• Octaves and third-octaves

    Octaves, thirds of octaves and decades are divisions of the spectrum used in audio, since they work in the same way as the ear, increasing their width in Hertz as frequency increases, and decreasing it when frequency decreases. They are used for the definition of the width of filters, as well as for the spectral analysis.

    What is an octave?

    An octave is a frequency band that lies between one frequency and another that is twice the frequency, such as 1000 to 2000 Hz, or 707 to 1414 Hz (the latter corresponding to the octave band centered at 1000 Hz1). As mentioned at the beginning, like third-octaves and decades, bandwidth of octave bands doubles when we double the frequency. Thus, the 1000 to 2000 Hz octave has 1000 Hz width (2000 minus 1000), and the 2000 to 4000 Hz octave has 2000 Hz width (4000 minus 2000), while the 4000 to 8000 Hz octave has 4000 Hz width (8000 minus 4000).

    An octave is also the interval between a musical note and the upper or lower octave of the scale, encompassing 12 semitones. To help give an idea of what an octave means to non-musicians, it is the interval between the first two notes of the chorus of the song 'Somewhere Over the Rainbow' (the word 'some-where'), and the same applies to David Bowie's Starman (that very word). For non-musicians, the musical notes are repeated every octave. For example, at 440 Hz there is the standard central A (in Anglo-Saxon notation), but at 220 and 880 Hz the notes are also A. When we play two notes with the same name in a piano spaced one octave (for example, the A with a fundamental frequency of 220 Hz and the 440 Hz), the sensation is to reinforce that note, while with any other combination of notes (for example, an A and a C), it is evident that there is a certain dissonance.

    Assuming that the audible spectrum ranges from 20 to 20,000 Hz, it could be divided into approximately ten octaves.

    What is one third of an octave?

    One third of an octave (third-octave) is a frequency band that corresponds to one third of an octave (and therefore, three adjacent bands of one third of an octave correspond to one octave band).

    And why a third and not a half or a quarter? The answer to this is to be found in the spectral behaviour of the ear, which we could approximate to a bank of third octave filters.

    Musically, on the equal temperament Western scale, one third of an octave is a 'third' interval, comprising four semitones (or almost, depending on the musical pitch and the way in which the frequency of the third of an octave is calculated 1).

    In spectrum analysis applications, fractional octave values such as thirds are often used, but also others such as half (1/2), 1/4, 1/8, 1/12, 1/16, 1/24 or 1/48 octave, either for a low resolution 'RTA' type analysis, or for the smoothing of a full resolution frequency response. Assuming that the audible spectrum ranges from 20 to 20,000 Hz, it could be divided into approximately 30 thirds of an octave (one-third octave equalizers such as the one shown in the photograph usually have 31 because their effect extends half a third-octave above and below the nominal audible band).

    As a curiosity, scaling the nominal frequencies of one-third octave can help calculate decibel differences without a calculator. For example, for power, we could remember 1250 and 1600 Hz and use them (divided by 1000) as multipliers (or dividers) to perform, an approximate conversion to relative decibels, namely add (or subtract) 1 or 2 decibels. The exact ratios are actually 1.259 and 1.589, respectively, but those nominal frequencies are close enough (specially considering that the ear has trouble distinguishing even 1 dB changes between levels). Similarly, these same frequencies can be used to calculate decibels of number of loudspeakers, voltage or pressure (or even distance to the loudspeaker), although in this case the result will be to add (or subtract) 2 or 4 dB. Naturally, the factor of 2 from 2000 Hz leads to 3 or 6 decibels; likewise, the factor 1 from 1000 Hz leads to 0 decibels.

    And what about decades?

    A decade is a frequency band that lies between a first frequency and another that is ten times greater than this, such as 1000 to 10000 Hz.

    Musically, on the western equal-temperament scale, this assumes an interval of 3 octaves and four semitones and, therefore, 10 thirds of an octave (or almost, depending on the musical tuning and how the frequency of the third of an octave is calculated 1). Assuming that the audible spectrum ranges from 20 to 20,000 Hz, it could be divided into three decades: 20-200 Hz, 200-2,000 Hz (which can be seen in the graph above) and 2,000-20,000 Hz (which, depending on the criteria used, could be labelled as 'low', 'medium' and 'high').

    The width of the different bandwidths can be seen in the graph above: decade, octave and third of octave. As a reference, we have added the keyboard of a standard 88-key piano. The green key corresponds to the central A (La, in italian notation) (of octave number four), with a frequency of 440 Hz, and the grey key corresponds to the central C (Do).

    Center frequencies for octave and third-octave filters

    The centre (midband) frequencies of the octave and octave fractions (third and half) filters are defined in ISO and ANSI standards, as well as and their 'nominal' (rounded) numerical representations.

    The table below shows, for the ten commonly used audio octave bands, the mathematical and nominal (rounded) centre frequencies for 1-octave width filters, in addition to the band limiting (bandedge) frequencies:

    In the case of centre frequencies for one-third octave wide filters, the above standards apply the so called 'preferred values' developed by Charles Renard at the end of the 19th century and which are also used to establish capacitor and resistor value spacing. As with one-octave filters, the centre frequencies and the 'nominal' (rounded) frequencies of the filters are standardized.

    The table below shows the center frequencies for the 31 third-octave bands commonly used in audio, for filters with one-third octave width, in addition to the bandedge frequencies):

    1 The centre frequencies of the octave and octave fraction filters are logarithmically spaced (i.e., they are further and further apart as we move up in frequency) and can be calculated in two standardized ways. The first is in base 10 and the second is in base 2, each with its advantages and disadvantages. We could say that the first form is more technical and the second is more musical. For the text of the article and the tables we have used base 2, the most common. As a reference, the central frequencies calculated in base 10 for octave band filters are: 31.62, 63.10, 125.9, 251.2, 501.2, 1000, 1995, 3981, 7943 and 15849 Hz. It can be observed that octave spacings do not correspond exactly to double or half. The octave third filters are: 25.12, 31.62, 39.81, 50.12, 63.10, 79.43, 100.0, 125.9, 158.5, 199.5, 251.2, 316.2, 398.1, 501.2, 631.0, 794.3, 1000, 1259, 1585, 1995, 2512, 3162, 3981, 5012, 6310, 7943, 10000, 12589, 15849 and 19953 Hz.

    2 Mathematically calculated frequencies have been rounded for ease of reading.

    References: ANSI S1.11: Specification for Octave-Band and Fractional-Octave-Band Analog and Digital Filters
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