The A# minor pentatonic scale is one of the most useful five-note toolkits you can have under your fingers. It's the classic rock and blues soloing sound, and the minor pentatonic is usually the first scale guitarists learn. Once you know it in A#, you can solo over a whole family of songs in this key, and the same shape moves to every other key too. Let's get it under your fingers.
"Pentatonic" just means five notes (penta = five, tonic = tone). The five notes of A# minor pentatonic are:
A#, C#, D#, E#, G#
After G#, you arrive back at A#, one octave higher. An octave is the same note at a higher pitch, twelve frets up the same string.
Musicians also describe scales by number, comparing each note to the root (the home note, here A#). The formula for minor pentatonic is:
Learn this formula once and you can build the scale from any root.
If E# looks odd, don't worry, you haven't misread anything. E# sounds exactly the same as F and sits on the same fret. So why not call it F? Because each scale uses each letter name only once in order. Starting from A#, the 5th of the scale has to be some kind of E, and the E that fits is E#. It keeps the theory tidy and makes chords easier to read. On the fretboard, just play it where you'd play F. You'll also often see this key written with flats instead, as Bb minor. Same sounds, different names.
Two distances do all the work here. A half step (H) is one fret. A whole step (W) is two frets. The minor pentatonic pattern is:
W+H - W - W - W+H - W
A "W+H" is a jump of three frets. Walk it from A#:
Try it on one string: put a finger on the low E string at fret 6 (A#) and move up 3, 2, 2, 3, 2 frets. You'll end at fret 18, A# again.
This scale comes from the A# natural minor scale: A#, B#, C#, D#, E#, F#, G#. Minor pentatonic is simply that scale with the 2nd (B#) and 6th (F#) removed. Those two notes are the ones that create the tightest, most tension-filled half-step rubs, so taking them out leaves a scale with no "wrong" notes. That's why beginners can improvise with it right away and still sound good.
The note that gives it its moody, bluesy colour is C#, the minor 3rd. Against the A# root, it's what makes the scale sound minor rather than bright. When you solo, try landing on C# and holding it. Listen for that slightly sad, soulful quality. Then bend or slide toward G#, the b7, for that classic rock feel.
One fun fact: the C# major pentatonic scale uses exactly the same five notes. Same notes, different home base. Centre on A# and it sounds minor; centre on C# and it sounds major.
Chords from the parent A# natural minor scale work well under this scale. The main ones to start with:
A simple progression to jam over is A#m - D#m - E#m - A#m, four beats on each chord. Record it on your phone or loop it, then improvise using only the five scale notes. For a rockier vibe, try A#m - G# - F# - G#. When the chord changes, try aiming for a scale note that's inside the new chord, and hear how it clicks into place.
In standard tuning, the root A# sits here:
The most popular place to play this scale is the box shape starting at fret 6 on the low E string. Use your first finger for fret 6 and your third or fourth finger for the higher frets:
Notice A# appears three times in this shape: low E fret 6, D string fret 8 and high E fret 6. Those are your safe landing spots. If you'd like to play lower, the A string at fret 1 is another root to build from, and fret 13 gives you the same sounds an octave up.
Spend five minutes a day playing the box shape slowly up and down, saying each note name out loud as you go. Then put on your A#m - D#m - E#m loop and play just three notes, A#, C# and D#, for a full minute. Limiting yourself like this forces you to make music with rhythm and feel. Add the other notes when you're ready, and before long you'll be soloing without thinking about the shape at all.
For the bigger picture, read The Minor Pentatonic Scale.