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As all observers, no matter their state of motion, agree about the distance between two points in Classical dynamics (and in "common sense") so, in Relativity theory, the interval between two events (such as two shots being fired) is commonly agreed. It is, for relativity theory, the invariant analogous to distance.

The reason for this is easy to see: The interval between two events is numerically equal to the proper time separating the events. For example, if a pistol, with a clock tied to it, moving at constant velocity, is fired twice: once when the clock reads t1 and again when it reads t2; then everyone must agree that the proper time separating the two explosions is t2 - t1. This equals the interval between the events.

If a particular reference frame measures the distance between two events to be x, y & z (i.e. forward, sideways & up) and the time gap to be t. Then the interval separating the two is:

s = sqrt(x2 + y2 + z2 - c2t2)

(Where c is the velocity of light in a vacuum.)

(Sporus is not the greatest expert in the universe on this one but the above seems reasonable enough.)

(Mathematics, other places where you need to talk about the real numbers:)
An interval is a subset IR of the real numbers with the property that if x,yI and z is between x and y, then z∈I too.

It's just a "contiguous" segment of the line (which may or may not be bounded on either end). Or, if you like, it's a convex subset of R -- 1-dimensional convexity is boring.

#### Examples:

Here are an open interval, a closed interval, a half-open interval (which is also half-closed) and 2 unbounded intervals:

(-2,3) = {x: -2<x<3}
[4,7] = {x: 4≤x≤7}
[11,17) = {x: 11≤x<17}
(-,9] = {x: x≤9}
(-∞,+∞) = R
Note: The symbol "∞" in an unbounded interval means precisely nothing: the only meaning here is carried by the entire symbol "(-∞,9]".

In music, an interval is the relative difference in pitch between two successive notes. Therefore we speak of ascending and descending intervals.

In traditional Western music, the notes are basically identical to what you find on the piano: it uses octaves equally divided in twelve. A selection of these notes - the white keys on the piano - roughly suffices to play most melodies (try Row, row, row your boat for an explanation), and the naming of intervals was based on this selection.

To be precise, the names indicate how many white keys are spanned in total. The typical distance between successive white keys is called a whole note; the distance between white and black, and between some white keys, is half of that, a half-note. As a result, we end up with:

the second ("secundus" in Latin)
the interval separated by one whole note, i.e., 1/6th of the octave; e.g., C-D on the piano
a third (tertius)
2 notes, i.e., 1/3rd of the octave, e.g., C-E
a fourth (quartus)
2.5 whole notes, or C-F
a fifth (quintus)
3.5 notes, or C-G
a sixth (sextus)
4.5 notes, or C-A
a seventh (septimus]
5.5 notes, or C-B
an eighth (octavus)
6 notes, 12 half-notes; C-C'
This leaves some gaps; these gaps are filled by speaking of diminished intervals, e.g. the interval of 4.0 tones is called a diminished sixth.

The only excuse for the weirdness of this naming scheme is its gradual development over many hundreds of years.

Combinations of more than two notes are known as chords. Musically, most chords contain a dominant interval (obtained by omitting all other notes); this explains why chords are named after intervals.

(basic music theory)

In music, the term "interval" is used to measure the distance between any two notes. An interval may be horizontal or "melodic," when the two notes are sounded at different times, or it may be vertical or "harmonic," when the two notes are sounded simultaneously.

# A music notation primer (of sorts)

In order to understand how intervals are spelled and understood, I will need to take you on a crash course through the basics of music notation:

```___________________________________________
| | | | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | | | |
| [_] [_] | [_] [_] [_] | [_] [_] | [_] [_]
|  |   |  |  |   |   |  |  |   |  |  |   |
|  |   |  |  |   |   |  |  |   |  |  |   |
[__[___[__[__[___[___[__[__[___[__[__[___]```

What you see above is a reasonable rendition of a piece of a typical keyboard, such as that for the piano. The horizontal lines and dashes form several smaller boxes; those boxes that are strictly rectangular and shorter than the rest are typically called the "black keys," and the L- and upside-down-T-shaped boxes are typically called the "white keys."

The white keys have a fixed naming system. In English we use the first seven letters of the alphabet and cycle through them:

```___________________________________________
| | | | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | | | |
| [_] [_] | [_] [_] [_] | [_] [_] | [_] [_]
|  |   |  |  |   |   |  |  |   |  |  |   |
|  |   |  |  |   |   |  |  |   |  |  |   |
[C_[D__[E_[F_[G__[A__[B_[C_[D__[E_[F_[G__]```

In the above diagram, each box has the name of the letter within its borders.

The names of the black keys, however, are not fixed and depend on context, typically taking on the name of one of their neighboring white keys with the addition of an accidental -- either # (sharp) or b (flat}. More on this in a moment.

The interval between any adjacent keys is exactly the same, the interval of the half step or semitone. This is true whether the two adjacent keys are of the same "color" or not. Note that no two black keys are a semitone apart, but some white keys are.

So now we can name the black keys. An accidental will change the pitch of the note to which it is assigned by one semitone, either up (as in the sharp (#)) or down (as in the flat (b)). On our diagram, moving "up" corresponds to moving to the right, while moving "down" corresponds to moving to the left. For example, the black key between G and A could be named "G#", since it is the first key to the right of G. It could also be named "Ab", since it is the first key to the left of A.

Another example: "B#" would be another name for the key we've already named C, just as "Cb" would be another name for the key we've already named B. It is perfectly acceptable for a key to have more than one name -- multiple note names that find themselves assigned to the same keys or pitches are said to be enharmonic.

# Building and naming intervals

The semitone is the most basic interval, with which we may build all other intervals. The whole step or whole tone is two half steps wide. We can find all whole steps in our above diagram by choosing a key and moving two semitones to the left or right. For example, B and C# are one whole tone apart, D and E are one whole tone apart, and Gb and Ab are one whole tone apart.

There are as many intervals on the keyboard as there are pairs of notes. Intervals larger than the whole tone often derive their names from their position in the major and minor scales.

To put it simply, major and minor scales are melodic and ordered arrangements of half and whole steps. To construct either scale, you first must choose a key on which to build the scale. Let us choose for now the key "C."

The next note in the scale depends on our ordering of half and whole steps. In the case of the major, our ordering is:

`whole - whole - half - whole - whole - whole - half`

So we proceed by moving up by one whole step from C -- "D". Next, "E", "F", "G", "A", "B", "C". Here is the scale with the whole/half step ordering, to clarify:

```C        D       E      F       G       A       B      C
whole    whole   half   whole   whole   whole   half```

You might take a moment to compare this ordering with the keyboard diagram.

The minor scale ordering is different; it yields the following:

```C        D      Eb       F      G       Ab      Bb      C
whole   half    whole   whole   half    whole   whole```

Now.

Each interval derives its name from these two basic scales. The distance between "C" and "D", for example, is a "second," since D is the second note in the C major and minor scales. Similarly,

```C-E and C-Eb are thirds
C-F is a fourth
C-G is a fifth
C-A and C-Ab are sixths
C-B and C-Bb are sevenths
C-C is an "octave"```

To distinguish between the different types of intervals that arise in both scales, we use certain qualifying terms. They are, not surprisingly, "major" and "minor." For example,

```C-E is a major third, C-Eb is a minor third
C-A is a major sixth, C-Ab is a minor sixth
C-B is a major seventh, C-Bb is a minor seventh```

You might take a moment to compare the names of these intervals with their occurence in the scales, above.

You'll also see a bit of a pattern here; when we take one of the major intervals and lower the second note by a semitone, the result is a minor interval of the same type. So, in the case of "C-D", which is the same between either scale, if we lower the "D" to give us the interval "C-Db", we say that we have a "minor second," while the "C-D" is "major second." If we were to lower the second note of a minor interval yet again, we would get a "diminished" interval. If we raise the second note of a major interval by a semitone, we get an "augmented" interval.

In the case of the fourth and fifth, when the intervals occur as they do in the scales, we say they are "perfect." When we lower the second note by a semitone, we get a "diminished" interval (not minor!), and if we raise the second note by a semitone, we get an "augmented" interval. Thus:

```C-G is a perfect fifth
C-Gb is a diminished fifth
C-F is a perfect fourth
C-F# is an augmented fourth```

So now we have names for every interval that can occur when C is the bottom note. But what if the bottom note is some other pitch? Simply construct the major and minor scales based on the bottom note, and figure out where in the scale the top note falls relative to the different notes of either scale.

This might seem a bit labor-intensive, but with practice you should learn to recognize intervals much more quickly and more intuitively.

In"ter*val (?), n. [L. intervallum; inter between + vallum a wall: cf. F. intervalle. See Wall.]

1.

A space between things; a void space intervening between any two objects; as, an interval between two houses or hills.

'Twixt host and host but narrow space was left, A dreadful interval. Milton.

2.

Space of time between any two points or events; as, the interval between the death of Charles I. of England, and the accession of Charles II.

3.

A brief space of time between the recurrence of similar conditions or states; as, the interval between paroxysms of pain; intervals of sanity or delirium.

4. Mus.

Difference in pitch between any two tones.

At intervals, coming or happening with intervals between; now and then. "And Miriam watch'd and dozed at intervals." Tennyson. -- Augmented interval Mus., an interval increased by half a step or half a tone.

In"ter*val (?), In"ter*vale (?), n.

A tract of low ground between hills, or along the banks of a stream, usually alluvial land, enriched by the overflowings of the river, or by fertilizing deposits of earth from the adjacent hills. Cf. Bottom, n., 7.

[Local, U. S.]

The woody intervale just beyond the marshy land. The Century.