ERB-Rate & Frequency Calculator
Convert between frequency in Hertz (Hz) and the psychoacoustic Equivalent Rectangular Bandwidth (ERB) scale.
Hz vs. ERB-rate Relationship
What is the ERB-rate?
The **ERB-rate** (Equivalent Rectangular Bandwidth rate) is a psychoacoustic measure that describes how humans perceive frequency. Unlike the Hertz (Hz) scale, which is linear, the ERB scale is designed to be perceptually uniform. This means that a one-unit step on the ERB scale corresponds to a roughly equal step in perceived pitch, regardless of where you are on the frequency spectrum. This concept is crucial for understanding human hearing, audio compression (like MP3s), and speech analysis. This **ERB-rate calculator** helps translate between the physical measurement of frequency (Hz) and its perceptual counterpart.
The unit for the ERB-rate scale is often called “Cams,” in honor of Cambridge, where much of the foundational research was conducted. A higher frequency in Hz will always result in a higher ERB-rate in Cams, but the relationship is not a straight line.
Clearing Up Confusion: ERB vs. ERB CTP
It’s important to clarify a common point of confusion. The term “ERB” in this calculator refers to **Equivalent Rectangular Bandwidth**, a concept from audio science. However, “ERB” is also the acronym for the **Educational Records Bureau**, an organization that administers a standardized test called the **CTP (Comprehensive Testing Program)**. The “ERB CTP” is an academic assessment for students and has no connection to the scientific calculator on this page.
If you were looking for information about the CTP test, please refer to official resources like the ERBlearn.org website. This tool is exclusively for psychoacoustic calculations.
The ERB-rate Calculator Formula
This calculator uses the widely accepted formulas developed by Glasberg and Moore (1990) to model the relationship between frequency and auditory filter bandwidth.
From Hertz to ERB-rate:
To convert a frequency f (in Hz) to the ERB-rate (in Cams), the following formula is used:
ERB-rate = 21.4 * log10(1 + 0.00437 * f)
ERB Bandwidth:
The calculator also provides the ERB bandwidth, which is the bandwidth of a single auditory filter at a given center frequency f. It is also measured in Hz.
ERB(f) = 24.7 * (1 + 0.00437 * f)
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| f | Center Frequency | Hertz (Hz) | 20 – 20,000 |
| ERB-rate | Perceptual Pitch Scale | Cams | ~2 – 40 |
| ERB(f) | Bandwidth of Auditory Filter | Hertz (Hz) | ~25 – 2,500 |
Practical Examples
Example 1: Converting a Middle C Frequency
- Input: Frequency = 261.63 Hz (Middle C)
- Calculation (ERB-rate):
21.4 * log10(1 + 0.00437 * 261.63)= 6.9 Cams - Calculation (ERB Bandwidth):
24.7 * (1 + 0.00437 * 261.63)= 53.0 Hz - Result: A frequency of 261.63 Hz corresponds to a perceptual ERB-rate of approximately 6.9 Cams and occurs within an auditory filter that has a bandwidth of 53.0 Hz.
Example 2: Converting a High Frequency
- Input: Frequency = 4000 Hz
- Calculation (ERB-rate):
21.4 * log10(1 + 0.00437 * 4000)= 27.1 Cams - Calculation (ERB Bandwidth):
24.7 * (1 + 0.00437 * 4000)= 455.4 Hz - Result: A 4000 Hz tone has an ERB-rate of 27.1 Cams and an auditory filter bandwidth of 455.4 Hz. Notice how much wider the filter bandwidth is at higher frequencies. Want to learn about other audio metrics? Check out our audio tools.
How to Use This ERB-rate Calculator
- Choose Your Input: Decide if you want to convert from Hertz (Hz) to ERB-rate (Cams) or vice-versa.
- Enter Your Value: Type your number into the appropriate input field. For example, to find the ERB-rate of a 1kHz tone, enter “1000” into the “Frequency (f)” field.
- View Real-Time Results: The calculator will automatically compute the corresponding value in the other field, along with the ERB bandwidth. No need to press a calculate button.
- Reset: Click the “Reset” button to clear all fields and start a new calculation.
- Interpret the Results: The main result is shown prominently, with the secondary ERB bandwidth calculation shown below it. You can explore our guides on signal processing for more context.
Key Factors That Affect ERB
- Center Frequency: This is the most significant factor. As the center frequency increases, both the ERB-rate and the ERB bandwidth increase.
- Logarithmic Scale: The relationship is logarithmic, not linear. This means a jump from 100 Hz to 200 Hz is a much larger perceptual leap than from 8000 Hz to 8100 Hz.
- Sound Level: The formulas used here are standardized for moderate sound levels. At very high sound levels, auditory filters tend to broaden slightly, which can change the exact ERB value.
- Individual Hearing: These formulas represent an average for a healthy young listener. Age-related hearing loss or other auditory conditions can significantly alter an individual’s auditory filter shapes.
- Masking Sounds: The presence of other sounds (maskers) can effectively change the shape and bandwidth of the auditory filter for a specific target sound.
- Model Used: While the Glasberg & Moore model is standard, other models exist that may produce slightly different results. This **ERB-rate calculator** sticks to the most common implementation. For more advanced modeling, see our advanced audio analysis page.
Frequently Asked Questions (FAQ)
- What does ‘Cams’ stand for?
- Cams is the unit for the ERB-rate scale. It was named as a tribute to Cambridge University, where much of the pioneering research in this area of psychoacoustics was conducted.
- Why not just use Hertz?
- The Hertz scale perfectly describes the physical properties of a sound wave, but it doesn’t reflect how our brains perceive it. The ERB-rate scale provides a more perceptually relevant measure, which is vital in fields like audiology, speech recognition, and audio data compression.
- What is the difference between ERB-rate and ERB bandwidth?
- ERB bandwidth is the width (in Hz) of a single auditory filter at a specific center frequency. ERB-rate is a cumulative scale; it represents the total number of ERB bandwidths stacked up from 0 Hz to the frequency of interest.
- What is a typical range for the ERB-rate?
- The range of human hearing (roughly 20 Hz to 20,000 Hz) corresponds to an ERB-rate of approximately 2 to 40 Cams.
- Can I use this calculator for any sound?
- Yes, this calculator can be used for any frequency within the range of human hearing. It is most accurate for pure tones at moderate listening levels.
- Is there an inverse formula to convert from ERB-rate back to Hz?
- Yes. This calculator uses the inverse formula automatically when you type in the ERB-rate field. The formula is:
f = (10^(ERB-rate / 21.4) - 1) / 0.00437. - Does this relate to the MEL scale?
- Yes, the ERB scale and the MEL scale are both psychoacoustic pitch scales designed to be more perceptually uniform than the linear Hertz scale. They are based on different experimental data but serve a similar purpose. Explore our MEL scale calculator for comparison.
- What does “calculator use erb ctp v” mean?
- This is a confusing search query that combines two unrelated topics. “ERB” can mean Equivalent Rectangular Bandwidth (used in this calculator) or the Educational Records Bureau, which makes the “CTP” test. This tool focuses only on the scientific meaning of ERB.
Related Tools and Internal Resources
If you found this ERB-rate calculator useful, you might also be interested in our other audio and signal processing tools.
- Audio Wavelength Calculator: Calculate the physical length of a sound wave in different mediums.
- Decibel Calculator: Work with sound pressure levels (SPL) and other decibel units.
- Frequency to Musical Note Converter: Find the closest musical note for a given frequency.