Octave (auxiliary unit)

Physical unit
Unit name	octave
Unit symbol	${\ displaystyle \ mathrm {O}}$
Physical quantity (s)	Frequency interval
Formula symbol	${\ displaystyle i \ ,, n \ ,, m \ ,, \ Delta}$
dimension	${\ displaystyle {\ mathsf {{\ frac {T ^ {- 1}} {T ^ {- 1}}} = 1}}}$
In SI units	${\ displaystyle 1}$
Named after	Greek ὄκτω , óktō = "eight"
See also: Cent , Millioctave , Neper , Savart

The octave ( symbol O) is an auxiliary unit that is used in music , acoustics and high frequency engineering. It acts as a reference word that refers to the interval with the frequency ratio 2: 1 in the dual or binary logarithmic representation of frequency intervals .

The name comes from the music where this frequency interval is called an octave ( lat. Octo = eight) because it describes the distance between the eighth tone of a diatonic scale and its root .

The frequency ratio belongs to a frequency interval of octaves ${\ displaystyle i}$

{\ displaystyle {\ frac {f_ {2}} {f_ {1}}} = 2 ^ {i}}

(with )

{\ displaystyle f_ {2} \ geq f_ {1}}

{\ displaystyle \ Leftrightarrow i = \ log _ {2} {\ frac {f_ {2}} {f_ {1}}} = \ operatorname {lb} {\ frac {f_ {2}} {f_ {1}} } = \ operatorname {lb} {\ frac {f_ {2}} {f_ {1}}} \, {\ text {O}} = \ operatorname {lb} {\ frac {f_ {2}} {f_ { 1}}} \, {\ text {octave (s)}}}

After converting the logarithm from two to ten, this becomes:

{\ displaystyle {\ begin {aligned} \ Leftrightarrow i & = {\ frac {1} {\ mathrm {lg} \, 2}} \ cdot \ mathrm {lg} \, {\ frac {f_ {2}} {f_ {1}}} \, \, {\ text {O}} \\ & \ approx 3 {,} 3219 \ cdot \ mathrm {lg} \, {\ frac {f_ {2}} {f_ {1}} } \, \, {\ text {O}} \ end {aligned}}}

A sub-unit of the octave is the cent :

{\ displaystyle 1 \, {\ text {Cent}} = {\ frac {1} {1200}} \, {\ text {octave}}}

use

The unit of measurement octave is used in high-frequency engineering when a bandwidth depends on the frequency and the precise cut-off frequencies are either variable or irrelevant. For example, a specific assembly with a bandwidth of 1 octave can be used in the frequency ranges

1 to 2 GHz or
6.2 to 12.4 GHz.

The specification of frequency ranges in octaves is particularly widespread in antenna technology, especially for horn radiators .

Remarks

↑ so referred to in Meyers Lexikon Technik und exact Naturwissenschaften , Bibliographisches Institut AG, Mannheim 1970, p. 1852.
↑ binary logarithm (lb for short), base 2 logarithm, also known as logarithm of two or dyadic logarithm (sometimes also with the abbreviation ld for logarithm dualis ).
↑ The addition (or omission) of the factors or is possible because purely mathematically they are nothing more than renaming the number 1 based on the formal definition: ${\ displaystyle {\ text {O}}}$ ${\ displaystyle {\ text {octave}}}$ ${\ displaystyle {\ text {octave}} = \ operatorname {lb} {\ frac {2} {1}} = \ operatorname {lb} 2 = 1}$

[1] so referred to in Meyers Lexikon Technik und exact Naturwissenschaften , Bibliographisches Institut AG, Mannheim 1970, p. 1852.

[2] ry logarithm (lb for short), base 2 logarithm, also known as logarithm of two or dyadic logarithm (sometimes also with the abbreviation ld for logarithm dualis ).

[3] The addition (or omission) of the factors or is possible because purely mathematically they are nothing more than renaming the number 1 based on the formal definition: ${\ displaystyle {\ text {O}}}$ ${\ displaystyle {\ text {octave}}}$ ${\ displaystyle {\ text {octave}} = \ operatorname {lb} {\ frac {2} {1}} = \ operatorname {lb} 2 = 1}$