Ideally, you should have been playing for a while (at least a year, give or take some depending on how dedicated you are and how quickly you've progressed) and have a decent grasp of the chalumeau (lower notes) and clarion (what you previously considered to be "high notes") registers. If you take private lessons, ask your teacher before you attempt to advance into the altissimo notes.
 While not required, a teacher can help you advance more quickly on your instrument and will help you spot problems and fix them, possibly saving you from a lot of frustration and/or spending a lot of unnecessary money on new equipment., Don't change anything major about your embouchure, but use more and faster air. To "warm up" your air, try saying the "E" syllable when you play. You will probably feel compelled to pinch your face or bite down harder if you have trouble, but don't do this. It creates a very bad habit that will only hinder your progress. Only try a few notes at a time, only advancing when you're comfortable with the previous notes., If doing this improves your tone in the altissimo register, chances are, you haven't had enough mouthpiece in all along, and should continue to do this regardless of how high you're playing. You should hear a good change in the tone quality of every note you play., If you're serious about playing clarinet, the money this will initially cost will be well worth it. Once you find a clarinet, mouthpiece, ligature, and reed combination that works with you, it will last a lifetime.

Try using different brands and/or strengths of reeds. If you're using relatively soft reeds (2, 2.5, or 3 in some brands), that may be part of your problem. Harder reeds, while a bit more difficult to play on, will support your high notes and improve your tone. A better quality (read: more expensive, but it's worth the investment) brand or a different type of reed may also help. For instance, Vandoren makes three popular types of reeds, each with a slightly different cut that makes it play differently. Try out many different kinds of reeds until you find a brand and strength that works for you.
Consider using a different mouthpiece. If you're still using the mouthpiece that came with your instrument, it may be time for an upgrade. Especially if your mouthpiece has no markings or brand names on it at all or says "Made in China" (meaning that it is a very cheap stock mouthpiece), there's a good chance that's what holding you back. Like reeds, all mouthpieces have very subtle differences that can make a huge difference in the quality, sound, and ease of your playing. Go to a music store and try several different mouthpieces, and choose the one that feels and sounds the best to you.
Try buying a new ligature. If you're still using the metal ligature that came with your clarinet, you definitely need to try a leather ligature. Rovner ligatures are relatively cheap, very popular, and last for quite a while, or you can go for an even more expensive brand. Some ligatures have interchangeable plates or other features that give you more control over your sound. Try several ligatures and pick one you like that is within your budget.
If all else fails, consider a new clarinet. This is a much more drastic step than your other options, but if you plan to continue in music, it will become necessary at some point. If you're playing on a plastic student model, a newer horn could make a lot of difference. Wooden clarinets have a much warmer, prettier sound, and a more expensive instrument will be designed specifically to sound better, be more consistent, and be easier to play. If your budget doesn't allow for a brand new wooden clarinet, consider buying a used one. There are lots of perfectly good used wooden clarinets available, if you know where to look. As always, try several clarinets before buying one and pick the one that's best for you.
 It takes years to fully master the altissimo register of the clarinet, but hopefully you're well on your way.
Know if you're ready to try to conquer the altissimo register.
Consider private lessons, if you aren't taking them already.
When you first try to play altissimo notes (which start with the C#6, or the first C# above the staff), just relax.
Try taking more mouthpiece in your mouth.
If you are still having trouble, consider investing in a new mouthpiece and/or ligature, different reeds, or even a new clarinet.
Keep practicing.
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