Classic Edition: the symbol ⍸ (Iota Underbar) is not available in Classic Edition, and Interval Index is instead represented by ⎕U2378.

X is an ordered non-scalar array that represents a set of intervals or ranges.

Note that the i^{th} interval starts at X[i], then includes all subsequent values up to but not including X[i+1].

For example, if X is (1 3 5) it defines 4 intervals numbered 0 to 3 as follows.

0 | less than 1 | <1 |

1 | between 1 and 3 | (≥1)∧(<3) |

2 | between 3 and 5 | (≥3)∧(<5) |

3 | greater than or equal to 5 | ≥5 |

If X is 'AEIOU' it defines 6 intervals numbered 0 to 5 as follows:

0 | before A | ⎕UCS 0,⍳64 |

1 | between A and E | ABCD |

2 | between E and I | EFGH |

3 | between I and O | IJKLMN |

4 | between O and U | OPQREST |

5 | U and after | UVWXYZ... |

Y is an array of the same type (numeric or character) as X.

The result R is an integer array that identifies into which interval the corresponding value in Y falls.

Like dyadic ⍳ (see Index Of), Interval Index works with major cells. For a vector these are its elements; for a matrix its rows, and so forth.

X and Y are compared using the same logic as monadic ⍋ (see Grade Up (Monadic)) which is independent of ⎕CT and ⎕DCT.

⎕IO is an implicit argument of Interval Index. In all the following examples, ⎕IO is 1.

#### Examples

10 20 30⍸11 1 31 21 1 0 3 2

In the above example:

- 11 is between X[1] and X[2] so the answer is 1.
- 1 is less than X[1] so the answer is 0
- 31 is greater than X[⍴X] so the answer is 3
- 21 is between X[2] and X[3] so the answer is 2.

'AEIOU' ⍸ 'DYALOG' 1 5 1 3 4 2

And in the alphabetic example above:

- "D" is between X[1] and X[2], so the answer is 1
- "Y" is after X[⍴X] so the answer is 5
- "A" is between X[1] and X[2], so the answer is 1
- as so on ...

##### Example (Classification)

Commercially, olive oil is graded as follows:

- if its acidity is less than 0.8%, as "Extra Virgin"
- if its acidity is less than 2%, as "Virgin"
- if its acidity is less than 3.3%, as "Ordinary"
- otherwise, as "Lampante"

grades←'Extra Virgin' 'Virgin' 'Ordinary' 'Lampante' acidity←0.8 2 3.3 samples←1.3 1.9 0.7 4 .6 3.2 acidity⍸samples 1 1 0 3 0 2 samples,⍪grades[1+acidity⍸samples] ┌───┬────────────┐ │1.3│Virgin │ ├───┼────────────┤ │1.9│Virgin │ ├───┼────────────┤ │0.7│Extra Virgin│ ├───┼────────────┤ │4 │Lampante │ ├───┼────────────┤ │0.6│Extra Virgin│ ├───┼────────────┤ │3.2│Ordinary │ └───┴────────────┘

#### Example (Data Consolidation by Interval)

x represents some data sampled in chronological order at timestamps t.

⍴x 200000 x 3984300 2020650 819000 1677100 3959200 2177250 3431800 ...

⍴t 200000 3

(10↑t) (¯10↑t) ┌─────┬────────┐ │0 0 0│23 59 54│ │0 0 0│23 59 55│ │0 0 0│23 59 56│ │0 0 0│23 59 56│ │0 0 0│23 59 58│ │0 0 2│23 59 58│ │0 0 3│23 59 59│ │0 0 3│23 59 59│ │0 0 4│23 59 59│ │0 0 5│23 59 59│ └─────┴────────┘

u represents timestamps for 5-minute intervals:

⍴u 288 3 (10↑u) (¯10↑u) ┌──────┬───────┐ │0 0 0│23 10 0│ │0 5 0│23 15 0│ │0 10 0│23 20 0│ │0 15 0│23 25 0│ │0 20 0│23 30 0│ │0 25 0│23 35 0│ │0 30 0│23 40 0│ │0 35 0│23 45 0│ │0 40 0│23 50 0│ │0 45 0│23 55 0│ └──────┴───────┘

Therefore, the expression (u⍸t){+/⍵}⌸x summarises x in 5-minute intervals.

u ⍸ t 1 1 1 1 1 1 1 1 1 1 ... 288 288 288 288 288 288 (u⍸t) {+/⍵}⌸ x 1339083050 1365108650 1541944750 1393476000 1454347100 ... (u⍸t) {(⍺⌷u),+/⍵}⌸ x 0 0 0 1339083050 0 5 0 1365108650 0 10 0 1541944750 0 15 0 1393476000 ... 23 45 0 1388823150 23 50 0 1453472350 23 55 0 1492078850

#### Higher-Rank Left Argument

If X is a higher rank array, the function compares sub-arrays in Y with the major cells of X, where a major cell is a sub-array on the leading dimension of X with shape 1↓⍴X. In this case, the shape of the result R is (1-⍴⍴X)↓⍴Y.

#### Example

x ← ↑ 'Fi' 'Jay' 'John' 'Morten' 'Roger' x Fi Jay John Morten Roger ⍴x 5 6

y ← x ⍪ ↑ 'JD' 'Jd' 'Geoff' 'Alpha' 'Omega' 'Zeus ' y Fi Jay John Morten Roger JD Jd Geoff Alpha Omega Zeus

x ⍸ y 1 2 3 4 5 1 2 1 0 4 5 y ,⍪ x⍸y Fi 1 Jay 2 John 3 Morten 4 Roger 5 JD 1 Jd 2 Geoff 1 Alpha 0 Omega 4 Zeus 5

#### Further Example

⍴x 5 6 ⍴y 3 3 6 x Fi Jay John Morten Roger y Fi Jay John Morten Roger JD Jd Geoff Alpha x⍸y 1 2 3 4 5 1 2 1 0

#### Nested Array Example

A card-player likes to sort a hand into suits spades, hearts, diamond, clubs (fortunately alphabetic) and high-to-low within each suit.

suits←'Clubs' 'Diamonds' 'Hearts' 'Spades' pack←,(⊂¨suits)∘.,1↓14 ⍝ 11=Jack ... 14=Ace hand←↑(,pack)[7?52] hand←hand[⍒hand;] hand ┌────────┬──┐ │Spades │12│ ├────────┼──┤ │Hearts │12│ ├────────┼──┤ │Hearts │7 │ ├────────┼──┤ │Hearts │2 │ ├────────┼──┤ │Diamonds│11│ ├────────┼──┤ │Diamonds│9 │ ├────────┼──┤ │Clubs │8 │ └────────┴──┘

Another card, the 10 of diamonds is dealt. Where must it go in the hand ?

(⊖hand)⍸'Diamonds' 10 ⍝ left arg must be sorted up 2 (¯2↓hand)⍪'Diamonds' 10⍪¯2↑hand ┌────────┬──┐ │Spades │12│ ├────────┼──┤ │Hearts │12│ ├────────┼──┤ │Hearts │7 │ ├────────┼──┤ │Hearts │2 │ ├────────┼──┤ │Diamonds│11│ ├────────┼──┤ │Diamonds│10│ ├────────┼──┤ │Diamonds│9 │ ├────────┼──┤ │Clubs │8 │ └────────┴──┘

Note that if (∧/Y∊X) and X is sorted and ⎕CT=0 ,then x⍸y is the same as x⍳y.