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{{About|the interval region|the octave as a just ratio|2/1}}
{{Wikipedia|Octave}}
A '''perfect octave''' ('''P8''') or '''octave''' ('''8ve''') is an [[interval]] that is approximately 1200 [[cent]]s in [[interval size measure|size]]. While a rough tuning range for octaves is sharper than 1170 cents according to [[Margo Schulter]]'s theory of interval regions, the term ''octave'' tends to imply a function within music that only works with intervals that corresponding to a [[just]] [[ratio]] of [[2/1]] or a close approximation thereof, usually preferred to be sharp-tempered if tempered. Other intervals are also classified as octaves, sometimes called '''wolf octaves''' or '''imperfect octaves''', if they are reasonably mapped to seven steps of the diatonic scale and twelve steps of the chromatic scale, reflecting the period and equave of both. Enharmonic intervals may be found at multiples of 12 steps along the chain of fifths, such as the diminished ninth (mapped to 8\7) and augmented seventh (6\7).


''This page is about the interval region. For the octave as a just ratio, see [[2/1]].''
The aforementioned function is the interval of equivalence, or [[equave]], because tones separated by an octave are perceived to have the same or similar [[pitch class]] to the average human listener. The reason for this phenomenon is probably due to the strong concordance of the octave or the strong amplitude of the second [[harmonic]] in most harmonic instruments.  As such, it is common practice to [[octave-reduce]] intervals so that they lie within the octave.  


'''This page is a work-in-progress.'''
Because of that, this page only covers intervals of 1200 cents and flatter, as sharper intervals octave-reduce to [[commas and dieses]].


An [[octave]] is an interval that is approximately 1200 [[Cent|cents]] in size. While a rough tuning range for octaves is sharper than 1140 cents, the term "octave" tends to imply a function within music that only works with intervals that are exactly (or almost exactly) 1200 cents.
For the sake of simplicity, this page also covers '''interseptimal seventh-octaves''', which are approximately 1150 cents in size and are the complements of [[Comma and diesis|dieses]]. Thus, the interval region considered as "octave" for the purpose of this page is 1140-1200 cents.


The aforementioned function is as the '''interval of equivalence''', because tones separated by an octave are perceived to have the same or similar pitch class to the average human listener. The reason for this phenomenon is probably due to the strong region of attraction of low [[harmonic entropy]], or the strong amplitude of the second [[harmonic]] in most harmonic instruments. As such, it is common practice to "[[Octave reduction|octave-reduce]]" intervals so that they lie within the octave.
== In just intonation ==


Because of that, this page only covers intervals of 1200 cents and flatter, as sharper intervals octave-reduce to [[Comma and diesis|commas and dieses]].
=== By prime limit ===
 
== In just intonation ==
The only "perfect" octave is the interval [[2/1]], which can be stacked to produce all other 2-limit intervals. It is 1200 cents in size, by definition. However, various "out-of-tune" octaves exist, usually flat or sharp of an octave by a small interval such as a comma.
The only "perfect" octave is the interval [[2/1]], which can be stacked to produce all other 2-limit intervals. It is 1200 cents in size, by definition. However, various "out-of-tune" octaves exist, usually flat or sharp of an octave by a small interval such as a comma.


Several notable ones are:
Several notable ones are:


* TBD
{| class="wikitable sortable center-all right-3"
|-
! class="unsortable" | Interval
! Prime <br>limit
! Distance <br>from 2/1
! Comma
|-
| [[1048576/531441]]
| 3
| 23.4600
| [[Pythagorean comma|531441/524288]]
|-
| [[160/81]]
| 5
| 21.5063
| [[81/80]]
|-
| [[125/64]]
| 5
| 41.0589
| [[128/125]]
|-
| [[125/63]]
| 7
| 13.7948
| [[126/125]]
|-
| [[63/32]]
| 7
| 27.2641
| [[64/63]]
|-
| [[49/25]]
| 7
| 34.9756
| [[50/49]]
|-
| [[96/49]]
| 7
| 35.6968
| [[49/48]]
|-
| [[35/18]]
| 7
| 48.7704
| [[36/35]]
|-
| [[64/33]]
| 11
| 53.2729
| [[33/32]]
|-
| [[33/17]]
| 17
| 51.6825
| [[34/33]]
|}


== In tempered scales ==
== In tempered scales ==
As the just octave of 2/1 is the interval being equally divided in [[EDO|EDOs,]] it is represented perfectly in all of them. The following table lists other octave-sized intervals (> 1140 cents) that exist in various significant EDOs.
As the just octave of 2/1 is the interval being equally divided in [[EDO]]s, it is represented perfectly in all of them. It is also represented perfectly in all octave-period MOSes. Note both of these statements assume the octave is untempered. The following table lists other octave-sized intervals (> 1140 cents) that exist in various significant EDOs.
 
{| class="wikitable"
{| class="wikitable"
|+
!EDO
!Suboctaves
|-
|-
|22
! EDO
|TBD
! Suboctaves
|-
|-
|24
| 22
|
| 1145{{c}}
|-
|-
|25
| 24
|
| 1150{{c}}
|-
|-
|26
| 25
|
| 1152{{c}}
|-
|-
|27
| 26
|
| 1154{{c}}
|-
|-
|29
| 27
|
| 1156{{c}}
|-
|-
|31
| 29
|
| 1159{{c}}
|-
|-
|34
| 31
|
| 1161{{c}}
|-
|-
|41
| 34
|
| 1165{{c}}
|-
|-
|53
| 41
|
| 1142{{c}}, 1171{{c}}
|-
| 53
| 1155{{c}}, 1177{{c}}
|}
|}
2/1 is also represented perfectly in most temperaments, or the most common tunings thereof, and is mainly involved in octave-reducing intervals (such as saying that, in meantone, four 3/2s (octave-reduced) stack to 5/4).
2/1 is also represented perfectly in most temperaments, or the most common tunings thereof, and is mainly involved in octave-reducing intervals (such as saying that, in meantone, four 3/2s (octave-reduced) stack to 5/4).
== See also ==
* [[Octave]] (disambiguation page)
{{Navbox intervals}}
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