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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 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'''.  


==The Schulter System==
== 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.


[[Margo_Schulter|Margo Schulter]] describes her system for categorizing intervals in [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum], which begins:
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]].


''"In naming categories of intervals, or regions of the spectrum in which they are found, there may be many valid and desirable schemes reflecting the diversity of viewpoints and styles to be found in world musics. What I describe here is merely one possible solution, and one influenced by my own musicmaking experience and philosophy which seeks an equitable and inclusive balance between intervals at or near simple integer ratios, and those having a more complex or active nature."''
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 ==
{{Main|Extended-diatonic interval names}}
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.
 
== Latitude ==
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 ==
[[Margo Schulter]] describes her system for categorizing intervals in ''[http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum]'', which begins:
 
: In naming categories of intervals, or regions of the spectrum in which they are found, there may be many valid and desirable schemes reflecting the diversity of viewpoints and styles to be found in world musics. What I describe here is merely one possible solution, and one influenced by my own musicmaking experience and philosophy which seeks an equitable and inclusive balance between intervals at or near simple integer ratios, and those having a more complex or active nature.


Schulter proposes the following categories and gives a tentative range of cents values for intervals that fall within those categories. In ''Regions'', she points out, "A main caution is that the borders are inevitably 'fuzzy,' so that one region shades into another and suggested values in cents are more illustrative than definitive."
Schulter proposes the following categories and gives a tentative range of cents values for intervals that fall within those categories. In ''Regions'', she points out, "A main caution is that the borders are inevitably 'fuzzy,' so that one region shades into another and suggested values in cents are more illustrative than definitive."
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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|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
|60-80
|-
|-
| | *small
|middle
| | 60-80
|80-100
|-
|-
| | *middle
|large
| | 80-100
|100-125
|-
|-
| | *large
| rowspan="3"|[[Neutral second|Neutral Second]]s
| | 100-125
| rowspan="3"| 125-170
|small
|125-135
|-
|-
| | [[Neutral_Second|Neutral Second]]s
|middle
| | 125-170
|135-160
|-
|-
| | *small
|large
| | 125-135
|160-170
|-
|-
| | *middle
| |[[Equable heptatonic|Equable Heptatonic]]
| | 135-160
|-
| | *large
| | 160-170
|-
| | 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|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|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|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|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
| | [[Interseptimal interval|Interseptimal]] (5-min6)
| | 709-732
|-
| | [[Interseptimal|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|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]]
| | 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
|
|
|}
|}
See: [[Interval_size_measure|interval size measure]], [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]


<pre>Pure unison (1:1)          0 cents
<pre>Pure unison (1:1)          0 cents
Line 348: Line 373:
 
 
Pure octave (2:1)        1200 cents</pre>
Pure octave (2:1)        1200 cents</pre>
== See also ==
* [[Mike Sheiman's Alternative Interval Categorizations]]
* [[SKULO interval names]]
* [[5L 2s/Interval categories]]
* [[Supermajor and subminor]]
* [[Interval size measure]]
* [[Table of MOSes]]
* [[Gallery of just intervals]]
* [[Universal solfege]] - [[solfege]] based on the Schulter system


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