# Anchor Intervals

## Note Intervals

Before we get started, we'll recall that we defined what the interval between two numbers is, so given two numbers X and Y, then the interval between them is the number you add to get from X to Y, namely Y - X.

Since our notes have numbers in them we can also do something similar, so given two notes X* and Y*, we define the interval between them to be Y - X, the result is a number which is not a note.

The interval between two notes could be any distance from 0 to 11 (we'll forget about negative numbers for now, since a negative is equivalent to one of the positive ones anyways). And in the standard system they have developed names for each of the different sizes of intervals, this will be useful so we understand what other musicians are talking about when the say an interval of a fourth, here is the conversion table:

Distance
Standard Name for Distance
0
Perfect Unison
1 Minor Second
2 Major Second
3 Minor Third
4 Major Third
5 Perfect Fourth
6 Tritone
7 Perfect Fifth
8 Minor Sixth
9 Major Sixth
11 Minor Seventh
12 Perfect Octave

## Anchor Intervals

In our first lesson we touched on considering notes with respect to some anchor note, for example if our anchor note was 8* (Ab) then the note 9* would be considered as 1. Whenever we specify a note like this, it will be called an anchor interval and no asterisk will be written.

More explicitly an anchor interval of a note is the interval between the anchor note and the note we're considering. Writing anchor interval over and over again takes up a lot of space, for that reason we'll shorten anchor interval to ai.

For a new example, let's set our anchor note to be 2* (D) and let's consider notes with respect to that anchor note. If we consider 4*, we can see that I(2*, 4*) = 2, so its ai is 2.

## Computing Anchor Intervals efficiently

All we have to do is use our idea from our article on numbers in a cyclic system which says that for any two numbers X and Y, the interval between them is Y - X, and that a fast way to figure that out is to do Y - s(X).

Moving that into our context, we have some anchor note A*, and some other note N*, then N* has the anchor interval of I(A*, N*) = N - A = N - s(A).

### The Process

• Take your anchor note and consider the smallest representation of that number. For example if my anchor note was 8*, then we would be thinking of the number -4.
• Now take your other note and subtract the small representation of the anchor note from your note, the result is the anchor interval. Note that if you're subtracting a negative number you can just add it. So in our case if our other note was 3*, then we consider 3 - (-4) = 3 + 4 = 7.

Now that you have the process, use the anchor interval trainer under the web applications until you are fast.