Common Intervals
Calculation Formulas
The Science Behind Semitone Calculation
A semitone represents a specific frequency ratio of approximately 1.0595:1, derived from the twelfth root of 2. This ratio creates the foundation for equal temperament tuning, where each octave divides into twelve equal logarithmic steps. The relationship is exponential rather than linear – the Hz difference between 100 Hz and 105.95 Hz (one semitone) differs from the Hz difference between 400 Hz and 423.81 Hz (also one semitone), yet both represent the same perceptual pitch interval.
The mathematical precision of semitone relationships allows accurate frequency manipulation in audio processing, acoustic analysis, and sound synthesis. Understanding these relationships enables precise control over pitch shifting, frequency distribution, and harmonic analysis across various applications.
Where n represents semitones, f₁ is the starting frequency, and f₂ is the target frequency.
Frequency Ratio Mathematics
Each semitone multiplies frequency by the constant 1.059463 (the twelfth root of 2). Twelve consecutive semitone steps produce an exact 2:1 frequency ratio – an octave. This exponential relationship means that semitone calculations require logarithmic functions rather than simple arithmetic.
| Semitones | Frequency Ratio | Example (440 Hz) | Result |
|---|---|---|---|
| 1 | 1.0595 | 440 × 1.0595 | 466.16 Hz |
| 3 | 1.1892 | 440 × 1.1892 | 523.25 Hz |
| 5 | 1.3348 | 440 × 1.3348 | 587.33 Hz |
| 7 | 1.4983 | 440 × 1.4983 | 659.25 Hz |
| 12 | 2.0000 | 440 × 2.0000 | 880.00 Hz |
To calculate semitones between two known frequencies, the formula inverts: n = 12 × log₂(f₂/f₁). This logarithmic calculation determines how many semitone steps separate any two frequencies, regardless of their absolute values.
Cents: Precision Measurement
Cents provide finer resolution than semitones, with 100 cents equaling one semitone. This scale allows precise measurement of small frequency deviations. A deviation of ±5 cents typically represents the threshold of human pitch perception accuracy under ideal conditions. Audio engineers use cent measurements for critical tuning adjustments where semitone resolution proves insufficient.
Semitone Calculator Operation
Two Frequencies Mode
Input two frequency values to calculate the semitone and cent interval between them. This mode works with any positive frequency values – from subsonic rumbles at 20 Hz to ultrasonic frequencies beyond 20,000 Hz. The calculator determines both the interval size and whether the shift is upward or downward.
Frequency + Interval Mode
Specify a starting frequency and desired semitone shift to calculate the target frequency. Positive semitone values shift upward, negative values shift downward. Decimal semitone inputs enable precise fractional shifts – useful when exact frequency relationships require sub-semitone precision.
Note Analysis Mode
Enter any frequency to identify its closest pitch reference and deviation in cents. The reference frequency field (defaulting to 440 Hz for A4) allows analysis against different tuning standards. Historical tunings used A4 = 415 Hz (Baroque), 432 Hz (alternative), or 442 Hz (some orchestras). Adjusting this reference enables analysis across different tuning systems.
Audio Engineering Applications
Pitch Shifting and Time Stretching
When audio samples require speed changes without pitch alteration, or pitch changes without timing alteration, semitone calculations determine the necessary frequency adjustments. A drum loop recorded at 90 BPM needs approximately +5.88 semitones to match 120 BPM when time-stretched. The calculator provides exact frequency targets for pitch-correction algorithms and time-stretching processors.
Frequency Spectrum Analysis
Sound designers use semitone relationships to ensure layered frequencies maintain proper spacing. When combining oscillators or samples, spacing them by specific semitone intervals prevents frequency masking and maintains clarity. A bass sound at 80 Hz paired with a sub-bass at 40 Hz (-12 semitones) creates clean octave separation.
Formant Preservation
Pitch-shifting algorithms often preserve formant frequencies (vocal characteristics) while shifting fundamental pitch. Understanding the semitone relationship between original and target pitch helps audio engineers compensate for formant shifts, maintaining natural sound quality during extreme pitch manipulation.
Extreme pitch shifts (>12 semitones) often introduce artifacts. When shifting by more than one octave, consider splitting the shift across multiple processing stages or using specialized algorithms designed for large interval changes.
Acoustic Frequency Analysis
Harmonic Series Relationships
Natural harmonic series follow specific frequency ratios that semitone calculations can quantify. The fundamental frequency and its harmonics (2x, 3x, 4x the fundamental) translate to semitone intervals of 12, 19, and 24 respectively. Analyzing these relationships helps identify harmonic content in complex waveforms.
For a 100 Hz fundamental:
| Harmonic | Frequency | Semitones from Fundamental |
|---|---|---|
| 1st (Fundamental) | 100 Hz | 0 |
| 2nd | 200 Hz | 12.00 |
| 3rd | 300 Hz | 19.02 |
| 4th | 400 Hz | 24.00 |
Beat Frequency Calculations
When two frequencies play simultaneously, they create beat frequencies at the difference between them. Semitone calculations help predict beat rates. Two oscillators 1 semitone apart at 440 Hz (440 Hz and 466.16 Hz) produce beats at 26.16 Hz – a useful technique in synthesis and acoustics.
Results From The Semitone Calculator
Semitone Values
Positive semitone values indicate upward frequency shifts, negative values indicate downward shifts. A result of -5 semitones means the second frequency is lower than the first by five semitone steps. The magnitude of the semitone value matters more than its sign for many applications – the absolute interval size determines processing requirements regardless of direction.
Frequency Ratios
The frequency ratio expresses the multiplicative relationship between two frequencies. A ratio of 1.5:1 means the second frequency is exactly 1.5 times the first. Certain ratios have special acoustic properties – 2:1 (octave), 3:2 (perfect fifth at 7 semitones), and 4:3 (perfect fourth at 5 semitones) represent simple integer ratios that human perception finds particularly consonant.
Cent Deviations
In note analysis mode, cent deviations show how far an input frequency sits from perfect tuning. A deviation of ±2 cents typically falls within acceptable tuning accuracy for most applications. Deviations beyond ±10 cents become clearly perceptible. Professional tuning aims for ±3 cents maximum deviation.
Practical Semitone Calculation Examples
Sample Pitch Correction
A recorded sample plays at 440 Hz but needs to match a reference at 432 Hz. Using the calculator:
Octave Verification
Verifying that 220 Hz and 440 Hz form a perfect octave:
Frequency Spacing
Determining if two oscillators are properly spaced:
Calculation Accuracy and Limitations
Floating Point Precision
The calculator uses double-precision floating point arithmetic, providing approximately 15-17 decimal digits of precision. This exceeds the accuracy requirements for virtually all audio and acoustic applications. Frequency values below 1 Hz or above 100,000 Hz may experience minor rounding effects, though these remain well within practical tolerances.
Equal Temperament Assumption
All calculations assume equal temperament tuning, where each semitone represents exactly 1/12 of an octave. Historical tuning systems (just intonation, Pythagorean tuning, meantone temperament) use different interval sizes. For analysis of non-equal-tempered systems, frequency ratios provide more accurate information than semitone values.
Perceptual vs. Mathematical Precision
While calculations provide cent-level precision, human pitch perception varies by frequency range and individual. Pitch discrimination typically ranges from 5-25 cents depending on frequency, training, and listening conditions. The calculator’s mathematical precision exceeds perceptual resolution in most cases.
Related Frequency Analysis
Semitone calculations complement other frequency analysis methods. For detailed pitch detection and frequency tracking across audio signals, the vocal range calculator provides real-time pitch analysis from microphone input. That tool applies the same semitone mathematics to identify sustained pitches and map frequency ranges.
Combining semitone calculations with spectral analysis, formant tracking, and harmonic analysis creates comprehensive frequency understanding across audio, acoustics, and sound synthesis applications.
Common Questions
Semitones represent perceptual pitch relationships better than absolute Hz values. A 10 Hz difference at 100 Hz sounds larger than a 10 Hz difference at 1000 Hz, but one semitone represents the same perceptual interval regardless of starting frequency.
Yes. Negative semitone values indicate downward frequency shifts. Shifting from 880 Hz to 440 Hz represents -12 semitones (one octave down).
The calculator provides cent accuracy to two decimal places, exceeding practical requirements. Most applications need accuracy to only whole cents or ±5 cents.
The mathematical relationship holds for any positive frequency values. Practical applications typically work within 20 Hz to 20,000 Hz (human hearing range), but the calculations remain valid outside this range.
Pitch perception follows logarithmic rather than linear scales. Doubling frequency (one octave) sounds like the same interval whether moving from 100 Hz to 200 Hz or 1000 Hz to 2000 Hz. This logarithmic relationship requires exponential mathematics, not simple addition.