# interval-class / ic

[Joe Monzo]

"Interval-class" or "ic" refers to the size of the pitch space between two different pitches, disregarding the effect of transposing either or both pitch any number of octaves (or other equivalence-interval).

An ic is ordinarily defined for 12-edo tuning, and is expressed as an ordinal integer describing the logarithmic linear frequency relationship as measured in degrees of 12-edo (for example, as used in an interval-vector). The pc integer of the lower note is subtracted from the pc integer of the higher note modulo 12, to obtain the ic integer. If the result is larger than 6 (i.e., 12/2), then that number is subtracted from 12, to obtain the ic integer. The concept of interval-class thus places an equivalence between a given interval and its octave-complement (or inversion). Below is an interval-class matrix showing all of the ic's for 12-edo:

### 12-edo ic matrix

```     higher:  C  |C#/Db|  D  |D#/Eb|  E  |  F  |F#/Gb|  G  |G#/Ab|  A  |A#/Bb|  B  |
0  |  1  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9  | 10  | 11  |
lower:   ===========================================================================
B     11 ||   1  |  2  |  3  |  4  |  5  |  6  |  5  |  4  |  3  |  2  |  1  |  0  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
A#/Bb 10 ||   2  |  3  |  4  |  5  |  6  |  5  |  4  |  3  |  2  |  1  |  0  |  1  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
A      9 ||   3  |  4  |  5  |  6  |  5  |  4  |  3  |  2  |  1  |  0  |  1  |  2  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
G#/Ab  8 ||   4  |  5  |  6  |  5  |  4  |  3  |  2  |  1  |  0  |  1  |  2  |  3  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
G      7 ||   5  |  6  |  5  |  4  |  3  |  2  |  1  |  0  |  1  |  2  |  3  |  4  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
F#/Gb  6 ||   6  |  5  |  4  |  3  |  2  |  1  |  0  |  1  |  2  |  3  |  4  |  5  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
F      5 ||   5  |  4  |  3  |  2  |  1  |  0  |  1  |  2  |  3  |  4  |  5  |  6  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
E      4 ||   4  |  3  |  2  |  1  |  0  |  1  |  2  |  3  |  4  |  5  |  6  |  5  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
D#/Eb  3 ||   3  |  2  |  1  |  0  |  1  |  2  |  3  |  4  |  5  |  6  |  5  |  4  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
D      2 ||   2  |  1  |  0  |  1  |  2  |  3  |  4  |  5  |  6  |  5  |  4  |  3  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
C#/Db  1 ||   1  |  0  |  1  |  2  |  3  |  4  |  5  |  6  |  5  |  4  |  3  |  2  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
C      0 ||   0  |  1  |  2  |  3  |  4  |  5  |  6  |  5  |  4  |  3  |  2  |  1  |
---------++------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
```

Below is a table showing the ic numbers corresponding to the traditional naming of intervals in 12-edo:

```   body of table indicates number of 12edo ic (interval-class)

+---------++-----------+---------------------------------+-----------+
|interval ||           |(imperfect): perfect :(imperfect)|           |
| name    || diminished|   minor   :         :   major   | augmented |
+=========++===========+===========:=========:===========+===========+
|         ||           |           :         :           |           |
| prime   ||      1    |    --     :    0    :    --     |      1    |
+---------++-----------+---------------------------------+-----------+
|  2nd    ||      0    |     1     :   --    :     2     |      3    |
+---------++-----------+---------------------------------+-----------+
|  3rd    ||      2    |     3     :   --    :     4     |      5    |
+---------++-----------+---------------------------------+-----------+
|  4th    ||      4    |    --     :    5    :    --     |      6    |
+---------++-----------+---------------------------------+-----------+
|  5th    ||      6    |    --     :    5    :    --     |      4    |
+---------++-----------+---------------------------------+-----------+
|  6th    ||      5    |     4     :   --    :     3     |      2    |
+---------++-----------+---------------------------------+-----------+
|  7th    ||      3    |     2     :   --    :     1     |      0    |
+---------++-----------+---------------------------------+-----------+
|  8ve    ||      1    |    --     :    0    :    --     |      1    |
+---------++-----------+---------------------------------+-----------+
```

It is possible to extend the concept to any other EDO, however, the ic number may be expressed either in relation to 12edo or in relation to the cardinality of the EDO. For example, in 24edo ("quarter-tones"), interval-class between the minor-3rd and the major-3rd (i.e., the "neutral-3rd") may be expressed either as 3.5 in reference to 12edo, or as 7 in reference to 24edo.

Because of the familiarity of theorists with the standard 12edo ic numbers, the former is generally preferred for 24edo. But for many other EDOs one decimal place will not express the interval size accurately enough to avoid the accumulation of rounding errors in performing ic math. One solution, only slightly better, is to use two decimal places (i.e., Monzo's unit of "Semitones"); a much better solution which eliminates rounding errors is to use a fraction to accompany the integer, expressing the relationship to 12edo exactly. However, this too can become unwieldy, and ultimately the best solution is probably just to use the method based on the cardinality of the EDO.

For example, in 46edo, the interval which represents the just major-3rd of ratio 5:4 encompasses 15 degrees of 46edo. The ic may thus be expressed as 3.91 (Semitones) or 3 21/23 in relation to 12edo, or simply as 15 where the reader knows that the cardinality is 46. Tables are given below for 19edo, showing the correspondence with regular interval names in both modulo-19 and modulo-12 (Semitones).

. . . . . . . . .

## ic matrices in non-12edo tunings

### 19-edo ic matrix, mod 19

```    higher: C | C#| Db| D | D#| Eb| E |E#/Fb| F | F#| Gb| G | G#| Ab| A | A#| Bb| B |B#/Cb|
0 | 1 | 2 | 3 | 4 | 5 | 6 |  7  | 8 | 9 |10 | 11| 12| 13| 14| 15| 16| 17| 18  |
lower:   ==================================================================================
B#/Cb 18 || 1 | 2 | 3 | 4 | 5 | 6 | 7 |  8  | 9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |  0  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
B     17 || 2 | 3 | 4 | 5 | 6 | 7 | 8 |  9  | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |  1  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
Bb    16 || 3 | 4 | 5 | 6 | 7 | 8 | 9 |  9  | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 |  2  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
A#    15 || 4 | 5 | 6 | 7 | 8 | 9 | 9 |  8  | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 |  3  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
A     14 || 5 | 6 | 7 | 8 | 9 | 9 | 8 |  7  | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 |  4  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
Ab    13 || 6 | 7 | 8 | 9 | 9 | 8 | 7 |  6  | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 |  5  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
G#    12 || 7 | 8 | 9 | 9 | 8 | 7 | 6 |  5  | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 |  6  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
G     11 || 8 | 9 | 9 | 8 | 7 | 6 | 5 |  4  | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |  7  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
Gb    10 || 9 | 9 | 8 | 7 | 6 | 5 | 4 |  3  | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |  8  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
F#     9 || 9 | 8 | 7 | 6 | 5 | 4 | 3 |  2  | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |  9  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
F      8 || 8 | 7 | 6 | 5 | 4 | 3 | 2 |  1  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |  9  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
E#/Fb  7 || 7 | 6 | 5 | 4 | 3 | 2 | 1 |  0  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 |  8  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
E      6 || 6 | 5 | 4 | 3 | 2 | 1 | 0 |  1  | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 8 |  7  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
Eb     5 || 5 | 4 | 3 | 2 | 1 | 0 | 1 |  2  | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 8 | 7 |  6  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
D#     4 || 4 | 3 | 2 | 1 | 0 | 1 | 2 |  3  | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 8 | 7 | 6 |  5  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
D      3 || 3 | 2 | 1 | 0 | 1 | 2 | 3 |  4  | 5 | 6 | 7 | 8 | 9 | 9 | 8 | 7 | 6 | 5 |  4  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
Db     2 || 2 | 1 | 0 | 1 | 2 | 3 | 4 |  5  | 6 | 7 | 8 | 9 | 9 | 8 | 7 | 6 | 5 | 4 |  3  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
C#     1 || 1 | 0 | 1 | 2 | 3 | 4 | 5 |  6  | 7 | 8 | 9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 |  2  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
C      0 || 0 | 1 | 2 | 3 | 4 | 5 | 6 |  7  | 8 | 9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 |  1  |
---------++---+---+---+---+---+---+---+-----+---+---+---+---+---+---+---+---+---+---+-----+
```

Below is a table showing the ic numbers corresponding to the traditional naming of intervals in 19-edo:

```   body of table indicates number of 19edo ic (interval-class), mod 19

+---------++-----------+---------------------------------+-----------+
|interval ||           |(imperfect): perfect :(imperfect)|           |
| name    || diminished|   minor   :         :   major   | augmented |
+=========++===========+===========:=========:===========+===========+
|         ||           |           :         :           |           |
| prime   ||      1    |    --     :    0    :    --     |      1    |
+---------++-----------+---------------------------------+-----------+
|  2nd    ||      1    |     2     :   --    :     3     |      4    |
+---------++-----------+---------------------------------+-----------+
|  3rd    ||      4    |     5     :   --    :     6     |      7    |
+---------++-----------+---------------------------------+-----------+
|  4th    ||      7    |    --     :    8    :    --     |      9    |
+---------++-----------+---------------------------------+-----------+
|  5th    ||      9    |    --     :    8    :    --     |      7    |
+---------++-----------+---------------------------------+-----------+
|  6th    ||      7    |     6     :   --    :     5     |      4    |
+---------++-----------+---------------------------------+-----------+
|  7th    ||      4    |     3     :   --    :     2     |      1    |
+---------++-----------+---------------------------------+-----------+
|  8ve    ||      1    |    --     :    0    :    --     |      1    |
+---------++-----------+---------------------------------+-----------+

body of table indicates number of 19edo ic (interval-class), mod 12

+---------++-----------+---------------------------------+-----------+
|interval ||           |(imperfect): perfect :(imperfect)|           |
| name    || diminished|   minor   :         :   major   | augmented |
+=========++===========+===========:=========:===========+===========+
|         ||           |           :         :           |           |
| prime   ||   0.63    |    --     :    0    :    --     |   0.63    |
+---------++-----------+---------------------------------+-----------+
|  2nd    ||   0.63    |   1.26    :   --    :   1.89    |   2.53    |
+---------++-----------+---------------------------------+-----------+
|  3rd    ||   2.53    |   3.16    :   --    :   3.79    |   4.42    |
+---------++-----------+---------------------------------+-----------+
|  4th    ||   4.42    |    --     :  5.05   :    --     |   5.68    |
+---------++-----------+---------------------------------+-----------+
|  5th    ||   5.68    |    --     :  5.05   :    --     |   4.42    |
+---------++-----------+---------------------------------+-----------+
|  6th    ||   4.42    |   3.79    :   --    :   3.16    |   2.53    |
+---------++-----------+---------------------------------+-----------+
|  7th    ||   2.53    |   1.89    :   --    :   1.26    |   0.63    |
+---------++-----------+---------------------------------+-----------+
|  8ve    ||   0.63    |    --     :    0    :    --     |   0.63    |
+---------++-----------+---------------------------------+-----------+
```
. . . . . . . . .

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