Interval region
There are infinite possible intervals (both tempered and just), even within a single 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 components, allowing us to find a composite interval's category through interval arithmetic.
The diatonic interval category system commonly used to categorize JI intervals consists of a quality and a diatonic scale degree.
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 and medial intervals.
Schulter system
Margo Schulter describes her system for categorizing intervals in 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."
| Interval Category | Approx. Cents Ranges | sub-category | |
|---|---|---|---|
| Pure Unison (1:1) | 0 | ||
| Commas | 0-30 | ||
| Dieses | 30-60 | ||
| Minor Seconds | 60-125 | small | 60-80 |
| middle | 80-100 | ||
| large | 100-125 | ||
| Neutral Seconds | 125-170 | small | 125-135 |
| middle | 135-160 | ||
| large | 160-170 | ||
| Equable Heptatonic | 160-182 | ||
| Major Seconds | 180-240 | small | 180-200 |
| middle | 200-220 | ||
| large | 220-240 | ||
| Interseptimal (Maj2-min3) | 240-260 | ||
| Minor Thirds | 260-330 | small | 260-280 |
| middle | 280-300 | ||
| large | 300-330 | ||
| Neutral Thirds | 330-372 | small | 330-342 |
| middle | 342-360 | ||
| large | 360-372 | ||
| Major Thirds | 372-440 | small | 372-400 |
| middle | 400-423 | ||
| large | 423-440 | ||
| Interseptimal (Maj3-4) | 440-468 | ||
| Perfect Fourths | 468-528 | small | 468-491 |
| middle | 491-505 | ||
| large | 505-528 | ||
| Superfourths | 528-560 | ||
| Tritonic Region | 560-640 | small | 560-577 |
| middle | 577-623 | ||
| large | 623-640 | ||
| Subfifths | 640-672 | ||
| Perfect Fifths | 672-732 | small | 672-695 |
| middle | 695-709 | ||
| large | 709-732 | ||
| Interseptimal (5-min6) | 732-760 | ||
| Minor Sixths | 760-828 | small | 760-777 |
| middle | 777-800 | ||
| large | 800-828 | ||
| Neutral Sixths | 828-870 | small | 828-840 |
| middle | 840-858 | ||
| large | 858-870 | ||
| Major Sixths | 870-940 | small | 870-900 |
| middle | 900-920 | ||
| large | 920-940 | ||
| Interseptimal (Maj6-min7) | 940-960 | ||
| Minor Sevenths | 960-1025 | small | 960-987 |
| middle | 987-1000 | ||
| large | 1000-1025 | ||
| Equable Heptatonic | 1018-1040 | ||
| Neutral Sevenths | 1030-1075 | small | 1030-1043 |
| middle | 1043-1065 | ||
| large | 1065-1075 | ||
| Major Sevenths | 1075-1140 | small | 1075-1100 |
| middle | 1100-1120 | ||
| large | 1120-1140 | ||
| Octave less diesis | 1140-1170 | ||
| Octave less comma | 1170-1200 | ||
| Pure Octave (2:1) | 1200 | ||
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