Interval category

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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 EDOs, EDONOIs, and other systems. However, it is helpful to consider interval categories, which there may be a finite number to consider.

The 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 Associated Ranges of Cents Values (borders are "fuzzy")
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

See: interval size measure, Gallery of Just Intervals

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