brasileira de Ensino de Física 34(2), 2012
- Anderson and Strong, The effect of inharmonic partials on pitch of piano tones, JASA 117(5), 2005
- Madsen and Oxenham, Mistuning perception in music is asymmetric and relies on both beats and inharmonicity, Communications Psychology 2024
- Madsen, Kreft, Purmalietis, Dau and Oxenham, Association between hearing loss but not age and mistuning perception in music, Hearing Research vol. 467, 2025
- Bank and Lehtonen, Perception of longitudinal components in piano string vibrations, JASA 128(3), 2010
- Galembo, Askenfelt, Cuddy and Russo, Effects of relative phases on pitch and timbre in the piano bass range, JASA 110(3), 2001
- Conklin, The cast-iron plate, Five Lectures on the Acoustics of the Piano, Royal Swedish Academy of Music 1990
- Wogram, Influence of string tension, Five Lectures on the Acoustics of the Piano 1990
- Choi, Yoon, Kim and Sung, Influence of inharmonicity on the tone of a piano, Seoul National University 2002
- Giordano, Evolution of music wire and its impact on the development of the piano, ASA 161st Meeting 2011
- Plath, Influence of long-term playing on the tonal characteristics of a concert piano, ISMA 2019
- Piano Technicians Guild, Humidity Control
- Piano Technicians Guild, Technical Bulletin 1: Pitch Raising (revised 1994)
- USDA Forest Products Laboratory, Wood Handbook FPL-GTR-282, Chapter 4: Moisture relations and physical properties of wood, 2021
- Data watch: published B ranges conflict. Hinrichsen (2012) gives B from 0.0002 in the bass up to 0.4 in the treble, while the 2021 Acta Acustica paper puts the highest notes at the order of 10⁻², so we use the Acta Acustica orders of magnitude. We could not read cent values for the extremes of the Railsback curve in any primary source available to us, so this roundup gives no single “cents of stretch at A0 and C8” figure; the numbers circulating online trace back to wikis and forums. The drift study and the Choi et al. tension table each come from one instrument. The Piano Technicians Guild bulletin was last revised in 1994, and the Wood Handbook moisture table is derived mainly from Sitka spruce.
Last updated: November 7, 2026. We update this roundup quarterly, and the next refresh is expected when the Journal of the Acoustical Society of America publishes its next piano acoustics papers.