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Octave (interval region): Difference between revisions

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{{interwiki
{{About|the interval region|the octave as a just ratio|2/1}}
| de = Oktave
| en = Octave
| es = Octava
| ja = オクターブ
}}
{{Infobox Interval
| Ratio = 2/1
| Name = octave, ditave, diapason
| Color name = w8, wa 8ve
| Sound = jid_2_1_pluck_adu_dr220.mp3
}}
{{Wikipedia|Octave}}
{{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).


The '''octave''' (interval ratio '''2/1''') is one of the most basic [[Gallery of just intervals|intervals]] found in musical systems throughout the entire world. It has a frequency ratio of 2/1 and a size of 1200 [[cent]]s. It is used as the standard of (logarithmic) measurement for all intervals, regardless if they are justly tuned or not.
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.  


Strangely enough, the Pelog and Slendro scales of the Javanese contain near-octaves even though Gamelan instruments exhibit inharmonic spectra. It is most likely reminiscent of an older musical system, or derived using the human voice instead of inharmonic instruments.
Because of that, this page only covers intervals of 1200 cents and flatter, as sharper intervals octave-reduce to [[commas and dieses]].


== Octave equivalence ==
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 octave is usually called 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.
== In just intonation ==


An article in ''Current Biology'' shows that octave equivalence might be a cultural phenomenon, it also includes an 8-minute video. <ref>[https://www.cell.com/current-biology/fulltext/S0960-9822(19)31036-X?_returnURL=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS096098221931036X%3Fshowall%3Dtrue Universal and Non&#45;universal Features of Musical Pitch Perception Revealed by Singing&#58; Current Biology]</ref>
=== By prime limit ===
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.


== Alternate names ==
Several notable ones are:


'''Ditave''' is an alternative name for the interval 2/1, which was proposed to neutralize the terminology against the predominance of 7-tone scales. The name is derived from the numeral prefix ''δι''- (''di-'', Greek for "two") in analogy to "[[tritave]]" (3/1). A brief but complementary description about it is [[:purdal:Ditave|here]].
{| 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]]
|}


'''Diapason''' is another term also sometimes applied to 2/1. It is also of Greek origin, but not related to the number two; instead it is formed from ''διά'' (''dia'') + ''πασων'' (''pason''), meaning something like "through all the notes".
== In tempered scales ==
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.


== See also ==
{| class="wikitable"
|-
! EDO
! Suboctaves
|-
| 22
| 1145{{c}}
|-
| 24
| 1150{{c}}
|-
| 25
| 1152{{c}}
|-
| 26
| 1154{{c}}
|-
| 27
| 1156{{c}}
|-
| 29
| 1159{{c}}
|-
| 31
| 1161{{c}}
|-
| 34
| 1165{{c}}
|-
| 41
| 1142{{c}}, 1171{{c}}
|-
| 53
| 1155{{c}}, 1177{{c}}
|}


* [[Prime interval]]
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).
* [[Gallery of Just Intervals]]
* [[Toctave]]
* [[EDO]]
* [[Octave reduction]]
* [[Octave complement]]


== References ==  
== See also ==
<references/>
* [[Octave]] (disambiguation page)


[[Category:Octave| ]] <!-- main article -->
{{Navbox intervals}}
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