Octave (interval region): Difference between revisions

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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]]. Other intervals are also classified as perfect octaves, sometimes called '''wolf octaves''' or '''imperfect octaves''', if they are reasonably mapped to 7\7 and [[24edo|24\24]] (precisely seven steps of the diatonic scale and twelve steps of the chromatic scale). The use of 24edo's 24\24 as the mapping criteria here rather than [[12edo]]'s 12\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].

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 ~~(which is represented in the model of [[harmonic entropy]] by a low region)~~ 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.

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.

Because of that, this page only covers intervals of 1200 cents and flatter, as sharper intervals octave-reduce to [[commas and dieses]].

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== In just intonation ==

=== 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.

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|-

|22

|~~TBD~~

|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

|}

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).

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 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]]. Other intervals are also classified as perfect octaves, sometimes called '''wolf octaves''' or '''imperfect octaves''', if they are reasonably mapped to 7\7 and [[24edo|24\24]] (precisely seven steps of the diatonic scale and twelve steps of the chromatic scale). The use of 24edo's 24\24 as the mapping criteria here rather than [[12edo]]'s 12\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].
-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 (which is represented in the model of [[harmonic entropy]] by a low region) 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.
+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.
 Because of that, this page only covers intervals of 1200 cents and flatter, as sharper intervals octave-reduce to [[commas and dieses]].
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 == In just intonation ==
+=== 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.
@@ Line 25: / Line 27: @@
 |-
 |22
-|TBD
+|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
 |}
 /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).