Octave complement - Revision history

Fredg999: Improve lead section, second Wikipedia box, remove duplicate links from see also section, misc. edits

2023-08-22T19:14:31Z

Improve lead section, second Wikipedia box, remove duplicate links from see also section, misc. edits

← Older revision		Revision as of 19:14, 22 August 2023
Line 1:		Line 1:
			{{Wikipedia\|Complement (music)}}
	{{Wikipedia\|Inversion (music) #Intervals}}		{{Wikipedia\|Inversion (music) #Intervals}}
	The '''octave complement''' or '''inverse interval''' of an [[interval]] is its ~~interval distance from~~ the [[octave]]. It can be seen as a binary symmetric relation over intervals. The concept is important in musical practice and most musical theories. Its use is typically restricted to [[octave-reduced]] intervals (including the octave).		The '''octave complement''' or '''inverse interval''' of an [[interval]] is its difference with the [[octave]]. It can be seen as a binary symmetric relation over intervals. The concept is important in musical practice and most musical theories. Its use is typically restricted to [[octave-reduced]] intervals (including the octave).

			The octave complement is often simply called '''complement''' in the context of octave-[[equivalent]] scales, but the broader concept of ''complement'' can also apply to other intervals, such as the [[fifth complement]], the [[fourth complement]] and the [[tritave complement]].

	== Calculation ==		== Calculation ==
Line 84:		Line 87:

	== See also ==		== See also ==
			* [[Ratio math]]
	* [[~~Octave reduction]]~~
	* [[Fifth complement]] -- the analogue for the [[3/2\|fifth (3/2)]]
	* [[Fourth complement]] -- the analogue for the [[4/3\|fourth (4/3)]]
	* [[Tritave complement]] -- the analogue for the [[tritave\|tritave (3/1)]]

	[[Category:Terms]]		[[Category:Terms]]

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T04:41:47Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 04:41, 5 March 2023
Line 91:		Line 91:

	[[Category:Terms]]		[[Category:Terms]]
	~~[[Category:Theory]]~~
	[[Category:Interval]]		[[Category:Interval]]
	[[Category:Octave]]		[[Category:Octave]]
	[[Category:Method]]		[[Category:Method]]

Fredg999: Added Wikipedia box, misc. edits

2021-09-02T01:13:42Z

Added Wikipedia box, misc. edits

← Older revision		Revision as of 01:13, 2 September 2021
Line 1:		Line 1:
	The '''octave complement''' or '''inverse interval''' of an [[interval]] is its interval distance from the [[octave]]. ~~The '''octave complement'''~~ can be seen as a binary symmetric relation over intervals. The concept important in musical practice and most musical theories. Its use is typically restricted to [[octave-reduced]] intervals (including the octave).		{{Wikipedia\|Inversion (music) #Intervals}}
			The '''octave complement''' or '''inverse interval''' of an [[interval]] is its interval distance from the [[octave]]. It can be seen as a binary symmetric relation over intervals. The concept is important in musical practice and most musical theories. Its use is typically restricted to [[octave-reduced]] intervals (including the octave).

	== Calculation ==		== Calculation ==
	Depending on the interval representation (name, ratio, monzo, edo steps, ~~cents~~), it's more or less easy to retrieve the complementary interval from a given interval.		Depending on the interval representation (name, [[ratio]], [[monzo]], [[edo]] steps, [[cent]]s), it's more or less easy to retrieve the complementary interval from a given interval.

	=== Classical interval names ===		=== Classical interval names ===
Line 61:		Line 62:

	=== Monzo ===		=== Monzo ===
	Intervals represented as ~~[[Monzos]]~~ can be transformed into their octave complement by inverting all arguments and increasing the 2-argument.		Intervals represented as monzos can be transformed into their octave complement by inverting all arguments and increasing the 2-argument.

	; Examples		; Examples
Line 69:		Line 70:

	=== Edo steps ===		=== Edo steps ===
	Octave-complement intervals represented as ''s\n'' meaning ''s'' steps of ''n''-EDO follow this relation <code>s1 + s2 = n</code>. For given s and n, the unknown x can be calculated by the formula <code>x := n-s</code>.		Octave-complement intervals represented as ''s''\''n'' meaning ''s'' steps of ''n''-EDO follow this relation <code>s1 + s2 = n</code>. For given ''s'' and ''n'', the unknown ''x'' can be calculated by the formula <code>x := n -s</code>.

	; Examples		; Examples
Line 76:		Line 77:

	=== Cents ===		=== Cents ===
	Octave-complement intervals represented as ''s~~¢~~'' follow this relation <code>s1 + s2 = 1200</code>. For given s, the unknown x can be calculated by the formula <code>x := 1200-s</code>.		Octave-complement intervals represented as ''s''¢ follow this relation <code>s1 + s2 = 1200</code>. For given s, the unknown x can be calculated by the formula <code>x := 1200 - s</code>.

	; Examples		; Examples
Line 88:		Line 89:
	* [[Fourth complement]] -- the analogue for the [[4/3\|fourth (4/3)]]		* [[Fourth complement]] -- the analogue for the [[4/3\|fourth (4/3)]]
	* [[Tritave complement]] -- the analogue for the [[tritave\|tritave (3/1)]]		* [[Tritave complement]] -- the analogue for the [[tritave\|tritave (3/1)]]
	* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]

	[[Category:Terms]]		[[Category:Terms]]

Aura at 17:48, 23 June 2021

2021-06-23T17:48:47Z

← Older revision		Revision as of 17:48, 23 June 2021
Line 86:		Line 86:
	* [[Octave reduction]]		* [[Octave reduction]]
	* [[Fifth complement]] -- the analogue for the [[3/2\|fifth (3/2)]]		* [[Fifth complement]] -- the analogue for the [[3/2\|fifth (3/2)]]
			* [[Fourth complement]] -- the analogue for the [[4/3\|fourth (4/3)]]
	* [[Tritave complement]] -- the analogue for the [[tritave\|tritave (3/1)]]		* [[Tritave complement]] -- the analogue for the [[tritave\|tritave (3/1)]]
	* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]		* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]

Keenan Pepper category edits: Moving from Category:Term to Category:Terms using Cat-a-lot

2020-12-04T22:44:58Z

Moving from Category:Term to Category:Terms using Cat-a-lot

← Older revision		Revision as of 22:44, 4 December 2020
Line 89:		Line 89:
	* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]		* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]

	[[Category:~~Term~~]]		[[Category:Terms]]
	[[Category:Theory]]		[[Category:Theory]]
	[[Category:Interval]]		[[Category:Interval]]
	[[Category:Octave]]		[[Category:Octave]]
	[[Category:Method]]		[[Category:Method]]

Xenwolf: /* See also */ reordering

2020-09-07T18:24:55Z

Xenwolf: /* See also */ +1

2020-09-07T18:22:07Z

Xenwolf: /* See also */ added wanted article

2020-06-13T11:59:48Z

Xenwolf: Created page with "The '''octave complement''' or '''inverse interval''' of an interval is its interval distance from the octave. The '''octave complement''' can be seen as a binary symm..."

2018-10-27T14:09:06Z

Created page with "The '''octave complement''' or '''inverse interval''' of an interval is its interval distance from the octave. The '''octave complement''' can be seen as a binary symm..."

New page

The '''octave complement''' or '''inverse interval''' of an [[interval]] is its interval distance from the [[octave]]. The '''octave complement''' can be seen as a binary symmetric relation over intervals. The concept important in musical practice and most musical theories. Its use is typically restricted to [[octave-reduced]] intervals (including the octave).

== Calculation ==
Depending on the interval representation (name, ratio, monzo, edo steps, cents), it's more or less easy to retrieve the complementary interval from a given interval.

=== Classical interval names ===
The intervals in western music have names derived from numerals, starting at 1 (''unison'' or ''prime'', ''second'', etc.). These names are prefixed with further size attributes (''just'', ''minor'', ''major'', etc.) which express relative size relations (the attribute ''just'' is often omitted for unison and octave). Complementary intervals are calculated by complementing both, name and attribute parts. For the name part the complement is calculated by subtracting the ordinal number from 9, the attribute part is negated, ''just'' is the negation of ''just'' . The following tables show names with ordinal numbers (#) and attributes with size hints (~).

{| 
| style="padding: 0 3em;" | 
{| class="wikitable"
|+ Interval name part
! colspan="2" | base value !! colspan="2" | complement
|-
! Name !! # !! # !! Name
|-
| unison || 1 || 8 || octave
|-
| second || 2 || 7 || seventh
|-
| third || 3 || 6 || sixth
|-
| fourth || 4 || 5 || fifth
|-
| fifth || 5 || 4 || fourth
|-
| sixth || 6 || 3 || third
|-
| seventh || 7 || 2 || second
|-
| octave || 8 || 1 || unison
|}
| style="vertical-align: top; padding: 0 3em;" | 
{| class="wikitable"
|+ Attribute parts
! base value !! ~ !! ~ !! complement
|-
| diminished || -2 || +2 || augmented
|-
| minor || -1 || +1 || major
|-
| just || 0 || 0 || just
|-
| major || +1 || -1 || minor
|-
| augmented || +2 || -2|| diminished
|}
|} 

; Examples
: minor third vs. major sixth
: (just) unison vs. (just) octave
: just fifth vs. just fourth

=== Ratio ===
Octave-complement intervals represented as ratios ''r'' follow the relation <code>r1*r2 = 2</code>. For given r the unknown x can be calculated by the formula <code>x := 2/r</code> or (for the ratio representation <code>r = a/b</code>) into <code>x := 2*b/a</code> (the result sometimes has to be reduced by the factor 2).

; Examples
: 5/4 vs. 2*4/5 = 8/5
: 4/3 vs. 2*3/4 = 6/4 = [[3/2]]

=== Monzo ===
Intervals represented as [[Monzos]] can be transformed into their octave complement by inverting all arguments and increasing the 2-argument.

; Examples
: {{Monzo|-1 1}} vs. {{Monzo|-(-1)+1 -(1)}} = {{Monzo|+2 -1}}
: {{Monzo|3 -3 1}} vs. {{Monzo|-(3)+1 -(-3) -(1)}} = {{Monzo|-2 3 -1}}
: {{Monzo|-2 2 1 0 -1}} vs. {{Monzo|-(-2)+1 -(2) -(1) -(0) -(-1)}} = {{Monzo|+3 -2 -1 0 +1}}

=== Edo steps ===
Octave-complement intervals represented as ''s\n'' meaning ''s'' steps of ''n''-EDO follow this relation <code>s1 + s2 = n</code>. For given s and n, the unknown x can be calculated by the formula <code>x := n-s</code>.

; Examples
: 7\12 vs. (12-7)\12 = 5\12
: 1\7 vs. (7-1)\7 = 6\7

=== Cents ===
Octave-complement intervals represented as ''s¢'' follow this relation <code>s1 + s2 = 1200</code>. For given s, the unknown x can be calculated by the formula <code>x := 1200-s</code>.

; Examples
: 333¢ vs. (1200-333)¢ = 867¢
: 701.955¢ vs. (1200-701.955)¢ = 498.045¢

== See also ==
* [[octave reduction]]
* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]

[[Category:Term]]
[[Category:Theory]]
[[Category:Interval]]
[[Category:Octave]]
[[Category:Method]]

← Older revision		Revision as of 18:24, 7 September 2020
Line 84:		Line 84:
	== See also ==		== See also ==

	* [[Fifth complement]]
	* [[Octave reduction]]		* [[Octave reduction]]
			* [[Fifth complement]] -- the analogue for the [[3/2\|fifth (3/2)]]
			* [[Tritave complement]] -- the analogue for the [[tritave\|tritave (3/1)]]
	* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]		* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]
	* [[Tritave complement]] -- the analogue for [[tritave\|tritave (3/1)]]

	[[Category:Term]]		[[Category:Term]]

← Older revision		Revision as of 18:22, 7 September 2020
Line 83:		Line 83:

	== See also ==		== See also ==
	* [[~~octave~~ reduction]]
			* [[Fifth complement]]
			* [[Octave reduction]]
	* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]		* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]
	* [[Tritave complement]] -- the analogue for [[tritave]]		* [[Tritave complement]] -- the analogue for [[tritave\|tritave (3/1)]]

	[[Category:Term]]		[[Category:Term]]

← Older revision		Revision as of 11:59, 13 June 2020
Line 85:		Line 85:
	* [[octave reduction]]		* [[octave reduction]]
	* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]		* [https://en.wikipedia.org/wiki/Inversion_(music)#Intervals Inversion (music) - Wikipedia #Intervals]
			* [[Tritave complement]] -- the analogue for [[tritave]]

	[[Category:Term]]		[[Category:Term]]