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It is well-known that there are infinite possible intervals, even if one confines one's view to a single octave. This includes the rational intervals of [[just intonation]] and the irrational intervals of [[EDO]]s, [[edonoi]]s, and other systems. However, it is helpful to consider '''interval regions''' (or '''interval categories'''), which there may be a finite number to consider.
There are infinite possible intervals (both tempered and just), even within a single [[2/1|octave]]. It can be helpful to group these intervals into a finite number of '''interval regions''' or '''interval categories'''.
 
== Concrete regions vs abstract categories ==
An ''interval region'' usually implies it is concrete, defined by concrete boundaries of interval sizes. The boundaries are usually fuzzy to allow some vagueness, in line with how we perceive them. Which region an interval falls into solely depends on the interval's size.
 
An ''interval category'' is usually meant to be abstract. It uses some [[mapping]] to determine which category an interval falls into, short-circuiting the question of where exactly to place the boundaries. It also takes account of an interval's [[prime factorization|prime components]], allowing us to find a composite interval's category through [[interval arithmetic]].
 
The [[5L 2s|diatonic]] interval category system commonly used to categorize JI intervals consists of a [[interval quality|quality]] and a diatonic scale degree.  


== Extended-diatonic interval names ==
== Extended-diatonic interval names ==
Line 5: Line 12:
Many interval naming systems extend the diatonic interval names by adding new [[interval qualities]] to the usual set. While some systems preserve the fifth-based structure entirely, other systems define regions based on the proximity to the intervals associated with the diatonic intervals, which are then divided into finer subregions.
Many interval naming systems extend the diatonic interval names by adding new [[interval qualities]] to the usual set. While some systems preserve the fifth-based structure entirely, other systems define regions based on the proximity to the intervals associated with the diatonic intervals, which are then divided into finer subregions.


== Mediants ==
== Latitude ==
When describing interval regions in terms of size relative to a (possibly tempered) fifth, it leads to the system of [[Mediant (interval measure)|mediants]] devised by Vector.
When describing interval regions in terms of size relative to a (possibly tempered) fifth, it leads to the system of [[Latitude|latitude and medial intervals]].


== Schulter system ==
== Schulter system ==
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|-
|-
! | Interval Category
! | Interval Category
! | Associated Ranges of Cents Values (borders are "fuzzy")
! | Approx. Cents Ranges
! colspan="2" |sub-category
|-
|-
| | [[Unison|Pure Unison]] (1:1)
| | [[Unison|Pure Unison]] (1:1)
| | 0
| | 0
|
|
|-
|-
| | [[Comma]]s
| | [[Comma]]s
| | 0-30
| | 0-30
|
|
|-
|-
| | [[Diesis|Dieses]]
| | [[Diesis|Dieses]]
| | 30-60
| | 30-60
|
|
|-
|-
| | [[Minor second|Minor Second]]s
| rowspan="3"| [[Minor second|Minor Second]]s
| | 60-125
| rowspan="3"| 60-125
|-
|small
| | *small
|60-80
| | 60-80
|-
|-
| | *middle
|middle
| | 80-100
|80-100
|-
|-
| | *large
|large
| | 100-125
|100-125
|-
|-
| | [[Neutral second|Neutral Second]]s
| rowspan="3"|[[Neutral second|Neutral Second]]s
| | 125-170
| rowspan="3"| 125-170
|small
|125-135
|-
|-
| | *small
|middle
| | 125-135
|135-160
|-
|-
| | *middle
|large
| | 135-160
|160-170
|-
|-
| | *large
| |[[Equable heptatonic|Equable Heptatonic]]
| | 160-170
|-
| | [[Equable heptatonic|Equable Heptatonic]]
| | 160-182
| | 160-182
|
|
|-
|-
| | [[Major second|Major Second]]s
| rowspan="3"|[[Major second|Major Second]]s
| | 180-240
| rowspan="3"| 180-240
|small
|180-200
|-
|-
| | *small
|middle
| | 180-200
|200-220
|-
|-
| | *middle
|large
| | 200-220
|220-240
|-
|-
| | *large
| |[[Interseptimal interval|Interseptimal]] (Maj2-min3)
| | 220-240
|-
| | [[Interseptimal interval|Interseptimal]] (Maj2-min3)
| | 240-260
| | 240-260
|
|
|-
|-
| | [[Minor third|Minor Third]]s
| rowspan="3"|[[Minor third|Minor Third]]s
| | 260-330
| rowspan="3"| 260-330
|small
|260-280
|-
|-
| | *small
|middle
| | 260-280
|280-300
|-
|-
| | *middle
|large
| | 280-300
|300-330
|-
|-
| | *large
| rowspan="3"| [[Neutral third|Neutral Third]]s
| | 300-330
| rowspan="3"| 330-372
|small
|330-342
|-
|-
| | [[Neutral third|Neutral Third]]s
|middle
| | 330-372
|342-360
|-
|-
| | *small
|large
| | 330-342
|360-372
|-
|-
| | *middle
| rowspan="3"|[[Major third|Major Third]]s
| | 342-360
|  rowspan="3"| 372-440
|small
|372-400
|-
|-
| | *large
|middle
| | 360-372
|400-423
|-
|-
| | [[Major third|Major Third]]s
|large
| | 372-440
|423-440
|-
|-
| | *small
| |[[Interseptimal interval|Interseptimal]] (Maj3-4)
| | 372-400
|-
| | *middle
| | 400-423
|-
| | *large
| | 423-440
|-
| | [[Interseptimal interval|Interseptimal]] (Maj3-4)
| | 440-468
| | 440-468
|
|
|-
|-
| | [[Perfect fourth|Perfect Fourth]]s
| rowspan="3"|[[Perfect fourth|Perfect Fourth]]s
| | 468-528
| rowspan="3"| 468-528
|small
|468-491
|-
|-
| | *small
|middle
| | 468-491
|491-505
|-
|-
| | *middle
|large
| | 491-505
|505-528
|-
|-
| | *large
| |[[Superfourth]]s
| | 505-528
|-
| | [[Superfourth]]s
| | 528-560
| | 528-560
|
|
|-
|-
| | [[Tritone|Tritonic Region]]
| rowspan="3"|[[Tritone|Tritonic Region]]
| | 560-640
| rowspan="3"| 560-640
|small
|560-577
|-
|-
| | *small
|middle
| | 560-577
|577-623
|-
|-
| | *middle
|large
| | 577-623
|623-640
|-
|-
| | *large
| |[[Subfifth]]s
| | 623-640
|-
| | [[Subfifth]]s
| | 640-672
| | 640-672
|
|
|-
|-
| | [[Perfect fifth|Perfect Fifth]]s
| rowspan="3"|[[Perfect fifth|Perfect Fifth]]s
| | 672-732
| rowspan="3"| 672-732
|small
|672-695
|-
|-
| | *small
|middle
| | 672-695
|695-709
|-
|-
| | *middle
|large
| | 695-709
|709-732
|-
| | *large
| | 709-732
|-
|-
| | [[Interseptimal interval|Interseptimal]] (5-min6)
| | [[Interseptimal interval|Interseptimal]] (5-min6)
| | 732-760
| | 732-760
|
|
|-
|-
| | [[Minor sixth|Minor Sixth]]s
| rowspan="3"|[[Minor sixth|Minor Sixth]]s
| | 760-828
| rowspan="3"| 760-828
|-
|small
| | *small
|760-777
| | 760-777
|-
| | *middle
| | 777-800
|-
| | *large
| | 800-828
|-
|-
| | [[Neutral sixth|Neutral Sixth]]s
|middle
| | 828-870
|777-800
|-
|-
| | *small
|large
| | 828-840
|800-828
|-
|-
| | *middle
| rowspan="3"|[[Neutral sixth|Neutral Sixth]]s
| | 840-858
|  rowspan="3"| 828-870
|small
|828-840
|-
|-
| | *large
|middle
| | 858-870
|840-858
|-
|-
| | [[Major sixth|Major Sixth]]s
|large
| | 870-940
|858-870
|-
|-
| | *small
| rowspan="3"|[[Major sixth|Major Sixth]]s
| | 870-900
|  rowspan="3"| 870-940
|small
|870-900
|-
|-
| | *middle
|middle
| | 900-920
|900-920
|-
|-
| | *large
|large
| | 920-940
|920-940
|-
|-
| | [[Interseptimal interval|Interseptimal]] (Maj6-min7)
| | [[Interseptimal interval|Interseptimal]] (Maj6-min7)
| | 940-960
| | 940-960
|
|
|-
|-
| | [[Minor seventh|Minor Seventh]]s
| rowspan="3"|[[Minor seventh|Minor Seventh]]s
| | 960-1025
| rowspan="3"| 960-1025
|-
|small
| | *small
|960-987
| | 960-987
|-
|-
| | *middle
|middle
| | 987-1000
|987-1000
|-
|-
| | *large
|large
| | 1000-1025
|1000-1025
|-
|-
| | [[Equable heptatonic|Equable Heptatonic]]
| | [[Equable heptatonic|Equable Heptatonic]]
| | 1018-1040
| | 1018-1040
|
|
|-
|-
| | [[Neutral seventh|Neutral Seventh]]s
| rowspan="3"|[[Neutral seventh|Neutral Seventh]]s
| | 1030-1075
| rowspan="3"| 1030-1075
|small
|1030-1043
|-
|-
| | *small
|middle
| | 1030-1043
|1043-1065
|-
|-
| | *middle
|large
| | 1043-1065
|1065-1075
|-
|-
| | *large
| rowspan="3"| [[Major seventh|Major Seventh]]s
| | 1065-1075
| rowspan="3"| 1075-1140
|small
|1075-1100
|-
|-
| | [[Major seventh|Major Seventh]]s
|middle
| | 1075-1140
|1100-1120
|-
|-
| | *small
|large
| | 1075-1100
|1120-1140
|-
| | *middle
| | 1100-1120
|-
| | *large
| | 1120-1140
|-
|-
| | Octave less diesis
| | Octave less diesis
| | 1140-1170
| | 1140-1170
|
|
|-
|-
| | Octave less comma
| | Octave less comma
| | 1170-1200
| | 1170-1200
|
|
|-
|-
| | [[Octave|Pure Octave]] (2:1)
| | [[Octave|Pure Octave]] (2:1)
| | 1200
| | 1200
|
|
|}
|}
<pre>Pure unison (1:1)          0 cents
 
Commas                  0-30 cents        (Section 11)
 
Dieses                  30-60 cents        (Section 11)
 
Minor seconds          60-125 cents        (Section 5)
        small              60-80 cents
        middle            80-100 cents
        large            100-125 cents
 
Neutral seconds      125-170 cents        (Section 6)
        small              125-135 cents
        middle            135-160 cents
        large              160-170 cents
 
Equable heptatonic
(heartland range)    160-182 cents        (Section 12)
 
Major seconds
(or tones)            180-240 cents        (Section 4)
        small              180-200 cents
        middle            200-220 cents
        large              220-240 cents
 
Interseptimal
(Maj2-min3)          240-260 cents        (Section 9)
 
Minor thirds          260-330 cents        (Section 2)
        small              260-280 cents
        middle            280-300 cents
        large              300-330 cents
 
Neutral thirds        330-372 cents        (Section 3)
        small              330-342 cents
        middle            342-360 cents
        large              360-372 cents
 
Major thirds          372-440 cents        (Section 2)
        small              372-400 cents
        middle            400-423 cents
        large              423-440 cents
 
Interseptimal        440-468 cents        (Section 9)
(Maj3-4)
 
Perfect fourths      468-528 cents        (Section 7)
        small              468-491 cents
        middle            491-505 cents
        large              505-523 cents
 
Superfourths          528-560 cents        (Section 10)
 
Tritonic region      560-640 cents        (Section 8)
        small              560-577 cents
        middle            577-623 cents
        large              623-640 cents
 
Subfifths            640-672 cents        (Section 10)
 
Perfect fifths        672-732 cents        (Section 7)
        small              672-695 cents
        middle            695-709 cents
        large              709-732 cents
 
Interseptimal        732-760 cents        (Section 9)
(5-min6)
 
Minor sixths          760-828 cents        (Section 2)
        small              760-777 cents
        middle            777-800 cents
        large              800-828 cents
 
Neutral sixths        828-870 cents        (Section 3)
        small              828-840 cents
        middle            840-858 cents
        large              858-870 cents
 
Major sixths          870-940 cents        (Section 2)
        small              870-900 cents
        middle            900-920 cents
        large              920-940 cents
 
Interseptimal        940-960 cents        (Section 9)
(Maj6-min7)
 
Minor sevenths      960-1025 cents        (Section 4)
        small              960-987 cents
        middle            987-1000 cents
        large            1000-1025 cents
 
Equable heptatonic  1018-1040 cents        (Section 12)
(heartland range)
 
Neutral sevenths    1030-1075 cents        (Section 6)
        small            1030-1043 cents
        middle          1043-1065 cents
        large            1065-1075 cents
 
Major sevenths      1075-1140 cents        (Section 5)
        small            1075-1100 cents
        middle          1100-1120 cents
        large            1120-1140 cents
 
Octave less diesis  1140-1170 cents        (Section 11)
 
Octave less comma  1170-1200 cents        (Section 11)
 
Pure octave (2:1)        1200 cents</pre>


== See also ==
== See also ==
; Interval region naming schemes
* [[Mike Sheiman's Alternative Interval Categorizations]]
* [[Mike Sheiman's Alternative Interval Categorizations]]
* [[SKULO interval names]]
* [[SKULO interval names]]
* [[User:VectorGraphics/Walker brightness notation|Walker brightness notation]]
* [[5L 2s/Interval categories]]
; Other related concepts
* [[Supermajor and subminor]]
* [[Interval size measure]]
* [[Interval size measure]]
* [[Table of MOSes]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
* [[Universal solfege]] - [[solfege]] based on the Schulter system
* [[Universal solfege]] - [[solfege]] based on the Schulter system


[[Category:Interval region| ]] <!-- main article -->
[[Category:Interval regions| ]]
<!-- main article -->
[[Category:Classification]]
[[Category:Classification]]
[[Category:Distance measure]]
[[Category:Distance measure]]
Retrieved from "https://en.xen.wiki/w/Interval_region"