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

1. so referred to in Meyers Lexikon Technik und exact Naturwissenschaften , Bibliographisches Institut AG, Mannheim 1970, p. 1852.
2. 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 ).
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}$