Savart: Logarithmic Pitch Unit for Frequency Ratios

The savart (symbol Sav) is a logarithmic unit that measures musical pitch intervals as ratios of acoustic frequency. One savart equals 1⁄1000 of a decade—log₁₀ frequency ratio of 10—and corresponds to approximately 3.9863 cents. Before the widespread adoption of the cent, acousticians, piano tuners, and organ builders used savarts to quantify temperament schemes and instrument response.

This guide reviews the savart’s mathematical definition, historical development, conversion formulas, and applications in modern acoustics. Use it alongside the hertz explainer and psychoacoustic metrics such as the sone article when designing experiments or tuning systems that span amplitude and frequency perception.

Definition and Conversions

A pitch interval expressed in savarts equals 1000 times the base-10 logarithm of the frequency ratio:

Sav = 1000 × log₁₀(f₂ / f₁).

Because many musical calculations rely on base-2 logarithms, conversions between savarts and cents (base-2 logarithmic units) are common. One octave (frequency ratio 2:1) equals 1000 × log₁₀(2) ≈ 301.03 savarts and exactly 1200 cents. Therefore, 1 savart ≈ 1200 / 301.03 ≈ 3.9863 cents, and 1 cent ≈ 0.2506 savarts. When converting savarts to hertz, apply the inverse relationship: f₂ = f₁ × 10^(Sav/1000).

Comparison with the Cent and the Decade

The savart’s definition arises from base-10 logarithms, while the cent uses base-2. Engineers working with spectrum analysers or logarithmic frequency sweeps often use decades (tenfold frequency changes) as a scaling unit. Viewing the decade as 1000 savarts clarifies how fine-grained savarts subdivide the same interval, offering a natural bridge between musical notation and instrumentation plots that display data per decade. When designing measurement graphs that overlay musical scales on Bode plots or waterfall charts, mapping each octave to 301 savarts ensures alignment between musical and engineering axes.

Historical Background

The unit honours French physicist Félix Savart (1791–1841), who, with Jean-Baptiste Biot, studied the relationship between frequency and perceived pitch. Savart’s experiments using rotating toothed wheels showed that pitch depends on vibration frequency and that logarithmic scaling matches how listeners perceive intervals. In 1826 he published tables listing frequency ratios in logarithmic steps, effectively defining what later became known as the savart. Instrument makers adopted the unit to compare temperaments and to chart how string length, air column dimensions, or tension adjustments affected pitch.

During the late 19th century, physicists such as Lord Rayleigh and Hermann von Helmholtz used savarts in treatises on acoustics and psychoacoustics. The cent emerged in the early 20th century through the work of Alexander J. Ellis, who defined it as one hundredth of a semitone in equal temperament. Because the savart is tied to decimal logarithms rather than musical intervals, it remained popular among engineers who favoured base-10 calculations. Understanding this history clarifies why historical tuning charts, organ builders’ notebooks, and early frequency analysers often annotate intervals in savarts.

Legacy in Instrument Making

French organ builder Aristide Cavaillé-Coll recorded voicing adjustments in savarts to ensure consistent tonal progression across pipe ranks. Violin makers compared plate resonances measured in savarts to monitor graduation and tap-tone adjustments. These practices persisted into the 20th century, when engineers designing electromechanical tonewheels and early synthesizers mapped gear ratios and capacitor values to savart increments.

Measurement Techniques

Today, savart intervals can be generated and measured with digital tools. Start with a reference frequency f₁ and compute f₂ using the exponential formula above. Audio workstations automate the process by applying pitch-shift processors in logarithmic steps or by adjusting oscillator tuning in high-resolution increments. When measuring live instruments, use a high-resolution frequency analyser or tuner that reports values in cents, then convert to savarts.

For research, record multiple takes, calculate average frequency, and document standard deviation in cents or savarts. Provide details about sampling rate, windowing function, and analysis length when publishing data. Because savarts rely on base-10 logs, confirm that rounding behaviour in your analysis software preserves the desired precision—especially for microtonal compositions where differences of 5 savarts (≈20 cents) carry musical significance.

Calibration and Reference Standards

Pitch measurements depend on accurate reference frequencies. Standard concert pitch sets A4 at 440 Hz, but ensembles may use 415, 432, or 444 Hz. Changing the reference shifts absolute frequencies but not the savart interval itself. Always document the tuning standard, measurement temperature, and instrument state (strings new or aged, reed condition, etc.). Reference tone generators, frequency counters, or time standards traceable to national metrology institutes provide repeatable baselines that align with SI frequency realizations.

Applications

Tuning Systems and Microtonal Music

Microtonal composers describe intervallic structures using savarts when they prefer decimal ratios or when they interoperate with engineering data. For example, dividing a decade (1000 savarts) into 53 equal steps produces intervals of 18.87 savarts (≈75 cents), close to the historical 53-tone equal temperament used to approximate just intonation. Electronic instrument designers map savart increments to control voltages or MIDI pitch-bend values, enabling seamless modulation between equal temperament and alternative tunings.

Acoustic Engineering

Loudspeaker designers plot frequency response per decade; overlaying savart grids reveals how resonances align with musical intervals. When crafting notch filters or equalisation curves, specifying centre frequencies in savarts clarifies spacing across logarithmic sweeps. Savarts also appear in aeroacoustics and machinery diagnostics, where tonal components are identified relative to rotational harmonics—expressed naturally as logarithmic ratios.

Psychoacoustic Research

Researchers examine just-noticeable differences (JNDs) in pitch by presenting tone pairs separated by small savart increments. Results inform auditory models and aid in designing cochlear implant strategies that balance electrode spacing and stimulation rates. Combining pitch JND data with loudness scaling—using metrics such as the sone—supports holistic hearing-aid tuning that respects both amplitude and pitch perception.

Instrument Diagnostics and Maintenance

Piano technicians evaluate stretch tuning by plotting string frequencies in savarts relative to equal-temperament targets. Deviations reveal whether scaling adjustments maintain pleasing inharmonicity. Wind-instrument makers monitor pad leaks or bore irregularities by tracking overtone series spacing in savarts; consistent intervals indicate healthy resonance structures. Coupling these diagnostics with tools like the acoustic dampening calculator helps studios balance instrument response with room treatment decisions.

Working with Savarts in Modern Workflows

Digital audio workstations (DAWs) often display pitch adjustments in cents, but custom scripts or MIDI processing tools can convert to savarts. Many synthesizers support high-resolution pitch control via MIDI Polyphonic Expression (MPE) or control voltage (CV) inputs; mapping these controls in savarts clarifies how modulation depth translates into frequency ratios. When designing user interfaces, consider offering savart and cent readouts simultaneously to accommodate both engineering and musical perspectives.

For educational content, overlay savart grids on spectrograms or waterfall plots to illustrate relationships between harmonics and musical intervals. Pair pitch analyses with timing tools such as the tempo to delay calculator so performers can synchronise rhythmic and harmonic design. Document software versions, sample rates, and plugin latency compensation when publishing datasets to ensure reproducibility.

Integration with Logarithmic Amplitude Metrics

Because both savarts and decibels are logarithmic, engineers frequently analyse them together. For example, when designing an equaliser that boosts harmonics spaced every 100 savarts (~398 cents), choose gain adjustments expressed in decibels and evaluate resulting loudness shifts using the decibel to power calculator. Maintaining consistent logarithmic frameworks simplifies interpretation and avoids mixing ratio-based and linear metrics.

Future Outlook

Interest in microtonality, adaptive tuning, and immersive audio continues to grow. Savart-based representations help developers of adaptive pitch systems, such as machine-learning-driven accompaniment tools, model intervallic change continuously across decades of frequency. Virtual and augmented reality applications use savarts to align spatial audio filters with head-related transfer functions, ensuring consistent pitch cues across binaural renderers.

Research into auditory perception across cultures benefits from savart metrics because they avoid cultural assumptions inherent in semitone-based systems. As standardisation bodies evaluate metadata schemas for digital scores and audio assets, including savart-compatible interval descriptors will enable interoperability among notation software, DAWs, and synthesis engines. Documenting savart usage alongside SI frequency measures ensures long-term accessibility of musical archives and engineering test records alike.

Key Takeaways

The savart measures pitch intervals as 1000 × log₁₀(frequency ratio), linking musical perception to engineering-friendly decade scaling.
One octave equals approximately 301.03 savarts; 1 savart corresponds to about 3.9863 cents, simplifying conversions between historical and modern tuning units.
Applications span microtonal composition, acoustic engineering, psychoacoustic research, and instrument maintenance, where savarts describe fine interval adjustments.
Combining savart analysis with decibel-based loudness tools and tempo-aligned calculators supports holistic design of musical experiences and audio technologies.