nthoct: n-th Octave Frequencies and Wavelengths (IEC 61260)

Complies with IEC 61260-2014, ISO 3, ISO 266.

nthoct complies with IEC 61260-2014.

Returns the exact and nominal values of the center frequencies, and the lower and upper band-egdes of the [n] times subdivided octaves between [lo] and [hi].

Note: In the context of this function, "frequency" is interchangable with "wavelength" or any logarithmic series of values.

Arguments:

n - (optional) octave subdivisions

lo - (optional) lowest center frequency

hi - (optional) highest center frequency

Return:

cx - exact center frequencies

cn - nominal center frequencies

bx - exact band-edges

bn - nominal band-edges

t - a table compiling centers and band-edges

Options:

ReferenceFrequency - reference frequency

Base - 2: perfect octaves, 10: perfect decades

IgnoreEvenDenominator - ignore IEC 61260-2014, 5.4.2 f (3)

IgnoreUniqueness - ignore IEC 61260-2014, E.3.2, E.3.3 & E.3.5

to make both cn AND bn unique

Reference Frequency:

Typically and with only few exceptions,

ReferenceFrequency = 1 kHz for technical applications,

ReferenceFrequency = 440 Hz for musical applications.

Base:

Base determines the usage of base 2 or 10.

Typically and with only few exceptions,

Base = 10 for technical application

Base = 2 for musical application (or when perfect octaves are required)

Typical n | Tech | Music (Base = 2)

----------+----------------+-------------------------

1 | (full) octave | octave

2 | half | tritone

3 | 3rd | major third

6 | 6th | whole step, major second

12 | 12th | half step, minor second

24 | ... | quarter step, 50 cents

Even denominators:

According to IEC 61260-2014 Section 5.4.2 formula (3),

G^((2x+1)/(2b)) is used, if n has an even denominator.

Consequently, the reference frequency IS NOT a center frequency but a band-edge.

With IgnoreEvenDenominator set true, this rule is ignored and, in turn,

the reference frequency becomes a center frequency.

Uniqueness of nominal values:

IEC 61260-2014, E.1 states that the nominal values shall be taken from the Renard numbers,

if n = 1 or 3.

IEC 61260-2014, E.2 states that the nominal values shall be the exact values

rounded to 3 significant digits, if n = 2.

IEC 61260-2014, E.3.2 and E.3.3 state that the nominal values shall be the exact values,

rounded to 3 significant digits, if the most significant one is 1 - 4

2 significant digits, if the most significant one is 5 - 9,

if 4 <= n <= 24

IEC 61260-2014, E.5 states that the nominal values shall be the exact values,

rounded to as many significant digits as needed to provide

unique mid-band values within a decade,

if n > 24

Examples: Electro-acoustic equipment often utilizes n-th octave bands,

but displays standardized nominal values with a

reference frequency of 1 kHz for user-friendlines.

(1) To obtain the audible frequencies in 3rd octaves, use

>> nthoct(3, 10, 20000, ...

>> ReferenceFrequency = 1000, Base = 10, ...

>> IgnoreEvenDenominator = false, IgnoreUniqueness = false)

Note: All arguments of nthoct are optional and these are the defaults.

To get the exact same returned values, use

>> nthoct()

(2) To obtain the frequencies of 6th octaves, use

>> nthoct(6)

Note: 6 is even. Therefore 1 kHz is not a center frequency, but a band-edge.

(3) To tune instruments to A = 440 Hz in equal 12-tone temperament, use

>> nthoct(12, 27.5, 4200, ReferenceFrequency = 440, Base = 2, IgnoreEvenDenominator = true)

Author: Axel Heusser