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One-third octave

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A one-third octave is a logarithmic unit of frequency ratio equal to either one third of an octave (1200/3 = 400 cents: major third) or one tenth of a decade (3986.31/10 = 398.631 cents: M3 Play^ⓘ).^[1] An alternative (unambiguous) term for one tenth of a decade is a decidecade.^[2]^[3]

Definitions

[edit]

Base 2

[edit]

ISO 18405:2017 defines a "one-third octave" (or "one-third octave (base 2)") as one third of an octave, corresponding to a frequency ratio of $2^{1/3}$ . A one-third octave (base 2) is precisely 400 cents.

Base 10

[edit]

IEC 61260-1:2014 and ANSI S1.6-2016 define a "one-third octave" as one tenth of a decade, corresponding to a frequency ratio of $10^{1/10}$ . This unit is referred to by ISO 18405 as a "decidecade" or "one-third octave (base 10)".^[4]

One decidecade is equal to 100 savarts (approximately 398.631 cents).

References

[edit]

^ Malcolm J. Crocker, Handbook of Acoustics (1997)
^ von Benda-Beckmann, A. M., Aarts, G., Sertlek, H. Ö., Lucke, K., Verboom, W. C., Kastelein, R. A., ... & Ainslie, M. A. (2015). Assessing the impact of underwater clearance of unexploded ordnance on harbour porpoises (Phocoena phocoena) in the Southern North Sea. Aquatic Mammals, 41(4), 503.
^ ISO 18405 Underwater Acoustics - Terminology (International Organization for Standardization, Geneva, 2017)
^ (This makes sense as, if we want one third of an octave, the ratio will be $f2/f1=2^{1/3}$ , and if we log10 both members of equation we have, $log{(f2/f1)}=log{(2^{1/3})}->log(f2/f1)=log(2)*1/3$ , which is approximately 0,1.

View on Grokipedia

from Grokipedia

A one-third octave is a logarithmic frequency band used in acoustics and audio engineering, defined such that the ratio of the upper band-edge frequency to the lower band-edge frequency is the cube root of 2 (approximately 1.26), spanning one-third of the logarithmic width of a full octave band.^[1] This subdivision allows for finer resolution in frequency analysis compared to full octave bands, where the upper-to-lower ratio is 2.^[2] The center frequency

f_0

of such a band is the geometric mean of its bounds, with the lower bound calculated as

f_0 / 2^{1/6}

and the upper bound as

f_0 \times 2^{1/6}

, following standards like ISO 266 for preferred frequencies.^[1] In practice, one-third octave bands divide the audible spectrum (typically 20 Hz to 20 kHz) into approximately 30 contiguous bands, enabling precise measurement of sound pressure levels across frequencies.^[3] Preferred center frequencies, standardized by ISO and ANSI, include values such as 25 Hz, 31.5 Hz, 40 Hz, up to 20 kHz, ensuring consistent analysis in applications like noise control and environmental monitoring.^[3] For example, the band centered at 1000 Hz extends from approximately 891 Hz to 1122 Hz.^[2] These bands are essential in fields such as acoustical engineering for tasks including noise source identification, room acoustics evaluation, and compliance with standards like NC (noise criteria).^[2] They provide greater detail than full octave bands (1/1 octave), which are broader and used for initial assessments, by splitting each octave into three segments for targeted frequency-specific interventions in hearing protection and vibration analysis.^[4]

Fundamentals

Octaves in Frequency

An octave represents a fundamental interval in frequency scales, defined as the range between two frequencies where the upper frequency is precisely double the lower one. For instance, the band from 100 Hz to 200 Hz spans exactly one octave.^[5] This ratio-based definition arises from the physical properties of sound waves, where doubling the frequency produces a perceptually distinct yet harmonically related tone. The concept of the octave originated in music theory, tracing back to ancient Greek terminology where it was known as the "diapason," denoting an all-embracing interval between the first and last notes of a scale. In modern usage, the term "octave" derives from the Latin word for eight, referring to the eight notes in the diatonic scale from a given note to its higher counterpart, such as C to the next C. This musical foundation extended to acoustics through the adoption of equal temperament, a tuning system that logarithmically divides the octave into 12 equal semitones to facilitate harmonic compatibility across instruments.^[6]^[7] Human perception of pitch follows a logarithmic scale with respect to frequency, such that equal multiplicative changes in frequency yield equal perceptual intervals, making the octave a natural perceptual unit. The octave ratio is mathematically expressed as

f_2 = 2 f_1,

where

f_1

is the lower frequency and

f_2

is the upper frequency. This logarithmic relationship ensures that the perceived distance between notes remains consistent across the audible spectrum, regardless of absolute frequency values.^[8]^[9] To determine the number of octaves between any two frequencies, the formula

n = \log_2 \left( \frac{f_2}{f_1} \right)

is applied, where

n

is the number of octaves. For example, between 250 Hz and 1000 Hz,

n = \log_2 (1000 / 250) = \log_2 4 = 2

, indicating two full octaves. This calculation underscores the octave's role as a logarithmic building block in frequency analysis.^[10] This octave interval forms the foundation for subdivisions like fractional octaves in more detailed acoustic measurements.

Fractional Octaves

Fractional octaves represent subdivisions of the octave interval on a logarithmic frequency scale, where the octave itself doubles the frequency ratio, serving as the parent unit for such divisions. These fractions, denoted as 1/n octaves, divide the octave into n equal parts logarithmically, ensuring that each sub-band spans a frequency ratio of

2^{1/n}

. For instance, a one-third octave corresponds to

n=3

, yielding a ratio of approximately 1.26, which allows for more granular segmentation of the spectrum compared to full octaves.^[11] The bandwidth of a fractional octave band is determined by this logarithmic division, with the relative bandwidth given by the formula

\Delta f / f_c = 2^{1/(2n)} - 2^{-1/(2n)}

, where

\Delta f

is the bandwidth and

f_c

is the center frequency. This expression captures the proportional width of the band relative to its center, facilitating consistent scaling across frequencies in acoustic analyses.^[11] Fractional octave bands offer enhanced resolution over full octaves, enabling finer analysis of non-uniform frequency responses in sound and vibration spectra by grouping energy into perceptually relevant intervals that approximate the human ear's sensitivity. This improved detail aids in identifying specific tonal components or resonances that might be obscured in broader bands, supporting applications in noise assessment and signal processing.^[11]^[12] The adoption of fractional octaves in engineering standards emerged post-World War II, driven by the need for precise noise measurement techniques amid industrial expansion and auditory health concerns. Early standardization efforts, such as ANSI S1.11-1966, formalized base-2 fractional bands like one-third octaves for acoustical measurements, influencing subsequent international guidelines.^[11]^[13]

Mathematical Definition

Band Width Calculation

The bandwidth of a one-third octave band is derived from the logarithmic division of a full octave, where a full octave spans a frequency ratio of 2 (from

f

2f

). Dividing this into three equal logarithmic parts yields a bandwidth ratio of

2^{1/3} \approx 1.2599

, meaning the upper frequency

f_u

to lower frequency

f_l

satisfies

f_u / f_l = 2^{1/3}

. The center frequency

f_c

is defined as the geometric mean of

f_u

and

f_l

, leading to the expressions

f_u = f_c \cdot 2^{1/6}

and

f_l = f_c \cdot 2^{-1/6}

, since

2^{1/6} \cdot 2^{1/6} = 2^{1/3}

and

(f_u \cdot f_l)^{1/2} = f_c

. This formulation ensures constant relative bandwidth across the spectrum, aligning with perceptual scaling in acoustics.^[1]^[11] The exact linear bandwidth is

\Delta f = f_u - f_l = f_c (2^{1/6} - 2^{-1/6})

, which evaluates to approximately

0.231 f_c

. For narrow bands at higher frequencies, a rough linear approximation is

\Delta f \approx f_c \cdot (2^{1/3} - 1) \approx 0.26 f_c

, but this overestimates slightly; more precise engineering contexts use

\Delta f \approx 0.23 f_c

to reflect the relative bandwidth proportion. The logarithmic nature must be emphasized for accuracy, as linear approximations degrade at low frequencies where the absolute bandwidth widens significantly relative to

f_c

, potentially leading to overlap or misrepresentation in spectral analysis.^[14]^[15] To illustrate, consider a center frequency of

f_c = 1000

Hz. First, compute

2^{1/6} \approx 1.12246

, so

f_l = 1000 \cdot 2^{-1/6} \approx 890.9

Hz. Then,

f_u = 1000 \cdot 1.12246 \approx 1122.5

Hz. The bandwidth ratio is

1122.5 / 890.9 \approx 1.260

, confirming

2^{1/3}

, and

\Delta f \approx 231.6

Hz, or about 23.2% of

f_c

. This step-by-step derivation highlights the precision required in applications like noise measurement. In standards, the band is often taken as 891 Hz to 1122 Hz.^[2]^[1] The base-2 logarithm is employed because an octave fundamentally represents a frequency doubling (

f

2f

), where

\log_2(2) = 1

, providing a natural unit for perceptual frequency scaling in acoustics and music. In contrast, base-10 logarithms define decades (frequency multiplication by 10,

\log_{10}(10) = 1

), which are more common in electrical engineering but less aligned with auditory octave divisions. This choice ensures one-third octave bands maintain perceptual uniformity across the audible spectrum.^[11]

Center Frequency Selection

In one-third octave band analysis, the center frequency

f_c

of each band is defined as the geometric mean of the lower band frequency

f_l

and the upper band frequency

f_u

, given by the formula

f_c = \sqrt{f_l \cdot f_u}.

fc=fl⋅fu

.
This choice ensures that the band spans equally on a logarithmic frequency scale, providing consistent proportional coverage across the audio spectrum and aligning with the perceptual nature of human hearing, where frequency differences are better represented logarithmically.^[11] The selection of center frequencies follows an iterative process to generate a geometric series that avoids overlaps and ensures contiguous bands. Starting from a reference frequency, typically 1,000 Hz as specified in international standards, subsequent center frequencies are obtained by multiplying the previous one by $2^{1/3} \approx 1.25992$ . This ratio corresponds to one-third of an octave interval on the base-2 logarithmic scale, allowing the entire frequency spectrum to be divided into non-overlapping bands. The process can extend bidirectionally: for the next higher band, $f_{c,next} = f_c \times 2^{1/3}$ ; for the previous lower band, $f_{c,prev} = f_c / 2^{1/3}$ . For example, beginning at 1,000 Hz, the next center frequency is approximately 1,259.9 Hz, and the previous is approximately 793.7 Hz. In standards like ISO 266, these are rounded to preferred values (e.g., 800 Hz and 1,250 Hz) from the R10 series.^[11] Special considerations apply to edge bands at low and high frequencies to maintain coverage of the audible spectrum, which typically ranges from about 20 Hz to 20,000 Hz. At these extremes, the iterative multiplication may result in frequencies outside practical measurement ranges, so the series is truncated accordingly. Additionally, calculated values are often rounded to preferred numbers based on standardized series (such as the R10 series) to ensure compatibility with measurement equipment and minimizing deviations—maximum less than 0.6% relative to the calculated geometric progression—while preserving the logarithmic spacing. This rounding facilitates practical implementation without significantly altering the spacing.^[16]

Standards and Implementation

ISO and ANSI Specifications

The development of standards for one-third octave bands traces its origins to the 1960s, when the International Electrotechnical Commission (IEC) published IEC 225:1966, titled "Octave, half-octave and third-octave band filters intended for the analysis of sounds and vibrations." This document established foundational specifications for passive and active bandpass filters used in acoustical analysis, emphasizing performance criteria for filter shapes and tolerances to ensure consistent measurement of sound and vibration spectra. Subsequent international standardization efforts culminated in ISO 266:1997, "Acoustics — Preferred frequencies," which serves as the primary reference for defining preferred center frequencies across a wide range from 1 Hz to 80 kHz. This standard adopts a geometric series based on the R10 preferred numbers from ISO 3, with a reference frequency of 1,000 Hz, to facilitate comparable acoustical measurements; in the audible range (approximately 20 Hz to 20 kHz), it encompasses 31 one-third octave bands. The standard was last confirmed in 2023.^[17] In parallel, the American National Standards Institute (ANSI) addressed implementation through ANSI/ASA S1.11-2014/Part 1/IEC 61260-1:2014 (R2023), "Electroacoustics — Octave-band and fractional-octave-band filters — Part 1: Specifications." This standard outlines performance requirements for bandpass filter sets, including one-third octave configurations, applicable to both analog and digital systems. It defines two classes of instruments: Class 1 for precision applications with stricter tolerances (e.g., ±0.5 dB ripple within the passband) and Class 2 for general purposes with relaxed limits (e.g., ±1 dB ripple), ensuring accurate spectral analysis while accommodating practical instrumentation constraints.^[18] Both standards emphasize constant percentage bandwidth, where each band's width is approximately 23% of its center frequency (derived from the one-third octave ratio of

2^{1/3}

), to maintain proportional resolution across the spectrum. Integration limits are specified as the lower and upper cutoff frequencies bounding each band, typically at

f_c \times 2^{-1/6}

and

f_c \times 2^{1/6}

relative to the center frequency

f_c

, enabling precise energy integration within defined intervals. Calibration requirements mandate periodic verification of filter responses against reference signals, often using standards like IEC 61672 for sound level meters, to achieve traceability and minimize measurement uncertainties in practical deployments.^[18]

Preferred Frequency Bands

The preferred one-third octave center frequencies are standardized nominal values defined in ISO 266 for consistent acoustic measurements across a wide frequency spectrum. These frequencies form a geometric series where each subsequent band is approximately 1.26 times the previous one (

2^{1/3} \approx 1.26

), ensuring uniform logarithmic spacing. The full set comprises approximately 50 bands, extending from sub-audible frequencies below 20 Hz to ultrasonic frequencies above 20 kHz. The audible human hearing range of 20 Hz to 20 kHz is covered by about 31 of these bands, providing detailed resolution for applications in sound analysis. Below 20 Hz, bands capture infrasonic content, while extensions above 20 kHz address ultrasonic phenomena, though practical implementations often focus on the core audible spectrum. The nominal center frequencies closely approximate the preferred number series from ISO 3, scaled relative to 1,000 Hz, suitable for logarithmic displays. The exact scaling follows

2^{k/3}

ratios relative to a reference frequency, with the approximation aligning due to

10^{1/10} \approx 2^{1/3}

. In practice, bands are numbered such that band 21 corresponds to the 1 kHz center frequency, facilitating easy reference in octave band charts; for non-standard ranges, additional bands can be derived by continuing the geometric progression beyond the preferred set.^[19] The following table lists the preferred nominal one-third octave center frequencies in Hz, from 0.8 Hz to 20 kHz, as commonly implemented per ISO 266:

Band Number	Nominal Center Frequency (Hz)
-10	0.8
-9	1
-8	1.25
-7	1.6
-6	2
-5	2.5
-4	3.15
-3	4
-2	5
-1	6.3
0	8
1	10
2	12.5
3	16
4	20
5	25
6	31.5
7	40
8	50
9	63
10	80
11	100
12	125
13	160
14	200
15	250
16	315
17	400
18	500
19	630
20	800
21	1000
22	1250
23	1600
24	2000
25	2500
26	3150
27	4000
28	5000
29	6300
30	8000
31	10000
32	12500
33	16000
34	20000

Applications

Noise and Vibration Analysis

In noise and vibration analysis, one-third octave bands play a crucial role in sound level meters, enabling detailed spectral decomposition of environmental noise for exposure assessment. These bands facilitate A-weighted one-third octave analysis, which approximates human hearing sensitivity while providing frequency-resolved data to evaluate environmental noise impacts on communities, as specified in ISO 1996-1 and ISO 1996-2 for basic quantities, measurement procedures, and determination of environmental noise levels. This approach allows for the identification of dominant frequency components contributing to overall exposure, supporting compliance with community noise regulations by integrating band-specific levels into metrics like the equivalent continuous sound level (L_eq). For vibration measurement, one-third octave bands are employed to isolate tonal components in road surface profiling, where they help characterize the acoustic and vibratory signatures of vehicle-tyre interactions on standardized test tracks. ISO 10844 specifies the essential characteristics of these test surfaces, ensuring reproducible measurements of noise emissions that indirectly inform vibration analysis by filtering out broadband noise and highlighting resonant frequencies in the 50 Hz to 10 kHz range. This band resolution is particularly useful in industrial acoustics to assess structure-borne vibrations from road vehicles, aiding in the design of quieter pavements and mitigation strategies for tonal excitations. A practical example in noise analysis involves calculating the overall sound pressure level by incoherently summing the levels from individual one-third octave bands, using the formula

L_{\text{total}} = 10 \log_{10} \sum 10^{L_i / 10},

where

L_i

represents the sound pressure level in each band; this method accounts for the logarithmic nature of decibels and is fundamental to aggregating spectral data in environmental assessments.^[20] In aircraft noise certification, one-third octave bands are integral to identifying dominant frequencies under FAR Part 36, where spectral analysis during flyover tests determines compliance with noise limits by computing the effective perceived noise level from band-resolved data. This process, outlined in Appendix A, ensures that tonal noise sources like engines are precisely quantified, contributing to regulatory limits that protect communities near airports. Preferred one-third octave bands serve as the standardized frequency set for such analyses.

Audio Equalization and Filtering

In audio equalization, one-third octave bands are widely employed in graphic equalizers to provide precise control over the frequency spectrum, enabling engineers to address room acoustics and shape tonal balance. A standard 31-band graphic equalizer spans the audible range from approximately 20 Hz to 20 kHz, with each band centered on ISO-defined frequencies spaced one-third of an octave apart, allowing for targeted adjustments that minimize unintended effects on neighboring frequencies.^[3]^[21] These equalizers are commonly used for room correction in live sound reinforcement and studio monitoring, where boosting or cutting specific bands compensates for acoustic anomalies like standing waves or excessive reverberation.^[22] In digital signal processing (DSP) implementations, one-third octave bands are approximated using finite impulse response (FIR) or infinite impulse response (IIR) filters to achieve the desired frequency selectivity. FIR filters offer linear-phase characteristics for phase-accurate equalization, while IIR filters provide computational efficiency with steeper roll-offs, both designed to emulate the constant-Q behavior of analog counterparts. The quality factor

Q

for these filters, defined as

Q = f_c / \Delta f

where

f_c

is the center frequency and

\Delta f

is the bandwidth, approximates 4.318 for one-third octave bands, ensuring the -3 dB points align with the band's fractional octave width.^[22]^[23]^[24] A practical example in audio mixing involves selectively boosting the 250 Hz one-third octave band to enhance perceived warmth in instruments or vocals, adding fullness to the low-midrange without broadening the effect across adjacent bands. This targeted adjustment leverages the narrow bandwidth of one-third octave filters to emphasize harmonic content associated with body and resonance, common in applications like vocal processing or acoustic guitar enhancement.^[25]^[26] Filter designs aligned with ISO specifications for center frequencies typically limit inter-band interaction to minimize phase distortion in both analog and digital systems, supporting applications from broadcast to high-fidelity playback.^[27]^[28]

Comparisons

Versus Full Octaves

Full octave bands span a frequency ratio of 2:1, meaning the upper frequency limit is twice the lower limit, resulting in a bandwidth of approximately 70.7% of the center frequency. For instance, a full octave band centered at 1000 Hz extends from about 707 Hz to 1414 Hz.^[3] In comparison, one-third octave bands have a narrower bandwidth with a frequency ratio of

2^{1/3} \approx 1.26

, offering roughly three times the frequency resolution of full octaves within the same overall spectrum.^[11] This difference in resolution leads to key trade-offs in application. Full octave bands provide a simpler, coarser analysis well-suited for broadband noise sources, where averaging over wider frequency ranges suffices for overall sound pressure level assessments without needing fine spectral detail.^[4] However, one-third octave bands excel in resolving narrowband features, such as tonal components in machinery noise, allowing for more precise identification and mitigation of specific frequency peaks that full octaves might obscure.^[29] In practice, spectrum plots illustrate this contrast: a full octave analysis often smears energy from a sharp tonal peak across its broader band, elevating adjacent levels and reducing the ability to pinpoint the source, whereas one-third octave plots maintain distinct separation, clearly delineating the peak's location and amplitude for targeted analysis.^[30] The adoption of one-third octave bands reflects a historical evolution in acoustic measurement. Mid-20th-century standards, such as those from the American Standards Association in 1953, primarily used full octave bands for their alignment with logarithmic frequency perception and simplicity in analog instrumentation.^[11] By the mid-1960s, fractional bands like one-third octaves gained prominence in national and international standards (e.g., BSI 1964 and IEC 1966) to enhance resolution and accuracy for complex noise environments, marking a shift toward more detailed spectral analysis in modern engineering practices.^[11]

Versus Other Band Resolutions

One-sixth octave bands offer a narrower frequency resolution compared to one-third octave bands, with a bandwidth ratio of

2^{1/6} \approx 1.122

, allowing for higher precision in applications such as room acoustics analysis where finer spectral details are needed.^[31] These bands result in approximately 66 divisions across the typical audible range (from 25 Hz to 16 kHz), enabling more granular examination of acoustic parameters like reverberation, though they require more computational resources and data processing than one-third octave bands.^[32] In contrast to linear frequency bands, which maintain a constant width in hertz (e.g., 100 Hz intervals), one-third octave bands use a logarithmic scale that aligns better with human auditory perception, providing perceptual uniformity by representing frequency changes as proportional ratios rather than absolute differences.^[33] Linear bands can distort perceptual analysis at higher frequencies, where the ear's sensitivity to relative changes is more pronounced, making logarithmic divisions like one-third octaves preferable for tasks involving noise assessment and audio processing.^[34] Compared to Fast Fourier Transform (FFT) methods, one-third octave bands provide fixed, standardized frequency groupings that simplify reporting in acoustics and vibration engineering, whereas FFT offers variable resolution determined by window size and sampling rate, allowing precise identification of narrowband components but at higher computational expense due to the need for extensive multiplications (scaling as

N \log_2 N

for

N

points).^[35] While FFT excels in detailed diagnostics like machine fault detection, one-third octave filtering via band-pass designs incurs lower processing costs for real-time applications, as energy is aggregated into predefined bands without requiring full spectral computation.^[35] One-third octave bands are selected in most engineering contexts as an optimal balance between spectral detail and practical simplicity, offering sufficient resolution for identifying dominant noise sources without the overhead of finer divisions like one-sixth octaves or the flexibility demands of FFT, particularly in standardized noise and vibration evaluations.^[30] This resolution serves as a refinement over broader full-octave bands, which provide a basic baseline for initial assessments but lack the precision for targeted interventions.^[2]

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One-third octave

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